# Fluid mechanics notes

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### Fluid mechanics notes

• 1. FLUID MECHANICS
• 2. 2 B.TECH. DEGREE COURSE SCHEME AND SYLLABUS (2002-03 ADMISSION ONWARDS) MAHATMA GANDHI UNIVERSITY KOTTAYAM KERALA Module 1 Introduction - Proprties of fluids - pressure, force, density, specific weight, compressibility, capillarity, surface tension, dynamic and kinematic viscosity- Pascal’s law-Newtonian and non-Newtonian fluids-fluid statics-measurement of pressure-variation of pressure-manometry-hydrostatic pressure on plane and curved surfaces-centre of pressure-buoyancy-floation-stability of submerged and floating bodies-metacentric height-period of oscillation. Module 2 Kinematics of fluid motion-Eulerian and Lagrangian approach-classification and representation of fluid flow- path line, stream line and streak line. Basic hydrodynamics-equation for acceleration-continuity equation-rotational and irrotational flow-velocity potential and stream function-circulation and vorticity- vortex flow-energy variation across stream lines-basic field flow such as uniform flow, spiral flow, source, sink, doublet, vortex pair, flow past a cylinder with a circulation, Magnus effect-Joukowski theorem-coefficient of lift. Module 3 Euler’s momentum equation-Bernoulli’s equation and its limitations-momentum and energy correction factors-pressure variation across uniform conduit and uniform bend-pressure distribution in irrotational flow and in curved boundaries-flow through orifices and mouthpieces, notches and weirs-time of emptying a tank-application of Bernoulli’s theorem-orifice meter, ventury meter, pitot tube, rotameter.
• 3. 3 Module 4 Navier-Stoke’s equation-body force-Hagen-Poiseullie equation-boundary layer flow theory-velocity variation- methods of controlling-applications-diffuser-boundary layer separation –wakes, drag force, coefficient of drag, skin friction, pressure, profile and total drag-stream lined body, bluff body-drag force on a rectangular plate- drag coefficient for flow around a cylinder-lift and drag force on an aerofoil- applications of aerofoil- characteristics-work done-aerofoil flow recorder-polar diagram-simple problems. Module 5 Flow of a real fluid-effect of viscosity on fluid flow-laminar and turbulent flow- boundary layer thickness-displacement, momentum and energy thickness-flow through pipes-laminar and turbulent flow in pipes-critical Reynolds number-Darcy- Weisback equation-hydraulic radius-Moody;s chart-pipes in series and parallel- siphon losses in pipes-power transmission through pipes-water hammer-equivalent pipe-open channel flow-Chezy’s equation-most economical cross section-hydraulic jump.
• 4. 4 DEFINITION OF FLUID  A fluid is a substance that deforms continuously in the face of tangential or shear stress, irrespective of the magnitude of shear stress .This continuous deformation under the application of shear stress constitutes a flow.  In this connection fluid can also be defined as the state of matter that cannot sustain any shear stress. Example : Consider Fig 1.1 Fig 1.1 Shear stress on a fluid body If a shear stress τ is applied at any location in a fluid, the element 011' which is initially at rest will move to 022', then to 033'. Further, it moves to 044' and continues to move in a similar fashion. In other words, the tangential stress in a fluid body depends on velocity of deformation and vanishes as this velocity approaches zero. A good example is Newton's parallel plate experiment where dependence of shear force on the velocity of deformation was established.
• 5. 5 FLUID PROPERTIES: Characteristics of a continuous fluid which are independent of the motion of the fluid are called basic properties of the fluid. Some of the basic properties are as discussed below. Property Symbol Definition Unit Density ρ The density p of a fluid is its mass per unit volume . If a fluid element enclosing a point P has a volume Δ and mass Δm (Fig. 1.4), then density (ρ)at point P is written as However, in a medium where continuum model is valid one can write - Fig 1.2 A fluid element enclosing point P kg/m3
• 6. 6 Specific Weight γ The specific weight is the weight of fluid per unit volume. The specific weight is given by γ= ρ g Where ‘g’ is the gravitational acceleration. Just as weight must be clearly distinguished from mass, so must the specific weight be distinguished from density. N/m3 Specific Volume v The specific volume of a fluid is the volume occupied by unit mass of fluid. Thus m3 /kg Specific Gravity s For liquids, it is the ratio of density of a liquid at actual conditions to the density of pure water at 101 kN/m2 , and at 4°C. The specific gravity of a gas is the ratio of its density to that of either hydrogen or air at some specified temperature or pressure. However, there is no general standard; so the conditions must be stated while referring to the specific gravity VISCOSITY (µ):  Viscosity is a fluid property whose effect is understood when the fluid is in motion.  In a flow of fluid, when the fluid elements move with different velocities, each element will feel some resistance due to fluid friction within the elements.
• 7. 7  Therefore, shear stresses can be identified between the fluid elements with different velocities.  The relationship between the shear stress and the velocity field was given by Sir Isaac Newton. Consider a flow (Fig. 1.3) in which all fluid particles are moving in the same direction in such a way that the fluid layers move parallel with different velocities. Fig 1.3 Parallel flow of a fluid Fig.1.4 Two adjusent layers of moving fluid
• 8. 8  The upper layer, which is moving faster, tries to draw the lower slowly moving layer along with it by means of a force F along the direction of flow on this layer. Similarly, the lower layer tries to retard the upper one, according to Newton's third law, with an equal and opposite force F on it (Figure 1.4).  Such a fluid flow where x-direction velocities, for example, change with y- coordinate is called shear flow of the fluid.  Thus, the dragging effect of one layer on the other is experienced by a tangential force F on the respective layers. If F acts over an area of contact A, then the shear stress τ is defined as τ = F/A NEWTON’S LAW OF VISCOSITY:  Newton postulated that τ is proportional to the quantity Δu/ Δy where Δy is the distance of separation of the two layers and Δu is the difference in their velocities.  In the limiting case of, Δu / Δy equals du/dy, the velocity gradient at a point in a direction perpendicular to the direction of the motion of the layer.  According to Newton τ and du/dy bears the relation Where, the constant of proportionality μ is known as the coefficient of viscosity or simply viscosity which is a property of the fluid and depends on its state. Sign of τ depends upon the sign of du/dy. For the profile shown in Fig. 1.4, du/dy is positive everywhere and hence, τ is positive.
• 9. 9 Defining the viscosity of a fluid is known as Newton's law of viscosity. Common fluids, viz. water, air, mercury obey Newton's law of viscosity and are known as Newtonian fluids. Other classes of fluids, viz. paints, different polymer solution, and blood do not obey the typical linear relationship, of τ and du/dy and are known as non-Newtonian fluids. In non-Newtonian fluids viscosity itself may be a function of deformation rate as you will study in the next lecture. KINEMATIC VISCOSITY: The ratio dynamic viscosity to density of the given fluid is known as kinematic viscosity. IDEAL FLUID:  Consider a hypothetical fluid having a zero viscosity ( μ = 0). Such a fluid is called an ideal fluid and the resulting motion is called as ideal or inviscid flow. In an ideal flow, there is no existence of shear force because of vanishing viscosity.  All the fluids in reality have viscosity (μ > 0) and hence they are termed as real fluid and their motion is known as viscous flow.  Under certain situations of very high velocity flow of viscous fluids, an accurate analysis of flow field away from a solid surface can be made from the ideal flow theory. NON-NEWTONIAN FLUIDS  There are certain fluids where the linear relationship between the shear stress and the deformation rate (velocity gradient in parallel flow) as expressed by  This not valid. For these fluids the viscosity varies with rate of deformation.
• 10. 10  Due to the deviation from Newton's law of viscosity they are commonly termed as non-Newtonian fluids. Figure 1.7 shows the class of fluid for which this relationship is nonlinear. Figure 1.7 Shear stress and deformation rate relationship of different fluids  The abscissa in Fig. 1.7 represents the behaviour of ideal fluids since for the ideal fluids the resistance to shearing deformation rate is always zero, and hence they exhibit zero shear stress under any condition of flow.  The ordinate represents the ideal solid for there is no deformation rate under any loading condition.  The Newtonian fluids behave according to the law that shear stress is linearly proportional to velocity gradient or rate of shear strain . Thus for these fluids, the plot of shear stress against velocity gradient is a straight line through the origin. The slope of the line determines the viscosity.  The non-Newtonian fluids are further classified as pseudo-plastic, dilatant and Bingham plastic.
• 11. 11 COMPRESSIBILITY  Compressibility of any substance is the measure of its change in volume under the action of external forces.  The normal compressive stress on any fluid element at rest is known as hydrostatic pressure p and arises as a result of innumerable molecular collisions in the entire fluid.  The degree of compressibility of a substance is characterized by the bulk modulus of elasticity E defined as   Where Δ and Δp are the changes in the volume and pressure respectively, and is the initial volume. The negative sign (-sign) is included to make E positive, since increase in pressure would decrease the volume i.e for Δp>0, Δ <0) in volume.  For a given mass of a substance, the change in its volume and density satisfies the relation m = 0, ρ ) = 0 using we get
• 12. 12  Values of E for liquids are very high as compared with those of gases (except at very high pressures). Therefore, liquids are usually termed as incompressible fluids though, in fact, no substance is theoretically incompressible with a value of E as .  For example, the bulk modulus of elasticity for water and air at atmospheric pressure are approximately 2 x 106 kN/m 2 and 101 kN/m 2 respectively. It indicates that air is about 20,000 times more compressible than water. Hence water can be treated as incompressible.  For gases another characteristic parameter, known as compressibility K, is usually defined , it is the reciprocal of E SURFACE TENSION OF LIQUIDS  The phenomenon of surface tension arises due to the two kinds of intermolecular forces (i) Cohesion: The force of attraction between the molecules of a liquid by virtue of which they are bound to each other to remain as one assemblage of particles is known as the force of cohesion. This property enables the liquid to resist tensile stress. (ii) Adhesion: The force of attraction between unlike molecules, i.e. between the molecules of different liquids or between the molecules of a liquid and those of a solid body when they are in contact with each other, is known as the force of adhesion. This force enables two different liquids to adhere to each other or a liquid to adhere to a solid body or surface.
