Conservation of Momentum Density

Consider now the conservation of momentum density (or linear momentum vector pv). First we write this law for the ideal (without viscosity) liquid in two different presentations. The Lagrange form of the equation of motion of the element of liquid coincide with the Newton form (md ldt= )  

Therefore, the total velocity in the reference point satisfies the equation 

Using identityi9p/i9xi = bydp/dxj with Kronecker symbol 8,y we may present the result in the compact form of the law of momentum conservation for the ideal liquid  

For viscous liquids the law for the mass conservation remains unchanged. As to the momentum density conservation, it keeps the same form (9.8) but tensor 11 should be changed to take the dissipation into account. Now we write 

for not very strong gradients, there should be linear relationship between 

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We can use the above relations to rewrite the law of conservation of momentum density in the form... 

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