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one-way-coupled magnetohydrodynamics with transient magnetic fields

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Old   September 27, 2013, 06:20
Default one-way-coupled magnetohydrodynamics with transient magnetic fields
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Christoph Kratzsch
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Hi everyone,

i 'm searching for a method to add the one-way-coupled MHD-equations with momentum source of transient magnetic fields.
This means quasi-(static) approximation for low magnetic Reynolds-numbers but not static .

I searched a lot in the literature, but can't find a reference regarding that problem.

Can everyone share some information, where i can find a solution for that?

Best regards ,

Christoph
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Old   September 30, 2013, 09:31
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Jonas T. Holdeman, Jr.
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I am not sure what you are asking. Do you mean adding a Lorentz force term to the fluid equations for specified magnetic fields? By one-way do you mean the magnetic field affects the fluid but the fluid does not affect the magnetic field?
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Old   September 30, 2013, 11:34
Default Sry for the coarse description ;-)
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Christoph Kratzsch
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Quote:
Originally Posted by Jonas Holdeman View Post
I am not sure what you are asking. Do you mean adding a Lorentz force term to the fluid equations for specified magnetic fields? By one-way do you mean the magnetic field affects the fluid but the fluid does not affect the magnetic field?
Hi Jonas,

yes i mean the addition of an lorentz-force-term to the Navier-Stokes-Equations.
At this time i'm using quasi-static approximation inside a solver:

\nabla \cdot \underline{u}=\nabla \cdot \underline{B}_0=0

\frac{\partial \underline{u}}{\partial t} +\left(\underline{u} \cdot \nabla\right) \underline{u}
 = -\frac{1}{\varrho}\nabla p + \nu \Delta \underline{u} 
+ \underline{g} 
+ \frac{1}{\varrho}\underline{j}\times\underline{B}_0


\underline{j} = \sigma\left(-\nabla\psi+\underline{u}\times\underline{B}_0\right)

\nabla^2\psi = \nabla \cdot \left(\underline{u}\times \underline{B}_0\right)


This is maintained if the magnetic reynolds number Re_{m}<<1. The magnetic fields reads: B=B_0. But i don't have the possibility to describe time-dependent magnetic fields.

In an two-way-coupling i have a time variation due to the coupling of velocity and magnetic field. Therefore B=B_0+b(t):

\nabla \cdot \underline{u}=0


\frac{\partial \underline{u}}{\partial t} 
+\left(\underline{u} \cdot \nabla\right) \underline{u}
 = -\frac{1}{\varrho}\nabla p + \nu \Delta \underline{u} 
+ \underline{g} 
-\frac{1}{2\varrho\mu}\nabla B^2
+\left(\frac{\underline{B}}{\varrho\mu}\cdot\nabla\right)\underline{B}

\frac{\partial\underline{B}}{\partial t} 
+ \left(\underline{u}\cdot\nabla\right) \underline{B}
= 
 \left(\underline{B}\cdot\nabla\right) \underline{u}
+ \frac{1}{\sigma\mu} \Delta \underline{B}

\nabla \cdot \underline{B} = 0

Now i want a time-varying magnetic field B=B_0(t) or B=B_0(\underline{x},t). This means rotating, travelling or pulsating magnetic fields without two-way-coupling.
Maybe i don't understand it right. So please can you explain why two-way-coupling is necessary if the initial magnetic fields become time-dependant and the induced magnetic fields will be neglected.

Best regards ,

Christoph
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