8.7: The Equations of Hydrodynamics

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We now derive the equations governing fluid flow. The equations of mass and momentum balance are $$\begin{aligned} {\pz \rho\over\pz t}+\bnabla\ncdot(\rho\,\BV)&=0\\ {\pz(\rho\, V\ns_\alpha)\over\pz t} + {\pz \RPi\ns_{\alpha\beta}\over\pz x^\beta}&=0\ ,\end{aligned}$$ where \[\RPi\ns_{\alpha\beta}=\rho\, V\ns_\alpha V\ns_\beta + p\,\delta\ns_{\alpha\beta} \, - \, \stackrel

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{\overbrace{\left\{\eta\Bigg( {\pz V\ns_\alpha\over\pz x\ns_\beta} + {\pz V\ns_\beta\over\pz x\ns_\alpha}-\frac{2}{3}\,\bnabla\!\cdot\!\BV\,\delta\ns_{\alpha\beta}\bigg) +\zeta\,\bnabla\!\cdot\!\BV\,\delta\ns_{\alpha\beta}\right\}}}\ .\] Substituting the continuity equation into the momentum balance equation, one arrives at $$\rho\,{\pz\BV\over\pz t} + \rho\,(\BV\ncdot\bnabla)\BV = -\bnabla p + \eta\,\nabla^2\BV + (\zeta+\third\eta)\bnabla(\bnabla\ncdot\BV)\ , \label{NSB}$$ which, together with continuity, are known as the Navier-Stokes equations. These equations are supplemented by an equation describing the conservation of energy, $$T{\pz s\over\pz T} + T\,\bnabla\ncdot(s\BV)={\tilde\sigma}\ns_{\alpha\beta}\,{\pz V\ns_\alpha\over\pz x^\beta} + \bnabla\ncdot(\kappa\bnabla T)\ .$$ Note that the LHS of Equation [NSB] is $\rho\,D\BV/Dt$, where $D/Dt$ is the convective derivative. Multiplying by a differential volume, this gives the mass times the acceleration of a differential local fluid element. The RHS, multiplied by the same differential volume, gives the differential force on this fluid element in a frame instantaneously moving with constant velocity $\BV$. Thus, this is Newton’s Second Law for the fluid.