Expansions at small Reynolds numbers for the flow past a porous circular cylinder

The purpose of this article is to use the method of matched asymptotic expansions (MMAE) in order to study the two-dimensional steady low Reynolds number flow of a viscous incompressible fluid past a porous circular cylinder. We assume that the flow inside the porous body is described by the continuity and Brinkman equations, and the velocity and boundary traction fields are continuous across the interface between the fluid and porous media. Formal expansions for the corresponding stream functions are used. We show that the force exerted by the exterior flow on the porous cylinder admits an asymptotic expansion with respect to low Reynolds numbers, whose terms depend on the characteristics of the porous cylinder. In addition, by considering Darcy's law for the flow inside the porous circular cylinder, an asymptotic formula for the force on the cylinder is obtained. Also, a porous circular cylinder with a rigid core inside is considered with Brinkman equation inside the porous region. Stress jump condition is used at the porous–liquid interface together with the continuity of velocity components and continuity of normal stress. Some particular cases, which refer to the low Reynolds number flow past a solid circular cylinder, have also been investigated.     

Simulation of Separating Flows over a Backward-Facing Step

Simulation results are reported for plane two-dimensional viscous incompressible flow in a channel with an abrupt expansion. The mathematical model is provided by the quasi-hydrodynamic equations in the incompressible fluid approximation. The computations are carried out in a range of Reynolds numbers including both laminar and turbulent flow. As the Reynolds number increases, the solution bifurcates and the steady laminar flow changes to time-dependent flow. The computation results are consistent with known experimental data. Turbulence models were not used for large Reynolds number computations.     

The motions of liquid films in a gas

The motions in a gas of thin films of a viscous incompressible liquid acted upon by capillary forces are considered. The surface tension depends on the impurity concentration of a surface-active material, soluble or insoluble in the liquid, and the liquid is non-volatile. The inertia of the liquid, viscous stresses, the Laplace pressure and the surface-tension gradients, impurity transfer and also the particular properties of super-thin films are taken into account. The motions of the films are described using the model of quasi-steady viscous flow. Systems of equations are obtained in the approximation of an ideal compressible medium and for small Mach numbers. The conditions for the incompressible film surface approximation to hold are obtained. The severe limitations of the gas-dynamic approximation in the case of a soluble impurity due to attenuation of the waves related to diffusion are investigated. A continuum model of the film as a compressible medium with a non-equilibrium pressure is constructed. The asymptotic form of the solutions of unsteady problems of impurity transfer in the limit of weak non-equilibrium is obtained. Integrals of the equations of motion of the films in steady one-dimensional problems are derived. Integral forms of the equations of momentum and its moment for an arbitrary contour of the film are presented, which hold for steady flows in a film and in quasi-statics. The boundary conditions for the solutions of the system of equations of motion of films are given.     

Asymptotic analysis of the Stokes flow in a thin cylindrical elastic tube

The purpose of this article is to perform an asymptotic analysis for an interaction problem between a viscous fluid and an elastic structure when the flow domain is a three-dimensional cylindrical tube. We consider a periodic, non-steady, axisymmetric, creeping flow of a viscous incompressible fluid through a long and narrow cylindrical elastic tube. The creeping flow is described by the Stokes equations and for the wall displacement we consider the Koiter's equation. The well posedness of the problem is proved by means of its variational formulation. We construct an asymptotic approximation of the problem for two different cases. In the first case, the stress term in Koiter's equation contains a great parameter as a coefficient and dominates with respect to the inertial term while in the second case both the terms are of the same order and contain the great parameter. An asymptotic analysis is developed with respect to two small parameters. Analysing the leading terms obtained in the second case, we note that the wave phenomena takes place. The small error between the exact solution and the asymptotic one justifies the below constructed asymptotic expansions.     

Modelling two-dimensional flow past arbitrary cylindrical bodies using boundary element formulations

This paper presents an efficient method of solving Queen's linearized equations for steady plane flow of an incompressible, viscous Newtonian fluid past a cylindrical body of arbitrary cross-section. The numerical solution technique is the well known direct boundary element method. Use of a fundamental solution of Oseen's equations, the ‘Oseenlet’, allows the problem to be reduced to boundary integrals and numerical solution then only requires boundary discretization. The formulation and solution method are validated by computing the net forces acting on a single circular cylinder, two equal but separated circular cylinders and a single elliptic cylinder, and comparing these with other published results. A boundary element representation of the full Navier-Stokes equations is also used to evaluate the drag acting on a single circular cylinder by matching with the numerical Oseen solution in the far field.     

