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Velocity Field


A velocity field is a time-dependent vector field v(x,t) that assigns a velocity to every position x at time t. A particle path x(t) transported by the field satisfies

 (dx)/(dt)=v(x(t),t).

In fluid mechanics, the material derivative measures change along such particle paths, and its spatial part is the convective term. The velocity field is the principal unknown in both Euler's equations of inviscid motion and the Navier-Stokes equations.


See also

Convective Term, Flow, Material Derivative, Vector Field

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References

Batchelor, G. K. An Introduction to Fluid Dynamics. Cambridge, England: Cambridge University Press, pp. 71-74, 1977.Quanta Magazine. "Why Navier-Stokes Pushes Math and Physics to the Edge." Sep. 17, 2026. https://www.youtube.com/watch?v=PsWR1rQEWYo.

Cite this as:

Weisstein, Eric W. "Velocity Field." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/VelocityField.html

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