Consider steady, incompressible, laminar flow of Newtonian fluid in the narrow gap between two infinite parallel plates (Figure 4). The top plate is moving at speed V, and the bottom plate is stationary. The distance between these two plates h, and gravity acts in the negative z-direction. There is applied pressure other than hydrostatic pressure due to gravity. This flow is called Couette flow: Find the velocity distribution; Draw the velocity profile for fully developed region of flow, Calculate the maximum velocity and the average velocity, Estimate the shear force per unit area acting on the bottom plate.
Note: You can solve this question either by writing shell momentum balance or by using the Navier Stokes equations:
Moving plate
Fluid: p. #
Fixed plate
Incompressible Flow Continuity Equation:
Jp ax (pov (pv ) az (pvj) = 0
Incompressible Flow Navier-Stokes Equations in Cartesian Coordinates: dv; OU , ": ax ay 7) 1 Pg , dvy 8v , +", Jx Gy ay 81? 3-4 08y dv ; dv _ dv . v, Jx vv Jy J2) Jz 0-9-9 Pgz