Asymptotic stability of linearly evolving non-stationary modes in a uniform hydromagnetic field over a rotating-disk flow with suction and injection

In this paper the linear stability properties of the magnetohydrodynamic flow of an incompressible, viscous and electrically conducting fluid are investigated for the boundary-layer due to an infinite permeable rotating-disk. The fluid is subjected to an external magnetic field perpendicular to the disk. The interest lies also in finding out the effects of uniform suction or injection. In place of the traditional linear stability method, a theoretical approach is adopted here based on the high-Reynolds-number triple-deck theory. It is demonstrated that the nonstationary perturbations evolve in accordance with an eigenrelation analytically obtained.     

Stability of a micropolar fluid layer heated from below

The stability of a layer of micropolar fluid heated from below is studied employing a linear theory as well as an energy method. It is proved that the principle of exchange of stability holds and the critical Rayleigh number is obtained. It is observed that the micropolar fluid layer heated from below is more stable as compared with the classical viscous fluid. The energy method is then used to study the stability under finite disturbances. A variational method is applied to obtain the sharp stability limit. It is found that no subcritical instability region exists and the critical Rayleigh number as derived from the energy method is identical to that of the linear limit.     

Convective and Absolute Instability in the Incompressible Boundary Layer on a Rotating Disk in the Presence of a Uniform Magnetic Field

The stability of a conducting fluid flow over a rotating disk with a uniform magnetic field applied normal to the disk, is investigated. It is assumed that the magnetic field is unaffected by the motion of the fluid. The mean flow and linear stability equations are solved for a range of magnetic field-strength parameters and the absolute/convective nature of the stability is investigated. It is found that increasing the magnetic field parameter is in general stabilizing.     

Linear stability of steady fluid flow through a constricted channel

Summary Considered in this paper is the linear stability of the steady symmetric flow of a Newtonian fluid through an abrupt planar constriction. Flows with Reynolds numbers up to 2,000 and various ratios of contraction were investigated. In all the cases considered asymmetric disturbances imposed on the steady flow were found to decay in time. The results indicate that the flow is linearly stable even for large Reynolds numbers. This is in agreement with experimental investigations which do not reveal the occurrence of any flow instabilities.     

Analysis of fibre wrinkling during squeezing flows of fibre-reinforced composites

An analysis of the stability of squeezing flows between flat plates (consolidation flows) of viscous liquids reinforced by continuous fibres is presented. The ideal linear fibre-reinforced fluid model is used to model the composite as an incompressible Newtonian fluid reinforced with inextensible fibres. The development of small fibre wrinkles initially present in the preimpregnated plies is analysed using linear stability theory. It is shown that when the flows are lubricated by resin rich layers, two perturbation modes are possible. In the first mode, the wrinkles are of the same form throughout the thickness of the sample while in the second mode they vary linearly with distance from the platens. In both cases the stability depends on the normal components of the applied stress. If the only traction acting in addition to hydrostatic pressure is that due to the squeezing force then the first perturbation mode is stable. This prediction is in agreement with experimental results.     

Suction-injection effects on the onset of Rayleigh-Bénard-Marangoni convection in a fluid with suspended particles

Summary The effect of Suction-Injection-Combination (SIC) on the linear stability of Rayleigh-Bénard Marangoni convection in a horizontal layer of an Boussinesq fluid with suspended particles confined between an upper free adiabatic boundary and a lower rigid isothermal/adiabatic boundary is considered. The Rayleigh-Ritz technique is used to obtain the eigenvalues. The influence of various parameters on the onset of convection has been analysed. It is found that the effect of Prandtl number on the stability of the system is dependent on the SIC being pro-gravity or anti-gravity. A similar Pe-sensitivity is found in respect of the critical wave number. It is observed that the fluid layer with suspended particles heated from below is more stable compared to the classical fluid layer without suspended particles. The problem has possible applications in microgravity situations.     

