Static Pressure along bounded channel

[Apnshin]

Homework Statement

[/B]
Note: There is NO z variation, everything is horizontal (imagine looking down at streamlines)

Homework Equations

Bernoulli's Equation along streamline:

P + 1/2*ρ*V² + ρ*g*h = Constant ... (1)

Bernoulli's Equation normal to streamline:

P + ρ*V²*(1/R) + ρ*g*h = Constant ... (2)

R: Radius of Curvature

Continuity Equation from Conservation of mass:

Needed to find out what is happening with velocity from inlet to outlet

ρ₁*A₁*V₁ = ρ₂*A₂*V₂ ... (3)[/B]

The Attempt at a Solution

For the plots of AF:
I think that the static pressure should be constant along the top wall as it is straight and there is no dynamic pressure. Also, I think that it should equal the total pressure.

GL: [/B]
Same as the plot of AF?

Plot of average static pressure in the streamwise direction:

Using ρ₁*A₁*V₁ = ρ₂*A₂*V₂, where 1 is the inlet and 2 is the outlet:

V₂ = A₁*V₁/A₂
A₁ > A₂
-->V₂ > V₁
--> P₂ < P₁

Given this, my assumption is that the pressure drops from the inlet to the outlet, however my main issue is understanding how the static pressure is affected during the concave up and down walls. From (2), I believe that the static pressure should increase as the radius of curvature increases. At the maximum radius of curvature (bottom wall) I think that the static pressure is therefore at its maximum and there will be no dynamic pressure along the bottom wall during the curved regions. My question is how will the average static pressure in the streamwise direction decrease as the fluid passes through the curved regions?

 

[Asim Riaz]

I am unsure how to determine the static pressure variation along the concave up and down walls. I know that Bernoulli's Equation is used to solve for the pressure variation, but I am not sure how to use it in this case. Any help would be greatly appreciated!