Flow of viscoelastic fluids through a porous channel—I

The flow of viscoelastic fluids through a porous channel with one impermeable wall is computed. The flow is characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. Three solutions are developed: (i) an exact numerical solution, (ii) a perturbation solution for small R, the cross-flow Reynold's number and (iii) an asymptotic solution for large R. The results from exact numerical integration reveal that the solutions for a non-Newtonian fluid are possible only up to a critical value of the viscoelastic fluid parameter, which decreases with an increase in R. It is further demonstrated that the perturbation solution gives acceptable results only if the viscoelastic fluid parameter is also small. Two more related problems are considered: fluid dynamics of a long porous slider, and injection of fluid through one side of a long vertical porous channel. For both the problems, exact numerical and other solutions are derived and appropriate conclusions drawn.     

Analysis of velocity equation of steady flow of a viscous incompressible fluid in channel with porous walls

Steady flow of a viscous incompressible fluid in a channel, driven by suction or injection of the fluid through the channel walls, is investigated. The velocity equation of this problem is reduced to nonlinear ordinary differential equation with two boundary conditions by appropriate transformation and convert the two‐point boundary‐value problem for the similarity function into an initial‐value problem in which the position of the upper channel. Then obtained differential equation is solved analytically using differential transformation method and compare with He's variational iteration method and numerical solution. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and sciences. Copyright © 2009 John Wiley & Sons, Ltd.     

Symmetry groups and similarity solutions for a free convective boundary-layer problem

The free convective boundary-layer problem due to the motion of an elastic surface into an electrically conducting fluid is studied with group-theoretical methods. The symmetry groups admitted by the corresponding boundary value problem are obtained. Particular attention is paid on the group of scaling which provides the similarity solution of the problem. Also, the admissible form of the data, in order to be conformed to the obtained symmetries, is provided. Finally, with the use of the entailed similarity solution the problem is transformed into a boundary value problem of ODEs and is solved numerically.     

Using fully implicit conservative statements to close open boundaries passing through recirculations

The numerical solution of the fluid flow governing equations requires the implementation of certain boundary conditions at suitable places to make the problem well‐posed. Most of numerical strategies exhibit weak performance and obtain inaccurate solutions if the solution domain boundaries are not placed at adequate locations. Unfortunately, many practical fluid flow problems pose difficulty at their boundaries because the required information for solving the PDE's is not available there. On the other hand, large solution domains with known boundary conditions normally need a higher number of mesh nodes, which can increase the computational cost. Such difficulties have motivated the CFD workers to confine the solution domain and solve it using artificial boundaries with unknown flow conditions prevailing there. In this work, we develop a general strategy, which enables the control‐volume‐based methods to close the outflow boundary at arbitrary locations where the flow conditions are not known prior to the solution. In this regard, we extend suitable conservative statements at the outflow boundary. The derived statements gradually detect the correct boundary conditions at arbitrary boundaries via an implicit procedure using a finite element volume method. The extended statements are validated by solving the truncated benchmark backward‐facing step problem. The investigation shows that the downstream boundary can pass through a recirculation zone without deteriorating the accuracy of the solution either in the domain or at its boundaries. The results indicate that the extended formulation is robust enough to be employed in solution domains with unknown boundary conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

Nonstationary ideal incompressible fluid flows: Conditions of existence and uniqueness of solutions

The problem of formulating minimal conditions on input data that can guarantee the existence and uniqueness of solutions of the boundary value problems describing non-one-dimensional ideal incompressible fluid flow is considered using as an example the initial boundary value problem in a space-time cylinder constructed on a bounded flow domain with the nonpenetration condition on its boundary (which corresponds to fluid flow in a closed vessel). The existence problems are considered only for plane flows, and the uniqueness issues for three-dimensional flows as well. The required conditions are obtained in the form of conditions specifying that the vorticity belongs to definite functional Orlicz spaces. The results are compared with well-known results. Examples are given of admissible types of singularities for which the obtained results are valid, which is a physical interpretation of these results. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 130–145, July–August, 2008.     

A class of singular perturbation solutions to semilinear equations of fourth order

A class of singularly perturbed boundary value problems for semilinear equations of fourth order with two parameters are considered. Under suitable conditions, using the method of lower and upper solutions, the existence and the asymptotic behavior of the solution to the boundary value problem are studied, In the present paper, the solution to the original singularly perturbed problem with two parameters has only one boundary layer.     

On the interactions of surface waves with immersed structures

The fluid forces resulting from wave interaction with large submerged structures may be calculated using numerical procedures based on the solution of the associated boundary-value problem. In this paper, the analysis of wave interaction with a fixed submerged object of arbitrary cross-section and infinite length using a two-dimensional boundary value formation based on linear diffraction theory is summarized. Subsequently, the application of the boundary element method to obtain a solution is presented. The numerical considerations are emphasized with particular reference to computational efficiency. Numerical results are presented in the form of dimensionless wave force plots for various structural shapes. In the case of a bottom-seated half cylinder, for which there exists a closed-form solution, comparisons are made between results generated using both boundary element and equivalent finite element approaches. In the case of a submerged cylinder, comparisons are made between boundary element derived values and experimental results. The boundary element results compare well with both the closed-form solution and the experimental values.     

