Compresible volumetric strain rate-fluid mechanics

[ozgurakkas]

This is not really part of my homework but I still want to drive this equation.

Continuity equation:
p=density
D/Dt=material derivative
Grad: Gradient
V=Velocity vector
v=volume

so;

(1/p)(Dp/Dt)= -Grad.V

and

p=m/v

then,

(1/(m/v))(D(m/v)/Dt)= -Grad.V

Following is what I can't seem to get correct.

(v/m)(m/v^2)(Dv/Dt)=(-1/v)(Dv/Dt)= -Grad.V

How do we obtain "(m/v^2)"?

I believe I am missing a rule on the derivatives. I just can't remember it and I can't forget this question and move on. Thank you for your help in advance.

 

[Valkarie]

The rule you are missing is the product rule of differentiation: (f*g)' = f'*g + f*g' So in this case, you can use the product rule to differentiate (m/v): (m/v)' = (m/v^2)*Dv/Dtand so(1/(m/v))(D(m/v)/Dt)=(1/(m/v))(m/v^2)(Dv/Dt)=(-1/v)(Dv/Dt)= -Grad.V