0.5 Equations of motion and energy in cartesian coordinates  (Page 3/12)

The left-hand term can be identified to be the rate of change of kinetic energy. The first term on the right-hand side is the rate of change of potential energy due to body forces. The second term is the rate at which surface stresses do work on the material volume. We will now focus attention on the last term.

The last term is the double contracted product of the stress tensor with the velocity gradient tensor. Recall that the stress tensor is symmetric for a nonpolar fluid and the velocity gradient tensor can be split into symmetric and antisymmetric parts. The double contract product of a symmetric tensor with an antisymmetric tensor is zero. Thus the last term can be expressed as a double contracted product of the stress tensor with the rate of deformation tensor. We will use the expression for the stress of a Newtonian fluid.

where $\Phi$ is the second invariant of the rate of deformation tensor. Thus the rate at which kinetic energy per unit volume changes due to the internal stresses is divided into two parts:

1. a reversible interchange with strain energy, ,
2. a dissipation by viscous forces,

Since ${\Theta }^2-2\Phi$ is always positive, this last term is always dissipative. If Stokes' relation is used this term is

for incompressible flow it is

(The above equation is sometimes written $-4\mu \Phi$ , where $\Phi$ is called the dissipation function. We have reserved the symbol $\Phi$ for the second invariant of the rate of deformation tensor, which however is proportional to the dissipation function for incompressible flow. $Y$ is the symbol used later for the negative of the dissipation by viscous forces.)

The energy equation

We need the formulation of the energy equation since up to this point we have more unknowns than equations. In fact we have one continuity equation (involving the density and three velocity components), three equations of motion (involving in addition the pressure and another thermodynamic variable, say the temperature) giving four equations in six unknowns. We also have an equation of state, which in incompressible flow asserts that $\rho$ is a constant reducing the number of unknowns to five. In the compressible case it is a relation

which increases the number of equations to five. In either case, there remains a gap of one equation, which is filled by the energy equation.

The equations of continuity and motion were derived respectively from the principles of conservation of mass and momentum. We now assert the first law of thermodynamics in the form that the increase in total energy (we shall consider only kinetic and internal energies) in a material volume is the sum of the heat transferred and work done on the volume. Let $\mathbf{q}$ denote the heat flux vector, then, since $\mathbf{n}$ is the outward normal to the surface, $-\mathbf{q}•\mathbf{n}$ is the heat flux into the volume. Let $U$ denote the specific internal energy, then the balance expressed by the first law of thermodynamics is