6. Consider a steady, laminar flow of an incompressible fluid through a channel of two infinitely long flat plates (in r

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6. Consider a steady, laminar flow of an incompressible fluid through a channel of two infinitely long flat plates (in r and directions), with the lower plate (at y = -b) stationary and the upper plate (at y = b) moving at a constant velocity of Vo in the positive r direction. A known, constant pressure gradient p/ is applied in the x direction. Neglect the effect of gravity. Let the dynamic viscosity of the fluid be and density be p. 1
Infinitely long and wide plates у 2b Fixed plate
(1) Obtain an expression for the mass-flow rate in terms of , b, V, and p. Is it possible to have net zero mass flow rate? Why or why not? If it is possible, under what condition would one get a zero mass flow rate? What would the velocity profile look like under this condition? (8) If y' < 0, that is we have a favorable pressure gradient (pressure decreases in positive x direction), sketch the velocity profile. If y) = 0, what would be the velocity profile? Similarly, if p > 0 (adverse pressure gradient, pressure increases in positive -direction), what would a typical profile look like? (h) Find the value of p/ for which the shear stress at y = -b will be zero? With no shear stress at the bottom plate, there is no shear force! How is that possible? Using physical insights, explain what is happening near the bottom plate at this pressure gradient?