Viscosity
Measurement
ER. FARUK BIN POYEN
FARUK.POYEN@GMAIL.COM

Content:
 Definition
 Coefficient
 Units
 Measuring Devices
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Viscosity – Definition and Formula
 Viscosity is the measure of a substance's resistance to motion under an
applied force.
 The formula for measuring viscosity is fairly simple:
𝑉𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 = μ = 𝑆ℎ𝑒𝑎𝑟 𝑆𝑡𝑟𝑒𝑠𝑠 𝑆ℎ𝑒𝑎𝑟 𝑅𝑎𝑡𝑒 = τ/γ
 Shear stress is the force per unit area required to move one layer of fluid
in relation to another.
 Shear rate is the measure of the change in speed at which intermediate
layers move with respect to one another.
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Viscosity – Coefficient of Viscosity
 Viscous Drag (Drag Force) (F) - The frictional backward force existing
between the two surfaces.
𝐹 ∝ 𝐴; 𝐹 ∝ 𝜕𝑉 𝜕𝑧
 Coefficient of Viscosity (μ)
𝜇 =
𝐹
𝐴
∗
𝜕𝑧
𝜕𝑉
A = Area of the layers; V = Velocity; 𝜕𝑉 𝜕𝑧 = velocity gradient;
z = distance between surfaces.
 Coefficient of Viscosity is the tangential backward force that acts
between two fluid layers of unit area, situated unit distance apart and
having unit relative velocity, when the fluid is in streamline motion.
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Viscosity – Newtonian and Non Newtonian Fluid
 Newtonian Fluid - At a given temperature and shear stress, the viscosity
of a fluid would remain constant regardless of changes to the shear rate.
 E.g., Petrol, kerosene, mineral oils, water, salt solutions etc.
 Non-Newtonian Fluid - Most fluids, however, have viscosities that
fluctuate depending on the shear rate. These are called Non-Newtonian
fluids.
 There are five types of non-Newtonian fluids: thixotropic, rheopectic,
pseudoplastic, dilatant, and plastic.
 E.g., Printer’s ink, starch, peanut butter, tar, chewing gum etc.
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Viscosity – Units of Viscosity
 CGS Unit – Poise (P).
 Centipoise (cP) = 0.01 Poise.
 SI Unit of μ – (kg/mt-s) = 10 Poise.
 SI Unit of Viscosity – (Newton*Sec/square mt.) = Poiseuille (PI).
 Other Unit – Pascal Second (Pa.s).
 1 PI = 10 Poise = 1000 cP
 1 cP = 1 mPa.s
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Viscosity - Types
 There are two different measurements of viscosity used to describe
fluids, dynamic and kinematic viscosities.
 These are interchangeable if the fluid density ρ is known.
 Dynamic Viscosity (μ) measures the ratio of the shear stress to the shear
rate for a fluid.
𝜇 =
𝑆ℎ𝑒𝑎𝑟𝑖𝑛𝑔 𝑆𝑡𝑟𝑒𝑠𝑠
𝑆ℎ𝑒𝑎𝑟𝑖𝑛𝑔 𝑅𝑎𝑡𝑒
=
𝜏
𝛾
= (𝐹 𝐴) ( 𝜕𝑉 𝜕𝑧)
 Kinematic viscosity (ν) measures the ratio of the viscous force to the
inertial force on the fluid. Kinematic viscosity is analogous to diffusivity
of mass and heat, being the diffusivity of momentum.
ν = 𝜇 𝜌
 Unit of Kinematic Viscosity in CGS is Stokes (St) = Square Mt/Sec.
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Viscosity – Hagen Poiseuille Formula
 Hagen Poiseuille Law is a physical law that gives the pressure drop in
an incompressible and Newtonian fluid in laminar flow flowing through
a long cylindrical pipe of constant cross section.
∆𝑃 =
8𝜇𝐿𝑄
𝜋𝑅4
Q = Volumetric flow rate; L = pipe length; R = pipe radius; ΔP = pressure difference;
μ = dynamic viscosity;
𝑄 𝑚𝑎𝑥 = 𝜋𝑅2
∆𝑃
𝜌
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Viscosity – Stokes Law
 Stokes Law delineates the frictional force, also called drag force (F)
exerted on spherical objects with very small Reynolds numbers in a
viscous fluid.
