A rheological model for fluid flows with arbitrary Reynolds numbers

An approach to constructing a phenomenological model of motion for a viscous fluid alternative to the Prandtl mixing-length hypothesis is suggested. The approach makes it possible to describe the motion of a fluid independently of the regime realized in a given region of the flow. On the basis of this approach, a differential one-parameter model for the flow of a viscous fluid applicable to any regimes of motion, called a uniform laminar-turbulent model, is constructed. For this purpose, the field of a scalar turbulence measure is introduced, which equals the ratio of the Reynolds stress to the total stress in the case of a simple shear flow. This makes it possible to write new expressions for turbulent viscosity. The influence of the turbulence measure field on the flow is taken into account by using an additional transport differential equation. The model is applicable to both compressible and incompressible fluids and makes it possible to obtain solutions in quadratures for steady simple shear flows. Various forms of the system of equations of motion and boundary conditions are given. Original Russian Text ￠ V.A. Pavlovskii, D.V. Nikushchenko, 2009, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2009, No. 1, pp. 104–112.     

On geometric properties of stochastic flows related to the Lyapunov spectrum

We study the geometric properties of two stochastic flows on spheres in Euclidean space. The underlying one-point motion in both cases is Brownian. Both flows arise from the action of a Lie group valued Brownian motion on a quotient. For both flows the curvature of a curve moving under the flow is shown to be a diffusion, null recurrent in one case and transient in the other. Received: 30 April 1999 / Revised version: 4 October 1999 / Published online: 8 August 2000     

On some unsteady flows of a non-Newtonian fluid

Some properties of unsteady unidirectional flows of a fluid of second grade are considered for flows impulsively started from rest by the motion of a boundary or two boundaries or by sudden application of a pressure gradient. Flows considered are: unsteady flow over a plane wall, unsteady Couette flow, flow between two parallel plates suddenly set in motion with the same speed, flow due to one rigid boundary moved suddenly and one being free, unsteady Poiseuille flow and unsteady generalized Couette flow. The results obtained are compared with those of the exact solutions of the Navier–Stokes equations. It is found that the stress at time zero on the stationary boundary for the flows generated by impulsive motion of a boundary or two boundaries is finite for a fluid of second grade and infinite for a Newtonian fluid. Furthermore, it is shown that for unsteady Poiseuille flow the stress at time zero on the boundary is zero for a Newtonian fluid, but it is not zero for a fluid of second grade.     

Low-Reynolds boundary driven cavity flows in a thin liquid shell

Low-Reynolds recirculating cavity flows are traditionally generated from lid-driven boundary motion at a solid-fluid interface or result from shear flow over an opening. Such flows are typically described by the equations of creeping motion, where viscous forces are dominant. We illustrate using Particle Image Velocimetry (PIV) an original family of boundary-driven cavity flows occurring, in contrast to classic configurations, at a liquid-gas interface: thermally-induced Marangoni flows in a thin liquid shell generate forced, steady-state recirculating flows inside the cavity. Forcing relies on viscous mechanisms at the boundary but resulting flow patterns are, however, inviscid. Here, the inviscid equations of fluid motion are not used as an approximation, but rather come as a result from the solution of the creeping motion equations in the region inside the sphere. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation of particles movement in cellular flows under the influence of magnetic forces

A numerical model of particle motion in fluid flow under the influence of hydrodynamic and magnetic forces is presented. As computational tool, a flow solver based on the Boundary Element Method is used. The Euler-Lagrange formulation of multiphase flow is considered. In the case of a particle with a magnetic moment in a nonuniform external magnetic field, the Kelvin body force acts on a single particle. The derived Lagrangian particle tracking algorithm is used for simulation of dilute suspensions of particles in viscous flows taking into account gravity, buoyancy, drag, pressure gradient, added mass and magnetophoretic force. As a benchmark test case the magnetite particle motion in cellular flow field of water is computed with and without the action of the magnetic force. The effect of the Kelvin force on particle motion and separation from the main flow is studied for a predefined magnetic field and different values of magnetic flux density. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Flows of a polymer fluid in domain with impermeable boundaries

Nonlinear boundary value problems modeling steady polymer flows in domains with impermeable solid walls are studied. The solvability of a nonhomogeneous boundary value problem for the equations governing a polymer flow in the case of an impermeable boundary is proved. The norms of solutions are estimated. The set of weak solutions is shown to be sequentially weakly closed. Additionally, explicit formulas are found for computing the solution of the boundary value problem describing the polymer flow induced by a stretching (shrinking) sheet.     

