Consider steady, incompressible, laminar flow of a Newtonian fluid in the narrow gap (h) between two infinite parallel plates. The top plate is moving at speed U, while the bottom plate is stationary as illustrated in the figure below. The gap between the plates is h. The fluid between the plates has a density p and viscosity . Consider a case where the flow is fully developed and laminar with pressure gradient (dp/dx -constant) in the direction of the flow. The liquid gap thickness, h compared to the length of the plate, L. is such that L. is much greater than h. Use the coordinate system attached to the lower plate. Uo Moving Plate h Fixed Plate L Given the Equations of motion in Cartesian coordinates system as the following: Continuity дри дро+дри + dp = 0 + дх ду Əz Momentum ди ди ди ar ди +1 +D + W P8x +μ ar ar ду dz дх до до до до +" ar +D. ду dr 292822 +w- =P8y +μ a dy dw aw +1 +W =PR др +4 ax ду a az (a) ar Use the physical conditions below (i), (ii), (iii), (iv), (v), and (vi) to reduce the continuity and momentu equations above to obtain the reduced differential equations which describe flow up the moving plate with a pressure gradient. Describe the boundary conditions used. (i) Steady flow (ii) Incompressible fluid (iii) No flow variation in the plane of the paper. Normally the surface is of unit depth into the paper (b-1m). (iv) Fully developed flow (v) No penetration at the walls (v (-1m)>>h(mm) (b) Determine an expression for the following: (i) sketch the velocity profile (ii) the velocity profile equation. (iii) the shear stress at the walls