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 1 (51.22 KiB) Viewed 84 times

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 = -) stationary and the upper plate (at y = 6) moving at a constant velocity of V in the positive r direction. A known, constant pressure gradient y' is applied in the direction. Neglect the effect of gravity. Let the dynamic viscosity of the fluid be and density be. (a) Write down the component of the momentum equation you will use to solve for the velocity profile wy). (b) It is argued that the flow is fully developed. Is it true? Explain (briefly) why or why not? Infinitely long and wide plates 1 Fixed plate (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 p, 6, Ve, and p (f) Obtain an expression for the mass-flow rate in terms of 4, 6, V, and / Is it possible to have net vero mass How 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 y/<0that is we have a favorable pressure gradient (pressure decreases in positive a 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 r-direction) what would a typical profile look like? (b) 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? a