One method for constructing exact solutions of equations of two-dimensional hydrodynamics of an incompressible fluid

We propose a simple algebraic method for constructing exact solutions of equations of two-dimensional hydrodynamics of an incompressible fluid. The problem reduces to consecutively solving three linear partial differential equations for a nonviscous fluid and to solving three linear partial differential equations and one first-order ordinary differential equation for a viscous fluid. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147,No. 1, pp. 64–72, April, 2006.     

Decay of potential vortex for a viscoelastic fluid with fractional Maxwell model

This paper is concerned with the exact analytic solutions for the velocity field and the associated tangential stress corresponding to a potential vortex for a fractional Maxwell fluid. The fractional calculus approach is taken into account in the constitutive relationship of a non-Newtonian fluid model. Exact analytic solutions are obtained by using the Hankel transform and the discrete Laplace transform of sequential fractional derivatives. The solutions for a Maxwell fluid appear as the limiting cases of our general solutions by setting α=1$\alpha =1$. The influence of fractional coefficient on the decay of vortex velocity is also analyzed by graphical illustrations.     

One kind of nth Beltrami spherical vortex tensor denotation and its classification

In this article, we discuss one kind of nth Beltrami Spherical Vortex (BSV) tensor denotation, axisymmetric and nonaxisymmetric, and its classification. We also find that chaos will occur in the dynamic system of nonaxisymmetric BSV. From these,we can say that the tensor denotation has more meaningful characters and nonaxisymmetric BSV is various and very complex.     

Melting heat and mass transfer in stagnation point micropolar fluid flow of temperature dependent fluid viscosity and thermal conductivity at constant vortex viscosity

Steady mixed convection micropolar fluid flow towards stagnation point formed on horizontal linearly stretchable melting surface is studied. The vortex viscosity of micropolar fluid along a melting surface is proposed as a constant function of temperature while dynamic viscosity and thermal conductivity are temperature dependent due to the influence of internal heat source on the fluid. Similarity transformations were used to convert the governing equation into non-linear ODE and solved numerically. A parametric study is conducted. An analysis of the results obtained shows that the flow-field is influenced appreciably by heat source, melting, velocity ratio, variable viscosity and thermal conductivity.     

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.     

Study of vortex breakdown in a stratified fluid

An analysis shows that nonsmooth solutions have to be considered. Weak solutions to the Euler equations describing an incompressible stratified fluid under gravity are defined and studied. The study makes use of a wave energy functional proposed for the nonlinear equations. It is shown that the Euler equations are insufficient for stating a well-posed generalized problem. Additional conditions based on physical considerations are proposed. One condition is energy conservation, and the other is a constraint imposed on the density, which is required for stability. A numerical method is developed that is used to analyze how wave breakdown in a stratified fluid depends on stratification. The numerical results are in satisfactory agreement with experiments.     

Steady vortex rings with swirl in an ideal fluid: asymptotics for some solutions in exterior domains

In this paper, the axisymmetric flow in an ideal fluid outside the infinite cylinder (rd) where (r, , z) denotes the cylindrical co-ordinates in ³ is considered. The motion is with swirl (i.e. the -component of the velocity of the flow is non constant). The (non-dimensional) equation governing the phenomenon is (Pd) displayed below. It is known from e.g. [9] that for the problem without swirl (f _q = 0 in (f)) in the whole space, as the flux constant k tends to 1) dist(0z, A) = O(k ^1/2); diam A = O(exp(–c ₀ k ^3/2));2) k^1/2)_k converges to a vortex cylinder U _m (see (1.2)).We show that for the problem with swirl, as k , 1) holds; if m q + 2 then 2) holds and if m > q + 2 it holds with U _q+2 instead of U _m. Moreover, these results are independent of f ₀, f _q and d > 0.     

Convergence of the point vortex method for 2-D vortex sheet

We give an elementary proof of the convergence of the point vortex method (PVM) to a classical weak solution for the two-dimensional incompressible Euler equations with initial vorticity being a finite Radon measure of distinguished sign and the initial velocity of locally bounded energy. This includes the important example of vortex sheets, which exhibits the classical Kelvin-Helmholtz instability. A surprise fact is that although the velocity fields generated by the point vortex method do not have bounded local kinetic energy, the limiting velocity field is shown to have a bounded local kinetic energy.

     

Thermocapillary motion of deformable fluid particles in tubes

The boundary integral technique is used to study the effect of deformation on the steady, creeping, thermocapillary migration of a fluid particle under conditions of axisymmetry, negligible thermal convection and an insulated tube wall. The spherical radius of the fluid particle (i.e. the radius as if the particle were a sphere, a ′= (3V _p ^′ /4π)^1/3, V _p ^′ is the particle volume) and that of the tube are denoted, respectively, by a′and b′. For small capillary numberCa = 0.05, only for a large fluid particle (a′/b′ = 0.8) is deformation significant. Fora′/b′= 0.8, hydrodynamic stresses squeeze the particle, reduce the interaction of the particle with the wall and thereby increase the terminal velocity. For small particles a′/b′< 0.8 and Ca = 0.05 the fluid particles translate as spheres, due to the fact that the fluid particle is too far away from the wall to be subject to distending hydrodynamic stresses. The deformable particle moves faster than a spherical one in the thermocapillary migration. The increase in velocity with capillary number is larger for thermocapillary motion than for buoyancy.     

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.     

Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion

The velocity field of generalized second order fluid with fractional anomalous diiusion caused by a plate moving impulsively in its own plane is investigated and the anomalous diffusion problems of the stress field and vortex sheet caused by this process are studied. Many previous and classical results can be considered as particular cases of this paper, such as the solutions of the fractional diffusion equations obtained by Wyss; the classical Rayleigh’s time-space similarity solution; the relationship between stress field and velocity field obtained by Bagley and co-worker and Podlubny’s results on the fractional motion equation of a plate. In addition, a lot of significant results also are obtained. For example, the necessary condition for causing the vortex sheet is that the time fractional diffusion index β must be greater than that of generalized second order fluid α; the establiihment of the vorticity distribution function depends on the time history of the velocity profile at a given point, and the time history can be described by the fractional calculus.     

Numerical stability analysis of the Taylor-Couette flow in the two-dimensional case

The stability of the laminar flow between two rotating cylinders (Taylor-Couette flow) is numerically studied. The simulation is based on the equations of motion of an inviscid fluid (Euler equations). The influence exerted on the flow stability by physical parameters of the problem (such as the gap width between the cylinders, the initial perturbation, and the velocity difference between the cylinders) is analyzed. It is shown that the onset of turbulence is accompanied by the formation of large vortices. The results are analyzed and compared with those of similar studies.     

Asymptotic behavior of vortices in axisymmetric flow without swirl

We investigate the confinement properties of bounded, nonnegative, compactly supported vortices of axisymmetric incompressible Euler flows without swirl. We show that along one direction of the symmetry axis, its support can grow no faster than O[(t log t)^1/2]. The rate at which it approaches the symmetry axis is also estimated. Together with the result of Maffei–Marchioro on the radial growth rate of the support, it is contained in a slowly expanding tubular region. The techniques of the above‐mentioned authors, Iftimie–Lopes–Nussenzveig and Iftimie–Sideris–Gamblin, are used. Copyright © 2008 John Wiley & Sons, Ltd.     

Linearized viscoelastic Oldroyd fluid motion in an almost periodic environment

In most of the linear homogenization problems involving convolution terms so far studied, the main tool used to derive the homogenized problem is the Laplace transform. Here we propose a direct approach enabling one to tackle both linear and nonlinear homogenization problems that involve convolution sequences without using Laplace transform. To illustrate this, we investigate in this paper the asymptotic behavior of the solutions of a Stokes–Volterra problem with rapidly oscillating coefficients describing the viscoelastic fluid flow in a fixed domain. Under the almost periodicity assumption on the coefficients of the problem, we prove that the sequence of solutions of our ?‐problem converges in L² to a solution of a rather classical Stokes system. One important fact is that the memory disappears in the limit. To achieve our goal, we use some very recent results about the sigma‐convergence of convolution sequences. Copyright © 2013 John Wiley & Sons, Ltd.     

Large Eddy Simulation of the vortex end in reverse-flow centrifugal separators

The present work is a study of the gas-flow phenomenon known as the “end of the vortex” (EoV), which spontaneously occurs at the lower end, or under, reverse-flow centrifugal separators such as cyclones or swirl tubes. Different CFD models of swirl tubes have been built to study and analyse this phenomenon in detail. The present numerical work is based on—and compared with—previous experimental observations of this phenomenon. The numerical models were built in complete agreement with the geometrical configurations and operating conditions used in these earlier experimental studies [1]. Two different configurations of swirl tubes were analyzed. One configuration was an in principle long tube with variable length in which the dependence on the vessel length of the behaviour of the vortex core in a simple, well-defined geometry was studied. The other configuration was equipped with a wide “dust collection vessel” at the bottom, the depth of which was varied, to study the behaviour of the vortex core in a widely-used geometry. 3-D LES simulations were carried out using the commercial CFD package Star-CD. The bending of the vortex core to the wall of the vessel and its precessional motion, constituting the phenomenon of the EoV, was seen in both configurations, and the obtained results are in very good agreement, both qualitatively and to an extent quantitatively, with previous experimental results [1].     

Temporal boundary value problems in interfacial fluid dynamics

We study problems in interfacial fluid dynamics which do not have well-posed initial value problems. We prove existence of solutions for these problems by considering instead boundary value problems, where boundary data is specified at two different times. We develop a general framework, for problems on the real line and for problems which are spatially periodic. A variety of boundary conditions are considered, including Dirichlet, Neumann and mixed conditions. The framework is applied to two specific problems from interfacial fluid dynamics: a family of generalizations of the Boussinesq equations developed by Bona, Chen and Saut, and the vortex sheet.     

The contact angle in inviscid fluid mechanics

We show that in general, the specification of a contact angle condition at the contact line in inviscid fluid motions is incompatible with the classical field equations and boundary conditions generally applicable to them. The limited conditions under which such a specification is permissible are derived; however, these include cases where the static meniscus is not flat. In view of this situation, the status of the many ‘solutions’ in the literature which prescribe a contact angle in potential flows comes into question. We suggest that these solutions which attempt to incorporate a phenomenological, but incompatible, condition are in some, imprecise sense ‘weak-type solutions’; they satisfy or are likely to satisfy, at least in the limit, the governing equations and boundary conditions everywhere except in the neighbourhood of the contact line. We discuss the implications of the result for the analysis of inviscid flows with free surfaces.     

Bifurcation analysis of the motion of a cylinder and a point vortex in an ideal fluid

We consider an integrable Hamiltonian system describing the motion of a circular cylinder and a vortex filament in an ideal fluid. We construct bifurcation diagrams and bifurcation complexes for the case in which the integral manifold is compact and for various topological structures of the symplectic leaf. The types of motions corresponding to the bifurcation curves and their stability are discussed.