 The force of viscosity on a small sphere moving through a viscous fluid
is given by
𝐹 = 6𝜋𝜇𝑅𝑣
F = drag force, μ = dynamic viscosity, R = radius of spherical object;
v = flow velocity relative to the object.
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Viscosity Reynolds Number
 Reynolds number is a dimensionless value which is applied in fluid
mechanics to represent whether the fluid flow in a duct is steady or
turbulent.
 This value is obtained by comparing the inertial force with the viscous
force.
 The Reynolds number id denoted by Re.
𝑅𝑒 =
𝜌𝑣𝐿
𝜇
 The Kind of flow is based on the value of Re
1. If Re < 2000, the flow is called Laminar
2. If Re > 4000, the flow is called turbulent
3. If 2000 < Re < 4000, the flow is called transition.
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Viscosity – Measurement Methods
 Viscosity measurement is carried out on the basis on any one of the
following three phenomena.
1. Flow through capillary tube.
2. Drag experienced by a falling ball through a fluid.
3. Drag experienced by one of the concentric cylinders carrying fluid
between them, when the outer cylinder is rotating.
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Viscosity – Flow through Capillary Tube
 The capillary viscometer measures the time between the volume of
liquid/sample to pass through the length of the capillary tubes.
 Viscosity is measured employing Hagen Poiseuille law.
𝜇 = 𝜋∆𝑃𝑅4
8𝑄𝐿
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Viscosity – Flow through Capillary Tube
 In a Capillary Tube Viscometer, the liquid to be tested is drawn into the
tube to a level above the top etched line.
 The time is then obtained for the liquid to drain to the bottom etched
line.
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Viscosity – Falling Sphere Viscometer
 Measures the viscosity by dropping a sphere of a specific weight &
density and measures the time it takes the sphere to reach designated
junctures.
 Weight brings the sphere down whereas the drag force and buoyancy
push the sphere upward.
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Viscosity – Falling Sphere Viscometer
 If the particles are falling in the viscous fluid by their own weight, then
a terminal velocity, also known as the settling velocity, is reached when
this frictional force combined with the buoyant force exactly balance the
gravitational force.
 The resulting settling velocity (or terminal velocity) is 𝑉𝑡𝑒𝑟 given by
𝑉𝑡𝑒𝑟 =
2
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∗
𝑅2 𝑔(𝜌𝑠 − 𝜌 𝑓)
𝜇
 Viscosity is measured by the following expression
𝜇 =
𝐷2 𝑔(𝜌𝑠 − 𝜌 𝑓)
18𝑉𝑡𝑒𝑟
D & R – Diameter & Radius of sphere respectively;
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Viscosity – Rotational or Concentric Cylinders
 Measures the torque required to revolve an object within the volume of
liquid.
 Rotation of the outer cylinder (typically) with inner cup being static,
generates shear on the fluid, causing the fluid to flow within the
viscometer.
 The rotational viscometer consists of two basic parts separated by the
fluid being tested.
 The two parts may be: concentric cylinders (cup and bob), parallel
plates, a low angle cone and plate, or a spindle inside of a cylinder.
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Viscosity
 Angular velocity of inner cylinder = ω; Tangential velocity = 𝑅 𝑜𝑢𝑡 𝜔;
 𝑆ℎ𝑒𝑎𝑟𝑖𝑛𝑔 𝑅𝑎𝑡𝑒 = γ =
𝜕𝑣
𝜕𝑧
=
𝑅 𝑜𝑢𝑡 𝜔
(𝑅 𝑜𝑢𝑡−𝑅 𝑖𝑛)
;
 μ = 𝑆ℎ𝑒𝑎𝑟 𝑆𝑡𝑟𝑒𝑠𝑠 𝑆ℎ𝑒𝑎𝑟 𝑅𝑎𝑡𝑒 = τ/γ => 𝜏 = 𝜇 ∗ 𝛾
 Polar Moment of Inertia 𝐽 =
𝜋𝑅4
2
 𝑇𝑜𝑟𝑞𝑢𝑒 𝑇 =
𝑆𝑡𝑟𝑒𝑠𝑠∗𝑃𝑜𝑙𝑎𝑟 𝑀𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝐼𝑛𝑒𝑟𝑡𝑖𝑎
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑐𝑒𝑛𝑡𝑒𝑟 𝑡𝑜 𝑠𝑡𝑟𝑒𝑠𝑠𝑒𝑑 𝑠𝑢𝑟𝑓𝑎𝑐𝑒
=
𝜏∗𝐽
𝑅
;
 𝐹𝑜𝑟 𝑘 > 0.99, 𝑆ℎ𝑒𝑎𝑟 𝑆𝑡𝑟𝑒𝑠𝑠 𝜏 = 𝑇 2𝜋𝐿𝑅𝑖𝑛
2
 𝑘 =
𝑖𝑛𝑛𝑒𝑟 𝑟𝑎𝑑𝑖𝑢𝑠
𝑜𝑢𝑡𝑒𝑟 𝑟𝑎𝑑𝑖𝑢𝑠
> 0.99
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Viscosity
 Viscosity for concentric cylinders is given by the expression
 𝜇 =
𝑇∗(𝑅 𝑜𝑢𝑡−𝑅 𝑖𝑛)
2𝜋𝑅 𝑖𝑛
2 𝑅 𝑜𝑢𝑡 𝜔𝐿
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Viscosity – Sybolt or Redwood Viscometer
 The Redwood (Sybolt – called in USA) viscometer consist of vertical
cylindrical oil cup with an orifice in the centre of its base . The orifice
can be closed by a ball .
