Effect of magnetic field dependent viscosity on ferroconvection in the presence of dust particles

This paper deals with the theoretical investigation of the effect of magnetic field dependent (MFD) viscosity on the thermal convection in a ferromagnetic fluid in the presence of dust particles. For a flat ferromagnetic fluid layer contained between two free boundaries, the exact solution is obtained using a linear stability analysis and a normal mode analysis method. For the case of stationary convection, dust particles always have a destabilizing effect, whereas the MFD viscosity has a stabilizing effect on the onset of convection. In the absence of MFD viscosity, the destabilizing effect of magnetization is depicted but in the presence of MFD viscosity, non-buoyancy magnetization may have a destabilizing or a stabilizing effect on the onset of convection. The critical wave number and critical magnetic thermal Rayleigh number for the onset of stationary convection are also determined numerically for sufficiently large values of buoyancy magnetization parameter M ₁. Graphs have been plotted by giving numerical values to the parameters to depict the stability characteristics. It is observed that the critical magnetic thermal Rayleigh number is reduced solely because the heat capacity of clean fluid is supplemented by that of the dust particles. The principle of exchange of stabilities is found to hold true for the ferromagnetic fluid heated from below in the absence of dust particles. The oscillatory modes are introduced due to the presence of the dust particles, which were non-existent in their absence. A sufficient condition for the non-existence of overstability is also obtained.     

Magneto hydrodynamic stability of self-gravitational fluid cylinder

The self-gravitating instability of a fluid cylinder pervaded by magnetic field and endowed with surface tension has been discussed. The dispersion relation is derived and some reported works are recovered as limiting cases from it. The capillary force is destabilizing only in the small axisymmetric domain and stabilizing otherwise. The magnetic field has a strong stabilizing effect in all modes of perturbation for all wavelengths. The self-gravitating force is destabilizing in the axisymmetric perturbation. However the magnetic field effect modified a lot the destabilizing character of the model and could overcome the capillary and self-gravitating instability of the model for all short and long wavelengths.     

The null spaces of elliptic partial differential operators in Rn

Overstability in a horizontal layer of a viscoelastic fluid is considered in the presence of a uniform magnetic field. The equations of motion appropriate to hydromagnetics in a Maxwellian fluid have been established and the analysis has been carried out in terms of normal modes. The proper solutions have been obtained for the case of two free boundaries. The dispersion relation obtained is found to be quite complex and involves the Prandtl number p₁, magnetic Prandtl number p₂, a parameter Q characterizing the strength of the magnetic field, and a parameter Γ which characterizes the elasticity of the fluid. Numerical calculations have been performed for different values of the parameters involved and the values of critical Rayleigh numbers, wave numbers, and frequencies for the onset of instability as overstability have been obtained. It is found that the magnetic field has a stabilizing influence on the overstable mode of convection in a viscoelastic fluid. Elasticity is found to have a destabilizing influence as in the absence of a magnetic field. Thus the effect of a magnetic field is the same as that for an ordinary viscous fluid.     

Effect of price elasticity of demand in monopolies with gradient adjustment

We study a monopolistic market characterized by a constant elasticity demand function, in which the firm technology is described by a linear total cost function. The firm is assumed to be boundedly rational and to follow a gradient rule to adjust the production level in order to optimize its profit. We focus on what happens on varying the price elasticity of demand, studying the effect on the equilibrium stability. We prove that, depending on the relation between the market size and the marginal cost, two different scenarios are possible, in which elasticity has either a stabilizing or a mixed stabilizing/destabilizing effect.     

Viscous potential flow analysis of magnetohydrodynamic interfacial stability through porous media

In the view of viscous potential flow theory, the hydromagnetic stability of the interface between two infinitely conducting, incompressible plasmas, streaming parallel to the interface and subjected to a constant magnetic field parallel to the streaming direction will be considered. The plasmas are flowing through porous media between two rigid planes and surface tension is taken into account. A general dispersion relation is obtained analytically and solved numerically. For Kelvin-Helmholtz instability problem, the stability criterion is given by a critical value of the relative velocity. On the other hand, a comparison between inviscid and viscous potential flow solutions has been made and it has noticed that viscosity plays a dual role, destabilizing for Rayleigh-Taylor problem and stabilizing for Kelvin-Helmholtz. For Rayleigh-Taylor instability, a new dispersion relation has been obtained in terms of a critical wave number. It has been found that magnetic field, surface tension, and rigid planes have stabilizing effects, whereas critical wave number and porous media have destabilizing effects.     