• 13. 13 Figure 1.8 the intermolecular cohesive force field in a bulk of liquid with a free surface A and B experience equal force of cohesion in all directions, C experiences a net force interior of the liquid The net force is maximum for D since it is at surface  Work is done on each molecule arriving at surface against the action of an inward force. Thus mechanical work is performed in creating a free surface or in increasing the area of the surface. Therefore, a surface requires mechanical energy for its formation and the existence of a free surface implies the presence of stored mechanical energy known as free surface energy. Any system tries to attain the condition of stable equilibrium with its potential energy as minimum. Thus a quantity of liquid will adjust its shape until its surface area and consequently its free surface energy is a minimum.  The magnitude of surface tension is defined as the tensile force acting across imaginary short and straight elemental line divided by the length of the line.  The dimensional formula is F/L or MT-2. It is usually expressed in N/m in SI units.  Surface tension is a binary property of the liquid and gas or two liquids which are in contact with each other and defines the interface. It decreases slightly
• 14. 14 with increasing temperature. The surface tension of water in contact with air at 20°C is about 0.073 N/m.  It is due to surface tension that a curved liquid interface in equilibrium results in a greater pressure at the concave side of the surface than that at its convex side. CAPILLARITY:  The interplay of the forces of cohesion and adhesion explains the phenomenon of capillarity. When a liquid is in contact with a solid, if the forces of adhesion between the molecules of the liquid and the solid are greater than the forces of cohesion among the liquid molecules themselves, the liquid molecules crowd towards the solid surface. The area of contact between the liquid and solid increases and the liquid thus wets the solid surface.  The reverse phenomenon takes place when the force of cohesion is greater than the force of adhesion. These adhesion and cohesion properties result in the phenomenon of capillarity by which a liquid either rises or falls in a tube dipped into the liquid depending upon whether the force of adhesion is more than that of cohesion or not (Fig.1.9).  The angle θ as shown in Fig. 1.9 is the area wetting contact angle made by the interface with the solid surface.  For pure water in contact with air in a clean glass tube, the capillary rise takes place with θ = 0 . Mercury causes capillary depression with an angle of contact of about 1300 in a clean glass in contact with air. Since h varies  Inversely with D as found from Eq. (), an appreciable capillary rise or depression is observed in tubes of small diameter only.
• 15. 15 Fig 1.9 Phenomenon of Capillarity VAPOUR PRESSURE: All liquids have a tendency to evaporate when exposed to a gaseous atmosphere. The rate of evaporation depends upon the molecular energy of the liquid which in turn depends upon the type of liquid and its temperature. The vapour molecules exert a partial pressure in the space above the liquid, known as vapour pressure. If the space above the liquid is confined (Fig. 1.10) and the liquid is maintained at constant temperature, after sufficient time, the confined space above the liquid will contain vapour molecules to the extent that some of them will be forced to enter the liquid. Eventually an equilibrium condition will evolve when the rate at which the number of vapour molecules striking back the liquid surface and condensing is just equal to the rate at which they leave from the surface. The space above the liquid then becomes saturated with vapour. The vapour pressure of a given liquid is a function of temperature only and is equal to the saturation pressure for boiling corresponding to that temperature. Hence, the vapour pressure increases with the increase in temperature. Therefore the phenomenon of boiling of a liquid is closely related to the vapour pressure. In fact, when the vapour pressure of a liquid becomes equal to the total pressure impressed on its surface, the liquid starts boiling. This concludes that
• 16. 16 boiling can be achieved either by raising the temperature of the liquid, so that its vapour pressure is elevated to the ambient pressure, or by lowering the pressure of the ambience (surrounding gas) to the liquid's vapour pressure at the existing temperature. Fig 1.10 FORCES ON FLUID ELEMENTS: Fluid Elements - Definition: Fluid element can be defined as an infinitesimal region of the fluid continuum in isolation from its surroundings. Two types of forces exist on fluid elements  Body Force: distributed over the entire mass or volume of the element. It is usually expressed per unit mass of the element or medium upon which the forces act. Example: Gravitational Force, Electromagnetic force fields etc.  Surface Force: Forces exerted on the fluid element by its surroundings through direct contact at the surface. Surface force has two components:
• 17. 17  Normal Force: along the normal to the area  Shear Force: along the plane of the area. The ratios of these forces and the elemental area in the limit of the area tending to zero are called the normal and shear stresses respectively. The shear force is zero for any fluid element at rest and hence the only surface force on a fluid element is the normal component. PASCAL'S LAW OF HYDROSTATICS PASCAL'S LAW The normal stresses at any point in a fluid element at rest are directed towards the point from all directions and they are of the equal magnitude. Fig 1.11 State of normal stress at a point in a fluid body at rest Derivation: The inclined plane area is related to the fluid elements (refer to Fig 3.1) as follows
• 18. 18 Substituting above values in equation 3.1- 3.3 we get CONCLUSION: The state of normal stress at any point in a fluid element at rest is same and directed towards the point from all directions. These stresses are denoted by a scalar quantity p defined as the hydrostatic or thermodynamic pressure. UNITS AND SCALES OF PRESSURE MEASUREMENT: Pascal (N/m2) is the unit of pressure. Pressure is usually expressed with reference to either absolute zero pressure (a complete vacuum) or local atmospheric pressure.  The absolute pressure: It is the difference between the value of the pressure and the absolute zero pressure.  Gauge pressure: It is the diference between the value of the pressure and the local atmospheric pressure(patm)  Vacuum Pressure: If p<patm then the gauge pressure becomes negative and is called the vacuum pressure.But one should always remember that hydrostatic pressure is always compressive in nature
• 19. 19 Fig 1.12 The Scale of Pressure At sea-level, the international standard atmosphere has been chosen as Patm = 101.32 kN/m2 MANOMETERS FOR MEASURING GAUGE AND VACUUM PRESSURE Manometers are devices in which columns of a suitable liquid are used to measure the difference in pressure between two points or between a certain point and the atmosphere.Manometer is needed for measuring large gauge pressures. It is basically the modified form of the piezometric tube. A common type manometer is like a transparent "U-tube" as shown in Fig. Fig 1.13 A simple manometer to measure gauge pressure
• 20. 20 Fig 1.14 A simple manometer to measure vacuum pressure One of the ends is connected to a pipe or a container having a fluid (A) whose pressure is to be measured while the other end is open to atmosphere. The lower part of the U-tube contains a liquid immiscible with the fluid A and is of greater density than that of A. This fluid is called the manometric fluid. The pressures at two points P and Q (Fig. 1.14) in a horizontal plane within the continuous expanse of same fluid (the liquid B in this case) must be equal. Then equating the pressures at P and Q in terms of the heights of the fluids above those points, with the aid of the fundamental equation of hydrostatics, we have Hence, Where p1 is the absolute pressure of the fluid A in the pipe or container at its centre line, and patm is the local atmospheric pressure. When the pressure of the fluid in the container is lower than the atmospheric pressure, the liquid levels in the manometer would be adjusted as shown in Fig. 4.5. Hence it becomes,
• 21. 21 ( MANOMETERS TO MEASURE PRESSURE DIFFERENCE A manometer is also frequently used to measure the pressure difference, in course of flow, across a restriction in a horizontal pipe. Fig 1.15 Manometer measuring pressure difference The axis of each connecting tube at A and B should be perpendicular to the direction of flow and also for the edges of the connections to be smooth. Applying the principle of hydrostatics at P and Q we have, where, ρ m is the density of manometric fluid and ρw is the density of the working fluid flowing through the pipe. We can express the difference of pressure in terms of the difference of heads (height of the working fluid at equilibrium).
• 22. 22 INCLINED TUBE MANOMETER:  For accurate measurement of small pressure differences by an ordinary u-tube manometer, it is essential that the ratio ρm/ρw should be close to unity. This is not possible if the working fluid is a gas; also having a manometric liquid of density very close to that of the working liquid and giving at the same time a well defined meniscus at the interface is not always possible. For this purpose, an inclined tube manometer is used.  If the transparent tube of a manometer, instead of being vertical, is set at an angle θ to the horizontal (Fig. 1.16), then a pressure difference corresponding to a vertical difference of levels x gives a movement of the meniscus s = x/sin𝜃 along the slope.  Fig 1.16 An Inclined Tube Manometer  If θ is small, a considerable mangnification of the movement of the meniscus may be achieved.  Angles less than 50 are not usually satisfactory, because it becomes difficult to determine the exact position of the meniscus.  One limb is usually made very much greater in cross-section than the other. When a pressure difference is applied across the manometer, the movement
• 23. 23 of the liquid surface in the wider limb is practically negligible compared to that occurring in the narrower limb. If the level of the surface in the wider limb is assumed constant, the displacement of the meniscus in the narrower limb needs only to be measured, and therefore only this limb is required to be transparent. INVERTED TUBE MANOMETER: For the measurement of small pressure differences in liquids, an inverted U-tube manometer is used. Fig 1.17 An Inverted Tube Manometer Here and the line PQ is taken at the level of the higher meniscus to equate the pressures at P and Q from the principle of hydrostatics. It may be written that where represents the piezometric pressure, (z being the vertical height of the point concerned from any reference datum). In case of a horizontal pipe (z1= z2) the difference in piezometric pressure becomes equal to the difference in the static pressure. If is sufficiently small, a large value of x may be obtained for a
• 24. 24 small value of . Air is used as the manometric fluid. Therefore, is negligible compared with and hence, Air may be pumped through a valve V at the top of the manometer until the liquid menisci are at a suitable level. HYDROSTATIC THRUSTS ON SUBMERGED PLANE SURFACE: Due to the existence of hydrostatic pressure in a fluid mass, a normal force is exerted on any part of a solid surface which is in contact with a fluid. The individual forces distributed over an area give rise to a resultant force. INCLINED PLANE SURFACES: Consider a plane surface of arbitrary shape wholly submerged in a liquid so that the plane of the surface makes an angle θ with the free surface of the liquid. We will assume the case where the surface shown in the figure below is subjected to hydrostatic pressure on one side and atmospheric pressure on the other side. Fig 1.18 Hydrostatic Thrust on Submerged Inclined Plane Surface
• 25. 25 Let p denotes the gauge pressure on an elemental area dA. The resultant force F on the area A is therefore Where h is the vertical depth of the elemental area dA from the free surface and the distance y is measured from the x-axis, the line of intersection between the extension of the inclined plane and the free surface. The ordinate of the centre of area of the plane surface A is defined as Hence from the above eqns, we get Where is the vertical depth (from free surface) of centre c of area. Implies that the hydrostatic thrust on an inclined plane is equal to the pressure at its centroid times the total area of the surface, i.e., the force that would have been experienced by the surface if placed horizontally at a depth hc from the free surface.
