Mass Governing Equations - Revision history

Admin at 00:38, 8 November 2010

2010-11-08T00:38:56Z

← Older revision		Revision as of 00:38, 8 November 2010
Line 53:		Line 53:
	</math>		</math>

	~~''What velocity is <math>\boldsymbol{v}</math>?''~~

	~~Mass-average velocity~~

	''Could we define our velocity to be different? Would this equation still hold?''		''What velocity is <math>\boldsymbol{v}</math>?'' Mass-average velocity

			''Could we define our velocity to be different? Would this equation still hold? (Why did we choose the mass-average velocity?''

			A material volume moves at a velocity that prevents the flux of the property (in this case, mass)

	== References ==		== References ==

Admin: /* Derivation */

2010-11-08T00:37:10Z

Derivation

@@ Line 6: / Line 6: @@
 The material volume moves at the velocity <math>\boldsymbol{v}_{mass}</math>, the average velocity of the mass of fluid.
 The conservation of mass stipulates that
@@ Line 42: / Line 44: @@
 \frac{ \partial \rho }{ \partial t } + \nabla \cdot \left( \rho \boldsymbol{v} \right) = 0
 </math>
 == References ==

Admin at 20:54, 7 November 2010

2010-11-07T20:54:25Z

← Older revision		Revision as of 20:54, 7 November 2010
Line 1:		Line 1:
			== Derivation ==

	The governing equation for total mass in a fluid flow can be obtained from [[Reynolds Transport Theorem]], by setting the extensive property <math>B</math> equal to total mass, <math>B = m</math>. The corresponding intensive property <math>b = 1</math>.		The governing equation for total mass in a fluid flow can be obtained from [[Reynolds Transport Theorem]], by setting the extensive property <math>B</math> equal to total mass, <math>B = m</math>. The corresponding intensive property <math>b = 1</math>.

Line 41:		Line 43:
	</math>		</math>

			== References ==

			<references />


	{{GoverningEquations}}		{{GoverningEquations}}

Admin at 20:53, 7 November 2010

2010-11-07T20:53:53Z

@@ Line 28: / Line 28: @@
 <math>
 \frac{ \partial \rho }{ \partial t } + \nabla \cdot \left( \rho \boldsymbol{v}_{mass} \right) = 0
 </math>

Admin: Created page with "The governing equation for total mass in a fluid flow can be obtained from Reynolds Transport Theorem, by setting the extensive property $B$ equal to total mass, <..."

2010-11-07T20:47:38Z

Created page with "The governing equation for total mass in a fluid flow can be obtained from Reynolds Transport Theorem, by setting the extensive property <math>B</math> equal to total mass, <..."

New page

The governing equation for total mass in a fluid flow can be obtained from [[Reynolds Transport Theorem]], by setting the extensive property <math>B</math> equal to total mass, <math>B = m</math>. The corresponding intensive property <math>b = 1</math>.

The material volume corresponding to the total mass is <math>V_{mass}^{M}(t)</math>, whose boundary moves in such a way that the mass flux across the boundary is zero. The surface area of this material volume is <math>S_{mass}^{M}</math>.

The material volume moves at the velocity <math>\boldsymbol{v}_{mass}</math>, the average velocity of the mass of fluid.

The conservation of mass stipulates that

<math>
\frac{dB}{dt} = 0
</math>

and therefore Reynolds Transport Theorem becomes

<math>
\iiint_{V_{mass}^{M}} \frac{\partial \rho}{\partial t} dV
+ \iint_{S_{mass}^{M}} \rho \boldsymbol{v}_{mass} \cdot \boldsymbol{n} dS = 0
</math>

This can be rewritten using Gauss's Theorem:

<math>
\iiint_{V_{mass}^{M}} \frac{\partial \rho}{\partial t} + \nabla \cdot \left( \rho \boldsymbol{v}_{mass} \right) dV = 0
</math>

This equation can be used in the finite volume formulation. Alternatively, the integral volume can be dropped to yield the differential form, applicable for finite difference:

<math>
\frac{ \partial \rho }{ \partial t } + \nabla \cdot \left( \rho \boldsymbol{v}_{mass} \right) = 0
</math>

{{GoverningEquations}}

Mass Governing Equations - Revision history

Admin at 00:38, 8 November 2010

Admin: /* Derivation */

Admin at 20:54, 7 November 2010

Admin at 20:53, 7 November 2010

Admin: Created page with "The governing equation for total mass in a fluid flow can be obtained from Reynolds Transport Theorem, by setting the extensive property B equal to total mass, <..."

Admin: Created page with "The governing equation for total mass in a fluid flow can be obtained from Reynolds Transport Theorem, by setting the extensive property $B$ equal to total mass, <..."