Hydromagnetic instability of two coaxial streaming cylinders with double perturbed interfaces

The magnetohydrodynamic (MHD) stability of a double interface perturbed streaming liquid cylinder coaxial with a streaming fluid mantle acting upon capillary, inertial, pressure gradient and electromagnetic forces has been developed. The problem is formulated, solved and the stability criterion of the model is estabilished. The latter is discussed analytically and the results are confirmed numerically and interpreted physically. Some reported works are recovered as limiting cases. The capillary force is stabilizing or not according to restrictions. The magnetic field has a strong stabilizing influence. The radii (liquid–fluid) ratio plays an important role in increasing the MHD stabilizing domains. The density of liquid–fluid ratio has a little stabilizing effect. The streaming has a destabilizing influence for all kinds of (non-) axisymmetric perturbation modes. However, if the magnetic field strength is so strong such that the Alfvén wave velocity is greater than the streaming velocity, then the destabilizing character due to capillary force or/and streaming is completely suppressed and stability sets in. In the absence of the magnetic field and we neglect the fluid inertial force, the present results are in good agreement with the experimental results of (Kendall J.M. Phys Fluids 1986;29:2086).     

Stability of an infinite horizontal layer of fluid with a spatial heat source in the presence of a magnetic field

The stability of an electrically conducting infinite layer of a viscous fluid which loses heat throughout its volume at a constant rate in the presence of a magnetic field is discussed. The value of the critical Rayleigh number R is found to decrease as the rate of heat loss increases which implies that the layer becomes more unstable. It is observed that the destabilizing effect of the heat source term Q is more prominent when the strength of the magnetic field is low. The aspect ratio ‘a’ increases with increasing Q showing that the horizontal dimension of the cells decreases with increasing Q.     

Toroidal magnetodynamic stability of cylindrical fluid jet

The hydromagnetic stability of a fluid jet under the combined influence of the electromagnetic (with toroidal varying field) and capillary forces has been developed. A general dispersion relation valid for all modes of perturbation is derived. The magnetic fields interior and exterior the fluid jet are stabilizing. The capillary force is destabilizing for small domain in the axisymmetric mode while it is stabilizing for all the rest. The magnetic fields increase the capillary stable domains and at the same time cause the shrinking those of instability. Above a certain value of the relative magnetic field strength, the capillary instability is depressed and then stability sets in. In spite of the field interior the fluid is non-uniform, however, we found that it has a strong stabilizing influence.     

Influence of induced magnetic field and heat transfer on peristaltic transport of a Carreau fluid

The effect of an induced magnetic field on peristaltic flow of an incompressible Carreau fluid in an asymmetric channel is analyzed. Perturbation solution to equations under long wavelength approximation is derived in terms of small Weissenberg number. Expressions have been constructed for the stream function, the axial induced magnetic field, the magnetic force function, the current density distribution and the temperature. Trapping phenomenon is examined with respect to emerging parameters of interest.     

On the non-orthogonal Stagnation flow of the Oldroyd-B fluid

This paper is concerned with a non-orthogonal stagnation flow of an Oldroyd-B fluid between two parallel plates. We reduce the problem to a set of ordinary differential equations (ODE's), which is then solved with finite differences using a parameter continuation method. Perturbation analyses are also carried out for small Reynolds numbers and small Weissenberg numbers respectively. The solution of the set of ODE's is discussed. It is known that for a Newtonian fluid, the stagnation point shifts from the potential flow case in the opposite direction of the tangential velocity. The effect of the fluid elasticity is to reduce this shift. It is also shown that the Oldroyd-B model has a limiting Weissenbeg number, depending on the angle of the injected flow.     

Rayleigh-Taylor instability of compressible finitely conducting rotating fluid in the presence of a magnetic field

The character of the equilibrium of a non-viscous, compressible finitely conducting rotating fluid in the presence of a vertical magnetic field along the direction of gravitational field has been investigated. It is shown that the solution is characterised by a variational principle. Based on the existence of variational principle, an approximate solution has been derived for the case of a fluid having exponentially varying density in the vertical direction. Due to finite resistivity of the medium it is found that potentially stable or unstable configuration retains its character. Further the growth rate of disturbance has been obtained corresponding to short and long wavelengths and it is found that electrical resistivity suppresses the growth rate for large wavelengths but it increases the same for small wavelengths. It is further shown that magnetic field has a destabilizing influence for large wavelengths and a stabilizing influence for small wavelengths.     