• 26. 26 Fig 1.19 Hydrostatic Thrust on Submerged Horizontal Plane Surface The point of action of the resultant force on the plane surface is called the centre of pressure . Let and be the distances of the centre of pressure from the y and x axes respectively. Equating the moment of the resultant force about the x axis to the summation of the moments of the component forces, we have Solving for yp we can write In the same manner, the x coordinate of the centre of pressure can be obtained by taking moment about the y-axis. Therefore,
• 27. 27 From which, The two double integrals in the numerators are the moment of inertia about the x-axis Ixx and the product of inertia Ixy about x and y axis of the plane area respectively. By applying the theorem of parallel axis Where, and are the moment of inertia and the product of inertia of the surface about the centroidal axes , and are the coordinates of the center c of the area with respect to x-y axes. With the help of above eqns it can be written as 1 2 The first term on the right hand side of the Eq. 1 is always positive. Hence, the centre of pressure is always at a higher depth from the free surface than that at which the centre of area lies. This is obvious because of the typical variation of hydrostatic pressure with the depth from the free surface. When the plane area is symmetrical about the y' axis, , and .
• 28. 28 HYDROSTATIC THRUSTS ON SUBMERGED CURVED SURFACES On a curved surface, the direction of the normal changes from point to point, and hence the pressure forces on individual elemental surfaces differ in their directions. Therefore, a scalar summation of them cannot be made. Instead, the resultant thrusts in certain directions are to be determined and these forces may then be combined vectorially. An arbitrary submerged curved surface is shown in Fig. 1.20. A rectangular Cartesian coordinate system is introduced whose xy plane coincides with the free surface of the liquid and z-axis is directed downward below the x - y plane. Fig 1.20 Hydrostatic thrust on a Submerged Curved Surface Consider an elemental area dA at a depth z from the surface of the liquid. The hydrostatic force on the elemental area dA is and the force acts in a direction normal to the area dA. The components of the force dF in x, y and z directions are
• 29. 29 Where l, m and n are the direction cosines of the normal to dA. The components of the surface element dA projected on yz, xz and xy planes are, respectively Substituting Eqs we can write Therefore, the components of the total hydrostatic force along the coordinate axes are Where zc is the z coordinate of the centroid of area Ax and Ay (the projected areas of curved surface on yz and xz plane respectively). If zp and yp are taken to be the coordinates of the point of action of Fx on the projected area Ax on yz plane, , we can write
• 30. 30 where Iyy is the moment of inertia of area Ax about y-axis and Iyz is the product of inertia of Ax with respect to axes y and z. In the similar fashion, zp ' and x p ' the coordinates of the point of action of the force Fy on area Ay, can be written as Where Ixx is the moment of inertia of area Ay about x axis and Ixz is the product of inertia of Ay about the axes x and z. We can conclude from the above eqns that for a curved surface, the component of hydrostatic force in a horizontal direction is equal to the hydrostatic force on the projected plane surface perpendicular to that direction and acts through the centre of pressure of the projected area., the vertical component of the hydrostatic force on the curved surface can be written as Where is the volume of the body of liquid within the region extending vertically above the submerged surface to the free surface of the liquid. Therefore, the vertical component of hydrostatic force on a submerged curved surface is equal to the weight of the liquid volume vertically above the solid surface of the liquid and acts through the center of gravity of the liquid in that volume.
• 31. 31 Buoyancy  When a body is either wholly or partially immersed in a fluid, a lift is generated due to the net vertical component of hydrostatic pressure forces experienced by the body.  This lift is called the buoyant force and the phenomenon is called buoyancy  Consider a solid body of arbitrary shape completely submerged in a homogeneous liquid as shown in Fig. 1.21. Hydrostatic pressure forces act on the entire surface of the body. Fig 1.21 Buoyant Forces on a Submerged Body
• 32. 32 To calculate the vertical component of the resultant hydrostatic force, the body is considered to be divided into a number of elementary vertical prisms. The vertical forces acting on the two ends of such a prism of cross-section dAz (Fig. 1.21) are respectively Therefore, the buoyant force (the net vertically upward force) acting on the elemental prism of volume is - Hence the buoyant force FB on the entire submerged body is obtained as Where is the total volume of the submerged body. The line of action of the force FB can be found by taking moment of the force with respect to z-axis. Thus Substituting for dFB and FB, the x coordinate of the center of the buoyancy is obtained as Which is the centroid of the displaced volume. It is found that the buoyant force FB equals to the weight of liquid displaced by the submerged body of volume . This phenomenon was discovered by Archimedes and is known as the Archimedes principle.
• 33. 33 ARCHIMEDES PRINCIPLE THE BUOYANT FORCE ON A SUBMERGED BODY  The Archimedes principle states that the buoyant force on a submerged body is equal to the weight of liquid displaced by the body, and acts vertically upward through the centroid of the displaced volume.  Thus the net weight of the submerged body, (the net vertical downward force experienced by it) is reduced from its actual weight by an amount that equals the buoyant force. THE BUOYANT FORCE ON A PARTIALLY IMMERSED BODY  According to Archimedes principle, the buoyant force of a partially immersed body is equal to the weight of the displaced liquid.  Therefore the buoyant force depends upon the density of the fluid and the submerged volume of the body.  For a floating body in static equilibrium and in the absence of any other external force, the buoyant force must balance the weight of the body STABILITY OF UNCONSTRAINED SUBMERGED BODIES IN FLUID  The equilibrium of a body submerged in a liquid requires that the weight of the body acting through its cetre of gravity should be colinear with an equal hydrostatic lift acting through the centre of buoyancy.  In general, if the body is not homogeneous in its distribution of mass over the entire volume, the location of centre of gravity G does not coincide with the centre of volume, i.e., the centre of buoyancy B.  Depending upon the relative locations of G and B, a floating or submerged body attains three different states of equilibrium- Let us suppose that a body is given a small angular displacement and then released. Then it will be said to be in
• 34. 34  STABLE EQUILIBRIUM: If the body returns to its original position by retaining the originally vertical axis as vertical.  UNSTABLE EQUILIBRIUM: If the body does not return to its original position but moves further from it.  NEUTRAL EQUILIBRIUM: If the body neither returns to its original position nor increases its displacement further, it will simply adopt its new position. STABLE EQUILIBRIUM Consider a submerged body in equilibrium whose centre of gravity is located below the centre of buoyancy (Fig. 1.22a). If the body is tilted slightly in any direction, the buoyant force and the weight always produce a restoring couple trying to return the body to its original position (Fig. 1.22b, 1.22c). Fig 1.22 A Submerged body in Stable Equilibrium UNSTABLE EQUILIBRIUM On the other hand, if point G is above point B (Fig. 1.23a), any disturbance from the equilibrium position will create a destroying couple which will turn the body away from its original position (1.23b, 1.23c).
• 35. 35 Fig 1.23 A Submerged body in Unstable Equilibrium NEUTRAL EQUILIBRIUM When the centre of gravity G and centre of buoyancy B coincides, the body will always assume the same position in which it is placed (Fig 1.24) and hence it is in neutral equilibrium. Fig 1.24 A Submerged body in Neutral Equilibrium Therefore, it can be concluded that a submerged body will be in stable, unstable or neutral equilibrium if its centre of gravity is below, above or coincident with the center of buoyancy respectively (Fig. 1.25).
• 36. 36 Fig 1.25 States of Equilibrium of a Submerged Body (a) STABLE EQUILIBRIUM (B) UNSTABLE EQUILIBRIUM (C) NEUTRAL EQUILIBRIUM STABILITY OF FLOATING BODIES IN FLUID  When the body undergoes an angular displacement about a horizontal axis, the shape of the immersed volume changes and so the centre of buoyancy moves relative to the body.  As a result of above observation stable equlibrium can be achieved, under certain condition, even when G is above B. Figure 1.26a illustrates a floating body -a boat, for example, in its equilibrium position. Fig 1.26 A Floating body in Stable equilibrium
• 37. 37 IMPORTANT POINTS TO NOTE HERE ARE a. The force of buoyancy FB is equal to the weight of the body W b. Centre of gravity G is above the centre of buoyancy in the same vertical line. c. Figure 1.26 b shows the situation after the body has undergone a small angular displacement 𝜃 with respect to the vertical axis. d. The centre of gravity G remains unchanged relative to the body (This is not always true for ships where some of the cargo may shift during an angular displacement). e. During the movement, the volume immersed on the right hand side increases while that on the left hand side decreases. Therefore the centre of buoyancy moves towards the right to its new position B'. Let the new line of action of the buoyant force (which is always vertical) through B' intersects the axis BG (the old vertical line containing the centre of gravity G and the old centre of buoyancy B) at M. For small values of the point M is practically constant in position and is known as met centre. For the body shown in Fig. 1.26, M is above G, and the couple acting on the body in its displaced position is a restoring couple which tends to turn the body to its original position. If M were below G, the couple would be an overturning couple and the original equilibrium would have been unstable. When M coincides with G, the body will assume its new position without any further movement and thus will be in neutral equilibrium. Therefore, for a floating body, the stability is determined not simply by the relative position of B and G, rather by the relative position of M and G. The distance of metacentre above G along the line BG is known as metacentric height GM which can be written as GM = BM -BG Hence the condition of stable equilibrium for a floating body can be expressed in terms of metacentric height as follows:
• 38. 38 GM > 0 (M is above G) Stable equilibrium GM = 0 (M coinciding with G) Neutral equilibrium GM < 0 (M is below G) Unstable equilibrium The angular displacement of a boat or ship about its longitudinal axis is known as 'rolling' while that about its transverse axis is known as "pitching". FLOATING BODIES CONTAINING LIQUID If a floating body carrying liquid with a free surface undergoes an angular displacement, the liquid will also move to keep its free surface horizontal. Thus not only does the centre of buoyancy B move, but also the centre of gravity G of the floating body and its contents move in the same direction as the movement of B. Hence the stability of the body is reduced. For this reason, liquid which has to be carried in a ship is put into a number of separate compartments so as to minimize its movement within the ship. PERIOD OF OSCILLATION The restoring couple caused by the buoyant force and gravity force acting on a floating body displaced from its equilibrium placed from its equilibrium position is . Since the torque equals to mass moment of inertia (i.e., second moment of mass) multiplied by angular acceleration, it can be written Where IM represents the mass moment of inertia of the body about its axis of rotation. The minus sign in the RHS of eqn arises since the torque is a retarding one and decreases the angular acceleration. If θ is small, sin θ=θ and hence Eq. can be written as
• 39. 39 The above equation represents a simple harmonic motion. The time period (i.e., the time of a complete oscillation from one side to the other and back again) equals to . The oscillation of the body results in a ﬂow of the liquid around it and this ﬂow has been disregarded here. In practice, of course, viscosity in the liquid introduces a damping action which quickly suppresses the oscillation unless further disturbances such as waves cause new angular displacements.
