Question

A laminar flow of constant density and constant viscosity at steady state flows thought the two flat and parallel infinite plate. The upper plate is mowing while lower plate is stationary. The flow upper plate: moving is driven by the pressure gradient in the x direction. The velocity profile is desired to be derived for this system. Then - What are the correct answers? 20 lower plate: stationary ay c) PS, 0 200 b- Write the equation to find out the velocity profile. X flow direction dy Examine any velocity change in flow direction. d- Reorganize the equation you wrote in choice B depending on your answers above. - Perform the differential process for reorganized equation to get velocity profile below dP Uy+cly + c2 Zuidx Then what are the boundary conditions for ciand c2? a) at y = 0, y = 0 b) at y = 0, v, - v C) at y = h, v, - 0 d) at y = h, v, v f- Find out the ci and c2 depending on your answer for choice E. <

Answer 1

(a) Since the width in z-direction is much larger compared to the distance separation of the plates (infinite parallel plate assumption), variation of fluid properties in z - direction can be neglected. Therefore, the given flow can be considered as two dimensional flow in x-y plane, Therefore dux 0 az Assuming gravitational force acts in y-direction (no component in x-direction) P9.x = 0 From the continuity equation it can be shown that, v, is not a function of x, which implies vx is a function of y alone. Therefore, aux 0 ду So the correct answers are aux b) > aux az = 0 e)p9x = 0 ay (b) The governing equation is given by the Navier-Stokes equation (momentum equation) in x-direction, which is as follows Vx δνα P + Vy อิน + Vy ду Ovx + V2 az ОР дх + (2²vx a²ux a²ux ax2 ay2 Oz2 + at ax (There's no body force in x-direction. Hence the body force term is omitted in the above equation) (c) Since the flow is steady Ovx at 0

Since there is no flow in y or z direction Vy = v, = 0 Continuity equation for an incompressible steady flow is given by Ovx Oy, 0v, + + дх ду az = 0 Substituting vy = v, = 0 in the continuity equation, we get aux = 0 Which implies V, is not a function of x, That's V, varies only with y. It follows that avy 0x2 = 0 Also, since the flow is approximated as a two dimensional flow in x-y pplane dv__02vx = 0 az2 az (d) Substituting the above results, the governing equation reduces to (av, ОР ax dy2 (e) The simplified governing equation can be rewritten as 22v, 1 (OP ay? ulax con Integrating the above equation with respect to y on both sides

δυ, ду 1 (OP ax y +cz Integrating again with respect to y V 1 OP ) y2 + C1y + C2 2u Cox Where, and C, are the constants of integration. Boundary conditions are given by a) at y = 0 V, = 0 d) at y = h 0x = v The above two boundary conditions follow from the no-slip boundary conditions at solid walls (i.e. at solid walls, the velocity of fluid with respect to solid wall must be zero). (f) Substituting the first boundary condition (a) in the derived equation of v, we get 1 (OP CO) (0)2 + G (0) +C2 = 0 2u lax or C2 = 0 Therefore U = zu le*) y2 +GY Now substituting the second boundary condition (d) in the above equation 1 OP ha + Gh= V 2u Cax)

Or C1 = v 1 OP h 2u lax h Substituting the values of C Vx = 1 OP y2 + 2u lax, + C 24 (3) )y V 1 aP h 2u lax Rearranging the above equation 1 OP V V = y + h (y2 – hy) 2u Cox