Viscous Rayleigh-Taylor and Richtmyer-Meshkov instabilities in the presence of a horizontal magnetic field

We first derive the exact dispersion relation for viscous Rayleigh-Taylor instability in the presence of a horizontal magnetic field using a decomposition method, and we find that the horizontal magnetic field contributes to the generation of vorticity inside the flow, thereby further distorting the velocity field. This differs from the previous view of the horizontal magnetic field behaving as a surface-tension-like force that does not produce any vorticity in inviscid flow. Vorticity transport is also investigated. The well-known approximate dispersion relation yields growth rates based on an irrotational approximation with a maximum error of 19% in comparison with the exact rates. Furthermore, we investigate the physics of the viscous Richtmyer-Meshkov instability in the presence of a magnetic field, and we find that the presence of the magnetic field leads to the generation of more eigenvalues, thereby modifying the motion of the interface. Comparisons confirm that the viscosity and magnetic field both play fundamental roles in interface behavior, and it is clarified that the behaviors of the interface for viscous Richtmyer–Meshkov instability become in agreement with the numerical simulations. The dependences of the eigenvalues on the viscosities and densities of the fluids, as well as on the magnetic field, are also discussed. Finally, we analyze the evolution of the decay modes to investigate the rotationality of the velocity fields.

Figure 1

Schematic description of the physical model. A magnetic field $H$ is directed along the $x$ axis, and SW denotes the vertical motion of a shock wave.

Figure 2

Dimensionless growth rate $\sigma$ as a function of dimensionless wave number $K$ for $A_T=0.5$, $S=0.1$ (a), 1 (b), and 5 (c), and $m={\mu }_b/{\mu }_a=0.1,1,10$ from top to bottom.

Figure 3

Dimensionless growth rate $\sigma$ as a function of dimensionless perturbation wave number $K$ for parameter $S=1$, viscous parameter $m={\mu }_b/{\mu }_a$ 0.1 (a), 1 (b), 10 (c), and Atwood number $A_T=0.2,0.5,0.8$ from bottom to top.

Figure 4

Dimensionless growth rate $\sigma$ as a function of dimensionless perturbation wave number $K$ for $m={\mu }_b/{\mu }_a=5$, $A_T=0.2$ (a), 0.5 (b), 0.8 (c), and $S=0.1,1,5$ from top to bottom.

Figure 5

Dimensionless damping factor $\gamma ={\rho }_{a}n/\left({\mu }_a{k}^2\right)$ as a function of $m$ for $A_T=0.5$.

Figure 6

$\gamma$ as a function of $A_T$ for viscous parameter $m={\mu }_b/{\mu }_a=0.5$.

Figure 7

$\gamma$ as a function of $T={\rho }_a\nu H^2/4\pi {\mu }_a^2{k}^2$ for $A_T=0.5$ and $m=2$.

Figure 8

$\gamma$ as a function of $m$ for $A_T=0.5$ and $T=2$.

Figure 9

$\gamma$ as a function of $A_T$ for $T=1$ and $m=2$: (a) first mode and (b) other modes.

Figure 10

Ratio of decay mode $q_a$ and wave number $k$ as a function of the parameter $K$ for $S=0$ and different $m$.

Figure 11

Ratio of decay mode $q_a$ and wave number $k$ as a function of the parameter $K$ for $m=1$ and different $S$.