• 40. 40 MODULE 2 KINEMATICS OF FLUID MOTION “Fluid kinematics is the branch of fluid mechanics which deals with the study of velocity and acceleration of the particles of fluid in motion and their distribution in space without considering the force caring them.” EULERIAN AND LAGRANGIAN APPROAH: EULERIAN APPROAH: This method describes the overall flow characteristics at various points as fluid particles pass. Thus it gives the various quantities such as velocity, acceleration, pressure, density etc. at any point in space and time, from a purely mathematical view point. The eulerian method is preferable as the resulting equation can be solved easily. The velocities at any point(x, y, z) can be written as U=f1(x, y, z, t) V=f2(x, y, z, t) W=f3(x, y, z, t) Identities of the particles are made by specifying their initial position at a given time. It is analytically represented as  0s s s , t r r Where s r is the position vector of a particle at a time with its initial position 0s r . LAGRANGIAN APPROACH: In this method, the fluid motion is described by tracing the kinematic behavior of each and every individual particle constituting the flow. Identities are made by specifying their initial position at a given time. The position of the particle at any instant of time then becomes a function of its identity and time.
• 41. 41 Analytical expression of the statement: (1) s Is the position vector of a particle (with respect to a fixed point of reference) at a time t. Is its initial position at a given time t =t0 Equation (1) can be written into scalar components with respect to a rectangular Cartesian frame of coordinates as: x = x(x0, y0, z0, t) z = z(x0, y0, z0, t) y = y(x0, y0, z0, t) Where, x0, y0, z0 are the initial coordinates and x, y, z are the coordinates at a time t of the particle. Hence in can be expressed as , , and are the unit vectors along x, y and z axes respectively. CLASSIFICATION OF FLUIDS: The fluid flow is classified as: a) Steady and unsteady flow. b) Uniform and non uniform flow. c) Laminar and turbulent flow. d) Compressible and incompressible flow.
• 42. 42 e) One, two, and three dimensional flow. a) Steady and Unsteady Flow: A steady flow is defined as a flow in which the various hydrodynamic parameters such as velocity, pressure, density, etc and fluid properties at any point do not change with time. u v p 0; 0, 0 t t t t t                                          An unsteady Flow is defined as a flow in which the hydrodynamic parameters such as velocity, pressure, density, etc and fluid properties changes with time. v p 0 0 t t                b) Uniform and Non-uniform Flow: Uniform Flow: The flow is defined as uniform flow when in the flow field the velocity and other hydrodynamic parameters do not change from point to point at any instant of time For a uniform flow, the velocity is a function of time only, which can be expressed in Eulerian description as Implication: 1. For a uniform flow, there will be no spatial distribution of hydrodynamic and other parameters. 2. Any hydrodynamic parameter will have a unique value in the entire field, irrespective of whether it
• 43. 43 Non-uniform Flow: When the velocity and other hydrodynamic parameters changes from one point to another the flow is defined as non-uniform. Important points: 1. For a non-uniform flow, the changes with position may be found either in the direction of flow or in directions perpendicular to it. 2. Non-uniformity in a direction perpendicular to the flow is always encountered near solid boundaries past which the fluid flows. FOUR POSSIBLE COMBINATIONS Type Example 1. Steady Uniform flow Flow at constant rate through a duct of uniform cross-section (The region close to the walls of the duct is disregarded) 2. Steady non-uniform flow Flow at constant rate through a duct of non- uniform cross-section (tapering pipe) 3. Unsteady Uniform flow Flow at varying rates through a long straight pipe of uniform cross-section. (Again the region close to the walls is ignored.) 4. Unsteady non-uniform flow Flow at varying rates through a duct of non- uniform cross-section.
• 44. 44 C) laminar and turbulent flow: Laminar flow: This term is generally used in pipe flow. A laminar flow is one in which the fluid particles move in layers or laminar with one layer slide over the other. Therefore there is no exchange of fluid particles from one layer to the other and hence no transfer of lateral momentum to the adjacent layers. Fig.1 laminar flow As the particles move in definite and observable paths or streamlines this flow is sometimes known as streamline flow. it is a smooth and regular flow. The velocity of flow is very small. The dimensionless Reynolds number is an important parameter in the equations that describe whether flow conditions lead to laminar or turbulent flow. In the case of flow through a straight pipe with a circular cross-section, Reynolds numbers of less than 2300 are generally considered to be of a laminar type .however, the Reynolds number upon which laminar flows become turbulent is dependent upon the flow geometry. When the Reynolds number is much less than 1, creeping motion or Stokes flow occurs. This is an extreme case of laminar flow where viscous (friction) effects are much greater than inertial forces. Turbulent flow: Fig.2 turbulent flow
• 45. 45 When the flow attains a certain velocity, it no longer remains steady and eddy currents appear. The velocities in such a flow vary from point to point in magnitude and direction as well as from instant to instant. All the fluid particles are disturbed and they mix with each other. On the basis of Reinhold’s number we can classify Laminar and turbulent flow. Re < 2000 flow in pipes is laminar Re > 4000 flow in pipes is turbulent Re between 2000 and 4000 flow may be laminar or turbulent. d) Compressible and incompressible flow: Compressible flow: Compressible flow is that type of flow in which the density of the fluid changes from point to point or in other words the density is not constant for the fluid. Thus mathematically, for compressible flow ρ ≠ constant Incompressible flow: Incompressible flow is that type of flow in which the density of the fluid is constant for the fluid flow. Liquids are generally incompressible while gases are compressible. Mathematically, for compressible flow, ρ =constant e) One, two, and three dimensional flow. Fluid flow is three-dimensional in nature. This means that the flow parameters like velocity, pressure and so on vary in all the three coordinate directions. Sometimes simplification is made in the analysis of different fluid flow problems by selecting the appropriate coordinate directions so that appreciable variation of the hydro dynamic parameters take place in only two directions or even in only one.
• 46. 46 One-dimensional flow: All the flow parameters may be expressed as functions of time and one space coordinate only. The single space coordinate is usually the distance measured along the centre-line (not necessarily straight) in which the fluid is flowing. Mathematically, u = f(x), v = 0,  = 0 Where u, v,  are velocity components in x, y and z directions. Example: The flow in a pipe is considered one-dimensional when variations of pressure and velocity occur along the length of the pipe, but any variation over the cross-section is assumed negligible. Fig.3 One-dimensional ideal flow along a pipe In reality, flow is never one-dimensional because viscosity causes the velocity to decrease to zero at the solid boundaries. If however, the non uniformity of the actual flow is not too great, valuable results may often be obtained from “one dimensional analysis” .The average values of the flow parameters at any given section (perpendicular to the flow) are assumed to be applied to the entire flow at that section. Two-dimensional flow: All the flow parameters are functions of time and two space coordinates (say x and y). No variation in z direction. The same streamline patterns are found in all planes perpendicular to z direction at any instant.