Effects of suction-injection-combination (SIC) on the onset of Rayleigh-Benard magnetoconvection in a micropolar fluid

The effects of suction-injection-combination (SIC) and magnetic field on the linear stability analysis of Rayleigh-Benard convection in a horizontal layer of an Boussinesq micropolar fluid is studied using a Rayleigh-Ritz techinque. The eigenvalues are obtained for free-free, rigid-free and rigid-rigid velocity boundary combinations with isothermal and adiabatic temperature conditions on the spin-vanishing boundaries. The eigenvalues are also obtained for lower rigid isothermal and upper free adiabatic boundaries with vanishing spin. The influence of various micropolar fluid parameters on the onset of convection has been analysed. It is found that the effect of Prandtl number on the stability of the system is dependent on the SIC being pro-gravity or anti-gravity. A similar Pe-sensitivity is found in respect of the critical wave number. It is observed that the micropolar fluid layer heated from below is more stable compared to the classical fluid layer.     

Control of Rayleigh-Bénard Convection in a Fluid Layer with Internal Heat Generation

The control of the onset of convection in a horizontal fluid layer with internal heat generation is studied. The horizontal boundaries of the system are cooled isothermally. The stability of the fluid layer is investigated on the basis of the linear stability theory and the resulting eigenvalues problem is solved numerically. Upon using a feedback proportional control, the heating power of the system is modulated in order to counteract any deviations of the temperature of the fluid from its conductive value. As a result, it is possible to postpone (or advance) significantly the onset of motion. The optimal positions of the thermal sensors can be predicted on the basis of the linear stability theory. The linear stability analysis also reveals the possible existence of Hopf’s bifurcations at the onset of motion. This type of bifurcation can be delayed using differential controllers. Two-dimensional numerical simulations of the full governing equations are carried out and found to agree well with the prediction of the linear stability theory.     

Parametric resonance of a composite cylindrical shell containing pulsatile flow of hot fluid

Vast literature is available for modeling the coupled fluid structure interaction involving the flow of a pulsatile fluid through pipes/shells under ambient temperature. In contrast it is found that there is no literature on such problems under the influence of temperature. This paper considers the evaluation of dynamic instability regions due to a flow of pulsating hot fluid through an insulated composite cylindrical shell. A coupled fluid structure interaction model in conjunction with uncoupled thermomechanical model is used for analysis. The system's equations of motion containing periodic coefficients are expressed in modal domain considering linear transformation. Fourth order Runge–Kutta method using Gill's coefficient is adopted to compute the state transition matrix which provides the stability information of the periodic system following the Floquet–Liapunov theory. Pulsatile flow of water at various magnitude of steady state temperature is considered and hence the effect of water temperature on the instability regions is analyzed. The influence of lamina fibre angle on the instability regions is also examined.     

On the stability of MHD flow of a viscoelastic fluid past a stretching sheet

Summary The effects of a magnetic field on the stability characteristics of a viscoelastic fluid flow due to the stretching sheet are investigated. A three-dimensional linear stability analysis is performed by means of the Method of Weighted Residuals for disturbances of the Taylor-Görtler type. It is found that the magnetic field exerts a stabilizing influence on the flow.     

Freckle formation in a solidifying binary alloy

The effect of solutal solidification on the unidirectional solidification of a binary alloy cooled from below is considered. Soon after the onset of convection a mushy region often forms, and is accompanied by vigorous convection in the molten alloy above. In contrast, the fluid in the mush appears unaffected by the neighbouring flow and remains essentially quiescent. This work considers the nonlinear convective stability of the mush and determines a criterion for channelling in the mush to occur. The basic state is a similarity solution, so that a quasi-static approximation must be applied in order to apply conventional stability theory. Moreover, although the model for solidification is relatively simple, an analytical expression for linear stability is not available. Thus the series of equations arising from the nonlinear stability analysis lead to a complicated set of symbolic and numerical calculations. Stable finite-amplitude solutions are found for Rayleigh numbers larger than critical for all values of the chosen superheat. The nonlinear solutions demonstrate the possibility of channelling within the mush dependent upon the strength of convection. These finite-amplitude solutions are extended further by the calculation of a numerical solution of the model equations. The evolution of the stream function and the mass fraction is followed, the onset of convection and freckling can then be deduced from the numerical simulation. The onset of convection in the mush is found in terms of the mush Rayleigh number, and compares favourably with linear stability theory and experimental data. The onset of freckling is also given in terms of a Rayleigh number, but is sensitive to the initial conditions. This appears to explain the large disagreement found in experiments aimed at finding a criterion for freckling.     

