Navier-Stokes equations - Revision history

Dduque: Other form for the mom. eq.

2010-05-17T15:32:36Z

Other form for the mom. eq.

← Older revision		Revision as of 17:32, 17 May 2010
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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.

			Another form of the equation, more similar in form to the continuity equation, stresses the fact that the '''momentum density''' is conserved. For each of the three Cartesian coordinates <math>\alpha=1,2,3</math>:

			:<math> \frac{\partial \rho v_\alpha}{\partial t} +
			\nabla \cdot {\rho v_\alpha \mathbf{v}} =
			-\frac{\partial p}{\partial x_\alpha} +
			\sum_\beta \frac{\partial }{\partial x_\beta} \mathbb{T}_{\beta\alpha} + f_\alpha. </math>

			In vector form:

			:<math> \frac{\partial \rho v_\alpha}{\partial t} +
			\nabla \cdot {\rho \mathbf{v} \mathbf{v}} =
			-\nabla p + \nabla\cdot\mathbb{T} + \mathbf{f}. </math>

			The term <math> \mathbf{v} \mathbf{v} </math> is a dyad (direct tensor product).

	==Stress==		==Stress==

Dduque: New section

2010-05-17T15:25:48Z

New section

← Older revision		Revision as of 17:25, 17 May 2010
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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,

Carl McBride: Added a category

2010-05-14T13:19:09Z

Added a category

← Older revision		Revision as of 15:19, 14 May 2010
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	:<math> \rho \left(\frac{D \mathbf{v}}{D t} \right) = -\nabla p + \mathbf{f} . </math>		:<math> \rho \left(\frac{D \mathbf{v}}{D t} \right) = -\nabla p + \mathbf{f} . </math>

			==References==
			<references/>
			[[Category: classical mechanics]]

Dduque: New page with NS equations

2010-05-13T09:54:14Z

New page with NS equations

New page

== Continuity ==

:<math> \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 </math>

or, using the [[substantive derivative]]:

:<math> \frac{D\rho}{Dt} + \rho (\nabla \cdot \mathbf{v}) = 0. </math>

For an incompressible fluid, <math>\rho</math> is constant, hence the velocity field must be divergence-free:

:<math> \nabla \cdot \mathbf{v} =0. </math>

== Momentum ==

(Also known as ''the'' Navier-Stokes equation.)

:<math> \rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \nabla \cdot\mathbb{T} + \mathbf{f}, </math>

or, using the [[substantive derivative]]:

:<math> \rho \left(\frac{D \mathbf{v}}{D t} \right) = -\nabla p + \nabla \cdot\mathbb{T} + \mathbf{f}, </math>

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.

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

:<math> \nabla \mathbb{T} = \mu \nabla^2 \mathbf{v}, </math>

with <math>\mu</math> being the (dynamic) viscosity.

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

:<math> \rho \left(\frac{D \mathbf{v}}{D t} \right) = -\nabla p + \mathbf{f} . </math>