Navier-Stokes Equations -- from Eric Weisstein's World of Physics

Navier-Stokes Equations

The Navier-Stokes equations are the fundamental partial differentials equations that describe the flow of incompressible fluids. Using the rate of stress and rate of strain tensors, it can be shown that the components of a viscous force F in a nonrotating frame are given by

(1)

(2)

(Tritton 1988, Faber 1995), where is the dynamic viscosity, is the second viscosity coefficient, is the Kronecker delta, Eric Weisstein's World of Math is the divergence, Eric Weisstein's World of Math is the bulk viscosity, and Einstein summation Eric Weisstein's World of Math has been used to sum over j = 1, 2, and 3.

Continuity Equation, Euler's Equation of Inviscid Motion, Navier-Stokes Equations--Rotational, Reynolds Number, Stokes Flow, Stokes Flow--Cylinder, Stokes Flow--Sphere

References

Clay Mathematics Institute. "Navier-Stokes Equations." http://www.claymath.org/Millennium_Prize_Problems/Navier-Stokes_Equations/.

Faber, T. E. Fluid Dynamics for Physicists. New York: Cambridge University Press, 1995.

Smale, S. "Mathematical Problems for the Next Century." In Mathematics: Frontiers and Perspectives 20000821820702 (Ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur). Providence, RI: Amer. Math. Soc., 2000.

Tritton, D. J. Physical Fluid Dynamics, 2nd ed. Oxford, England: Clarendon Press, pp. 52-53 and 59-60, 1988.