where the first and second terms on the right-hand side are the net internal and kinetic energies, respectively. Here, is the internal (i.e., thermal) energy per unit mass of the fluid. The energy flux across , and out of , is [cf., Equation (1.29)]

where use has been made of the tensor divergence theorem. According to the first law of thermodynamics, the rate of increase of the energy contained within , plus the net energy flux out of , is equal to the net rate of work done on the fluid within , minus the net heat flux out of : that is,

(1.61) |

where is the net rate of work, and the net heat flux. It can be seen that is the effective energy generation rate within [cf., Equation (1.31)].

The net rate at which volume and surface forces do work on the fluid within is

where use has been made of the tensor divergence theorem.

Generally speaking, heat flow in fluids is driven by temperature gradients. Let the be the Cartesian components of the heat flux density at position and time . It follows that the heat flux across a surface element , located at point , is . Let be the temperature of the fluid at position and time . Thus, a general temperature gradient takes the form . Let us assume that there is a linear relationship between the components of the local heat flux density and the local temperature gradient: that is,

(1.63) |

where the are the components of a second-rank tensor (which can be functions of position and time). In an isotropic fluid we would expect to be an isotropic tensor. (See Section B.5.) However, the most general second-order isotropic tensor is simply a multiple of . Hence, we can write

(1.64) |

where is termed the

(1.65) |

or, equivalently,

Moreover, it is a matter of experience that heat flows down temperature gradients: that is, . We conclude that the net heat flux out of volume is

where use has been made of the tensor divergence theorem.

Equations (1.59)-(1.62) and (1.67) can be combined to give the following energy conservation equation:

(1.68) |

However, this result is valid irrespective of the size, shape, or location of volume , which is only possible if

(1.69) |

everywhere inside the fluid. Expanding some of the derivatives, and rearranging, we obtain

(1.70) |

where use has been made of the continuity equation, (1.40). The scalar product of with the fluid equation of motion, (1.53), yields

(1.71) |

Combining the previous two equations, we get

(1.72) |

Finally, making use of Equation (1.26), we deduce that the

Here,

is the rate of heat generation per unit volume due to viscosity. When written in vector form, Equation (1.73) becomes

According to the previous equation, the internal energy per unit mass of a co-moving fluid element evolves in time as a consequence of work done on the element by pressure as its volume changes, viscous heat generation due to flow shear, and heat conduction.