Flow past an elliptic cylinder

A theoretical investigation of the unsteady two-dimensional flow of a viscous, incompressible fluid normal to a thin elliptic cylinder is described. The cylinder, which is started impulsively from rest in an open field, continues to move with uniform velocity for the remainder of the problem. Using a vorticity-streamfunction formulation of the full Navier-Stokes equations, transformation techniques are employed to find the initial flow. Strategies which employ boundary layer theory and series expansions of the flow variables to find flow solutions for small values of time are outlined.     

On the well‐posedness of a mathematical model describing water‐mud interaction

In this paper, we consider a mathematical model describing the two‐phase interaction between water and mud in a water canal when the width of the canal is small compared with its depth. The mud is treated as a non‐Newtonian fluid, and the interface between the mud and fluid is allowed to move under the influence of gravity and surface tension. We reduce the mathematical formulation, for small boundary and initial data, to a fully nonlocal and nonlinear problem and prove its local well‐posedness by using abstract parabolic theory. Copyright © 2012 John Wiley & Sons, Ltd.     

The contact problem for hollow and solid cylinders with stress-free faces

The contact problem for hollow and solid circular cylinders with a symmetrically fitted belt and stress-free faces is considered. Homogeneous solutions corresponding to zero stresses on the cylinder faces are obtained. The generalized orthogonality of homogeneous solutions is used to satisfy the modified boundary conditions. In the final analysis the problem is reduced to a system of integral equations in the functions describing the displacement of the outer and inner surfaces of the cylinders. These functions are sought in the form of the sum of a trigonometric series and a power function with a root singularity. The ill-posed infinite systems of algebraic equations obtained as a result, are regularized by introducing small positive parameters [Ref. Kalitkin NN. Numerical Methods. Moscow: Nauka; 1978] and, after reduction, have stable regularized solutions. Since the elements of the matrices of the system are given by poorly converging numerical series, an effective method of calculating the residues of these series is developed. Formulae for the distribution function of the contact pressure and the integral characteristic are obtained. Since the first formula contains a third-order derivative of the functional series, a numerical differentiation procedure is employed when using it [Refs. Kalitkin NN. Numerical Methods. Moscow: Nauka; 1978; Danilina NI, Dubrovskaya NS, Kvasha OP et al. Numerical Methods. A Student Textbook. Moscow: Vysshaya Shkola; 1976]. Examples of the analysis of a cylindrical belt are given.     

Global existence,singular solutions,and ill‐posedness for the Muskat problem

The Muskat, or Muskat‐Leibenzon, problem describes the evolution of the interface between two immiscible fluids in a porous medium or Hele‐Shaw cell under applied pressure gradients or fluid injection/extraction. In contrast to the Hele‐Shaw problem (the one‐phase version of the Muskat problem), there are few nontrivial exact solutions or analytic results for the Muskat problem. For the stable, forward Muskat problem, in which the higher‐viscosity fluid expands into the lower‐viscosity fluid, we show global‐in‐time existence for initial data that is a small perturbation of a flat interface. The initial data in this result may contain weak (e.g., curvature) singularities. For the unstable, backward problem, in which the higher‐viscosity fluid contracts, we construct singular solutions that start off with smooth initial data but develop a point of infinite curvature at finite time. © 2004 Wiley Periodicals, Inc.     

Numerical solution of the free‐surface viscous flow on a horizontal rotating elliptical cylinder

The numerical solution of the free‐surface fluid flow on a rotating elliptical cylinder is presented. Up to the present, research has concentrated on the circular cylinder for which steady solutions are the main interest. However, for noncircular cylinders, such as the ellipse, steady solutions are no longer possible, but there will be periodic solutions in which the solution is repeated after one full revolution of the cylinder. It is this new aspect that makes the investigation of noncircular cylinders novel. Here we consider both the time‐dependent and periodic solutions for zero Reynolds number fluid flow. The numerical solution is expedited by first mapping the fluid film domain onto a rectangle such that the position of the free‐surface is determined as part of the solution. For the time‐dependent case a simple time‐marching method of lines approach is adopted. For the periodic solution the discretised nonlinear equations have to be solved simultaneously over a time period. The resulting large system of equations is solved using Newton's method in which the form of the Jacobian enables a straightforward decomposition to be implemented, which makes matrix inversion manageable. In the periodic case all derivatives have been approximated pseudospectrally with the time derivative approximated by a differentiation matrix which has been specially derived so that the weight of fluid is algebraically conserved. Of interest is the solution for which the weight of fluid is at its maximum possible value, and this has been obtained by increasing the weight until a consistency break‐down occurs. Time‐dependent solutions do not produce the periodic solution after a long time‐scale but have protuberances which are constantly appearing and disappearing. Periodic solutions exhibit spectral accuracy solutions and maximum supportable weight solutions have been obtained for ranges of eccentricity and angular velocity. The maximum weights are less than and approximately proportional to those obtained for the circular case. The shapes of maximum weight solutions is distinctly different from sub‐maximum weight solutions. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Finite element method to fluid–solid interaction problems with unbounded periodic interfaces