Singular perturbations for the exterior three-dimensional slow viscous flow problem

In this paper the rigorous justification of the formal asymptotic expansions constructed by the method of matched inner and outer expansions is established for the three-dimensional steady flow of a viscous, incompressible fluid past an arbitrary obstacle. The justification is based on the series representation of the solution to the Navier-Stokes equations due to Finn, and it involves the reductions of various exterior boundary value problems for the Stokes and Oseen equations to boundary integral equations of the first kind from which existence as well as asymptotic error estimates for the solutions are deduced. In particular, it is shown that the force exerted on the obstacle by the fluid admits the asymptotic representation F = A₀ + A₁Re + O(Re² ln Re⁻¹) as the Reynolds number Re → 0⁺, where the vectors A₀ and A₁ can be obtained from the method of matched inner and outer expansions.     

Asymptotic approximation of the solution of Stokes equations in a domain with highly oscillating boundary

We consider a viscous incompressible flow in an infinite horizontal domain bounded at the bottom by a smooth wall and at the top by a rough wall. The latter is assumed to consist in a plane wall covered with periodically distributed asperities which size depends on a small parameter, and with a fixed height. We assume that the flow is governed by the stationary Stokes equations. Using a boundary layer corrector we derive and analyze a first order asymptotic approximation of the flow.      

Nonlinear stability of a parabolic velocity profile in a plane periodic channel

An inviscid or viscous incompressible flow with a general parabolic velocity profile in an infinite plane periodic channel with parallel walls that can move is considered with the impermeability conditions (for the Euler equations) or the no-slip conditions (for the Navier-Stokes equations). The nonlinear (for the original equations) and nonlocal (for all Reynolds numbers) stability of the unperturbed flow with respect to arbitrary two-dimensional smooth perturbations of the initial velocity field is established.     

A note on plane Stokes flow past a shear free impermeable cylinder

Summary The slow steady two-dimensional motion of a viscous incompressible fluid in the unbounded region exterior to a shear free circular cylinder which is impermeable is examined. It is shown that the above problem requires a certain consistency condition for the existence of a solution. In addition, a circle theorem for the biharmonic equation is presented, for the above plane Stokes flow. Some examples are also given.     

Solvability of a nonstationary problem of a rigid body motion in the flow of a viscous incompressible fluid in a pipe of arbitrary cross-section

The existence of a generalized weak solution is proved for the nonstationary problem of motion of a rigid body in the flow of a viscous incompressible fluid filling a cylindrical pipe of arbitrary cross-section. The fluid flow conforms to the Navier–Stokes equations and tends to the Poiseuille flow at infinity. The body moves in accordance with the laws of classical mechanics under the influence of the surrounding fluid and the gravity force directed along the cylinder. Collisions of the body with the boundary of the flow domain are not admitted and, by this reason, the problem is considered until the body approaches the boundary.     

Steady two-dimensional flow past a normal flat plate

A vorticity/stream function formulation is used to obtain a numerical simulation of steady two-dimensional flow of a viscous incompressible fluid past a normal flat plate for a range of Reynolds numbers. A method of Fornberg [J. Fluid Mech. 98, 819 (1980)] is used to determine upstream and downstream boundary conditions on the stream function. Special care is taken in the neighbourhood of the singularities in vorticity at the plate edges and this is very important because any errors introduced are swept downstream and severely affect such quantities as the length and width of the attached eddies. The computed results are compared with those of a laboratory experiment in which a plane strip is drawn through water and ethylene glycol for the range of Reynolds numbers for which the experimental flow is stable.     

Viscous incompressible free-surface flow down an inclined perturbed plane

The plane stationary free boundary value problem for the Navier-Stokes equations is studied. This problem models the viscous fluid free-surface flow down a perturbed inclined plane. For sufficiently small data the solvability and uniqueness results are proved in Hölder spaces. The asymptotic behavior of the solution is investigated.     

