The Developments and the Applications of the Numerical Algorithms in Simulating the Incompressible Magnetohydrodynamics with Complex Boundaries and Free Surfaces pp 27-31 | Cite as

# Governing Equations

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## Abstract

The set of equations governing the viscous incompressible fluid motion under the influence of an external MF are the momentum and mass conservation equations, given by where \(\mathbb {S}\) is the viscous stress tensor written as with density \(\rho \), pressure

$$\begin{aligned} \rho \left( \frac{\partial {\varvec{u}}}{\partial t}+{\varvec{u}}\cdot \nabla {\varvec{u}}\right) = -\nabla p +\nabla \cdot \mathbb {S}+{\varvec{F}}_{s}+{\varvec{F}}_{l}+{\varvec{S}} \end{aligned}$$

$$\begin{aligned} \nabla \cdot {\varvec{u}}=0 \end{aligned}$$

$$\begin{aligned} \mathbb {S}=2\mu \mathbb {D}=2\mu \left( \frac{1}{2}(\nabla {\varvec{u}}+\nabla {\varvec{u}}^{T})\right) \end{aligned}$$

*p*and dynamics viscosity \(\mu \). \({\varvec{F}}_{s}\) stands for the surface tension which just acts on the interface and is formulated as a volume force based on the CSF technique.## Keywords

Incompressible Viscous Fluid Motions Normal Stress Balance Marangoni Effect Surface Tension Coefficient Marangoni Number
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

- 1.Müller U, Bühler L (2013) Magnetofluiddynamics in channels and containers. Springer Science and Business Media, BerlinGoogle Scholar
- 2.Ni MJ, Munipalli R, Morley NB et al (2007) A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part I: on a rectangular collocated grid system. J Comput Phys 227(1):174–204MathSciNetCrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2019