Energy Conservation

Consider a fixed volume $V$ surrounded by a surface $S$ . The total energy content of the fluid contained within $V$ is

$$E = \int_V \rho\,{\cal E}\,dV + \int_V\frac{1}{2}\,\rho\,v_i\,v_i\,dV,$$ (1.59)

where the first and second terms on the right-hand side are the net internal and kinetic energies, respectively. Here, ${\cal E}({\bf r},t)$ is the internal (i.e., thermal) energy per unit mass of the fluid. The energy flux across $S$ , and out of $V$ , is [cf., Equation (1.29)]

$${\mit\Phi}_E = \oint_S \rho\left({\cal E} + \frac{1}{2}\,v_i\,v_i... ...ial x_j}\! \left[\rho\left({\cal E} + \frac{1}{2}\,v_i\,v_i\right)v_j\right]dV,$$ (1.60)

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 $V$ , plus the net energy flux out of $V$ , is equal to the net rate of work done on the fluid within $V$ , minus the net heat flux out of $V$ : that is,

$$\frac{dE}{dt} + {\mit\Phi}_E = \dot{W} -\dot{Q},$$ (1.61)

where $\dot{W}$ is the net rate of work, and $\dot{Q}$ the net heat flux. It can be seen that $\dot{W}-\dot{Q}$ is the effective energy generation rate within $V$ [cf., Equation (1.31)].

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

$$\dot{W} = \int_V v_i\,F_i\,dV + \oint_S v_i\,\sigma_{ij}\,dS_j = \int_V\left[v_i\,F_i+ \frac{\partial(v_i\,\sigma_{ij})}{\partial x_j}\right]dV,$$ (1.62)

where use has been made of the tensor divergence theorem.

Generally speaking, heat flow in fluids is driven by temperature gradients. Let the $q_i({\bf r},t)$ be the Cartesian components of the heat flux density at position ${\bf r}$ and time $t$ . It follows that the heat flux across a surface element $d{\bf S}$ , located at point ${\bf r}$ , is ${\bf q}\cdot d{\bf S} = q_i\,dS_i$ . Let $T({\bf r},t)$ be the temperature of the fluid at position ${\bf r}$ and time $t$ . Thus, a general temperature gradient takes the form $\partial T/\partial x_i$ . 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,

$$q_i = A_{ij}\,\frac{\partial T}{\partial x_j},$$ (1.63)

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

$$A_{ij} = -\kappa\,\delta_{ij},$$ (1.64)

where $\kappa({\bf r},t)$ is termed the thermal conductivity of the fluid. It follows that the most general expression for the heat flux density in an isotropic fluid is

$$q_i = -\kappa\,\frac{\partial T}{\partial x_i},$$ (1.65)

or, equivalently,

$${\bf q} = -\kappa\,\nabla T.$$ (1.66)

Moreover, it is a matter of experience that heat flows down temperature gradients: that is, $\kappa>0$ . We conclude that the net heat flux out of volume $V$ is

$$\dot{Q} = -\oint_S\kappa\,\frac{\partial T}{\partial x_i}\,dS_i =... ...artial}{\partial x_i}\!\left(\kappa\,\frac{\partial T}{\partial x_i}\right) dV,$$ (1.67)

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:

$$\int_V\left\{\frac{\partial}{\partial t}\!\left[\rho\left({\cal E... ...x_j}\!\left[\rho\left({\cal E}+\frac{1}{2}\,v_i\,v_i\right)v_j\right]\right\}dV$$

$$= \int_V\left[v_i\,F_i + \frac{\partial}{\partial x_j}\!\left(v_i\,\sigma_{ij} + \kappa\,\frac{\partial T}{\partial x_j}\right)\right]dV.$$ (1.68)

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

$$\frac{\partial}{\partial t}\!\left[\rho\left({\cal E}+ \frac{1}{2... ...l x_j}\!\left(v_i\,\sigma_{ij} + \kappa\,\frac{\partial T}{\partial x_j}\right)$$ (1.69)

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

$$\rho\,\frac{D}{D t}\!\left({\cal E}+ \frac{1}{2}\,v_i\,v_i\right)... ... x_j}\!\left(v_i\,\sigma_{ij} + \kappa\,\frac{\partial T}{\partial x_j}\right),$$ (1.70)

where use has been made of the continuity equation, (1.40). The scalar product of ${\bf v}$ with the fluid equation of motion, (1.53), yields

$$\rho\,v_i\,\frac{D v_i}{Dt} = \rho\,\frac{D}{Dt}\!\left(\frac{1}{2}\,v_i\,v_i\right) = v_i\,F_i + v_i\,\frac{\partial \sigma_{ij}}{\partial x_j}.$$ (1.71)

Combining the previous two equations, we get

$$\rho\,\frac{D{\cal E}}{Dt} = \frac{\partial v_i}{\partial x_j} \,... ...{\partial}{\partial x_j}\!\left(\kappa\,\frac{\partial T}{\partial x_j}\right).$$ (1.72)

Finally, making use of Equation (1.26), we deduce that the energy conservation equation for an isotropic Newtonian fluid takes the general form

$$\frac{D{\cal E}}{Dt} = - \frac{p}{\rho}\,\frac{\partial v_i}{\par... ...l}{\partial x_j}\!\left( \kappa\,\frac{\partial T}{\partial x_j}\right)\right].$$ (1.73)

Here,

$$\chi = \frac{\partial v_i}{\partial x_j}\,d_{ij} = 2\,\mu\left(e_... ...3}\,\frac{\partial v_i}{\partial x_i}\,\frac{\partial v_j}{\partial x_j}\right)$$ (1.74)

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

$$\frac{D{\cal E}}{Dt} = - \frac{p}{\rho}\,\nabla\cdot{\bf v} + \frac{\chi}{\rho} +\frac{\nabla\!\cdot( \kappa\,\nabla T)}{\rho}.$$ (1.75)

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.