Slow rotation of a sphere with source at its centre in a viscous fluid

In this note, the problem of a sphere carrying a fluid source at its centre and rotating with slow uniform angular velocity about a diameter is studied. The analysis reveals that only the azimuthal component of velocity exists and is seen that the effect of the source is to decrease it. Also, the couple on the sphere is found to decrease on account of the source.     

On a resolvent problem for the linearized system from the dynamical system describing the compressible viscous fluid motion

The linearized initial boundary value problem describing the motion of the viscous compressible fluid is studied under Dirichlet zero condition in bounded and unbounded domains. The resolvent estimate for the corresponding operator is proved in the L_p framework and the sharp inner estimate of the resolvent set is obtained. Copyright © 2004 John Wiley Sons, Ltd.     

On the motion of rigid bodies in a viscous fluid

We consider the problem of motion of several rigid bodies in a viscous fluid. Both compressible and incompressible fluids are studied. In both cases, the existence of globally defined weak solutions is established regardless possible collisions of two or more rigid objects.     

A boundary integral formulation for elastically deformable particles in a viscous fluid

This paper reports a formulation and implementation of a mixed (both direct and indirect) boundary element method using the double layer and its adjoint in a form suitable for solving Stokes flow problems involving elastically deformable particles. The formulation is essentially the Completed Double Layer Boundary Element Method for solving an exterior traction problem for the surrounding fluid or solid phase, followed by an interior displacement, and a mobility problem (if required) for the elastic particles. At the heart of the method is a deflation procedure that allows iterative solution strategies to be adopted, effectively opens the way for large-scale simulations of suspensions of deformable particles to be performed. Several problems are considered, to illustrate and benchmark the method. In particular, an analytical solution for an elastic sphere in an elongational flow is derived. The stresslet calculations for an elastic sphere in shear and elongational flows indicate that elasticity of the inclusions can potentially lead to positive second normal stress difference in shear flow, and an increase in the tensile resistance in elongational flow.This work is supported by a grant from the Australian Research Grant Council. X-J F wishes to acknowledge the support of the National Natural Science Foundation of China.     

Vibrations of a cylinder in a concentric vessel filled with a two-layer fluid

A hybrid vibrational system containing a solid (a cylinder) with an elastic connection to a coaxial cylindrical cavity, completely filled with a heavy ideal stably stratified two-layer fluid, is considered. The combined self-consistent vibrations of the body and the fluid (of the internal waves) are studied. An explicit solution of the internal boundary value problem of an inhomogeneous liquid in an annular domain for a specified motion of the body is obtained. An integrodifferential equation of the Newton type is constructed on the basis of this. This equation describes the self-consistent oscillations of the cylinder. In the case of weak coupling of the interaction between the solid and the medium, an approximate solution is obtained using asymptotic methods and an analysis is carried out. Qualitative effects of the mutual effect of the motions of the cylinder and the fluid are found.     

On the existence of hydrodynamic instability in single diffusive bottom heavy systems with permeable boundaries

We utilize the reformulated equations of the classical theory, as derived by Banerjee et al. (J. Math. Anal. Appl. 175 (1993) 458), to establish mathematically, the existence of hydrodynamic instability in single diffusive bottom heavy systems, when considered in the more general framework of the boundary conditions of the type specified by Beavers and Joseph (J. Fluid Mech. 30 (1967) 197), in the parameter regime T ₀ α ₂ > 1, where T ₀ and α ₂ being some properly chosen mean temperature and coefficient of specific heat (at constant volume) variation due to temperature variation respectively.     

Scattering of oblique waves by bottom undulations in a two-layer fluid

Scattering of waves obliquely incident on small cylindrical undulations at the bottom of a two-layer fluid wherein the upper layer has a free surface and the lower layer has an undulating bottom, is investigated here assuming linear theory. There exists two modes of time-harmonic waves propagating at each of the free surface and the interface. Due to an obliquely incident wave of a particular mode, reflected and transmitted waves of both the modes are created in general by the bottom undulations. For small undulations, a simplified perturbation analysis is used to obtain first-order reflection and transmission coefficients of both the modes due to oblique incidence of waves of again both modes, in terms of integrals involving the shape function describing the bottom. For sinusoidal undulations, these coefficients are plotted graphically to illustrate the energy transfer between the waves of different modes induced by the bottom undulations.     