Consider a time‐harmonic acoustic plane wave incident onto a doubly periodic (biperiodic) surface from above. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid fluid of constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by its Lamé constants. This article is concerned with a variational approach to the fluid–solid interaction problems with unbounded biperiodic Lipschitz interfaces between the domains of the acoustic and elastic waves. The existence of quasiperiodic solutions in Sobolev spaces is established at arbitrary frequency of incidence, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. A finite element scheme coupled with Dirichlet‐to‐Neumann mappings is proposed and the convergence analysis is performed. The Dirichlet‐to‐Neumann mappings are approximated by truncated Rayleigh series expansions. Finally, numerical tests in 2D are presented to confirm the convergence of solutions and the energy balance formula. In particular, the frequency spectrum of normally reflected signals is plotted for water–brass and water–brass–water interfaces. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 5–35, 2016     

On an incompressible model in radiation hydrodynamics

We consider a simplified model arising in radiation hydrodynamics based on the incompressible Navier–Stokes–Fourier system describing a macroscopic fluid motion coupled to a transport equation modeling the propagation of radiative intensity. We establish global‐in‐time existence for the associated initial‐boundary value problem in the framework of weak solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

Stokeswaves with vorticity

The existence of periodic waves propagating downstream on the surface of a two-dimensional infinitely deep body of water under the force of gravity is established for a general class of vorticities. When reformulated as an elliptic boundary value problem in a fixed semi-infinite cylinder with a parameter, the operator describing the problem is nonlinear and non-Fredholm. A global connected set of nontrivial solutions is obtained via singular theory of bifurcation. The proof combines a generalized degree theory, global bifurcation theory, and Whyburn’s lemma in topology with the Schauder theory for elliptic problems and the maximum principle.     

The free-parameter solution of the convection equations in a vertical cylinder with a volume heat source

An exact solution of the free-convection equations is constructed in the Oberbeck–Boussinesq approximation, describing the flow of a viscous heat-conducting fluid in a vertical cylinder of large radius when heated by radiation. The initial problem is reduced to an operator equation with an extremely non-linear operator, satisfying Schauder's theorem in C[0,1]. An iteration procedure is proposed for determining the independent parameter, that occurs in the solution, which enables three different values to be obtained and, correspondingly, three classes of solution of the initial problem. The linear stability of all the solutions obtained is investigated and it is shown that, for chosen values of the problem parameters, the most dangerous one is the plane wave mode and two instability mechanisms are present. The flow structure and the type of instability depend considerably on the values of the free parameter.     

Wall effects on the slow steady motion of a particle in a viscous incompressible fluid

In this paper, asymptotic expansions with respect to small Reynolds numbers are proved for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid past a single plane wall. The flow problem is modelled by a certain boundary value problem for the stationary, nonlinear Navier-Stokes equations. The coefficients of these expansions are the solutions of various, linear Stokes problems which can be constructed by single layer potentials and corresponding boundary integral equations on the boundary surface of the particle. Furthermore, some asymptotic estimates at small Reynolds numbers are presented for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid between two parallel, plane walls and in an infinitely long, rectilinear cylinder of arbitrary cross section. In the case of the flow problem with a single plane wall, the paper is based on a thesis, which the author has written under the guidance of Professor Dr. Wolfgang L. Wendland.     

Trapped modes along a periodic array of freely floating obstacles

We consider the coupled problem describing the motion of a linear array of three‐dimensional obstacles floating freely in a homogeneous fluid layer of finite depth. The interaction of time‐harmonic waves with the floating objects is analyzed under the usual assumptions of linear water‐wave theory. Quasi‐periodic boundary conditions and a simplified reduction scheme turn the system into a linear spectral problem for a self‐adjoint operator in Hilbert space. Based upon the operator formulation, we derive a sufficient condition for the nonemptiness of its discrete spectrum. Various examples of obstacles that generate trapped modes are given. Copyright © 2014 John Wiley & Sons, Ltd.     

A note on decay of potential vortex in an Oldroyd-B fluid through a porous space

The problem of decay of a potential vortex in an Oldroyd-B fluid filling the porous space is studied. The flow problem is first modeled and then solved by employing the Hankel transform. Analytical expressions of the velocity field and the associated tangential tension are developed. The well known solutions for a Newtonian fluid as well as those corresponding to a Maxwell fluid and a second grade one, appear as limiting cases of the present solutions. Finally, some graphical results describing the influence of porous space parameters are sketched and interpreted.     

