Angular Momentum in a Viscous Fluid

Abstract

The initial differential equations valid for any fluid flow are the balances of mass, impulse and energy for differential volume dx ³:

$$\frac{{\partial \rho }}{{\partial t}} + \frac{{\partial \rho {u_j}}}{{\partial {x_j}}} = 0,$$

(1.1.1)

$$\frac{{\partial \rho {u_i}}}{{\partial t}} + \frac{{\partial \rho {u_i}{u_j}}}{{\partial {x_j}}} = \frac{{\partial {\sigma _{ij}}}}{{\partial {x_j}}} + {F_i},$$

(1.1.2)

$$\frac{\partial }{{\partial t}}\rho (E + \frac{{{u_i}{u_i}}}{2}) + \frac{\partial }{{\partial {x_j}}}\rho (E + \frac{{{u_i}{u_j}}}{2}){u_j} = \frac{{\partial {\sigma _{ij}}{u_j}}}{{\partial {x_j}}} + {F_i}{u_i} - \frac{{\partial {q_j}}}{{\partial {x_j}}} + Q,$$

(1.1.3)

Here ρ is a density, u _i is a local velocity, σ _ij is a stress tensor, F _i is a mass force, E is a specific internal energy, q is a heat flow, Q is a bulk heat source.