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.     

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

Propagation of small perturbations in a two-layer inviscid stratified uniformly rotating fluid is studied assuming that the higher and lower density fluids occupy unbounded lower and upper half-spaces, respectively. The source of excitation is a plane wave travelling along the interface of the fluids. An explicit analytical solution of the problem is constructed, and its existence and uniqueness is proved. The long-time wave pattern developing in the fluids is analyzed.     

Oscillations of a rotating stratified fluid with its free surface excited by moving sources

Propagation of small perturbations in a weakly stratified inviscid fluid rotating at a constant angular velocity in the lower half-space is studied. The source of excitation is a plane wave traveling on the free surface of the fluid. An explicit analytical solution to the problem is constructed. Existence and uniqueness theorems are proved. The long-time wave pattern in the fluid is analyzed.     

On oscillations of a semi-infinite rotating liquid with its free surface excited by moving sources

Propagation of small perturbations in a homogeneous inviscid liquid rotating with a constant angular velocity in the lower half-space is considered. The source of excitation is a plane wave traveling on the free surface of the liquid. The explicit analytical solution to the problem is constructed. Uniqueness and existence theorems are proved. The wave pattern in the liquid at large times is examined.     

Propagation of perturbations in fluids excited by moving sources

A large series of A.A. Dorodnicyn’s works deals with rigorous mathematical formulations and development of efficient research techniques for mathematical models used in inhomogeneous fluid dynamics. Numerous problems he studied in these directions are closely related to stratified fluid dynamics, which were addressed in a series of works having been published in this journal by this paper’s authors and their coauthors since 1980. This paper describes the results of a series of works analyzing the propagation of small perturbations in various stratified and/or uniformly rotating inviscid fluids. It is assumed that each of the fluids either occupies an unbounded lower half-space with a free surface or is a semi-infinite two-component fluid layer. The perturbations are excited by a moving source specified as a periodic plane wave traveling along the interface of the fluids. Problems for five mathematical fluid models are formulated, their explicit analytical solutions are constructed, and their existence and uniqueness are discussed. The asymptotics of the solution as t → +∞ are studied, and the long-time wave patterns developing in five fluid models are compared.     

Stretching surface in rotating viscoelastic fluid

The boundary layer flow over a stretching surface in a rotating viscoelastic fluid is considered. By applying a similarity transformation, the governing partial differ-ential equations are converted in...     

Wave motions in a compressible stratified rotating fluid

The dynamics equations for a stratified rotating fluid with a random distribution of stratification are considered. These equations are reduced to a scalar equation using two potential functions. The solvability of the initial-boundary value problems of the wave theory is established.     

Wave motions in a stratified electrically conducting rotating fluid

The 3D dynamics equations for the stratified superconducting rotating fluid are studied. These equations are reduced to a scalar equation by representing the magnetic and density fields by a superposition of the unperturbed fields corresponding to the steady state of the fluid and the induced fields appearing due to the wave motion; the reduction also uses two auxiliary functions. The analysis of the scalar equation enables us to prove the solvability of the initial-boundary value problems of the wave theory for electrically conducting rotating fluids with nonhomogeneous density.     

Hydromagnetic rotating flow of a fourth‐order fluid past a porous plate

The rotating flow in the presence of a magnetic field is a problem belonging to hydromagnetics and deserves to be more widely studied than it has been to date. In the non‐linear regime the literature is scarce. We develop the governing equations for the unsteady hydromagnetic rotating flow of a fourth‐order fluid past a porous plate. The steady flow is governed by a boundary value problem in which the order of differential equations is more than the number of available boundary conditions. It is shown that by augmenting the boundary conditions based on asymptotic structures at infinity it is possible to obtain numerical solutions of the nonlinear hydromagnetic equations. Effects of uniform suction or blowing past the porous plate, exerted magnetic field and rotation on the flow phenomena, especially on the boundary layer structure near the plate, are numerically analysed and discussed. The flow behaviours of the Newtonian fluid and second‐, third‐ and fourth‐order non‐Newtonian fluids are compared for the special flow problem, respectively. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Flow of a Reiner‐Rivlin fluid between two infinite coaxial rotating disks

Present study deals with the steady flow and heat transfer of a non‐Newtonian Reiner‐Rivlin fluid between two coaxially rotating infinite disks. Using similarity transformations, the governing equations are reduced to a set of nonlinear, highly coupled ordinary differential equations and by means of an effective analytical method called homotopy analysis method; analytical solutions are constructed in series form. Different cases, such as, when one disk is at rest and the other is rotating with constant angular velocity, two disks rotating with different angular velocities in same as well as opposite sense, two disks rotating with same angular velocities in opposite sense, are discussed. The effects of non‐Newtonian parameter, Reynolds number, are also discussed, and results are presented graphically.     