A Steady Three-Dimensional Noncompact Free Boundary-Value Problem for the Navier-Stokes Equations

A steady three-dimensional flow of a viscous incompressible fluid with a noncompact free boundary above a fixed unbounded bottom is studied. It is assumed that the motion of the fluid is generated by sources and sinks situated in a bounded part of the bottom and having zero total flux. The existence of a unique solution to this problem with small data is proved and the asymptotic of the solution is constructed. Bibliography: 33 titles.This paper is dedicated to Prof. V. A. Solonnikov’s 70th birthday__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 134–164.     

Blasius flow of thixotropic fluids: A numerical study

In the present work, laminar, two-dimensional flow of an incompressible thixotropic fluid obeying Harris rheological model is investigated above a fixed semi-infinite plate- the so-called Blasius flow. Assuming that the flow is occurring at high Reynolds number, use will be made of the boundary layer theory to simplify the equations of motion. The equations so obtained are then reduced to a single fourth-order ODE using a suitable similarity variable. It is shown that Harris fluids do not render themselves to a self-similar solution in Blasius flow. A local similarity solution is found which enabled investigating the effects of the model parameters on the velocity profile and wall shear stress at a given location above the plate. Numerical results show that for the Harris model to represent thixotropic fluids, the sign and magnitude of the material parameters appearing in this fluid model cannot be arbitrary.     

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.     

Lower Branch Modes of the Compressible Boundary Layer Flow Due to a Rotating-Disk

In this article, a theoretical study is pursued to investigate the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the compressible boundary layer flow due to a rotating-disk. Special attention is focused on to the short-wavelength stationary/nonstationary compressible crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies. Following closely the asymptotic framework introduced in [ 1 ] for the incompressible stationary modes, it is demonstrated here that the compressible modes having sufficiently long time scale can also be described by an asymptotic expansion procedure based on the triple-deck approach. Making use of this rational asymptotic technique, which rigorously takes into account the nonparallel effects, the asymptotic structure of the nonstationary modes is shown to be adjusted by a balance between viscous and Coriolis forces, and resulted from the fact of vanishing shear stress at the disk surface, as in the incompressible Von Karman's flow. As a consequence of matching successive regions in the asymptotic procedure, it is found that the wavenumber and the orientation of the compressible lower branch modes are governed by an eigenrelation, which is akin to the one obtained previously in [ 1 ] for the incompressible stationary mode and in [ 2 ] for the compressible stationary modes. The nonparallel influences are toward destabilizing all the modes, though the wall insulation and heating are relatively stabilizing for the modes in the vicinity of the stationary mode, unlike the wall cooling. The asymptotic compressible data obtained at high Reynolds number limit compares fairly well with the numerical results generated directly solving the linearized compressible system with usual parallel flow approximation.     

Numerical study of spherical Couette flows for certain zenith-angle-dependent rotations of boundary spheres at low Reynolds numbers

The numerical method with splitting of boundary conditions developed previously by the first and third authors for solving the stationary Dirichlet boundary value problem for the Navier-Stokes equations in spherical layers in the axisymmetric case at low Reynolds numbers and a corresponding software package were used to study viscous incompressible steady flows between two con-centric spheres. Flow regimes depending on the zenith angle ?? of coaxially rotating boundary spheres (admitting discontinuities in their angular velocities) were investigated. The orders of accuracy with respect to the mesh size of the numerical solutions (for velocity, pressure, and stream function in a meridional plane) in the max and L ₂ norms were studied in the case when the velocity boundary data have jump discontinuities and when some procedures are used to smooth the latter. The capabilities of the Richardson extrapolation procedure used to improve the order of accuracy of the method were investigated. Error estimates were obtained. Due to the high accuracy of the numerical solutions, flow features were carefully analyzed that were not studied previously. A number of interesting phenomena in viscous incompressible flows were discovered in the cases under study.     

Quasistationary Approximation in the Rotating Ring Problem

We consider the planar rotation-symmetric motion by inertia of a viscous incompressible fluid in a ring with free boundary. We reduce the corresponding initial-boundary value problem for the Navier–Stokes equations to some problem for a coupled system of one parabolic equation and two ordinary differential equations. We suppose that the coefficient of the derivatives of the sought functions with respect to time (the quasistationary parameter) is small; so the system is singularly perturbed. In this article we construct an asymptotic expansion for a solution to the rotating ring problem in a small quasistationary parameter and obtain a smallness estimate for the difference between the exact and approximate solutions.     