Computation of viscoelastic fluid flows using continuation methods

The numerical simulation of viscoelastic fluid flow becomes more difficult as a physical parameter, the Weissenberg number, increases. Specifically, at a Weissenberg number larger than a critical value, the iterative nonlinear solver fails to converge, a phenomenon known as the high Weissenberg number problem. In this work we describe the application and implementation of continuation methods to the nonlinear Johnson–Segalman model for steady-state viscoelastic flows. Simple, natural, and pseudo-arclength continuation approaches in Weissenberg number are investigated for a discontinuous Galerkin finite element discretization of the equations. Computations are performed for a benchmark contraction flow and, several aspects of the performance of the continuation methods including high Weissenberg number limits, are discussed.     

RETRACTED ARTICLE: Effect of electromagnetic field on the stability of viscoelastic fluid film flowing down an inclined plane

The stability of thin electrically conducting viscoelastic fluid film flowing down on a non-conducting inclined plane in the presence of electromagnetic field is investigated under induction-free approximation. Surface evolution equation is derived by long-wave expansion method. The stabilizing role of Hartman number M (magnetic field) and the destabilizing role of the viscoelastic property \({\varGamma}\) and the electric parameter E on such fluid film are established through the linear stability analysis of the surface evolution equation. Investigation shows that at small values of Hartman number (M), the influence of electric parameter (E) on the viscoelastic parameter \({(\varGamma)}\) is insignificant, while for large values of M, E introduces more destabilizing effect at low values of \({\varGamma}\) than that at high values of \({\varGamma }\). An interesting result also perceived from our analysis is that the stabilizing effect of Hartman number (M) is decreasing with the increase of the values of \({\varGamma}\) and E, even it gives destabilizing effect after a certain high value of the electric field depending on the high value of \({\varGamma}\). The weakly nonlinear study reveals that the increase of \({\varGamma}\) decreases the explosive and subcritical unstable zones but increases the supercritical stable zone keeping the unconditional zone almost constant.     

Stability analysis of an interface between immiscible liquids in Hele-Shaw flow in the presence of a magnetic field

Hydrodynamic instabilities may occur when a viscous fluid is driven by a less viscous one through a porous medium. These penetrations are common in enhanced oil recovery, dendrite formation and aquifer flow. Recent studies have shown that the use of magnetic suspensions allow the external control of the instability. The problem is nonlinear and some further improvements of both theory and experimental observations are still needed and continue being a current source of investigation. In this paper we present a generalized Darcy law formulation in order to examine the growth of finger instabilities as a magnetic field is applied to the interface between the fluids in a Hele-Shaw cell. A new linear stability analysis is performed in the presence of magnetic effects and provides a stability criterion in terms of the non-dimensional physical parameters of the examined flow and the wavenumber of the finger disturbances. The interfacial tension inhibits small wavelength instabilities. The magnetic field contributes to the interface stability for moderate wavelength as it is applied parallel to the liquid-interface. In particular, we find an explicit expression, as a function of the susceptibility, for a critical angle between the interface and the magnetic field direction, in which its effect on the interface is neutral. We have developed a new asymptotic solution for the flow problem in a weak nonlinear regime. The first correction captures the second order nonlinear effects of the magnetic field, which tends to align the fingers with the field orientation and have a destabilizing effect. The analysis predicts that the non-linear effects at second order can counterbalance the first order stabilizing effect of a parallel magnetic field which results in a loss of effectiveness for controlling the investigated finger instabilities. The relevant physical parameters for controlling these finger instabilities are clearly identified by our non-dimensional analysis.     

Magnetoviscous potential flow analysis of Kelvin–Helmholtz instability with heat and mass transfer

A linear analysis of the Kelvin–Helmholtz instability of interface between two viscous and magnetic fluids has been carried out where there was heat and mass transfer across the interface while the fluids have been subjected to a constant magnetic field parallel to the streaming direction. The viscous potential flow theory has been used for the investigation. A dispersion relation has been obtained and a stability criterion is given by a critical value of relative velocity as well as the critical value of magnetic field. The resulting plots show the effect of various physical parameters such as wave number, viscosity ratio, ratio of magnetic permeabilities and heat transfer coefficient. It has been observed that heat and mass transfer has a destabilizing effect whereas the horizontal magnetic field stabilizes the system.     

