In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:[1][2]
{\displaystyle \underbrace {\rho T{Ds \over {Dt}}} _{\text{Heat Gain}}=\underbrace {\nabla \cdot (\kappa \nabla T)} _{\text{Thermal Conduction}}+\underbrace {{\mu \over {2}}\left({\partial v_{i} \over {\partial x_{j}}}+{\partial v_{j} \over {\partial x_{i}}}-{2 \over {3}}\delta _{ij}\nabla \cdot {\bf {v}}\right)^{2}+\zeta (\nabla \cdot {\bf {v}})^{2}} _{\text{Viscous Dissipation}}}
is the del operator used to characterize the gradient and divergence, and
If the flow velocity is negligible, the general equation of heat transfer reduces to the standard heat equation.
It may also be extended to rotating, stratified flows, such as those encountered in geophysical fluid dynamics.
[3] For a viscous, Newtonian fluid, the governing equations for mass conservation and momentum conservation are the continuity equation and the Navier-Stokes equations:
is the viscous stress tensor, with the components of the viscous stress tensor given by:
In an ideal fluid, as described by the Euler equations, the conservation of energy is defined by the equation:
However, for conservation of energy to hold in a viscous fluid subject to thermal conduction, the energy flux due to advection
must be supplemented by a heat flux given by Fourier's law
and a flux due to internal friction
Then the general equation for conservation of energy is:
Note that the thermodynamic relations for the internal energy and enthalpy are given by:
We may also obtain an equation for the kinetic energy by taking the dot product of the Navier-Stokes equation with the flow velocity
{\displaystyle \rho {Dk \over {Dt}}=-{\bf {v}}\cdot \nabla p+v_{i}{\partial \sigma _{ij} \over {\partial x_{j}}}}
The second term on the righthand side may be expanded to read:
With the aid of the thermodynamic relation for enthalpy and the last result, we may then put the kinetic energy equation into the form:
{\displaystyle \rho {Dk \over {Dt}}=-\rho {\bf {v}}\cdot \nabla h+\rho T{\bf {v}}\cdot \nabla s+\nabla \cdot (\sigma \cdot {\bf {v}})-\sigma _{ij}{\partial v_{i} \over {\partial x_{j}}}}
Now expanding the time derivative of the total energy, we have:
{\displaystyle {\partial \over {\partial t}}\left[\rho (k+\varepsilon )\right]+\nabla \cdot \left[\rho {\bf {v}}(k+h)-\sigma \cdot {\bf {v}}\right]=\rho T{Ds \over {Dt}}-\sigma _{ij}{\partial v_{i} \over {\partial x_{j}}}}
Now adding the divergence of the heat flux due to thermal conduction to each side, we have that:
{\displaystyle {\partial \over {\partial t}}\left[\rho (k+\varepsilon )\right]+\nabla \cdot \left[\rho {\bf {v}}(k+h)-\kappa \nabla T-\sigma \cdot {\bf {v}}\right]=\rho T{Ds \over {Dt}}-\nabla \cdot (\kappa \nabla T)-\sigma _{ij}{\partial v_{i} \over {\partial x_{j}}}}
However, we know that by the conservation of energy on the lefthand side is equal to zero, leaving us with:
The product of the viscous stress tensor and the velocity gradient can be expanded as:
Thus leading to the final form of the equation for specific entropy production:
In the case where thermal conduction and viscous forces are absent, the equation for entropy production collapses to
- showing that ideal fluid flow is isentropic.
This equation is derived in Section 49, at the opening of the chapter on "Thermal Conduction in Fluids" in the sixth volume of L.D.
Landau and E.M. Lifshitz's Course of Theoretical Physics.
[1] It might be used to measure the heat transfer and air flow in a domestic refrigerator,[4] to do a harmonic analysis of regenerators,[5] or to understand the physics of glaciers.