• 47. 47 Fig.4 Flow past a car antenna is approximately two-dimensional, Two dimensional flow – flow parameter such as velocity is a function of time and two rectangular space co-ordinates is called 2-D flow. Mathematically u = f1 (x, y) v = f2 (x, y) w = 0 Three dimensional flow: The hydrodynamic parameters are functions of three space coordinates and time. It is that type of flow in which the velocity is a function of time and three mutually perpendicular directions. Mathematically v = f(x, y, z, t) - unsteady flow v = f(x, y, z) - steady flow REPRESENTATION OF FLUID FLOW: Path line: A path line is the actual path traversed by a given (marked) fluid particle. A path line is hence also an integrated pattern. A path line represents an integrated history of where a fluid particle has been. In general it is a curve in a 3 – D space. If the flow is 2D, the curve becomes two dimensional
• 48. 48 Fig.5 path line Stream line: A streamline is a line that is everywhere tangent to the velocity vector at a given instant of time. A streamline is hence an instantaneous pattern. Fig.6 stream line Equation for a streamline Following points about streamlines are worth noting. a) A streamline cannot intersect itself or two streamlines cannot cross. Because the fluid is moving in the same direction as the streamlines, fluid cannot cross
• 49. 49 a streamline. Streamlines cannot cross each other. If they were to cross, this would indicate two different velocities at the same point. This is not physically possible. The above point implies that any particles of fluid starting on one streamline will stay on that same streamline throughout the fluid b) There cannot be any movement of the fluid mass across the streamlines. c) Streamline sparing varies inversely as the velocity converging of streamlines at any particular direction shows accelerated flow in that direction. d) Where as path line indicates the path of one particular particle at successive instants of time, streamline indicates the direction of flow of a number of particles. e) Series of streamlines represent the flow pattern at an instant. In steady flow, streamlines remains invariant with time. The path lines and streamlines will then be identical. In unsteady flow, pattern of streamline may not be same. The mathematical expression of a streamline can also be obtained from Where V is the fluid velocity vector and dris a tangential vector along the streamline. The above cross product is zero since the two vectors are in the same direction. A useful technique in fluid flow analysis is to consider only a part of the total fluid in isolation from the rest. This can be done by imagining a tubular surface formed by streamlines along which the fluid flows. This tubular surface is known as a stream tube, which is a tube whose walls are streamlines. Since the velocity is tangent to a streamline, no fluid can cross the walls of a stream tube. Stream tube: Is a fluid mass bounded by a group of stream lines or is an imaginary tube formed by a group of stream lines passing through a small closed curve which may or may not be circular. The contents of stream tube is known as current filament. Eg:- pipes and nozzles
• 50. 50 a) Stream tube has finite dimensions b) Stream surface functions as if it were a solid wall. c) Shape of a stream tube changes from one instant to another because of the change in the position of streamlines. Fig.7 Stream tube Streak line: FIG.8 Streak line:
• 51. 51 A streak line is an instantaneous line whose points are occupied by particles which have earlier passed through a prescribed point in space. A streak line is hence an integrated pattern. A streak line can be formed by injecting dye continuously into the fluid at a fixed point in space. As time marches on, the streak line gets longer and longer, and represents an integrated history of the dye streak. Eg:- path taken by smoke coming out of chimney. For steady flow, streamline, streak line and path line coincides. For unsteady flow, streamline, streak line and path line are all different Three basic principles are used in the analysis of the problems of fluid in motion: a) Principle of conservation of mass states that mass can neither be created nor destroyed. On the basis of this principle, the continuity equation is derived. b) Principle of conservation of energy – States that energy can neither be destroyed nor be created. On the basis of this principle energy equation is derived. c) Principle of conservation of momentum or Impulse momentum principle states that the impulse of the resultant force, or the product of the force and time increment during which it cuts, is equal to the change in momentum of the body. On the basis of this principle momentum equation is derived. CONTINUITY EQUATION The continuity equation is a statement of the conservation of mass in a system. A most general expression on the basis of this principle may be obtained by considering a fixed region within the flowing fluid. Since the fluid is neither created nor destroyed in this region, it may be stated that “Rate of increase of fluid mass within the region must be equal to the rate at which the fluid mass enters the region and the rate at which the fluid mass leaves the region.”
• 52. 52 Consider a pipe which is uniform in diameter at both ends, but has a constriction between the ends as in figure. This is called a venturi tube. Stations 1 and 2 have cross-sectional areas A1 and A2, respectively. Let V1 and V2 be the average flow speeds at these cross sections (one-dimensional flow). A further assumption is that there are no leaks in the pipe nor is fluid being pumped in through the sides. The continuity equation states that the fluid mass passing station 1 per unit time must equal the fluid mass passing station 2 per unit time Fig.9 venturi tube (Mass rate)1 = (Mass rate)2 (1) Mass rate = Density x Area x Velocity This equation (1) reduces to pl AlV1 = p2A2V2 (2) Since the fluid is assumed to be incompressible, p [Greek letter rho] is a constant and equation (2) reduces to AlV1 = A2V2 (3)
• 53. 53 This is the simple continuity equation for inviscid, incompressible, steady, one- dimensional flow with no leaks. CONTINUITY EQUATION IS CARTESIAN CO-ORDINATES: Consider a rectangular parallelopiped element with sides of length x, y, z. Let p(x, y, z) the centre of parallelopiped, where the velocity components in x, y, z direction v, v1 and  respectively and  be the mass density. Fig.10 CARTESIAN CO-ORDINATES Mass of fluid passing per unit time through the face with area y z normal to x-axis through point  is (y, y, z) Mass of fluid flowing per unit time through face ABCD is   x u y z u y z x 2                  . Mass of fluid flowing per unit time through force A B C D    is   x u y z u y z x 2                 
• 54. 54 Net mass of the fluid remaining in the paralleopiped per unit time through the force ABCD and A B C D    is     x x u y z u y z u y z u y z x 2 x 2                                       =  u y z x x        =  u y z x x        (1) By applying the same method for the faces DCCD and ABBA and ADDA and BCCB we obtain, =  u x y z y        (2) =  y x y z z        (3) By adding (1), (2) and (3) net total mass of fluid that has remained in the parallelopiped is      u v x y z x y z                   Since fluid is neither created nor destroyed in the paralleopiped, any increase in the mass of the fluid contained in this space per unit time is equal to the net total mass of fluid that remained in this parallelopiped per unit time. The mass of the fluid in parallelopiped is x y z    and its rate of increase with time is  x y z x y z t t             So      u v 2 w x y z x y z x y z t                      
• 55. 55      u v x y z t                        y v v 0 t x y z               (4) (4) Represents continuity equation in Cartesian co-ordinates which is applicable for steady and unsteady flow, uniform and non-uniform flow and compressible as well as incompressible flow. For steady flow t   = 0 So (4) reduces to      u v w 0 x y z             (5) For incompressible fluid, 0 x y z t             So (4) reduces to u v w 0 x y z          (6) CONTINUITY EQUATION IN CYLINDRICAL POLAR CO- ORDINATES: Consider any point p(r, , 2) in space. Let r,  and z be the small increments in the direction r, , z respectively. So that 1PS r PQ r , PT z      . Construct an elementary parallelopiped. Lt Vr, V and Vz be the components of velocity V in the direction of r, , z at point . Let be the mass density of fluid at point P.
• 56. 56 Fig.11 CYLINDRICAL POLAR CO-ORDINATES Consider the forces PSTQ and PSTQ, the mass of fluid entering paralleopiped per unit time through the face  PST Q PVz r r     Mass of the fluid leaving the paralleopiped per unit time through the face P S TQ      2V r r r V r . r z z             Wet mass of fluid that has remained in the parallelopiped per unit time through this pair of faces. =  2. V r, r z z        Similarly the net mass of fluid that remains in the parallelopiped per unit time through the pair of faces PTQ S and P T QS   . =  r zV ,        and that through the pair of faces PQS T  and P Q ST  =  r zV r , r       
• 57. 57 Net total mass of fluid that has remained in the parallelopiped per unit time through all the three pair of faces =      r 2 r z VV . r V r r r r z                  The mass of fluid in parallelopiped, =  r z r     and its rate of increase with time =  r r t       =  r z r t       Thus equating the two expressions      r 2 r z VV . r V . r r r z                  =  r z r t      Dividing both sides by volume of paralleopiped and taking the limit so as to reduce the paralleopiped to point P, continuity equation is obtained as      r 2 r VV .r V 0 t r r z                EQUATION FOR ACCELERATION: The Cartesian vector form of a velocity filed can be written as: The flow filed is the most important variable in the fluid mechanics, i.e., knowledge of the velocity vector filed is equivalent to solving a fluid flow problem. The acceleration vector field can be calculated:
• 58. 58 Where the compact dot products is: With a similar approach, we obtain the total acceleration vector: The term is called the local acceleration and defined as the rate of increase of velocity with respect to time at a given point in the flow field. It vanishes if the flow is steady. The three terms in the parentheses are called the convective acceleration and defined as the rate of change of velocity due to the change of position of fluid particles in a fluid flow. It rises when the particles move through regions of spatially varying velocity, e.g. nozzle.
• 59. 59 ROTATIONAL FLOW: In this type of flow the fluid element has net rotation about its own axis which is perpendicular to the plane of motion. For example the fluid in the rotating tank. Fig: 12 rotational flow IRROTATIONAL FLOW: Fig: 13irrotational flow In this type of flow the fluid element has no net rotation about its own axis which is perpendicular to the plane of motion. However the fluid element may be deformed and distorted. VELOCITY POTENTIAL: The flow takes place in the pipe if there is a difference of pressure. The direction of flow will take place from a higher to the lower pressure. Thus the velocity of flow in
• 60. 60 a certain direction will depend upon the potential difference which is known as velocity potential. STREAM FUNCTION: It is defined as the scalar function of space and time such that its partial derivative with respect to any direction gives the velocity component at right angles to that direction. It is denoted by ψ (psi) and defined only for two dimensional flow. Mathematically for steady flow it is defined as ψ=f (x, y) such that Properties: a) If the stream function (ψ) exists, it is a possible case of fluid flow which may be rotational or irrotational. b) If stream function (ψ) satisfies the Laplace equation it is possible case of an irrotational flow. CIRCULATION: Circulation for a fluid element as shown in Fig is defined as the line integral of the tangential component of velocity and is given by Fig:13 circulation
• 61. 61 Where ds is an element of contour and the loop through the integral sign signifies that the contour is closed. VORTEX FLOW: Vortex flow is defined as the flow of a fluid along a curved path or the flow of a rotating mass of fluid .the vortex flow is of two types namely: a) forced vortex flow b) Free vortex flow. FORCED VORTEX FLOW: Forced vortex flow is defined as that type of vortex flow, in which some external torque is required to rotate the fluid. The mass in this type of flow rotates at constant angular velocity ω. The tangential velocity of any fluid particle is given by V= ω*r r=radius of particle from the axis of rotation. Example: Flow of liquid inside the impeller of a centrifugal pump. Flow of water through the runner of the turbine. FREE VORTEX FLOW: When no external torque is required to rotate the fluid mass that type of flow is called free vortex flow. Thus the liquid in case of free vortex is rotating due to the rotation which is imparted to the fluid previously. Examples: A whirlpool in a river. Flow of fluid in a centrifugal pump casing.
• 62. 62 VORTICITY: It is defined as the value twice of rotation and hence it is given as=2 ω. BASIC FLOW: Uniform flow: In uniform flow the velocity of the fluid remains constant. All the fluid particles are moving with the same velocity. The uniform flow may be: a) Parallel to x- axis. b) Parallel to y axis. Fig. 14 uniform flow Source flow: Fig.15 source flow The source flow is the flow coming from a point (source) and moving out radially in all directions of a plane at uniform rate. The strength of a source is defined as the volume flow rate per unit depth. The unit of strength of the source is m2/s.it is represented by q. Let ur= radial velocity of flow at a radius r from the source.
