Two layers of different incompressible Newtonian fluids are contained between tw

Question

Two layers of different incompressible Newtonian fluids are contained between two flat plates separated by a distance d. Both layers are of the same thickness (d/2). The fluid in the top layer has a viscosity ?II, and the fluid in the bottom layer has a viscosity ?I. The bottom plate is kept stationary. To move the top plate with speed V in the horizontal direction, it is necessary (at steady state) to exert on the top plate a force F per unit plate area in the horizontal direction. An effective viscosity ?? for the system is defined by

Derive a formula which expresses ?? as a function of ?I and ?II only. Assume that ?p/?x = 0 everywhere and that the flow is laminar.

Explanation / Answer

>> Now, due to Bottom Plate, The shear force trying to oppose the motion of top plate ,

                                         = (uI)*V/d

So, Force Required = (uI)*V/d *A,

A = Area of Cross Section = L*d/2

=> Force Require = [(uI)*V/d] * Ld/2 = (uI)*LV/2

So, comparing , with F = u*V/d

u* = (uI)Ld/2            .....ANSWER.....

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