Inextensible flows of curves and developable surfaces

The flow of a curve or surface is said to be inextensible if, in the former case, the arclength is preserved, and in the latter case, if the intrinsic curvature is preserved. Physically, inextensible curve and surface flows are characterized by the absence of any strain energy induced from the motion. In this paper we investigate inextensible flows of curves and developable surfaces in ${\mathbb{R}}^3$. Necessary and sufficient conditions for an inextensible curve flow are first expressed as a partial differential equation involving the curvature and torsion. We then derive the corresponding equations for the inextensible flow of a developable surface, and show that it suffices to describe its evolution in terms of two inextensible curve flows.     

Numerical Simulation of Highly Viscoelastic Polymer Melts using VF and DEVSS Methods

Viscoelastic polymer melts are usually modeled with a macro approch. This is done using an anisotropic mobility tensor to generalize the Maxwell model. In recent works the use of micro approches is increasing, e.g. the so-called POM-POM model. In this work a frame independent flow type property is introduced in order to help visualizing the elongation and shear regions in a 3D flow. Which in turn helps to choose the right material model, since the model parameters are adjusted with 1D shear flows and the elongation properties are mostly neglected. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Symmetries in equations of incompressible viscoelastic Maxwell medium<Superscript>*</Superscript>

We consider unsteady flows of incompressible viscoelastic Maxwell medium with upper, low, and Jaumann convective derivatives in the rheological constitutive law. We give characteristics of a system of equations that describe a three-dimensional motion of such a medium for all three types of convective derivative. In the general case, due to the incompressibility condition, the equations of motion have both real and complex characteristics. We study group properties of this system in the two-dimensional case. On this basis, we choose submodels of the Maxwell model that can be reduced to hyperbolic ones. The properties of the hyperbolic submodels obtained depend on the choice of the invariant derivative in the rheological relation. We also present concrete examples of invariant solutions.     

The critical cross-section of a vortex

Summary This paper discusses the problem of critical-flow cross-sections in vortex flows. It is shown that there are two different types of vortex flows, A-type and B-type vortices (say). An A-type vortex approaches its critical flow state as its cross-sectional area increases and departs from the critical state as the cross-sectional area is decreased. This property is associated with the particular dependence of total pressure and circulation on the stream function, and it holds for both subcritical and supercritical A-type vortices. On the other hand, both subcritical and supercritical B-type vortices approach their critical flow states as their cross-sectional areas are decreased and depart from their critical states for increasing cross-sectional area. As was shown by Benjamin, setting the first variation of the flow force with respect to the stream function equal to zero leads to Euler's equation of motion. The second variation also vanishes if the corresponding flow state is critical. In this case the sign of the third variation decides whether the flow is an A-type or a B-type vortex. Within the framework of inviscid-fluid flow theory the type of a vortex is preserved unless vortex breakdown occurs. Making use of the knowledge that vortex flows are controlled by two different types of critical-flow cross-sections a variety of vortex flow phenomena are investigated, including the two types of inlet vortices that are observed upstream of jet engines, the behavior of vortex valves, the flow characteristics of liquid-fuel atomizers and the bath tub vortex.     

Some analytical solutions for second grade fluid flows for cylindrical geometries

This paper deals with some unsteady flow problems of a second grade fluid. The flows are generated by the sudden application of a constant pressure gradient or by the impulsive motion of a boundary. The velocities of the flows are described by the partial differential equations. Exact analytic solutions of these differential equations are obtained. The well known solutions for a Navier–Stokes fluid in the hydrodynamic case appear as the limiting cases of our solutions.     

