Fourier Principle

1. Fluid is in local thermal equilibrium
2. Fluid obeys Stokes Principle
3. Energy flux in Newtonian fluid by molecular conduction
4. If thre is no temperature gradient, the energy flux by molecular conduction is zero

From 1: We can define at every local point a temperature T. Thus we write e=f(V,T) then

In most fluids is very small. Note that is not a constant. Also note that h=e+PV and hence,

then

and are respectively known as the specific heats at constant volume and at constant pressure. From 2 & 3:

From 4: If then , thus

for example:

From 2: Istropic tensor of second order

the minus sign is just conventional, so that k>0 Thus

The heat is conducted along the direction of the temperature gradient. Here k is the thermal conductivity and is a coeffcient of viscous conducting fluid. Substitution of de=cvdT and gives

this is the Fourier Energy Equation. The thermal conductivity, k, is determined from experiments like . From continuity equation and

then

Substitution of the above relation into the enrgy equation gives