Fluid Flow Continuity in Control Volume Analysis for Shallow Channels

[Trenthan]

Ey guys, girls

Trying to work out what I've attached below

Now i can get the right part of my expersion to match however the left I am not 100% sure how to change d/dt from control volume analysis to delta(h)/delta(t)

I think I am more stuck with the maths here than anything, not the theory.

From my guess i would say that height and width do not vary with time, along the length of the control volume. but according to what i should get height does vary with time...

I should also add this is for small amplitude waves in shallow channels

once i get that term I am after i simply divide by dx and density to get the required solution
i haven't attached the working for the right term in the expression, since I've derived that bit

Any idea's, hints?

Cheers Trent

 

[Valkarie]

The expression you're trying to work out is most likely derived from the continuity equation, which states that the rate of change of mass within a control volume is equal to the net rate of flow of mass across its boundaries. This can be written as: d/dt [ρ * A * h] = ∇ . (ρ * v * A) Where ρ is the density, A is the area, h is the height and v is the velocity. To solve for the left term, you can use the chain rule to express the time derivative as a function of the change in height with respect to time, and the area and density as constants. This results in the expression: d/dt [ρ * A * h] = ρ * A * (Δh/Δt) Hope this helps!