A two-dimensional channel of width H has a slot of width h as shown. Fluid is injected through the slot at an angle Psi to the horizontal. The fluid is incompressible with density rho and body forces can be neglected. Assume velocity and pressure are constant across the channel. a Derive an equation governing mass for the channel, b Derive an equation governing x momentum for the channel. c Now, assume Psi = 60 Degree such that cos Psi = 1/2. Combine your results of Parts (a) and (b) with Bernoulli\'s equation to determine U_2 as a function of U_1, H and h. Solution Mass of channel before jiont =pAv 1 A= AREA = H*D D= depth of channel mass of channel after joint = mass before joint + mass added by small channel = pHDV 1 + phD*V MOMENTUM = MASS+ VELOCITY TOTEL MOMENTUM IN X DIRECTION BEFORE JOINT =   pHDV 1 2 + phD*V 2 *Cos velocity of fluid after jiont is = (A 1 V 1 +A 2 V)/A 1 = (HD*V 1 + hDV)/HD = V 1 + hV/H MOMENTUM AFTER JOINT IN X DIRECTION = pHD(V 1 + hV/H) 2 .

1. A two-dimensional channel of width H has a slot of width h as shown. Fluid is injected through
the slot at an angle Psi to the horizontal. The fluid is incompressible with density rho and body
forces can be neglected. Assume velocity and pressure are constant across the channel. a Derive
an equation governing mass for the channel, b Derive an equation governing x momentum for
the channel. c Now, assume Psi = 60 Degree such that cos Psi = 1/2. Combine your results of
Parts (a) and (b) with Bernoulli's equation to determine U_2 as a function of U_1, H and h.
Solution
Mass of channel before jiont =pAv 1
A= AREA = H*D D= depth of channel
mass of channel after joint = mass before joint + mass added by small channel
= pHDV 1 + phD*V
MOMENTUM = MASS+ VELOCITY
TOTEL MOMENTUM IN X DIRECTION BEFORE JOINT = Â Â pHDV 1
2
+ phD*V 2
*Cos
velocity of fluid after jiont is = (A 1 V 1 +A 2 V)/A 1 = (HD*V 1 + hDV)/HD = V 1 + hV/H
MOMENTUM AFTER JOINT IN X DIRECTION = pHD(V 1 + hV/H) 2