Momentum and heat transfer over a continuously moving surface with a parallel free stream in a viscoelastic fluid

The flow and heat transfer characteristics for a continuous moving surface in a viscoelastic fluid are investigated. Constitutive equations of viscoelastic fluid obey the second‐grade model. Analytic expressions to velocity and temperature have been developed by employing homotopy analysis method. The criterion to the convergence of the solution is properly discussed. Furthermore, the values of skin friction coefficient and the local Nusselt number have been computed and discussed. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Effect of vertical vibrations on a two-layer system with a deformable interface

The effect of single- and two-frequency vibrations on the behavior of a system consisting of two homogeneous viscous fluids bounded by rigid walls is analyzed. It is assumed that the system as a whole is under vertical vibrations obeying a certain law. An eigenvalue problem is obtained in order to analyze the stability of the relative equilibrium. The case of finite frequencies and arbitrary modulation amplitudes is treated along with the case of high frequencies and small modulation amplitudes. In the former case, the parametric resonance domains are examined depending on the parameters of the system. In the latter case, the high-frequency vibration is shown to create effective surface tension, thus flattening the interface, and can suppress instability when the heavy fluid is over the light one.     

On parametric instability of viscous two-layer fluid in a vessel with a permeable bottom

A linear problem of parametric oscillations of a low-viscous two-layer fluid in a closed vessel partially filled with a porous medium is studied. An asymptotic solution is constructed on the basis of combined application of boundary functions and averaging methods. Approximate formulas for boundaries of instability domains in the case of subharmonic and harmonic resonances are derived.     

Unsteady hydromagnetic motion of a rotating conducting fluid

The combined effects of stratification and magnetic field on the unsteady motion of a viscous, electrically conducting fluid between two rotating disks are analysed. Solutions are obtained for the linearized equations under Boussinesq approximation and steady state solutions are deduced from them. The results are compared with those obtained by Loper and Benton and Balanet al. Graphs are presented for the steady state velocity, magnetic field and temperature distributions.     

Three‐dimensional Stokes flow caused by a translating–rotating sphere bisected by a free surface bounding a semi‐infinite micropolar fluid

Explicit velocity and microrotation components and systematic calculation of hydrodynamic quasistatic drag and couple in terms of nondimensional coefficients are presented for the flow problem of an incompressible asymmetrical steady semi‐infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi‐infinite micropolar fluid. Two asymmetrical cases are considered for the motion of the sphere: parallel translation to the free surface and rotation about a diameter which is lying in the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. A linear slip, Basset‐type, boundary condition has been used. The variation of the resistance coefficients is studied numerically and plotted versus the micropolarity parameter and slip parameter. The two limiting cases of no‐slip and perfect slip are then recovered. Copyright © 2008 John Wiley & Sons, Ltd.     

Electrohydrodynamic instability of a rotating layer of a viscoelastic fluid heated from below

The stability of a rotating layer of viscoelastic dielectric liquid (Walters liquid B) heated from below is considered. Linear stability theory is used to derive an eigenvalue system of ten orders and exact eigenvalue equation for a neutral instability is obtained. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. Critical Rayleigh heat numbers and wavenumber for the onset of instability are presented graphically as function of the Taylor number for various values of electric Rayleigh number and the elastic parameters.     

Dynamics of a rotating layer of an ideal electrically conducting incompressible fluid

A system of nonlinear partial differential equations is considered that models perturbations in a layer of an ideal electrically conducting rotating fluid bounded by spatially and temporally varying surfaces with allowance for inertial forces. The system is reduced to a scalar equation. The solvability of initial boundary value problems arising in the theory of waves in conducting rotating fluids can be established by analyzing this equation. Solutions to the scalar equation are constructed that describe small-amplitude wave propagation in an infinite horizontal layer and a long narrow channel.     

On two‐dimensional unsteady incompressible fluid flow in an infinite strip

In this paper, we study two‐dimensional incompressible fluid flow in an infinite strip. The stream function form of Navier–Stokes equation is considered, which keeps the physical boundary condition and avoids some difficulties in numerical simulations. The existence and uniqueness of global solution are proved. Some results on the regularity of solution are obtained. Copyright © 2000 John Wiley & Sons, Ltd.