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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:

(1.54) 
This equation is generally known as the NavierStokes equation, and is named after ClaudeLouis Navier (17851836)
and George Gabriel Stokes (18191903).
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
NavierStokes equation simplifies
somewhat to give

(1.55) 
When expressed in vector form, the previous expression becomes

(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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Richard Fitzpatrick
20160331