• 63. 63 q= volume flow rate per unit depth. r=radius. The radial velocity ur at any radius r is given by, ur =q/(2*π*r) The above equation shows that with increase of r, the radial velocity decreases. And at a large distance away from the source, the velocity will be approximately equal to zero. Sink flow: The sink flow is the flow in which fluid moves radially inwards towards a point where it disappears at a constant rate. The flow is just opposite to the source flow. The fig shows a sink flow. Fig.16 sink flow The pattern of stream lines and equi potential lines of a sink is the same as that of a source flow. all the equations derived for a source flow shall hold to good for sink flow also except that in sink flow equations,q is to be replaced by(-q) . Doublet: The flow pattern due to a doublet is obtained when a source and a sink of equal strength q are considered to approach one another under such conditions that their strengths increase, as the distance between them approaches zero.
• 64. 64 Fig.17 doublet FLOW PAST A CYLINDER WITH A CIRCULATION: Fig.18 flow past a cylinder with a cylinder A cylinder of infinite length kept across a uniform flow of fluid horizontally, the stream lines are symmetrical on the top and bottom of the cylinder. For every point on the upper side of the cylinder there is a corresponding point on the lower side having the same pressure and velocity and so there can be no lift force on the cylinder. If the cylinder is rotated about its longitudinal axis anti-clock wise, the stream lines will take up a path as shown in the fig. at the upper surface of the cylinder the force will be slightly greater velocity than at the lower part. The addition of velocity on the upper half is due to the fact that the rotating cylinder adds a little velocity to the air adjacent to it. At the same time the rotating surface in the lower part retards the flow. The velocity at the lower half has a magnitude given by the difference between the velocity of flow and the velocity imparted due to rotation. The region of lower velocity will exert more pressure than the region of higher velocity there by providing a lifting force on the cylinder considered.
• 65. 65 MAGNUS EFFECT: When a body (such as a sphere or circular cylinder) is spinning in a fluid, it creates a boundary layer around itself, and the boundary layer induces a more widespread circular motion of the fluid. If the body is moving through the fluid with a velocity V the velocity of the fluid close to the body is a little greater than V on one side, and a little less than V on the other. This is because the induced velocity due to the boundary layer surrounding the spinning body is added to V on one side, and subtracted from V on the other. In accordance with Bernoulli's principle, where the velocity is greater the fluid pressure is less; and where the velocity is less, the fluid pressure is greater. This pressure gradient results in a net force on the body, and subsequent motion in a direction perpendicular to the relative velocity vector Fig.18 Magnus effect COEFFICENT OF LIFT: The lift coefficient (CL or CZ) is a dimensionless coefficient that relates the lift generated by an airfoil, the dynamic pressure of the fluid flow around the airfoil, and the plan form area of the airfoil. It may also be described as the ratio of lift pressure to dynamic pressure.
• 66. 66 QUESTIONS 1. Find the direction of stream lines and equipotential lines in uniform flow when uniform flow is along – (1) x – axis (2) y – axis (3) an angle to x-axis. Case I. Uniform flow along x-axis with constant velocity u. u U y x         v 0 x x         Uy and Ux    Hence stream lines are parallel to x-axis and equipotential lines are parallel to y-axis. Case 2. Uniform flow parallel to y axis ie, u = 0 and v = U u 0 y x        v U y y         Ux    and Uy  Hence stream lines are parallel to y-axis and equipotential lines are parallel to x-axis. Case 3. A uniform flow inclined at angle ‘’ to x-axis. Such flow can be considered to flow at velocity U cos  in x-direction and U sin  in y-direction. In case cylindrical co-ordinates are used, then x = r cos  and y = r sin .  rU sin      and  rU cos     Stream lines and equipotential lines in uniform flow (u = U cos  & v = U sin )
• 67. 67 2. What is a source flow? What is the source strength? Find the direction of stream lines and equipotential lines in a source. Source: In two-dimensional flow, a source is a point from which fluid discharge radially outwards in all directions. A point source is plane source (two-dimensional). A line source as compared to a plane source is three –dimensional. In fact a cross-section of a line source is a plane point source. The point source remains at the origin ‘O’ and fluid flows radially outwards from the source. The radial velocity (ur) of the flow at a distance ‘r’ from the origin is given by: r q k u 2 r r    where q = discharge per unit length and k = source strength. or q k 2   Source strength (k). The source strength is a measure of the radial velocity at unit radius. It is defined as volume flow per unit depth q k 2       . r k u r  when r = 1, then ur = k Stream function for source flow r I u r    and u    but r q u 2 r   1 q r 2 r     or q 2     or 1 q C 2     
• 68. 68 But 0  when  = 0, hence C1 = 0 q 2      The stream function is a function of angle ‘’. Equipotential function of source flow rU r    and 1 U r     r q u 2 r r       q 1 or . r 2 r     q r 2 r      q logr 2    Hence equipotential function is the function of ‘r’ (the distance from the source origin) 3. Derive an expression of pressure distribution in source flow We consider two points in the source flow to derive an expression of pressure distribution in source flow. First point at r = 1 and other point at large distance away from the source where radial velocity ur = 0. Now apply Bernoulli’s equation – 2 2 2 1 1 2 2 1 2 p u p u z z g 2g g 2g        Now as flow is in one plane, hence 1 2z z , P1 = P. ie, pressure at point r = 1 and u1 = ur. P2 = P0 ie, pressure at large distance from source where u2 = ur = 0. 2 0 rP P u g 2g   
• 69. 69 2 r 0 u P P 2     Now r q u 2 r   2 0 q P P 2 2 r           2 2 2 q 8 r    Hence the pressure increases inversely proportional to the square of the radius from the source. 4. What is a sink flow? What is the pressure variation in a sink flow? A sink is a source with a negative strength. Fluid flows radially inwards in a plane towards a point where the fluid disappears at a constant rate. For a sink flow – r q u kr 2 r     where q k 2    = sink strength The pressure variation in a plane sink flow is given by the same expression as applicable in the source flow – 2 2 0 2 2 2 k q P P 2r 8 r       5.What is the Langrangian method of describing fluid motion? In Langrangian method, the motion of each particle is defined with respect to its location in space and time from a fixed position before the start of the motion. The movement of single particle is observed which gives the path followed by the particle. The position of the fluid particle in space (x, y, z) at any time from its fixed position (a, b, c) at t = 0, shall be x = f1(a, b, c, t) y = f2 (a, b, c, t) z = f3 (a, b, c, t)
• 70. 70 From the position, velocities, u, v and w and acceleration ax, ay, and az can be given as - (1) Velocities x y x u ,v & w t t t          (2) Accelerations 2 2 2 x y z2 2 2 x y x a ,a &a t t t          The resultant of velocities and acceleration can be calculated. The resultant of pressure and densities can also be worked out. 6. What is the Eulerian method of describing fluid motion? In this method, instead of concentrating on single fluid particle and its movement, concentration is made on all fluid particles passing through a fixed point. The changes of velocity, acceleration, pressure and density of the fluid particles as they pass through the fixed point are observed. Hence the Eulerian method describes the overall flow characteristics at various points as fluid particles pass through them. The velocity at any point (x, y, z) can be given as u = f1(x, y, z, t), v = f2(x, y, z, t), w = f3(x, y, z, t) 7. What is the difference between Langrangian and Eulerian method? Langrangian Eulerian 1. Observer concentrates on the movement of single particle. 2. Observer has to move with the fluid particle to observe its movement. 3. The path and changes in velocity, acceleration, pressure and density of a single particle are described. 4. Not commonly used. 1. Observer concentrates on various fixed point particles 2. Observer remains stationary and observes changes in the fluid parameters at the fixed point only. 3. The method describes the overall flow characteristics at various points as fluid particles pass. 4. Commonly used.
• 71. 71 8. What are various lines of flow to describe the motion of the fluid? The various lines of flow are: (1) path line, (2) stream line, (3) stream tube, (4) streak line or filament line and (5) potential lines. 9. What is a path line in fluid motion? What happens to path line during steady and unsteady flow? The path traced or taken by a single fluid particle in motion over a period of time is called its path line. Hence path line is a locus of a fluid particle as it moves along. The path line shows the direction of velocity of the particle as it moves ahead. During steady flow, path line concides with the stream line as there is no fluctuation in velocity. However the path line fluctuates between different stream lines during an unsteady flow. 10. What is a stream line? A stream line is an imaginary line drawn through a flowing fluid such that the tangent at each point on the line indicates the direction of the velocity of the fluid particle at that point. For example, if ABC is a stream line, then the direction of velocities of A, B and is C as shown 11. Explain the following (1) the value of velocity at right angle to the stream line (2) can there be any flow across the stream line and (3) can two stream lines cross each other? Since velocity of the fluid particle at any point on the stream line is tangential to the stream line, there cannot by any component of velocity normal or right angle to the stream line i.e. the component of velocity at right angle to the stream line is zero. There cannot be any flow across the stream line as the flow is always tangential to the stream line. Two stream lines cannot cross each other as otherwise there would be two velocities at that point, one each tangential to the stream lines. This is inconsistent with the definition of a stream line.