Propagation of harmonic waves in prestressed fiber viscoelastic thick tubes filled with a viscous dusty fluid

In this work, propagation of harmonic waves in initially stressed cylindrical viscoelastic thick tubes filled with a Newtonian fluid is studied. The tube, subjected to a static inner pressure P_i and a positive axial stretch λ, will be considered as an incompressible viscoelastic and fibrous material. The fluid is assumed as an incompressible, viscous and dusty fluid. The field equations for the fluid are obtained in the cylindrical coordinates. The governing differential equations of the tube’s viscoelastic material are obtained also in the cylindrical coordinates utilizing the theory of small deformations superimposed on large initial static deformations. For the axially symmetric motion the field equations are solved by assuming harmonic wave solutions. A closed form solution can be obtained for equations governing the fluid body, but due to the variability of the coefficients of resulting differential equations of the solid body, such a closed form solution is not possible to obtain. For that reason, equations for the solid body and the boundary conditions are treated numerically by the finite-difference method to obtain the effects of the thickness of the tube on the wave characteristics. Dispersion relation is obtained using the long wave approximation and, the wave velocities and the transmission coefficients are computed.     

The coupled problem of a solid oscillating in a viscous fluid under the action of an elastic force

The torsional oscillations are studied of a solid of revolution under the action of elastic torque inside a container with a viscous incompressible fluid. We prove the asymptotic stability of the static equilibrium. We use the two approaches: the direct Lyapunov and linearization methods. The global asymptotic stability is established using a one-parameter family of Lyapunov functionals. Then small oscillations are studied of the fluid-solid system. The linearized operator of the problem of a solid oscillating in a fluid can be realized as an operator matrix obtained by appending two scalar rows and two columns to the Stokes operator. This operator is therefore a two-dimensional bordering of the Stokes operator and inherits many properties of the latter; in particular, the spectrum is discrete. The eigenvalue problem for the linearized operator is reduced to solving a dispersion equation. Inspection of the equation shows that all eigenvalues lie inside the right (stable) half-plane. Basing on this, we justify the linearization. Using an abstract theorem of Yudovich, we prove the asymptotic stability in a scale of function spaces, the infinite differentiability of solutions, and the decay of all their derivatives in time.     

On the explicit parametric description of waves in periodic media

A method for the parameterization of the one-dimensional wave equation is proposed that makes it possible to find its solution by quadratures under an arbitrary dependence of the refraction index on the current wave phase. The form of the solution found is used to investigate the structure of the wave function for a periodic refraction index. Explicit expressions for the fundamental system of solutions and for the Floquet index are obtained. Examples of applying the proposed method to the optimal synthesis of multilayer interference mirrors and Bragg waveguides are discussed.     

Convective flow and heat transfer of a viscous heat generating fluid in the presence of a moving,infinite, vertical,porous plate

The analysis of convective flow and heat transfer of a viscous heat generating fluid past a uniformly moving, infinite, vertical, porous plate has been made systematically with a view to throw adequate light on the effects of the plate-motion and the presence of heat generation/absorption on the flow and heat transfer characteristics. The equations of conservation of momentum and energy which govern the flow and heat transfer of the said problem have been solved numerically by the method of Runge-Kutta-Gill. The numerical results thus obtained for the flow and heat transfer characteristics have revealed many an interesting behaviour, of the skin friction and the rate of heat transfer coefficient at the plate.     

Radiation instability of a circular cylinder in a uniform flow of a two-layer fluid

A solution of the plane linear problem of the oscillations of a horizontal circular cylinder in a uniform flow of a two-layer unbounded fluid is obtained using the method of multipole expansions. The direction of the flow is perpendicular to the cylinder axis. The whole cylinder Ges in the upper or lower layer. The fluid is assumed to be ideal and incompressible, the flow in each layer being a potential one. All the components of the radiation load (the apparent masses and damping coefficients) are determined and the regions of existence of radiation instability are found, depending on the flow velocity for a cylinder suspended by horizontal and vertical elastic links. By solving the integro-differential equation numerically the relative oscillations of the body under specified initial conditions are found.     

Stochastic non-Newtonian fluid motion equations of a nonlinear bipolar viscous fluid

An analysis based on the Galerkin method is developed to examine the behaviour of a nonlinear bipolar viscous fluid mathematically modelled by stochastic non-Newtonian fluid motion equations. Existence and uniqueness of solutions to the stochastic equations are derived.     

