Navier-Stokes equations: Difference between revisions

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where <math>\mathbf{f} </math> is a volumetric force (e.g. <math>\rho g</math> for gravity), and <math>\mathbb{T} </math> is the stress tensor.
where <math>\mathbf{f} </math> is a volumetric force (e.g. <math>\rho g</math> for gravity), and <math>\mathbb{T} </math> is the stress tensor.
==Stress==


The vector quantity <math> \nabla \cdot\mathbb{T} </math> is the ''shear stress''. For a Newtonian incompressible fluid,
The vector quantity <math> \nabla \cdot\mathbb{T} </math> is the ''shear stress''. For a Newtonian incompressible fluid,

Revision as of 16:25, 17 May 2010

Continuity

or, using the substantive derivative:

For an incompressible fluid, is constant, hence the velocity field must be divergence-free:

Momentum

(Also known as the Navier-Stokes equation.)

or, using the substantive derivative:

where is a volumetric force (e.g. for gravity), and is the stress tensor.


Stress

The vector quantity is the shear stress. For a Newtonian incompressible fluid,

with being the (dynamic) viscosity.

For an inviscid fluid, the momentum equation becomes Euler's equation for ideal fluids:

References