Integral forms of Momentum and Energy Equations

Main Question or Discussion Point

Hi

I was reading a book that introduced momentum and energy in integral forms and I had some confusion regarding what the terms meant. All integrals below are closed integrals

For the momentum equation, the result was:

F = d(mV)/dt = ∫∫ρ(V[dot]dS)V + ∫∫∫∂(ρV)/∂tdV

From product rule, d(mV)/dt = dm/dt*V + mdV/dt. I can see how dm/dt*V is reflected in ∫∫ρ(V[dot]dS)V but I do not see how mdV/dt is reflected in ∫∫∫∂(ρV)\∂tdV?

For the energy equation, the result was:

∫∫∫qdotρdV - ∫∫pV[dot]dS + ∫∫∫ρ(f[dot]V)dV = ∫∫∫∂(ρ(e+V^2/2))/∂t + ∫∫ ρ(e+V^2/2)V[dot]dS. p is pressure, e is specific internal energy, f is body force per unit mass
, V is velocity, and qdot is heat transfer per unit mass

from another book, another form of this equation was:

Qdot - Wdot = dEcv/dt + mdotout (hout+V^2/2) - mdotin (hout+V^2/2)

I'm basically wondering, is the mdot*h (the enthalpy term) reflected in the terms ∫∫ ρ(e+V^2/2)V and ∫∫pV[dot]dS or in other words is mdot*(e + V^2/2)-> ∫∫ ρ(e+V^2/2)V and mdot*p*v -> ∫∫pV[dot]dS?

Thanks

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In the momentum equation, the terms are as follows:
$$\iiint\rho\vec{v} (\vec{v}\cdot d\vec{s})$$
represents the mass flux across the surface of the control volume.
$$\iint\dfrac{\partial (\rho\vec{v})}{\partial t}dV$$
is the time rate of change of momentum in the control volume. Usually there is also a body force term and a surface force term (pressure and viscosity if you are doing viscous flows).

For your energy equation, the terms are as follows:
$$\iint\dot{q}\rho\;dV$$
is the rate of heat added to the control volume.
$$-\iint p \vec{v}\cdot d\vec{s}$$
is the rate of work done on the fluid in the control volume by pressure forces.
$$\iiint\rho(\vec{f}\cdot\vec{v})\;dV$$
is the rate of work done on the fluid in the control volume by body forces.
$$\iiint\dfrac{\partial}{\partial t}\left[ \rho\left( e + \dfrac{|\vec{v}|^2}{2} \right) \right]dV$$
is the the time rate of change of energy inside the control volume.
$$\iint\rho\left( e +\dfrac{|\vec{v}|^2}{2}\right)\vec{v}\cdot d\vec{s}$$
is the net energy flux across the surface of the control volume.