Universal stability of thermo-magneto-micropolar fluid motions

Criteria for universal stability of the unsteady motion of an incompressible, electrically conducting linear micropolar fluid with heat transfer in the presence of an arbitrary magnetic field, and in an arbitrary time dependent domain are established. The model of the micropolar fluid employed is essentially the one proposed by Eringen. The interaction between the flow field and the magnetic field is manifested through the Lorentz force and the coupling between the flow and temperature field arises through the Boussinesq equation of state.The stability method employed is an energy technique due to James Serrin. Certain uniqueness theorems for the unsteady and steady flows of thermo-magneto-micropolar fluid are also established.The theorems established for the stability and uniqueness are universal in the sense that they may be applied to any geometry of bound domains and any distribution of the basic field variables.     

Universal stability of magneto-micropolar fluid motions

Criteria for universal stability of the unsteady motion of an incompressible, electrically conducting linear micropolar fluid, i.e. with rigid microinclusions, in the presence of an arbitrary magnetic field, and in an arbitrary bounded time dependent domain are established. The model of the micropolar fluid employed is essentially the one proposed by Eringen. The interaction between the flow field and the magnetic field is manifested through the body force and body couple. Relativistic, Hall, and temperature effects are neglected. The stability method employed is an energy technique due to James Serrin. Certain uniqueness theorems for the unsteady and steady flows of magneto-micropolar fluid are also established.The theorems established for the stability and uniqueness are universal in the sense that they may be applied to any geometry of bounded domains and any distribution of the basic field variables. The present problem finds application in MHD generators with neutral fluid seedings in the form of rigid microinclusions.     

Electroviscoelastic stability of a Kelvin fluid layer

The electroviscoelastic stability of a Kelvin fluid layer under the effect of a constant tangential electric field is discussed. The linear perturbation is considered. The analysis includes all possible modes of perturbations. A fourth-order partial differential equation is introduced to control the fluid motion. Applications of the boundary conditions lead to two simultaneous ordinary differential equations. A transcendental dispersion equation is obtained. A case of large viscosity is considered in order to simplify the complexity of the dispersion relation. The growth rate with respect to the retardation time parameter is carried out. The necessary and sufficient conditions for stability are introduced. It is found that the retardation time parameter has a stabilizing effects as well as the viscosity parameter, in the presence of the field. It is also, shown that viscosity and elastic parameters play against the density destabilization effects. It is shown that the tangential electric field plays a dual role in the stability criteria.     

On the stability of a hot rotating layer of micropolar fluid

The convective stability of a horizontal layer of incompressible micropolar fluid heated from below and rotating about a vertical axis has been investigated on the basis of linear theory, using normal mode analysis. The boundaries are assumed to be free. After introducing the corrections to the basic equations considered by Sastry and Rao [1], it has been found that the rotation has a destabilizing effect which contradicts the earlier assertion presented in [1].Moreover, microinertia, which does not affect the stability of a hot horizontal layer of incompressible micropolar fluid in the absence of rotation [2], is found to have destabilizing effect.     

The onset of transient Marangoni convection in a liquid layer subjected to rotation about a vertical axis

In view of the interesting possibilities of controlling surface tension-driven convection, anticipated in space experiments involving fluid interfaces, the problem of the stability of a thin horizontal fluid layer subjected to rotation about a vertical axis, when the thermal (or concentration) gradient is not uniform is examined by linear stability analysis. Attention is focussed on the situation where the critical Marangoni number is greater than that for the case of uniform thermal gradient and the convection is not, in general, maintained. The case of adiabatic boundary condition is examined because it brings out the effect of surface tension at the free surfaces and allows a simple application of the Galerkin technique, which gives useful results. Numerical results are obtained for special cases and some general conclusions about the destabilizing effects of various basic temperature profiles and the stabilizing effect of coriolis force are presented. The results indicate that the most destabilizing temperature gradient is one for which the temperature gradient is a step function of the depth. Increase in Taylor number and the inverted parabolic basic temperature profile suppress the onset of convection.     

