How Can I Form a Differential Equation for a Sphere's Flow Rate?

[vigintitres]

Homework Statement

I just need to get these eqn's together to form a DE

Homework Equations

Q = CA*sqrt(2gH) = -dV/dt

and

V = pi/3 * H^2 * (3*r - H)

These eqn's refer to a sphere which has a flow Q of fluid out the bottom of said sphere. The radius of the sphere is r, the height H is from the bottom of the sphere (where the hole is) to the top of the liquid. Also, C is a constant and A is just the area pi*r^2

The Attempt at a Solution

I need to get a differential eqn in terms of H but I've exhausted, from what I can see, any algebraic manipulation. The final story to be told will use Heun's method to determine how long it takes to drain the liquid out of the sphere (I fully understand the method, it is just this preliminary step which I am confused about)

 

[srmeier]

The cross-sectional area multiplied by the height from the hole at the bottom to the water level at the top is the volume of water contained within the sphere at any given time, is it not?

$$A_{cross}=\frac{V(H)}{H(t)}$$ - (assuming of course V is a function of H, & H is a function of t.)

see if that helps you at all

 

[vigintitres]

yes, that looks great. I didn't even consider this and as such, it is a reminder to KIS! thanks

 

[vigintitres]

ah, but actually in order to use Heun's method, I'll need to evaluate a "preliminary" V which means I'll really need to use a preliminary H in the equation involving V(H) (i.e. solving a cubic polynomial, which does not fly because I have to code this problem into MATLAB shortly...), so I guess I'm stuck again