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Energy Conservation

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

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

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

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

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

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

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

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

or, equivalently,

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

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

$\displaystyle \int_V\left\{\frac{\partial}{\partial t}\!\left[\rho\left({\cal E...
...x_j}\!\left[\rho\left({\cal E}+\frac{1}{2}\,v_i\,v_i\right)v_j\right]\right\}dV$    
$\displaystyle = \int_V\left[v_i\,F_i + \frac{\partial}{\partial x_j}\!\left(v_i\,\sigma_{ij} + \kappa\,\frac{\partial T}{\partial x_j}\right)\right]dV.$   (1.68)

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

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

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

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

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

Combining the previous two equations, we get

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

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

Here,

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

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

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


next up previous
Next: Equations of Incompressible Fluid Up: Mathematical Models of Fluid Previous: Navier-Stokes Equation
Richard Fitzpatrick 2016-03-31