 The cylindrical cup is surrounded by the water bath .
 The water bath maintain the temperature of the oil to be tested at
constant temperature .
 The oil is heated by heating the water bath by means of an immersed
electric heater in the water bath
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Viscosity - Sybolt or Redwood Viscometer
 This viscometer is used to determine the kinematic viscosity of the oil.
 Kinematic Viscosity (ν) is given in centistokes as
 ν = 0.226∗t −
195
𝑡
𝑓𝑜𝑟 𝑡 < 100 𝑠; 0.200 ∗ 𝑡 −
135
𝑡
𝑓𝑜𝑟 𝑡 > 100 𝑠
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Viscosity – Zahn Cup
 A Zahn cup is a viscosity measurement device widely used in the paint industry.
 It is commonly a stainless steel cup with a tiny hole drilled in the centre of the bottom
of the cup.
 Measures by observing the time it takes the volume of liquid to empty the cup though a
small hole in the bottom of a container/cups.
 There is also a long handle attached to the sides.
 To determine the viscosity of a liquid, the cup is dipped and completely filled with the
substance.
 After lifting the cup out of the substance the user measures the time until the liquid
streaming out of it breaks up, this is the corresponding "efflux time".
 Efflux is defined as the rate at which a liquid flows out of a system.
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Viscosity - Vibrational Viscometer
 By measuring the vibrational waves using a vibrating rod submerged in
fluid, viscosity is calculated by analyzing the dampening of the
vibration.
 The vibro viscometer works by sending uniform frequency vibrations
out from two sensor plates which sit submerged in the sample fluid.
 The thicker (more viscous) the substance the more driving current is
required to maintain the vibration frequency, the viscosity is then
worked out by the positive correlation between the driving electric
current and the viscosity.
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Viscosity - Vibrational Viscometer
 The viscosity is measured by the following equation.
 𝑅 𝑧 =
𝐹
𝑉𝑒 𝑖𝜔𝑡 = 𝐴 𝜋𝑓𝜇𝜌𝑅 𝑧
𝑅 𝑧 = Viscous Resistance
F = Force
f = vibrational freq.
A = planar dimensions
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Viscosity - VROC Viscometer
 Viscometer-Rheometer-on-a-Chip) combines microfluidic and MEMS
(Micro-Electro-Mechanical Systems) technologies to measure dynamic
viscosity over a wide dynamic range of operation.
 This viscometer is pressure driven using a pumping system that the
laminar flow to push the liquid into a rectangular slit with pressure
sensors, measuring the viscosity of a fluid through the change in
pressure after passing each pressure sensor within the microchip.
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Reference:
 https://www.cscscientific.com/viscosity#targetText=
 https://blog.rheosense.com/different-ways-to-measure-
viscosity#targetText
 https://www.odinity.com/stokes-law-reynolds-number-measuring-liquid-
viscosity/
 https://www.chegg.com/homework-help/
 http://encyclopedia.che.engin.umich.edu/Pages/ProcessParameters/Visco
meters/Viscometers.html
 http://www.area4.info/Area4%20Informations/Instrumentation-15.htm
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