Analytical solutions to Stokes-type flows of inhomogeneous fluids

In this paper, we study the unsteady motion of an inhomogeneous incompressible viscous fluid, where the viscosity varies spatially according to various models. We study the Stokes-type flow for these types of fluids where in the first case the flow between two parallel plates is examined with one of the plates oscillating and in the second case when the flow is caused by a pulsatile pressure gradient. A general argument establishes the existence of oscillatory solutions to our problem. Exact solutions are obtained in terms of some special functions and comparisons are made with the cases of constant viscosity and the slow flow regimes.     

Fluid dynamics of an underwater electric discharge on the axis of a shell immersed in a fluid

We study the motion of a fluid during an electrical discharge on the axis of an elastic cylindrical shell immersed in the fluid. We estimate the influence of the parameters of the shell on the dynamics of a gasvapor chamber. We point out several properties of the computation of the pulsed flows in an infinite volume of fluid (the choice of the method of constructing the grid, the limitation of the computational region). We obtain the pressure distribution over space at large distances from the shell. We analyze the influence of the shell on the parameters of the flow in the interior and exterior regions. Four figures. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 120–126.     

Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form

In this paper we consider a model for the motion of incompressible viscous flows proposed by Ladyzhenskaya. The Ladyzhenskaya model is written in terms of the velocity and pressure while the studied model is written in terms of the streamfunction only. We derived the streamfunction equation of the Ladyzhenskaya model and present a weak formulation and show that this formulation is equivalent to the velocity–pressure formulation. We also present some existence and uniqueness results for the model. Finite element approximation procedures are presented. The discrete problem is proposed to be well posed and stable. Some error estimates are derived. We consider the 2D driven cavity flow problem and provide graphs which illustrate differences between the approximation procedure presented here and the approximation for the streamfunction form of the Navier–Stokes equations. Streamfunction contours are also displayed showing the main features of the flow.     

Rossby Elevation Waves in the Presence of a Critical Layer

In a previous paper, we investigated the solitary-wave-like development of small-amplitude Rossby waves propagating in a zonal shear current, for the particular case when the Rossby wave speed equals the mean-flow velocity at a certain latitude in the β-plane. We presented a general theory for the nonlinear critical-layer theory, and illustrated it by explicitly describing the motion of a depression solitary wave (D-wave). Here, we report a continuation of that study and consider the more complex case of an elevation solitary wave (E-wave). The method involves matched asymptotic expansions between the outer flow away from the critical layer and the inner flow inside the latter, both these flows having different scalings. We showed previously that the critical-layer flow expansion diverged in the case of the E-wave on the separatrices bounding the open and closed streamlines, which led us to defer a detailed E-wave study. Thus, in this paper, we examine the motion in the additional layer located along the separatrices where this singularity is removed by using a third scaling and find that the previous undesirable distortions are discarded. The evolution equation is derived and is a Korteveg-de-Vries type-equation modified by new nonlinear terms generated by the nonlinear interactions occuring in the critical layer. This equation supports a family of E-waves provided that the mean flow obeys certain conditions. The energy exchange that occurs between the mean flow and the D or E-wave during the critical-layer formation is evaluated in the quasi-steady régime assumption.     

Regular and quasiregular isometric flows on Riemannian manifolds

We study the nontrivial Killing vector fields of constant length and the corresponding flows on smooth Riemannian manifolds. We describe the properties of the set of all points of finite (infinite) period for general isometric flows on Riemannian manifolds. It is shown that this flow is generated by an effective almost free isometric action of the group S ¹ if there are no points of infinite or zero period. In the last case, the set of periods is at most countable and generates naturally an invariant stratification with closed totally geodesic strata; the union of all regular orbits is an open connected dense subset of full measure.     

