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
$$\begin{aligned} \rho \left( \frac{\partial {\varvec{u}}}{\partial t}+{\varvec{u}}\cdot \nabla {\varvec{u}}\right) = -\nabla p +\nabla \cdot \mathbb {S}+{\varvec{F}}_{s}+{\varvec{F}}_{l}+{\varvec{S}} \end{aligned}$$
$$\begin{aligned} \nabla \cdot {\varvec{u}}=0 \end{aligned}$$
where \(\mathbb {S}\) is the viscous stress tensor written as
$$\begin{aligned} \mathbb {S}=2\mu \mathbb {D}=2\mu \left( \frac{1}{2}(\nabla {\varvec{u}}+\nabla {\varvec{u}}^{T})\right) \end{aligned}$$
with density \(\rho \), pressure 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.


Incompressible Viscous Fluid Motions Normal Stress Balance Marangoni Effect Surface Tension Coefficient Marangoni Number 
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    Müller U, Bühler L (2013) Magnetofluiddynamics in channels and containers. Springer Science and Business Media, BerlinGoogle Scholar
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    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

Authors and Affiliations

  1. 1.Chinese Academy of SciencesBeijingChina

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