Magnetoconvection in a Bidispersive Porous Media

Thermal instability of magnetoconvection in a horizontal bidispersive porous layer, uniformly heated from below, is analyzed. To study the linear stability theory, we perturbed the basic state with small-amplitude disturbances. Then, the governing dimensionless equations are solved using the normal modes. By employing the one-term Galerkin weighted residuals method, the critical values of Rayleigh numbers for the onset of stationary and oscillatory instability, have been determined. The effect of Chandrasekhar number on the system was analyzed.

Hopf bifurcations in plasma layers between rigid planes in thermal magnetohydrodynamics, via a simple formula

The phenomenon produced by the Hopf bifurcations is of notable importance. In fact, a Hopf bifurcation—guaranteeing the existence of an unsteady periodic solution of the linearized problem at stake—is also an optimum limit cycle candidate of the nonlinear associated problem and, if non linearly globally attractive, is an absorbing set and an effective limit cycle. The present paper deals with the onset of Hopf bifurcations in thermal magnetohydrodynamics (MHD). Precisely, it is devoted to characterization—via a simple formula—of the Hopf bifurcations threshold in horizontal plasma layers between rigid planes, heated from below and embedded in a constant transverse magnetic field. This problem, remarked clearly and notably by the Nobel Laureate Chandrasekhar (Nature 175:417–419, 1955), constitutes a difficulty met by him and—for plasma layers between rigid planes electricity perfectly conducting—is, as far as we know, still not removed. Let $$m_0$$ m 0 be the thermal conduction rest state and let $$P_r, P_m, R, Q$$ P r , P m , R , Q , be the Prandtl, the Prandtl magnetic, the Rayleigh and the Chandrasekhar number, respectively. Recognized (according to Chandrasekhar) that the instability of $$m_0$$ m 0 via Hopf bifurcation can occur only in a plasma with $$P_m>P_r$$ P m > P r , in this paper it is shown that the Hopf bifurcations occur if and only if $$\begin{aligned} Q>Q_c=\displaystyle \frac{4\pi ^2[1+P_r(\mu /2\pi )^4]}{P_m-P_r}, \end{aligned}$$ Q > Q c = 4 π 2 [ 1 + P r ( μ / 2 π ) 4 ] P m - P r , with $$ \mu =7.8532$$ μ = 7.8532 . Moreover, the critical value of R at which the Hopf bifurcation occurs is characterized via the smallest zero of the second invariant of the spectrum equation governing the most destabilizing perturbation. The critical value of Q, in the free-rigid and rigid-free cases is shown to be $$\displaystyle \frac{1}{4}$$ 1 4 of the previous value.

The convective instability of a Maxwell–Cattaneo fluid in the presence of a vertical magnetic field

We study the instability of a Bénard layer subject to a vertical uniform magnetic field, in which the fluid obeys the Maxwell–Cattaneo (MC) heat flux–temperature relation. We extend the work of Bissell ( Proc. R. Soc. A 472, 20160649 (doi:10.1098/rspa.2016.0649)) to non-zero values of the magnetic Prandtl number p m . With non-zero p m , the order of the dispersion relation is increased, leading to considerably richer behaviour. An asymptotic analysis at large values of the Chandrasekhar number Q confirms that the MC effect becomes important when C Q 1/2 is O (1), where C is the MC number. In this regime, we derive a scaled system that is independent of Q . When CQ 1/2 is large, the results are consistent with those derived from the governing equations in the limit of Prandtl number p → ∞ with p m finite; here we identify a new mode of instability, which is due neither to inertial nor induction effects. In the large p m regime, we show how a transition can occur between oscillatory modes of different horizontal scale. For Q ≫ 1 and small values of p , we show that the critical Rayleigh number is non-monotonic in p provided that C > 1/6. While the analysis of this paper is performed for stress-free boundaries, it can be shown that other types of mechanical boundary conditions give the same leading-order results.

Heat transfer and flow regimes in quasi-static magnetoconvection with a vertical magnetic field

Numerical simulations of quasi-static magnetoconvection with a vertical magnetic field are carried out up to a Chandrasekhar number of $Q=10^{8}$ over a broad range of Rayleigh numbers $Ra$. Three magnetoconvection regimes are identified: two of the regimes are magnetically constrained in the sense that a leading-order balance exists between the Lorentz and buoyancy forces, whereas the third regime is characterized by unbalanced dynamics that is similar to non-magnetic convection. Each regime is distinguished by flow morphology, momentum and heat equation balances, and heat transport behaviour. One of the magnetically constrained regimes appears to represent an ‘ultimate’ magnetoconvection regime in the dual limit of asymptotically large buoyancy forcing and magnetic field strength; this regime is characterized by an interconnected network of anisotropic, spatially localized fluid columns aligned with the direction of the imposed magnetic field that remain quasi-laminar despite having large flow speeds. As for non-magnetic convection, heat transport is controlled primarily by the thermal boundary layer. Empirically, the scaling of the heat transport and flow speeds with $Ra$ appear to be independent of the thermal Prandtl number within the magnetically constrained, high-$Q$ regimes.