Nonstationary mass transfer of a reacting spherical particle in laminar stream of a viscous fluid : PMM vol. 38, n 6, 1974, pp. 1041–1047

The process of the formation of a stationary mass transfer mode for a moving reacting particle is examined. An analytic expression valid for a nonstationary distribution of the concentration of matter in a steady stream of viscous fluid, flowing past a spherical particle, was obtained for the case when at a certain instant a chemical reaction of the first order begins at the surface of the sphere. The problem is solved for small finite Reynolds and Péclet numbers. The solution of the corresponding stationary problem has been obtained in [1]. Paper [2] examined a nonstationary heat transfer of a fluid spherical drop in an inviscid flow with spasmodic change of initial temperature at high Péclet numbers. Paper [3] contains an analysis of the problem of a nonstationary heat transfer of a rigid spherical particle for small Reynolds and Péclet numbers at spasmodic change of temperature of the particle surface. The results obtained in [3] can be used to describe the mass transfer for a moving reacting particle only in the case of a diffusion mode of the chemical reaction.     

Continuum models in problems of the hypersonic flow of a rarefied gas around blunt bodiest

All possible continuum (hydrodynamic) models in the case of two-dimensional problems of supersonic and hypersonic flows around blunt bodies in the two-layer model (a viscous shock layer and shock-wave structure) over the whole range of Reynolds numbers, Re, from low values (free molecular and transitional flow conditions) up to high values (flow conditions with a thin leading shock wave, a boundary layer and an external inviscid flow in the shock layer) are obtained from the Navier-Stokes equations using an asymptotic analysis. In the case of low Reynolds numbers, the shock layer is considered but the structure of the shock wave is ignored. Together with the well-known models (a boundary layer, a viscous shock layer, a thin viscous shock layer, parabolized Navier-Stokes equations (the single-layer model) for high, moderate and low Re numbers, respectively), a new hydrodynamic model, which follows from the Navier-Stokes equations and reduces to the solution of the simplified (“local”) Stokes equations in a shock layer with vanishing inertial and pressure forces and boundary conditions on the unspecified free boundary (the shock wave) is found at Reynolds numbers, and a density ratio, k, up to and immediately after the leading shock wave, which tend to zero subject to the condition that (k/Re)^1/2 → 0. Unlike in all the models which have been mentioned above, the solution of the problem of the flow around a body in this model gives the free molecular limit for the coefficients of friction, heat transfer and pressure. In particular, the Newtonian limit for the drag is thereby rigorously obtained from the Navier-Stokes equations. At the same time, the Knudsen number, which is governed by the thickness of the shock layer, which vanishes in this model, tends to zero, that is, the conditions for a continuum treatment are satisfied. The structure of the shock wave can be determined both using continuum as well as kinetic models after obtaining the solution in the viscous shock layer for the weak physicochemical processes in the shock wave structure itself. Otherwise, the problem of the shock wave structure and the equations of the viscous shock layer must be jointly solved. The equations for all the continuum models are written in Dorodnitsyn--Lees boundary layer variables, which enables one, prior to solving the problem, to obtain an approximate estimate of second-order effects in boundary-layer theory as a function of Re and the parameter k and to represent all the aerodynamic and thermal characteristic; in the form of a single dependence on Re over the whole range of its variation from zero to infinity.

An efficient numerical method of global iterations, previously developed for solving viscous shock-layer equations, can be used to solve problems of supersonic and hypersonic flows around the windward side of blunt bodies using a single hydrodynamic model of a viscous shock layer for all Re numbers, subject to the condition that the limit (k/Re)^1/2 → 0 is satisfied in the case of small Re numbers. An aerodynamic and thermal calculation using different hydrodynamic models, corresponding to different ranges of variation Re (different types of flow) can thereby, in fact, be replaced by a single calculation using one model for the whole of the trajectory for the descent (entry) of space vehicles and natural cosmic bodies (meteoroids) into the atmosphere.