Numerical modeling of the movement of a rigid particle in viscous fluid

Modeling the movement of a rigid particle in viscous fluid is a problem physicists and smathematicians have tried to solve since the beginning of this century. A general model for an ellipsoidal particle was first published by Jeffery in the twenties. We exploit the fact that Jeffery was concerned with formulae which can be used to compute numerically the velocity field in the neighborhood of the particle during his derivation of equations of motion of the particle. This is our principal contribution to the subject. After a thorough check of Jeffery's formulae, we coded software for modeling the flow around a rigid particle based on these equations. Examples of its applications are given in conclusion. A practical example is concerned with the simulation of sigmoidal inclusion trails in porphyroblast.     

Propagation of perturbations in a two-layer stratified fluid with an interface excited by moving sources

Propagation of small perturbations in a two-layer inviscid stratified fluid is studied. It is assumed that the higher density fluid occupies the lower unbounded half-space, while the lower density fluid occupies the upper unbounded half-space. The source of the excitation is a plane wave traveling along the interface of the fluids. An explicit analytical solution to the problem is constructed, and its existence and uniqueness are proved. The long-time wave pattern developing in the fluids is analyzed.     

Heat transfer in a visco-elastic fluid past a stretching sheet with viscous dissipation and internal heat generation

This paper deals with the study of heat transfer characteristics in the laminar boundary layer flow of a visco-elastic fluid over a linearly stretching continuous surface with variable wall temperature subjected to suction or blowing. The study considers the effects of frictional heating (viscous dissipation) and internal heat generation or absorption. An analysis has been carried out for two different cases of heating processes namely: (i) Prescribed surface temperature (PST) and (ii) Prescribed wall heat flux (PHF) to get the effect of visco-elastic parameter for various situations. Further increase of visco-elastic parameter is to decrease the skin friction on the sheet. The solutions for the temperature and the heat transfer characteristics are obtained in terms of Kummers function. Received: June 16, 2004; revised: February 8, 2005     

MHD mixed convection of a viscous dissipating fluid about a permeable vertical flat plate

The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a vertical plate is studied numerically, taking into account the effects of Ohmic heating and viscous dissipation. A uniform magnetic field is applied perpendicular to the plate. The resulting governing equations are transformed into the non-similar boundary layer equations and solved using the Keller box method. Both the aiding-buoyancy mode and the opposing-buoyancy mode of the mixed convection are examined. The velocity and temperature profiles as well as the local skin friction and local heat transfer parameters are determined for different values of the governing parameters, mainly the magnetic parameter, the Richardson number, the Eckert number and the suction/injection parameter, f_w. For some specific values of the governing parameters, the results agree very well with those available in the literature. Generally, it is determined that the local skin friction coefficient and the local heat transfer coefficient increase owing to suction of fluid, increasing the Richardson number, Ri (i.e. the mixed convection parameter) or decreasing the Eckert number. This trend reverses for blowing of fluid and decreasing the Richardson number or decreasing the Eckert number. It is disclosed that the value of Ri determines the effect of the magnetic parameter on the momentum and heat transfer.     

Propagation of perturbations in a two-layer rotating fluid with an interface excited by moving sources

Propagation of small perturbations in a two-layer inviscid fluid rotating at a constant angular velocity is studied. It is assumed that the lower density fluid occupies the upper unbounded half-space, while the higher density fluid occupies the lower unbounded half-space. The source of excitation is a plane wave traveling along the interface of the fluids. An explicit analytical solution to the problem is constructed, and its existence and uniqueness are proved. The long-time wave pattern developing in the fluids is analyzed.     

On asymptotic stability and instability with respect to a fading stochastic perturbation

We develop necessary and sufficient conditions for the a.s. asymptotic stability of solutions of a scalar, non-linear stochastic equation with state-independent stochastic perturbations that fade in intensity. These conditions are formulated in terms of the intensity function: roughly speaking, we show that as long as the perturbations fade quicker than some identifiable critical rate, the stability of the underlying deterministic equation is unaffected. These results improve on those of Chan and Williams; for example, we remove the monotonicity requirement on the drift coefficient and relax it on the intensity of the stochastic perturbation. We also employ different analytic techniques.