Difference between revisions of "Navier-Stokes equations"
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| − | The Navier-Stokes equation is an equation in [[ | + | The Navier-Stokes equation is an equation in [[fluid mechanics]] that states: |
<math>\rho \frac{D \mathbf{V}}{D t} = -\nabla p + \mu \nabla^2 \mathbf{V} + \rho \mathbf{g}</math> | <math>\rho \frac{D \mathbf{V}}{D t} = -\nabla p + \mu \nabla^2 \mathbf{V} + \rho \mathbf{g}</math> | ||
| − | where <math>\nabla p</math> is the pressure difference (expressed as the partial derivative of pressure in each dimension), <math>\frac{D \mathbf{V}}{D t}</math> is the total derivative of velocity, <math>\mu \,</math> is the kinematic viscosity of the fluid, <math>\rho \,</math> is the density of the fluid, and <math>\mathbf{g}</math> is the gravitational acceleration. <ref>A.J. Smits, "A Physical Introduction to Fluid Mechanics," John Wiley & Sons, ISBN 0-471-25349-9</ref> | + | where <math>\nabla p</math> is the pressure difference (expressed as the partial derivative of pressure in each dimension), <math>\frac{D \mathbf{V}}{D t}</math> is the total derivative of velocity, <math>\mu \,</math> is the kinematic viscosity of the fluid, <math>\rho \,</math> is the density of the fluid, and <math>\mathbf{g}</math> is the gravitational acceleration.<ref>A.J. Smits, "A Physical Introduction to Fluid Mechanics," John Wiley & Sons, ISBN 0-471-25349-9</ref> |
| + | == In Incompressible flow == | ||
| + | |||
| + | <math>\boldsymbol \tau = \mu (\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T})</math> | ||
== References == | == References == | ||
<references></references> | <references></references> | ||
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
Latest revision as of 16:30, July 13, 2016
The Navier-Stokes equation is an equation in fluid mechanics that states:
<math>\rho \frac{D \mathbf{V}}{D t} = -\nabla p + \mu \nabla^2 \mathbf{V} + \rho \mathbf{g}</math>
where <math>\nabla p</math> is the pressure difference (expressed as the partial derivative of pressure in each dimension), <math>\frac{D \mathbf{V}}{D t}</math> is the total derivative of velocity, <math>\mu \,</math> is the kinematic viscosity of the fluid, <math>\rho \,</math> is the density of the fluid, and <math>\mathbf{g}</math> is the gravitational acceleration.[1]
In Incompressible flow
<math>\boldsymbol \tau = \mu (\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T})</math>
References
- ↑ A.J. Smits, "A Physical Introduction to Fluid Mechanics," John Wiley & Sons, ISBN 0-471-25349-9