• 72. 72 12. How does a stream line behave in the vicinity of a solid surface? The stream line in the vicinity of a solid surface conforms to the outline of the boundary surface. For example, the stream lines ariund a solid cylinder and within a closed conduit are as shown in the figure 13. What happens to stream lines during steady and steady flow conditions? In steady flow, the pattern of stream lines remains unchanging with time. However the pattern of stream lines may or may not remain same with time for unsteady flow. If unsteady flow is due to the change of magnitude of velocity, the stream line pattern remains invariant (unchanging) with time. However if the unsteadiness due to the change in the direction of the velocity, the stream line pattern cannot remain same. 14. What is a stream tube? The stream tube consists of stream lines forming its boundary surface. The stream tube is defined as a circular space formed by the collection of stream lines passing through the perimeter of a closed curve in a steady flow. As stream tube is bounded on all sides by stream lines therefore no fluid can enter or leave the stream tube from the sides except from the ends. Hence stream tube behaves as a solid surface tube. The general equation of continuity can be applied on stream tube though it has no solid boundaries. Stream tube may be of regular or irregular shape. The velocity is uniform throughout the cross-section of the stream tube which necessitates small cross-section for the stream tube. In steady flow with uniform velocity, all stream lines are straight and parallel. The contents of stream tube are called current-filaments. 15. What do you understand from streak line or filament line? It is an instantaneous picture of the positions of all fluid particles in the flow which have passed or emerged from a given point. The line of smoke from a cigarette or from a chimney is nothing but a streak line. 16. What is a potential line? The lines of equal velocity potential are called potential lines. These
• 73. 73 potential lines cut stream lines orthogonally i.e. a stream line and a potential line are at right angle to each other. 17. Distinguish between stream lines, streak lines and path lines Stream lines Streak lines Path lines 1. Imaginary lines showing positions of various fluid particles 2. Particle may change stream line depending on type of flow 3. Stream lines cannot intersect each other, they are always parallel 4. No flow across stream line 1. Real line showing instantaneous positions of various particles 2. May change from instant to instant 3. Streak line changes with time. Two streak lines may intersect each other 4. Flow across streak line is possible 1. Real line showing successive position of one particle 2. Particle may cross its path line 3. Two path lines for two particles may intersect each other 4. Flow across a path line is possible by other particles 18. What are different types of fluid flow? The fluid flows can be: (1) steady or unsteady (2) uniform or non-uniform (3) laminar or turbulent (4) compressible or incompressible (5) rotational or irrotational (6) one-, two- or three-dimensional 19. Define steady and unsteady flow with one particle example The flow in which fluid characteristics like velocity, acceleration, pressure or density do not change with time at any point in the fluid is called steady flow. A flow of water with constant discharge through a pipeline is an example of steady flow. For steady flow, we have the following: 2 2 v v dP 0, 0, 0, 0 t t tt           The flow in which fluid characteristics like velocity, acceleration, pressure
• 74. 74 and density change with time at any point is called unsteady flow. The water with varying discharge through a pipe is an unsteady flow. For unsteady flow, we have: 2 2 v v dP 0, 0, 0,& 0 t t tt           In steady flow, the path line and stream line will concide. In unsteady fluid flow, the path line of successive particle will be different and stream line pattern of the flow will be changing at every instant. 20. Define laminar and turbulent flow with one practical example. A laminar flow is one in which the fluid particles move in layers with each layer sliding over the other. There is no movement of fluid particles from one layer to another. Flow of blood in small veins and oil flow in bearings are examples of laminar flow. It is a smooth regular flow where velocity of flow is very small The flow in which adjacent layers cross each other and the layers do not move along the well defined path is called turbulent flow. The flow through rivs or canals, smoke from chimney or cigarette are turbulenter flow. 21. How can flows be classified laminar or turbulent or transit flow? The flow can be classified by Reynolds number. The Reynolds number is the ratio of inertia force to viscous force. The viscous force tends to make the motion of fluid in parallel layers while inertia force (mass x acceleration) tends to diffuse the fluid particles. Higher viscous force (higher increase in velocity or inertial force) are turbulent flow which have higher values of Reynolds number. Reynolds number = Inertiaforce VL Viscousforce    For flow in pipes, the flow can be: (a) Laminar flow if Reynolds number < 2100 (b) Transit flow if Reynolds number is in between 2100 to 4000 (c) Turbulent flow if Reynolds number > 4000
• 75. 75 22. What do you understand by compressible and incompressible flows? The flow in which the volume and thereby the density of fluid does not remain constant during the flow is called compressible flow. Gases are most compressible. The compressibility related problems are observed when gases flow through orifice, nozzle, turbine, compressor and in flight of planes. The flow in which the changes in volume and thereby in density of the fluid are insignificant, the flow is said is to be incompressible. Flow of liquids are incompressible flow. For gases, in subsonic aerodynamic conditions air flow can be considered incompressible. 23. Differentiate one-, two- and three-dimensional flow. 1. One-dimensional flow: One -dimensional flow is a flow in which the velocity of the flow is a function of time and one space coordinate (x, y, or z). The flow through a pipe is one-dimensional flow. For one-dimensional flow, we have for (a) Steady flow: V = f(x) (b) Unsteady flow: V = f(x, t) Fig. 2. Two-dimensional flow: Two-dimensional flow is a flow in which the velocity of the flow is a function of two space coordinates (xy, xz or yz) and time. The flows between two parallel plates of large extent, flow over a long spillway and flow at the middle part of the wing of an aircraft are two-dimensional flows. For two- dimensional flow, we have for: (a) Steady flow: v = f(x, y) (b) Unsteady flow: v = f(x, y, t) Fig. 3. Three-dimensional flow: Three-dimensional flow is a flow in which the velocity of the flow is a function of three space coordinates (x, y, z) and time. Flow in converging or diverging pipe section is a three-dimensional flow. For three-dimensional flow, we have for -
• 76. 76 (a) Steady flow: v = f(x, y, z) (b) Unsteady flow: v=f(x, y, z, t) 24. What is discharge of rate of flow (Q)? The discharge is the amount of fluid flow per unit time. Hence discharge is Q = Volume Time = Area length Time  = Area  velocity = A  V The discharge can also be defined as cross-sectional area of the flow multiplied by the velocity of flow. 25. What is the principle of conservation of mass? The principle of conservation of mass states that matter cannot be created or destroyed in non-nuclear processes. 26. What is the equation of continuity of flow? The equation of continuity of flow is based on the principle of conservation of mass. It states that the mass of fluid entering the stream tube from one end must be equal to the mass of fluid leaving the stream tube at the other end per unit time and there is no accumulation of fluid in the stream tube. 27. What do you understand from rotation and Vorticity of fluid particles? A flow is said to be rotational if fluid particles are rotating about their own mass centres. The rotation of a fluid particle at a point is specified by funding the average angular velocity of two perpendicular linear elements of fluid particle. Rotation of the fluid particle takes place about an axis which is perpendicular to the plane formed by these two elements of the fluid. The sense of rotation is given by the right hand rule for the motion of a right-hand screw. Vorticity of a rotation flow has twice the value of the rotation of the flow. Both rotation and Vorticity are vectors.
• 77. 77 28. What are the conditions for irrotational flow? The irrotational flow will have both rotation and vorticity as zero. w = 0 and 0  Or x y zw w w 0,   and x y z 0      The above gives the conditions of irrotation of the flow as: w v y z      u w z x      w u x y      Note: The conditions in polar coordinates is 0  which gives rv V 1 V 0 r r r         29. What is circulation? How is it related to vorticity? Circulation is defined as the line integral of the tangential velocity of the fluid around a closed curve  Circulation = c V ds  Ñ Where V = tangential velocity & ds = small length of curve. Fig. Consider a number of stream lines in a closed curve. Let tangential and normal components of velocity V are V = V cos  and rV = V sin . If we calculate circulation around a rectangular rotating element of sizedx dy , then we get - c Vcos ds   Ñ
• 78. 78 v u udx v dx dy u dy dx vdy y y                    v u dx.dy x y          z dx dy    z   Area enclosed by closed curve in plane at right angle to z-axis Hence vorticity is the circulation per unit surface area.  A    Where A = are enclosed by closed curved Or v u dxdy x y             30. Explain the physical significance and use of the term ‘stream function’ The flow rate (q) between two stream lines per unit thickness and time is known as stream function (). Physical significance of stream function Consider a flow between two stream lines AA and BB which is two- dimensional steady and incompressible. The flow (q) per unit time and thickness between these stream lines AA and BB is stream function . The flow across points A and B will remain same =  = q irrespective of their joining line which may be AB or ACB or ADB. If we assume  = 0 at point A, then  at point B will be flow per unit time and thickness = q =  Stream function is also defined as the scalar function of space and time whose partial derivative with respect to any direction gives the velocity component at right angles to that direction. Hence in steady two-dimensional flow,  f x,y  which gives v x    and u y   
• 79. 79 . 31. Show that stream function satisfies the equation of continuity. The equation of continuity for a two-dimensional flow is u v 0 x y       (1) As per stream function, we have u y     v x    Put the values of u and v in the continuity equation (eqn. (1)) 0 x y y x                Or 2 2 0 x y y x            The above is satisfying as LHS = RHS. Hence stream function satisfies the equation of continuity. 32. How and  in polar coordinates is used to find velocity components in radial and tangential directions.  = f (r, ) Radial velocity ru = r 1 u r    Tangential velocity u = r    f r,   Radial velocity ru r   
