An incompressible fluid flows past an impermeable flat plate, as in the figure,
ID: 1862051 • Letter: A
Question
An incompressible fluid flows past an impermeable flat plate, as in the figure, with a uniform inlet profile u=U0 and an exit velocity profile of u =A?sin(By+C). Here A, B, and C are constants to be determined. Use proper boundary conditions to determine A, B, and C. Compute the volume flow Q across the top surface of the control volume.
An incompressible fluid flows past an impermeable flat plate, as in the figure, with a uniform inlet profile u=U0 and an exit velocity profile of u =A?sin(By+C). Here A, B, and C are constants to be determined. Use proper boundary conditions to determine A, B, and C. Compute the volume flow Q across the top surface of the control volume.Explanation / Answer
In order for u=U0 at y=? and 0 at y=0 then A*sin(B*y+C) needs to equal sin(pi/2) at y=? and sin(0) at y=0
Let's start with y=0 in the case A*sin(B*y+C), if you make C=0 then at y=0 A*sin(B*0+0)=0 no matter what A and B are.
Now for y=? if we go ahead and make A=U0 then we just need B*y to equal pi/2 since sin(pi/2)=1 we know y=? then if we set B=pi/(2*?) then the ?'s will cancel out and we're left with U0*sin(pi/2)=U0
Therefore A=U0 B=pi/(2*?) and C=0
u=U0*sin(pi/(2*?)*y+0)
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