Question

4. A laminar, one-dimensional flow far from the entrance is occurring between two parallel plates, as in question 3. Given the following velocity profile, 11(y)= 4ux with h = 0.05m, Umax = 4m/s. u=0.0010 kg/m-s and plate lengths L = 10m, obtain (a) a relation for the drag force applied by the fluid on a section of the plates of length L and (b) calculate the actual drag force. Assume a unit plate width (i.e. width = 1m).

Answer 1

Laminar flow b/w fined parallel plates Ч y Assumptions :- is very large as compared to gap between the plates heel and width 0 plate size is @ the flow between the plateralan be concidered fully developed flow (1- Dimensional) u = f(y) , [No change in 'u w.r.to & & 3)

& At any Section pressure is assumed to be uniform P= P(x) (No change wor.to ① flow is steady. Giver 1-Dimensional flow far from the entrance is occuring between two parallel plates U = 4 Umar u Velocity at ang Section at 'yl distance from bottom Fles h2 plate h - M ty Uman - Maximum Velocity Length of the plates between the plates gap dynamic viscosi 1

w = width of the plates Red In - Drag force on plate at any Section T = shear stress We know that, Fo= TXA A = Area of the plate contact withe flow From Newton's law of viscosity We know that, P = M dy dy dy - velocity gradient dy Let us first find out, algo (94 Yumax lu dy dy dy

du 4 Umae (A - 4 dy T = Mx du dy = Mx 4. Umax (t T = 4M Umar ( 24 ) : Drag force on any in plates on section Fo=PXA Area = length of plate x width of the plate A = Low

plates : Fo=YM umor xLxW (6) Actual drag force on Given, Umax = 4 m/s h = 0.05m all=0.0010 kg/m pec L=10m W=lm ( Given that to usume) Drag force on top Heate For top plate, y = 0.05m FD = 4 x 0.001 - -210.01) x10 0.05 • 6005) top 3.2N toe top

1 I represents the drag force is in x-axis direction negative 1- om Drag force on bottom plate For the bottom plate, y= = 4x0.001oxy -2(0) oos to bottom 0.05 10.05) xlox to bottom 3.2N Nelocity Variation given is parabolic The shear stress Variation is linear ie. Shear Stress will be maximum at top and bottom points at Velocity of radient is high at top and bottom point.

the Is celmost zero. goo As at centre of the There shear stress is zero Tmare Uman T=0 صلى uz Grim man u= 4 Uman (h чу h h2 Fotop - 3-2 N Fobottom= 3.20