Instability of Poiseuille flow of two immiscible liquids with different viscosities in a channel

We study the stability of plane Poiseuille flow of two immiscible liquids of different viscosities and equal densities. The problem is like one considered by C. S. Yih who found that flow in two layers of equal thickness was always unstable. We find regions of stability when there are three layers with one of the fluids centrally located. We view our contribution as a study of selection of stable steady flow from a nonunique continuum of Poiseuille flows all of which satisfy the steady Navier-Stokes and which differ from another in the number and thickness of layers of different viscosity. Experiments have shown that there is a tendency for the less viscous fluid to encapsulate the more viscous one. This arrangement of components, with the more viscous fluid in the center of the channel maximizes the mass flux for a fixed pressure gradient. A linear stability analysis of centrally located configuration to long waves is carried out by the analytic methods introduced by Yih [1]. The stability results depend on the viscosity and volume ratio in a fairly complicated way. The flow with the high viscosity fluid centrally located is always stable. Centrally located layers of less viscous fluid, called fingering flows, are always unstable.     

On the linear problem of swelling porous elastic soils with incompressible fluid

In this paper we consider the linear theory of swelling porous elastic soils in the case that the fluid is incompressible. The formulation belongs to the theory of mixtures for porous elastic solids filled with fluid and gas. It proposes some new mathematical difficulties. Continuous dependence results on initial conditions and supply terms are obtained in the general case. Logarithmic convexity method is used in case of fluid saturation. Structural stability with respect two constitutive coefficients is also obtained in the case of fluid saturation.     

Robust and provably second‐order explicit–explicit and implicit–explicit staggered time‐integrators for highly non‐linear compressible fluid–structure interaction problems

An explicit–explicit staggered time‐integration algorithm and an implicit–explicit counterpart are presented for the solution of non‐linear transient fluid–structure interaction problems in the Arbitrary Lagrangian–Eulerian (ALE) setting. In the explicit–explicit case where the usually desirable simultaneous updating of the fluid and structural states is both natural and trivial, staggering is shown to improve numerical stability. Using rigorous ALE extensions of the two‐stage explicit Runge–Kutta and three‐point backward difference methods for the fluid, and in both cases the explicit central difference scheme for the structure, second‐order time‐accuracy is achieved for the coupled explicit–explicit and implicit–explicit fluid–structure time‐integration methods, respectively, via suitable predictors and careful stagings of the computational steps. The robustness of both methods and their proven second‐order time‐accuracy are verified for sample application problems. Their potential for the solution of highly non‐linear fluid–structure interaction problems is demonstrated and validated with the simulation of the dynamic collapse of a cylindrical shell submerged in water. The obtained numerical results demonstrate that, even for fluid–structure applications with strong added mass effects, a carefully designed staggered and subiteration‐free time‐integrator can achieve numerical stability and robustness with respect to the slenderness of the structure, as long as the fluid is justifiably modeled as a compressible medium. Copyright © 2010 John Wiley & Sons, Ltd.     

The effect of a uniform vertical magnetic field on the onset of oscillatory marangoni convection in a horizontal layer of conducting fluid

Summary In this paper we use classical linear stability theory to undertake a detailed investigation of the effect of a uniform vertical magnetic field on the onset of oscillatory thermocapillary-driven Marangoni convection in a horizontal layer of quiescent, electrically conducting fluid heated from below. For simplicity we restrict our attention to the simplest case of a fluid layer with a non-deformable free surface and perfectly electrically conducting boundaries in which the onset of convection is always steady in the absence of the magnetic field. The present numerical calculations show that the presence of the magnetic field can cause the preferred mode of instability to be oscillatory rather than steady convection. Nevertheless, in all the cases investigated the effect of the magnetic field is always to stabilise the layer relative to the case of no magnetic field.