NS with variable body force-Momentum conserved?

CFDtoy

Hello,

In a fluid momentum equation with variable body force term, say

d(rhoU)/dt+ Del(rhoUU)=-gradP + del (mu)delU + F(alpha)

My viscosity mu = function of variable alpha and i have a body force F = F(alpha).

and as far as alpha,

I solve a scalar equation for alpha (something like a species) following U eqn.

is the fluid momentum conserved in this scenario?

Thanks,

CFDtoy