Magnetohydrodynamics flow of a Burgers’ fluid in an orthogonal rheometer

The magnetohydrodynamics flow of an electrically conducting, incompressible Burgers’ fluid in an orthogonal rheometer is investigated. An exact solution is obtained. The effects of various dimensionless parameters existing in the model on the velocity field, vorticity and traction are studied graphically. It is noted that boundary layers form for a variety of reasons. It form as the Reynolds number increases. Also, as the Weissenberg number increases a distinct boundary layer formation is observed. It can develop at low Reynolds number provided the Weissenberg number is sufficiently high, however, it is not possible in the case of a Newtonian fluid. It is shown that no torque is exerted by the fluid on one of the disks. Results are compared with Oldroyd-B fluid.     

MHD flows of UCM fluids above porous stretching sheets using two-auxiliary-parameter homotopy analysis method

The performance of a two-auxiliary-parameter homotopy analysis method (HAM) is investigated in solving laminar MHD flow of an upper-convected Maxwell fluid (UCM) above a porous isothermal stretching sheet. The analysis is carried out up to the 20th-order of approximation, and the effect of parameters such as elasticity number, suction/injection velocity, and magnetic number are studied on the velocity field above the sheet. The results will be contrasted with those reported recently by Hayat et al. [Hayat T, Abbas Z, Sajid M. Series solution for the upper-convected Maxwell fluid over a porous stretching plate. Phys Lett A 358;2006:396–403] obtained using a third-order one-auxiliary-parameter homotopy analysis method. It is concluded that the flow reversal phenomenon as predicted by Hayat et al. (2006) may have arisen because of the inadequacies of using just one-auxiliary-parameter in their analysis. That is, no flow reversal is predicted to occur if instead of using one-auxiliary-parameter use is made of two auxiliary parameters together with a more convenient set of base functions to assure the convergence of the series used to solve the highly nonlinear ODE governing the flow.     

Nonlinear stability of an axial electric field: Effect of interfacial charge relaxation

We introduce a nonlinear perturbation technique to third order, to study the stability between two cylindrical inviscid fluids, subjected to an axial electric field. The study takes into account the relaxation of electrical charges at the interface between the two fluids. At first order, a linear dispersion relation is obtained. Analytical and numerical results for the overstability and incipient instability conditions are given. For perfect dielectric fluids, the electric field has a stabilizing influence, while for leaky dielectric fluids, the electric field can have either a stabilizing or a destabilizing influence depending on the conductivity and permittivity ratios of the two fluids. At higher order, a nonlinear dispersion relation (nonlinear Ginzburg–Landau equation) is derived, describing the evolution of wave packets of the problem. For leaky dielectric fluids near the marginal state, a nonlinear diffusion equation (nonlinear incipient instability) is obtained. For perfect dielectric fluids, two cubic nonlinear Schrödinger equations are obtained. One of these equations to determine a nonlinear cutoff electric field separating stable and unstable disturbance, whereas the other is used to analyze the stability of the system. It is found that the nonlinear stability criterion depending on the ratio of permittivity, Such effects can only be explained successfully in the nonlinear sense, as the linear analysis unsuccessful to inform about them.     

Entropy generation in nanofluid flow of Walters-B fluid with homogeneous-heterogeneous reactions

Research on optimization of entropy generation in nanofluid flow gained much interest. In this study, the Walter's-B nanofluid flow is considered to analyze the irreversibility in cubic autocatalysis. Fluid motion is considered in presence of viscous dissipation, magnetohydrodynamics (MHD), radiation, and heat generation absorption. Homotopy analysis method (HAM) is employed to solve nonlinear ordinary differential system. Results show that fluid flow reduces for larger Weissenberg and Hartman numbers. Temperature gradually enhances for larger Weissenberg number and radiation parameter. For higher estimation of thermophoresis parameter, the temperature and concentration are enhanced. Opposite impact of Hartman and Weissenberg numbers is noticed for entropy generation and Bejan number. Disorderedness and Bejan number are reduced near the sheet, while the opposite trend is seen away from the sheet.     

Revisited Hastings and Powell model with omnivory and predator switching

The effect of omnivory in predator–prey system is debatable regarding its stabilizing or destabilizing characteristics. Earlier theoretical studies predict that omnivory is stabilizing or destabilizing depending on the condition of the system. The effect of omnivory in the food chain system is not yet properly understood. In the present paper, we study the effect of omnivory in a tri-trophic food chain system on the famous Hastings and Powell model. Omnivory enhances the chance of predator switching between prey and middle predator. The novelty of this paper is to study the effect of predator switching of the top predator which is omnivorous in nature. Our results suggest that in the absence of switching, an increase of omnivory stabilizes the system from chaotic dynamics, however, if we further increase the strength of omnivory, the system becomes unstable and middle predator goes to extinction. It is also observed that the predator switching enhance the stability and persistence of all populations.