• 80. 80 Tangential velocity u = 1 r   Note: Following may be used as simple way to remember - (1) ru u ,v u  (2) dx rd ,dy dr   33. Enumerate the properties of stream function . The properties of stream function are: (1)  is constant at all places at a stream line. (2) The flow around any path in the fluid is zero (3) The velocity vector at any point on the stream line can be found out by differentiating its stream function i.e. u y     and v x    (4) The discharge between the stream lines is equal to difference of their stream functions i.e. 2 1q     (5) The rate of change of stream function with distance is proportion to the component of velocity normal to that direction (6) If two stream lines superimposed, then  1 21 2 s s s           34. What do you understand from the velocity potential? The flow takes place as in pipeline when there is a difference of pressure. The flow always takes place from higher to lower pressure side. This potential difference resulting into the flow of fluid is known as velocity potential and it is denoted by  . The velocity potential is defined as a scalar function of space and time such that its negative derivative with respect to any direction gives the fluid velocity in that direction. Hence for steady flow, if velocity potential is given by  f x,y,z  , then
• 81. 81 u ,v x y        And w z     (Note: negative sign indicates that flow is taking place in direction in which velocity potential decreases.) 35. Derive the Laplace equation for the velocity potential. For steady flow, the continuity equation is: u v w 0 x y z          Now u ,v x y        and w z     . Put these values in continuity equation. 0 x x y y z z                               This is Laplace equation 2 2 2 2 2 2 0 x y z              36. What are the properties of potential function? The properties of potential function are: (1) Flow having potential function  are irrotational (2) Flows are steady, incompressible and irrotation if velocity potential satisfies the Laplace equation. 37. What is the relation between stream function   and velocity potential   ? As per definition, the derivatives of stream and velocity potential give components of velocities as under: u ,v y x       Also u ,v x y      
• 82. 82  u , y x        And v , x y        Hence, we can write - x y      and y x      38. Prove equipotential line and stream line are orthogonal to each other.  f x,y constant   d .dx dy 0 x y         u x     and v y     , we get  udx vdy 0   or dy u dx v   It is slope 1m of equipotentail line Now stream function along a stream line is constant.   f x,y constant   Or d .dx .dy 0 x y         As v x    and u y     , we get vdx udy 0 
• 83. 83  dy v dx u  Hence the above is slope 2m of stream line Now 1 2 u v m m 1 v u       Hence equipotentail line and stream line are orthogonal. 39. What is a flow net? What is the condition of flow net to be a set of square? A flow net is a graphical representation of stream lines and equipotential lines. The stream lines in flow net show the direction of flow which the equipotential lines join the equal velocity potential   points in the flow. Flow net provides a simple graphical technique for studying flows which are two dimensional and irrotational as compared to mathematical calculation techniques which are difficult and tedious. In flow net, the stream lines are spaced such that the rate of flow q is same between each successive pair of streamline. The flow net can be drawn for irrotational flows. is constant along stream line and  is If q is the flow and 1 2V &V are velocities along streamline 1 2&   1 1 2 2q V n V n    = constant Where 1n and 2n are the distances between two streamlines  1 2 2 1 V n V n    But 1 1 2 2V s V s     = constant  1 2 2 1 V s V s    2 2 1 1 n s n s      or 2 1 2 1 n n s s     
• 84. 84 or n s   = constant In case n s   = 1, then n = s . We get a set of square in the flow net 40. What are the methods used for drawing the flow nets? Three methods are used for drawing the flow nets which are: (1) analytical method (2) graphical method (3) electrical analogy method. In analytical method, flow net can be drawn be finding the expression for  and  in terms of x and y which can be plotted. In graphical method, the flow passage is equally devided to draw stream lines and then equally potential lines are drawn orthogonally to the stream line. In electric analogy method, equipotential lines are drawn using null method. Stream lines are later drawn orthogonally. 41. Explain graphical method of drawing flow net. The graphical method is carried out as under: (1) The flow passage between the boundaries is derived into suitable number of equal parts and a set of stream lines are drawn enclosing these equal parts. (2) Equipotential lines are drawn orthogonally to those stream lines such that each block of flow net is almost a square. 42. At a point on a stream line, the velocity is 3 m/s and the radius of curvature is 9 m. If the rate of increase of velocity along the stream line at this point is 1 m/s2, then the total acceleration at this point would be: (a) 1 m/s2 (b) 3 m/s2 (c) 21 m/s 3 (d) 2 2 m /s 2 2 n V 9 a 1 m/s r 9    2 a 1 m/s  2 2 2 na a a 1 1 2 m/s     Answer (d) is correct
• 85. 85 43. If u and v are the components of velocity in the x and y directions of flow given by u = ax + by, v = cx + dy. Then the condition to be satisfied is: (a) a + c = 0 (b) b + d = 0 (c) a + b + c + d = 0 (d) a + d = 0 u = ax + by v = cx + dy u a x    v d y    Flow has to satisfy continuity equation u v 0 x y       a + d = 0 Answer (d) is correct 44. x component of velocity in a two-dimensional incompressible flow is given by u = y2 + 4xy. If Y-component of velocity v equals zero at y = 0, the expressions for v is given by – (a) uy (b) 2y2 (c) –2y2 (d) 2xy u = y2 + 4xy u 4y x    From continuity equation u v 0 x y       v u 4y y x         2 24y v c 2y c 2        Condition y = 0, v = 0. c 0  2 v 2y  
• 86. 86 Answer (c) is correct. 45. Given the x-component of velocity u = 6xy –x2, the y-component of flow v is given by, (a) 6y2 – 5xy (b) –6xy + 2x2 (c) –6x2 – 2xy (d) 4xy – 3y2 The continuity equation is: u v 0 x y       2 u 6xy 2x  u 6y 4x x     v u 6y 4x y x           If we carryout integration, we get – 2 y v 6 4xy c 2     2 3y 4xy c    Answer (d) is correct. 46 The velocity component representing the irrotational flow is: (a) u = x + y, v = 2x – y (b) u = 2x + 3y, v = – 2y2 + x (c) u = x2, v = –2xy (d) u = –2x, v = 2y For irrotation flow, u v 0 y x       For choice (a) : u 1 y    , and v 2 x    u v 1 2 1 0 y x            For choice (b) u v 3 and 1 y x       
• 87. 87 u v 3 1 2 0 y x           For choice (c) u v 0, 2y y x        u v 2y 0 y x          For choice (d) u v 0, 0 y x       u v 0 y x       Also v u 2 2 0 x y          Hence flow is irrotational. Answer (d) is correct. 47. Which of the following is rotational flow? (a) u = y, v = 3/2 x (b) u = xy2, v = x2y (c) Both (a) and (b)(d) none of the other For choice (a), we have u v 1, and 3/ 2 y x       u v 1 3/ 2 0 y x          It is rotational flow For choice (b), we have u 2xy y    and v 2xy x    u v 2xy 2xy 0 y x          It is an irrotational flow. Answer (a) is correct.
• 88. 88 MODULE 3 Forces acting on Fluid in motion: The various forces that may influence the motion are due to gravity, pressure, viscosity, turbulence, surface tension and compressibility. Alike mechanics of solids, dynamics of fluid is also governed by Newton’s second law of motion. It states that the resultant force on any fluid element must be equal to the product of mass and the acceleration of the element. In Mathematical form the law is expressed as F Ma  (1) Therefore if the mass of fluid is influenced by all the above mentioned forces then we can write the equation as:- g p v s t cF F F F F F Ma      (2) By resolving the various forces and acceleration in x, y and z directions, the following equations of motion may be obtained x x x x x xax g p v s t eM F F f F F F      y y y y y yay g p v s t eM F F F F F F      (2.1) z z z z z z za g p v s t eM F F F F F F      In most of the fluid flow cases, the surface tension forces & the compressibility forces are not significant. Hence they may be neglected. Then equations 2.0 and 2.1 becomes a g p v tM F F F F    (3) and x x x xax g p v tM F F F F    y y y yay g p v tM F F F F    (3.1) z z z zaz g p v tM F F F F   
• 89. 89 Equations 3.1 are known as Reynold’s equation of motion which is useful in turbulent flow analysis. For Laminar or viscous, turbulent forces become less significant and hence these may be neglected. Thus equation 3 and 3.1 can be modified as a g p rM F F F   (4.0) and ax gx px vxM F F F   ay gy py vyM F F F   (4.1) az gz pz vzM F F F   The equation 4.0 and 4.1 are known as Navier stoke’s equation which are useful in viscons flows. Further if viscous forces are also little significant, then the equations can be modified as a g pM F F  (5. 0) and x xax g pM F F  y yay g pM F F  (5.1) z z za g pM F F  The equation 5.0 and 5.1 are known as Euler’s equation of motion.
• 90. 90 Euler’s Equation of Motion: Fig.1 cylindrical fluid element In Euler’s equation of motion, the forces due to gravity and pressure are considered. Consider a cylindrical element of cross section dA and length dS. The forces acting in the cylindrical element are 1. Presence force PdA in the direction of flow. 2. Preserve force P P dA S       opposite to direction of flow. 3. Weight of element gd AdS Let  be the angle between the direction of flow and the line of action of weight of element. P PdA P dS dA gdA dScos dAds as S            Where as – acceleration in the direction of s Now dV as dt  , where v = f(s, t) v ds v v v ds v v s dt t s t dt                  Q
• 91. 91 If the flow is steady, v t   = 0 s v a v s     ie, P V pdA p dS dA g dA ds cos dA ds .v. S S              Dividing by dA ds 1 p v gcos v s s          dz v 1 p dz cos , v g 0 ds s s ds           ie, p gdz vdv 0      (I) (I) is known as Euler’s equation of motion. Integrating (I), dp g dz vdv     = constant If the flow is in compressible,  = constant. ie, 2 p v gz 2    = constant ie, 2 p v z g g     = constant (II) Where p g - Pressure energy per unit weight of fluid. z – Potential energy per unit weight of fluid. 2 v g - K.E. per unit weight of fluid. (II) Is known as Bernoulli’s equation. Following are the assumptions made in the derivation of Bernoulli’s Equation.
• 92. 92 (i) The fluid is ideal, viscosity is zero. (ii) The flow is steady (iii) The flow is incompressible (iv) The flow is irrotational The Bernoulli’s equation was derived on the assumption that fluid is invisicid (non- viscous) and therefore frictionless. But all the real fluids are viscous and hence after resistance to flow. Hence there are always some losses in fluid flows and leave in the application of Bernoulli’s equation there losses have to be taken into consideration. Thus the Bernoulli’s equation for real fluids between any two points 1 and 2 is given as 2 2 1 1 2 2 1 2 L p v p v z z h g 2g g 2g         Where hL is the loss energy between point 1 and 2. Kinetic Energy Corrections factor: Kinetic Energy correction factor is defined as the ratio of the kinetic energy of the flow per second based on actual velocity across the section to the kinetic energy of the flow per second based on average velocity across the section. It is denoted by 2. EE/sec bsed on actual velocity KE/sec based an average velocity   Momentum Correction Factor: It is defined as the ratio of momentum of flow per second based on actual velocity to the momentum of flow per second based an average velocity across a section. It is denoted by . Momentum per second based an actual velocity Momentum/ second based an average velocity   Statement of Bernoulli’s theorem: It states that in a steady, ideal flow of an in compressible fluid, the total energy at any