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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)Vbut 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,fis body force per unit mass

,Vis 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)Vand ∫∫pV[dot]dSor in other words is mdot*(e + V^2/2)-> ∫∫ ρ(e+V^2/2)Vand mdot*p*v-> ∫∫pV[dot]dS?

Thanks

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# Integral forms of Momentum and Energy Equations

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