6 consider a steady laminar flow of an incompressible fluid through a
6. Consider a steady, laminar flow of an incompressible fluid through a channel of two infinitelylong flat plates (in x and z directions), with the lower plate (at y = −b) stationary and theupper plate (at y = b) moving at a constant velocity of V₁ in the positive a direction. A known,constant pressure gradient p' is applied in the x direction. Neglect the effect of gravity. Letthe dynamic viscosity of the fluid be µ and density be p. (a) Write down the component of the momentum equation you will use to solve for thevelocity profile u(y). (b) It is argued that the flow is fully developed. Is it true? Explain (briefly) why or not?why
(c) Is it safe to assume that the flow is two-dimensional? Why? What does it mean in terms of simplifying the equation? (d) Using appropriate assumptions, simplify the momentum equation clearly indicating which-terms drop out and why. Clearly indicate the boundary conditions needed to solve the-equation. (e) Solve for u in terms of µ, b, Vo, and p'. (f) Obtain an expression for the mass-flow rate in terms of µ, b, Vo, 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? (g) If p′ < 0, that is we have a favorable pressure gradient (pressure decreases in positive x direction), sketch the velocity profile. If p' = 0, what would be the velocity profile?Similarly, if p'> 0 (adverse pressure gradient, pressure increases in positive x-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?
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