A calculation of the nonlinear state of the m=n=1 internal kink mode within the ideal magnetohydrodynamic approximation is presented. The nonlinear problem is divided into a layer around the q=1 surface and an outer region. An examination of the nonlinear terms from the two regions shows that within the optimal ordering of the linear theory, nonlinear terms from the outer region are formally bigger than the corresponding terms of the layer. However, because of the rigid shift nature of the linear radial displacement, these terms are actually weaker than the terms from the layer. The contribution from the nonlinear terms from the outer region has been calculated and it is shown that it produces a small correction in the final expression. This is essential for the calculation of Avinash et al. [Phys. Rev. Lett. 59, 2647 (1987)] and Bussac and Pellat [Phys. Rev. Lett. 59, 2650 (1987)] to be correct. Further, the case of arbitrary resonant q profiles has been considered and the relevance of these results to recently observed sawtooth activity in tokamaks is discussed.
Some stationary solutions of two-fluid magnetohydrodynamic equations are constructed using generalized helicity invariants. Solutions corresponding to the Z pinch, Bennett pinch, θ pinch, etc., are constructed. The Z-pinch-like solution is identical to Weibel’s solution [Phys. Fluids 2, 52 (1959)], while the Bennett-pinch-like solution contains nonuniform axial drifts. By constructing the same solutions from the Vlasov–Maxwell system of equations, it is shown that the results obtained here are consistent with those of Mahajan [Phys. Fluids B 1, 43 (1989)]. Similarly, new θ-pinch-like solutions are constructed and the relation of these to earlier work is discussed.
The influence of thermal radiation on a two-dimensional non-Newtonian fluid flow past a slendering stretching surface is investigated theoretically. Casson and Williamson fluid models are considered with Soret and Dufour effects. The transformed ODEs are solved numerically using the bvp5c Matlab package and dual solutions are executed for Casson and Williamson fluid cases. The influence of various parameters, namely, thermal radiation parameter, cross diffusion parameters and slip parameters on velocity, thermal and concentration distributions are discussed with the assistance of graphs. The local Nusselt and Sherwood numbers are computed and presented through tables. It is observed that the influence of cross diffusion is higher on Williamson flow when equated with the Casson flow.
This study covers a numerical investigation of gyrotactic microorganisms contained MHD flow over a vertical plate bearing thermal radiation, thermophoresis, Brownian motion, chemical reaction and inclined magnetic field effects. With the assistance of similarity transforms, the derived governed equations are transformed as set of ODEs and solved numerically by R-K and Newton’s methods. Graphs are exhibited and explained for various parameters of interest. For engineering interest, we mainly talked about the Skin friction coefficient, reduced Sherwood, Nusselt numbers and density of motile microorganisms. We noticed a rise in the heat transfer rate of motile microorganisms for rising values of the thermophoresis and Brownian motion parameters. Increasing values of the aligned angle hikes the drag force.
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