Question

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a) As shown in the Figure below, a fluid flows down an inclined surface. Show that the velocity distribution of that fluid is u=[pgsin0/(2)]/(2hy - y²)) by using Navier-Stokes equations. (Show assumptions that you make and every each step) [13 points) b) Laminar viscous flow between two parallel plates are shown in the figure below. Both bottom plate and top plate moving in the same direction, their velocities are U,Ut respectively and they are not equal to each other. Assume that pressure gradient between point A and point B is zero. By using Navier Stokes equations find the shear stress distribution and velocity profile for that fluid. Plot both velocity profile and shear distribution. (Show assumptions that you make and every each step) [12 points] U B

Answer 1

a) The Navier Stokes equation for a direction : mlu my tu veut waktu 2 ) -Lum potent on a bed] trga (سا Assumption : (i)Encomprusible flow (1) planas, fully developed flow (iii) Steady flow (iv) parallel flow ie V=o (v) Pressure = constant wat 2 Isina No applied pressure gradient (400) (vi) Velocity field is Two dimensional a W=0 (vii) Gravity due to Du & (.احر vi W) (Voi2 (vi) 100+ az of any velocity component ik zou. acts in negative ż direction 9x=g sino Flow takes place gravity. (iv) of aw (vii) Apply continuity equation co - In Joy Joz ie ucrfly only (1) Reduce to slu u od of .24 pos Ton V o = MD²u g sino dy Integrate twice U=-8g sino y²+G4+(2 2M -22 y=0 - U=0 (in y h =) du = 0 dy you + + az " pt A trga dyz + sino rg js. Boundary conditions

- 0 = 0 + ( x (0) + 6 Put dil co ВС (i) BC (1) C₂ 0 du و - Sino ar (ay) + (1) G dy at y= h du =0 =0=-vg Sinerh). C dy ам Cr= regsino k M Substitute C C2 12 -2 ) U= - Pg sino yat(eg smo h) to ам U-trg sino y Cahay) ar U= pasino Cahy-ya) U = U (y) only. ам