The induced magnetic field is neglected under the assumption of a small magnetic Reynolds number
. The top and bottom disks start to execute non-torsional oscillations in their own planes with the velocities U and -U, respectively, where [??]sinnt, n is the frequency of the oscillation and t is the time.
The fluid is assumed to be slightly conducting, so that the magnetic Reynolds number
is much less than unity and hence the induced magnetic field is negligible in comparison with the applied magnetic field.
Russian physicists Zel'dovich (1914-87) and Ruzmaikin discuss some topics in hydromagnetic dynamo theory in the astrophysical context of large magnetic Reynolds number
, define criteria for field generation in a state of near-complete freezing-in, and offer an account of certain qualitative aspects of a turbulent dynamo operating through non-uniform rotation of a conducting medium subject to random motions with helicity.
The magnetic Reynolds number
is small and the induced magnetic field is negligible.
We will assume that the magnetic Reynolds number
for the flow is small so that the induced magnetic field can be neglected.
The magnetic Reynolds number
is takentobesmall andtherefore theinduced magnetic field is neglected.
The ratio between b and [B.sub.0] is proportional to the magnetic Reynolds number
, defined as
In Table 1 we give also the magnetic Reynolds number
, computed by taking as typical length the horizontal length [l.sub.h]
This assumption is valid for low magnetic Reynolds number
Furthermore the following assumptions are considered: (i) fluid has constant kinematic viscosity and thermal diffusivity and the Boussinesq approximation may be adopted for steady laminar flow, (ii) the particle diffusivity is constant, (iii) the concentration of particles is sufficiently dilute that particle coagulation in the boundary layer is negligible, and (iv) the magnetic Reynolds number
is small so that the induced magnetic field is negligible in comparison with the applied magnetic field.
A theoretical analysis of the MHD equations steady state solutions in the incompressible case is given as a function of three parameters: the Reynolds number [R.sub.e], the magnetic Reynolds number
[R.sub.m] and Alfvenic Mach number MA for some of significant asymptotic limits.
The topics include a posteriori error estimation via nonlinear error transport with application to shallow water, enforcing discrete mass conservation in incompressible flow simulations with continuous velocity approximation, the stability of partitioned methods for magnetohydrodynamic flows at small magnetic Reynolds number
, an immersed finite element method of lines for moving interface problems with non-homogeneous flux jumps, and full Eulerian modeling and effective numerical studies for the dynamic fluid-structure interaction problem.