Viscous two‐fluid flows in perturbed unbounded domains

Two stationary plane free boundary value problems for the Navier‐Stokes equations are studied. The first problem models the viscous two‐fluid flow down a perturbed or slightly distorted inclined plane. The second one describes the viscous two‐fluid flow in a perturbed or slightly distorted channel. For sufficiently small data and under certain conditions on parameters the solvability and uniqueness results are proved for both problems. The asymptotic behaviour of the solutions is investigated. For the second problem an example of nonuniqueness is constructed. Computational results of flow problems that are very close to the above problems are presented. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

D'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon equation

In this paper, we construct a weakly‐nonlinear d'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon (BKG) equation. Similarly to our earlier work based on the use of spatial Fourier series, we consider the problem in the class of periodic functions on an interval of finite length (including the case of localized solutions on a large interval), and work with the nonlinear partial differential equation with variable coefficients describing the deviation from the oscillating mean value. Unlike our earlier paper, here we develop a novel multiple‐scales procedure involving fast characteristic variables and two slow time scales and averaging with respect to the spatial variable at a constant value of one or another characteristic variable, which allows us to construct an explicit and compact d'Alembert‐type solution of the nonlinear problem in terms of solutions of two Ostrovsky equations emerging at the leading order and describing the right‐ and left‐propagating waves. Validity of the constructed solution in the case when only the first initial condition for the BKG equation may have nonzero mean value follows from our earlier results, and is illustrated numerically for a number of instructive examples, both for periodic solutions on a finite interval, and localized solutions on a large interval. We also outline an extension of the procedure to the general case, when both initial conditions may have nonzero mean values. Importantly, in all cases, the initial conditions for the leading‐order Ostrovsky equations by construction have zero mean, while initial conditions for the BKG equation may have nonzero mean values.     

Solvability,asymptotic analysis and regularity results for a mixed type interaction problem of acoustic waves and piezoelectric structures

In the paper, we investigate the mixed type transmission problem arising in the model of fluid–solid acoustic interaction when a piezoceramic elastic body (Ω⁺) is embedded in an unbounded fluid domain (Ω^?). The corresponding physical process is described by the boundary‐transmission problem for second‐order partial differential equations. In particular, in the bounded domain Ω⁺, we have a 4×4 dimensional matrix strongly elliptic second‐order partial differential equation, while in the unbounded complement domain Ω^?, we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations based on the Wiener–Hopf factorization method, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. We derive asymptotic expansion of solutions, and on the basis of asymptotic analysis, we establish optimal Hölder smoothness results for solutions. Copyright © 2017 John Wiley & Sons, Ltd.     

Accelerated slip flow past a cylinder

This study deals with boundary layer flow along the entire length of a stationary semi-infinite cylinder under a steady, accelerated free-stream. Considering flow at reduced dimensions, the no-slip boundary condition is replaced with a Navier boundary condition. Asymptotic series solutions are obtained for the shear stress coefficient in terms of the Bingham number that corresponds to prescribed values of both the slip coefficient and the index of acceleration. By investigating motion at small and large axial distances, the series solutions are presented. For flow in the intermediate distances, exact and interpolated numerical solutions are obtained. Using these results, the shear stress along the entire cylinder wall is evaluated in terms of the parameters of acceleration and slip.     

Accelerated slip flow past a cylinder

This study deals with boundary layer flow along the entire length of a stationary semi-infinite cylinder under a steady, accelerated free-stream. Considering flow at reduced dimensions, the no-slip boundary condition is replaced with a Navier boundary condition. Asymptotic series solutions are obtained for the shear stress coefficient in terms of the Bingham number that corresponds to prescribed values of both the slip coefficient and the index of acceleration. By investigating motion at small and large axial distances, the series solutions are presented. For flow in the intermediate distances, exact and interpolated numerical solutions are obtained. Using these results, the shear stress along the entire cylinder wall is evaluated in terms of the parameters of acceleration and slip.     

Unsteady helical flows of Oldroyd-B fluids

The unsteady helical flow of an Oldroyd-B fluid, in an infinite circular cylinder, is studied by using finite Hankel transforms. The motion is produced by the cylinder that, at time t = 0⁺, is subject to torsional and longitudinal time-dependent shear stresses. The solutions that have been obtained, presented under series form, satisfy all imposed initial and boundary conditions. The corresponding solutions for Maxwell, second grade and Newtonian fluids are obtained as limiting cases of general solutions. Finally, the influence of the pertinent parameters on the fluid motion is underlined by graphical illustrations.     

Slow magnetohydrodynamic flow past a circular cylinder

Oseen’ approximations are used to study the slow motion of a viscous, incompressible, electrically conducting fluid past a circular cylinder in the presence of a uniform aligned magnetic field. Using series truncation method, the analytical solutions for the first three terms in the Fourier sine series expansion of the stream function are obtained. Numerical values of the tangential drag for different values of magnetic interaction parameter and viscous Reynolds number are calculated.     

On Bounded Solutions of the Emden-Fowler Equation in a Semi-cylinder

Bounded solutions of the Emden-Fowler equation in a semi-cylinder are considered. For small solutions the asymptotic representations at infinity are derived. It is shown that there are large solutions whose behavior at infinity is different. These solutions are constructed when some inequalities between the dimension of the cylinder and the homogeneity of the nonlinear term are fulfilled. If these inequalities are not satisfied then it is proved, for the Dirichlet problem, that all bounded solutions tend to zero and have the same asymptotics as small solutions.