MHD Kelvin-Helmholtz Instability in Magnetized Flows#
Study the stability of two fluid layers with different velocities and magnetic fields, leading to the development of the Kelvin-Helmholtz instability.
Mathematical Model#
MHD Equations:
\[\begin{split}
\begin{align*}
\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) &= 0 \\
\rho \frac{D\mathbf{v}}{Dt} &= -\nabla p + \rho \mathbf{g} + \nabla \cdot \boldsymbol{\tau} + \mathbf{J} \times \mathbf{B} \\
\frac{\partial \mathbf{B}}{\partial t} &= \nabla \times (\mathbf{v} \times \mathbf{B} - \eta \nabla \times \mathbf{B}) \\
\mathbf{J} &= \sigma (\mathbf{E} + \mathbf{v} \times \mathbf{B}) \\
\rho C_p \frac{DT}{Dt} &= -p \nabla \cdot \mathbf{v} + \nabla \cdot (k \nabla T) + \frac{1}{\sigma}(\mathbf{J} \cdot \mathbf{E})
\end{align*}
\end{split}\]
Initial and Boundary Conditions:
Initial condition: \(\rho(\mathbf{x},0) = \rho_0, \mathbf{v}(\mathbf{x},0) = \mathbf{v}_0, \mathbf{B}(\mathbf{x},0) = \mathbf{B}_0, T(\mathbf{x},0) = T_0\)
Boundary conditions: Appropriate conditions on \(\mathbf{v}, \mathbf{B}, T\) at the interfaces.