Fluid Mechanics: Rain fall on a roof

[fogel1497]

Homework Statement

The exact problem can be seen in the attached jpeg, and is summarized without figures here:

Homework Equations

Volumetric flow rate: Q = A*V
Continuity equation: (d/dt)*(Integral Of:density*dVolume) + (Integral Of:density*velocity*dA)
Velocity of rain fall down the roof, (parallel to the roof): v = y(h-y)

The Attempt at a Solution

a. Merely wrote the continuity equation down.
b. Since Q=A*V, and I have an expression for the velocity of rainfall down the roof, ( v=y(h-y) ), I merely multiplied that by A to get:

Q = y*h*A - y*A

c. This is as far as I've gotten. By looking at the control volume you can see that mass enters from the top of the control volume at an angle, and at the left hand side directly normal to the control volume, and leaves the control volume at the right hand side normal to the control volume. So:

M_in1: (density)*(dx*1)*(cos theta)*(V)
M_in2: (density)*(h*1)*(v)
M_out: M_in1 + M_in2

but if the question wants dQ then it implies that the volumetric flow rate is changing. I am a little confused as to what exactly is wanted in this section, and how to go about solving it.

d. I know that the shear stress Tau_yx = dv/dy * viscoscity , but your not given viscosity so even if I had an explicit expression for dv/dy i wouldn't be able to find the shear stress on the roof. I assume I need to apply some kind of momentum analysis to find the change in momentum along the axis parallel to the roof, and attribute this momentum change to the shear stress applied on the fluid by the roof. To do this I define a control volume encompassing the whole roof and need to sum up the momentum of the fluid going 'into' the system in the x-direction, and the momentum of the fluid 'leaving' the system in the x-direction:

(mass_out * velocity_{out_x}) - (mass_in*velocity_{in_x})

(mass_out * velocity_{out_x}) - (density)(L*1)(V)(cos theta) * (V)(sin theta)

e. I would need to know how h varies with x in order to integrate the density over dX to find the weight, which is my problem in a lot of the preceding parts of this question.


Any help is much appreciated!

 

[nilesh_pat]

§§ COMNo equations were necessary for part e) of the problem. The weight of rain falling on the roof can be found by simply multiplying the density of water, the width of the roof, and the height at a given point along the roof. W = rho*L*h