A kinetic formulation for a model coupling free surface and pressurised flows in closed pipes

The aim of this paper is to present a kinetic formulation of a model for the coupling of transient free surface and pressurised flows. Firstly, we revisit the system of Saint-Venant equations for free surface flow: we state some properties of Saint-Venant equations, we propose a kinetic formulation and we verify that this kinetic formulation leads to a Gibbs equilibrium that minimises (in some general case) an energy and preserves the still water steady state. Secondly, we propose a model for pressurised flows in a Saint-Venant-like conservative formulation. We then propose a kinetic formulation and we verify that this kinetic formulation leads to a Gibbs equilibrium that minimises in any case an energy and preserves the still water steady state. Finally, we propose a dual model that couples these two types of flow.     

On existence and multiplicity of similarity solutions to a nonlinear differential equation arising in magnetohydrodynamic Falkner–Skan flow for decelerated flows

Previously, existence and uniqueness of a class of monotone similarity solutions for a nonlinear differential equation arising in magnetohydrodynamic Falkner–Skan flow were considered in the case of accelerating flows. It was shown that a solution satisfying certain monotonicity properties exists and is unique for the case of accelerated flows and some decelerated flows. In this paper, we show that solutions to the problem can exist for decelerated flows even when the monotonicity conditions do not hold. In particular, these types of solutions have nonmonotone second derivatives and are, hence, a distinct type of solution from those studied previously. By virtue of this result, the present paper demonstrates the existence of an important type of solution for decelerated flows. Importantly, we show that multiple solutions can exist for the case of strongly decelerated flows, and this occurs because of the fact that the solutions do not satisfy the aforementioned monotonicity requirements. Copyright © 2015 John Wiley & Sons, Ltd.     

Modeling unsteady flow characteristics using smoothed particle hydrodynamics

Smoothed particle hydrodynamics (SPH) method has been extensively used to simulate unsteady free surface flows. The works dedicated to simulation of unsteady internal flows have been generally performed to study the transient start up of steady flows under constant driving forces and for low Reynolds number regimes. However, most of the fluid flow phenomena are unsteady by nature and at moderate to high Reynolds numbers. In this study, first a benchmark case (transient Poiseuille flow) is simulated to evaluate the ability of SPH to simulate internal transient flows at low and moderate Reynolds numbers (Re = 0.05, 500 and 1500). For this benchmark case, the performance of the two most commonly used formulations for viscous term modeling is investigated, as well as the effect of using the XSPH variant. Some points regarding using the symmetric form for pressure gradient modeling are also briefly discussed. Then, the application of SPH is extended to oscillating flows imposed by oscillating body force (Womersley type flow) and oscillating moving boundary (Stokes’ second problem) at different frequencies and amplitudes. There is a very good agreement between SPH results and exact solution even if there is a large phase lag between the oscillating pressure difference and moving boundary and the movement of the SPH particles generated. Finally, a modified formulation for wall shear stress calculations is suggested and verified against exact solutions. In all presented cases, the spatial convergence analysis is performed.     

Some Self-Similar Two-Phase Flows of Non-Newtonian Fluids Through a Porous Medium

Pascal This paper addresses the question of the rheological effects of non-Newtonian fluids in a flow system, in which a two-phase flow zone is coupled to a single-phase flow zone by a moving fluid interface. This flow system is involved in a technique for oil displacement in a porous medium, where a non-Newtonian displacing fluid (a polymer solution) is used to displace a non-Newtonian heavy oil. The self-similar solutions of the equations governing the dynamics of the moving interface, separating the displacing and displaced fluids, are obtained for the one-dimensional and plane radial flows. The effects associated with the presence of a two-phase flow zone, behind the moving interface, on the interface movement are analyzed. The existence of a pressure front ahead of the moving interface, moving with a finite velocity, is also shown. The relevance of this result to the propagation of pressure disturbances in a non-Newtonian fluid flowing through a porous medium is discussed with regard to interpretation of the transient pressure response in an injection well for polymer-solution floods.