Question

3 PL 23.5 Laminar slit flow with a moving wall ("plane Couette flow"). Extend Problem 23.4 by allow- ing the wall at x B to move in the positive z direction at a steady speed 0 Obtain (a) the shear-stress distribution, and (b) the velocity distribution.

Answer 1

S ; Given that, 1. The fluid flows in Z-direction. whese, Velocity = function of x -> VZ = vz6327 Here: - z momentum in Positive x-direction = 0xz and -> z momentum in positive Z-direction =0 zz So, we write the boundary conditions of; i-> at x=-B ; VZ =0 ;-) at x =B ; v2=vo

Then the Pressure in Z-directions P=P(z) Given diagram Y =WA -- 90 - yao ct øzzle ------ ZEL L T Direction of gravity THAL 50, momentum in (z=o) = (wasc) Ø zz = خي (هلی) = (1-2) انه »More ab momentum in (x) = (WL) Oxz momentum cut (tax) = (We) Øxz Force due to gravity (2) = (WLAX) eg. so now we write the basic momentum balance equation for z Component at Steady state is; -> momentum in - momentum aut + Force due to gravity o Now, → (wax) • Zz - Cwox) ozz + (WL)& X2 -(WL) Óxz + (wL92)69=0 Then, - (wax (0 Zz-8 Zz) - WL (Øxz - 8x2) + (Whax) Lg=0 Now, we divide by w Lase, then we get OzZ -ZZ Øtz - Øxz L+Pa =0 So, Ax -70 Then, → Øzz - Øzz , dør2 2 doctlg -0; 0x2 = 0; XZ TXZ +lUXvz=

L Now -> gzz = 8zz +zz + luz vz = P(z) +evz? -> (760) - PC) + Pg 1) - dotz o - PL07-2960 - [PCD=691] - dpsz -. As, we know here; → P2 = P(2) – Pg2 ; -> dyke - (DO PL) Then according to Newton's law of viscosity -> 787 =-je (dez - (Po-PD) Now, on double integration we get ear of; NZ C) = - (PC -PL32 Y ان | - ML 2 ULICH HC 20 - 2ML Given that, so, from boundary conditions we write; > C= v Los ; C2 = vo + (Po - P.) 132 Then , > VZGC) = -(Po-P2) 22/24L + Vox/213 +00/2 + CPO-P1732 puli 50 → vě - Poz? (BP_x2) +mon (1+) - Poule B2 (-) + (14) ZUL

i. vz = 2UL :V = (R: )? [-(41)+ Y2 (14) As, Taza-M (duz (de); we get - Txz = -4 (- Po-PL x + y ) 2. Finally we get : Tzz = Po-Pe oc - MV y x=-B oc x = Miro 2B (P.-P) v. Paabolic velocity distribution V6 [213² (Po-P+ ML572 cg B2ML (BR) VON Txze P-PL.3. b rity Lineur Shasz-stress disto sution Txz = P-PL B - Auto