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Equations (1.24), (1.26), and (1.53) can be combined to give the equation of motion
of an isotropic, Newtonian, classical fluid:
![$\displaystyle \rho \,\frac{D v_i}{Dt}= F_i - \frac{\partial p}{\partial x_i} + ...
...partial x_i}\!\left(\frac{2}{3}\,\mu\,\frac{\partial v_j}{\partial x_j}\right).$](img279.png) |
(1.54) |
This equation is generally known as the Navier-Stokes equation, and is named after Claude-Louis Navier (1785-1836)
and George Gabriel Stokes (1819-1903).
In situations in which there are no strong temperature gradients in the fluid, it is a good approximation to treat viscosity as a spatially uniform quantity, in which case the
Navier-Stokes equation simplifies
somewhat to give
![$\displaystyle \rho \,\frac{D v_i}{Dt}= F_i - \frac{\partial p}{\partial x_i} +\...
..._j}+ \frac{1}{3}\,\frac{\partial^{\,2} v_j}{\partial x_i\,\partial x_j}\right).$](img280.png) |
(1.55) |
When expressed in vector form, the previous expression becomes
![$\displaystyle \rho\,\frac{D{\bf v}}{Dt}\equiv \rho\!\left[\frac{\partial {\bf v...
...\mu\left[\nabla^{\,2} {\bf v} + \frac{1}{3}\,\nabla(\nabla\cdot{\bf v})\right],$](img281.png) |
(1.56) |
where use has been made of Equation (1.39). Here,
Note, however, that the previous identities are only valid in Cartesian coordinates. (See Appendix C.)
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Previous: Momentum Conservation
Richard Fitzpatrick
2016-01-22