Numerical Solution of Natural Convective Heat Transfer Under Magnetic Field Effect

In this study, non-Newtonian pseudoplastic fluid flow equations for 2-D steady, incompressible, the natural convective heat transfer are solved numerically by pseudo time derivative. The stability properties of natural convective heat transfer in an enclosed cavity region heated from below under magnetic field effect are investigated depending on the Rayleigh and Chandrasekhar numbers. Stability properties are studied, in particular, for the Rayleigh number from 104 to 106 and for the Chandrasekhar number 3, 5 and 10. As a result, when Rayleigh number is bigger than 106 and Chandrasekhar number is bigger than 10, the instability occurs in the flow domain. The results obtained for natural convective heat transfer problem are shown in the figures for Newtonian and pseudoplastic fluids. Finally, the local Nusselt number is evaluated along the bottom wall.

Hydromagnetic Stability of Metallic Nanofluids (Cu-Water and Ag-Water) Using Darcy-Brinkman Model

Thermal convection of a nanofluid layer in the presence of imposed vertical magnetic field saturated by a porous medium is investigated for both-free, rigid-free, and both-rigid boundaries using Darcy-Brinkman model. The effects of Brownian motion and thermophoretic forces due to the presence of nanoparticles and Lorentz’s force term due to the presence of magnetic field have been considered in the momentum equations along with Maxwell’s equations. Keeping in mind applications of flow through porous medium in geophysics, especially in the study of Earth’s core, and the presence of nanoparticles therein, the hydromagnetic stability of a nanofluid layer in porous medium is considered in the present formulation. An analytical investigation is made by applying normal mode technique and Galerkin type weighted residuals method and the stability of Cu-water and Ag-water nanofluids is compared. Mode of heat transfer is through stationary convection without the occurrence of oscillatory motions. Stability of the system gets improved appreciably by raising the Chandrasekhar number as well as Darcy number whereas increase in porosity hastens the onset of instability. Further, stability of the system gets enhanced as we proceed from both-free boundaries to rigid-free and to both-rigid boundaries.

MHD Flow Of Walters’ Liquid B Over A Nonlinearly Stretching Sheet

The paper discusses the boundary layer flow of a weak electrically conducting viscoelastic Walters’ liquid B over a nonlinearly stretching sheet subjected to an applied transverse magnetic field, when the liquid far away from the surface is at rest. The stretching is assumed to be a quadratic function of the coordinate along the direction of stretching. An analytical expression is obtained for the stream function and velocity components as a function of the viscoelastic parameter, the Chandrasekhar number and stretching related parameters. The results have possible technological applications in liquid based systems involving stretchable materials.

Effect of Boussinesq-Stokes Suspension over an Exponentially Stretching Sheet in a Hydromagnetic Flow

The paper presents the study of velocity profiles in a hydromagnetic flow of Boussinesq-Stokes suspension over an exponentially stretching impermeable sheet. The basic equations governing the flow are in the form of partial differential equations. The equations have been transformed to nonlinear ordinary differential equation by applying a suitable local similarity transformation. The solution of the transformed equation is obtained by using differential transform method (DTM) with assistance from the Newton-Raphson method in obtaining the unknown initial values. The solution is obtained as a power series with assured convergence. The effects of local Chandrasekhar number and couple stress parameter on velocity profiles are studied. The findings of the study are represented graphically.

Effect of magnetic field on parametrically driven surface waves

Stability analysis of parametrically driven surface waves in liquid metals in the presence of a uniform vertical magnetic field is presented. Floquet analysis gives various subharmonic and harmonic instability zones. The magnetic field stabilizes the onset of parametrically excited surface waves. The minima of all the instability zones are raised by a different amount as the Chandrasekhar number is raised. The increase in the magnetic field leads to a series of bicritical points at a primary instability in thin layers of a liquid metal. The bicritical points involve one subharmonic and another harmonic solution of different wavenumbers. A tricritical point may also be triggered as a primary instability by tuning the magnetic field.

Compositional convection in the presence of a magnetic field. II. Cartesian plume

The analysis of part I, dealing with the morphological instability of a single interface in a fluid of infinite extent, is extended to the case of a Cartesian plume of compositionally buoyant fluid, of thickness 2 x 0 , enclosed between two vertical interfaces. The problem depends on six dimensionless parameters: the Prandtl number, σ ; the magnetic Prandtl number, σ m ; the Chandrasekhar number, Q c ; the Reynolds number, Re ; the ratio, B v , of vertical to horizontal components of the ambient magnetic field and the dimensionless plume thickness. Attention is focused on the preferred mode of instability, which occurs in the limit Re ≪1 for all values of the parameters. This mode can be either sinuous or varicose with the wavenumber vector either vertical or oblique , comprising four types. The regions of preference of these four modes are represented in regime diagrams in the ( x 0 , σ ) plane for different values of σ m , Q c , B v . These regions are strongly dependent on the field inclination and field strength and, to a lesser extent, on magnetic diffusion. The overall maximum growth rate for any prescribed set of the parameters σ m , Q c , B v , occurs when 1.3< x 0 <1.7, and is sinuous for small σ and varicose for large σ . The magnetic field can enhance instability for a certain range of thickness of the plume. The enhancement of instability is due to the interaction of the field with viscous diffusion resulting in a reverse role for viscosity. The dependence of the helicity and α -effect on the parameters is also discussed.