Heat and mass transfer in stagnation-point flow towards a stretching surface in the presence of buoyancy force and thermal radiation

In this paper an analysis has been made to study heat and mass transfer in two-dimensional stagnation-point flow of an incompressible viscous fluid over a stretching vertical sheet in the presence of buoyancy force and thermal radiation. The similarity solution is used to transform the problem under consideration into a boundary value problem of nonlinear coupled ordinary differential equations containing Prandtl number, Schmidt number and Sherwood number which are solved numerically with appropriate boundary conditions for various values of the dimensionless parameters. Comparison of the present numerical results are found to be in excellent with the earlier published results under limiting cases. The effects of various physical parameters on the boundary layer velocity, temperature and concentration profiles are discussed in detail for both the cases of assisting and opposing flows. The computed values of the skin friction coefficient, local Nusselt number and Sherwood number are discussed for various values of physical parameters. The tabulated results show that the effect of radiation is to increase skin friction coefficient, local Nusselt number and Sherwood number.     

Simulation of viscous fluid flow with consideration of boundary outflow and pressure drop

The problem of modeling a viscous fluid flow over the surface of a plate is considered when the pressure changes along the longitudinal coordinate according to a linear law. The corresponding boundary conditions are formulated for this problem. The Navier-Stokes equations are solved exactly in the problem of flow past the plate for the case of fluid outflow and a longitudinal pressure drop. Several formulas to determine the velocity profile are derived. The limiting cases are analyzed to study the consistency of various models. The corresponding pressure conditions are proposed for the case when the Navier-Stokes system has a known exact solution.     

Analytic Solution of an Inhomogeneous Kinetic Equation in the Problem of Second-Order Thermal Creep

An analytic solution of the problem of second-order thermal creep is obtained. A method for solving the half-space boundary value problem for an inhomogeneous linearized kinetic BGK equation forms the basis of the solution. The general solution of the input equation is constructed in the form of an expansion of the corresponding characteristic equation in terms of the eigenfunctions. Substitution of the solution in the boundary conditions leads to a Riemann boundary value problem. The unknown thermal creep velocity is found from the condition of solvability of the boundary value problem. The numerical analysis performed confirms the existence of negative thermophoresis (in the direction of the temperature gradient) for high-conductivity aerosol particles at low Knudsen numbers.     

Finite analytic boundary condition for a higher‐order finite analytic scheme

A higher‐order finite analytic scheme based on one‐dimensional finite analytic solutions is used to discretize three‐dimensional equations governing turbulent incompressible free surface flow. In order to preserve the accuracy of the numerical scheme, a new, finite analytic boundary condition is proposed for an accurate numerical solution of the partial differential equation. This condition has higher‐order accuracy. Thus, the same order of accuracy is used for the boundary. Boundary conditions were formulated and derived for fluid inflow, outflow, impermeable surfaces and symmetry planes. The derived boundary conditions are treated implicitly and updated with the solution of the problem. The basic idea for the derivation of boundary conditions was to use the discretized form of the governing equations for the fluid flow simplified on the boundaries and flow information. To illustrate the influence of the higher‐order effects at the boundaries, another, lower‐order finite analytic boundary condition, is suggested. The simulations are performed to demonstrate the validity of the present scheme and boundary conditions for a Wigley hull advancing in calm water. Copyright © 2005 John Wiley & Sons, Ltd.     

Boundary condition formulation for viscous fluid flow problems

In the solution of the Navier-Stokes equations by difference methods in infinite regions, the question arises as to the nature of the approximate boundary conditions at those portions of the computational region boundary where these conditions are not determined directly by the formulation of the basic problem. In certain cases of practical importance, these boundary conditions may be obtained by coupling the N-S equations with equations which are similar to the boundary-layer equations.In the present paper, we propose boundary conditions for the case of viscous incompressible fluid flow. Their application is illustrated for the problem of flow past the leading edge of a semi-infinite flat plate.The author wishes to thank I. Yu. Brailovskaya and L. A. Chudov for helpful suggestions in the course of this investigation.     

Singular perturbation for the weakly nonlinear reaction diffusion equation with boundary perturbation

In this paper, a class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation are considered under suitable conditions. Firstly, by dint of the regular perturbation method, the outer solution of the original problem is obtained. Secondly, by using the stretched variable and the expansion theory of power series the initial layer of the solution is constructed. And then, by using the theory of differential inequalities, the asymptotic behavior of the solution for the initial boundary value problems is studied. Finally, using some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.     

State space approach to magnetohydrodynamic flow of perfectly conducting micropolar fluid with stretch

In this work we introduce a model of the boundary layer equations for a perfect conducting micropolar fluid with stretch, bounded by an infinite vertical flat plane surface of a constant temperature. This model is applied to study the effects of free convection currents on the flow of the fluid in the presence of a constant magnetic field. The state space technique is adopted for the solution of a one‐dimensional problem for any set of boundary conditions. The resulting formulation together with the Laplace transform techniques are applied to a thermal shock problem. The inversion of the Laplace transforms is carried out using a numerical approach. Numerical results are given and illustrated graphically for the problem. Copyright © 2011 John Wiley & Sons, Ltd.     

Algorithms for interface treatment and load computation in embedded boundary methods for fluid and fluid–structure interaction problems

Embedded boundary methods for CFD (computational fluid dynamics) simplify a number of issues. These range from meshing the fluid domain, to designing and implementing Eulerian‐based algorithms for fluid–structure applications featuring large structural motions and/or deformations. Unfortunately, embedded boundary methods also complicate other issues such as the treatment of the wall boundary conditions in general, and fluid–structure transmission conditions in particular. This paper focuses on this aspect of the problem in the context of compressible flows, the finite volume method for the fluid, and the finite element method for the structure. First, it presents a numerical method for treating simultaneously the fluid pressure and velocity conditions on static and dynamic embedded interfaces. This method is based on the exact solution of local, one‐dimensional, fluid–structure Riemann problems. Next, it describes two consistent and conservative approaches for computing the flow‐induced loads on rigid and flexible embedded structures. The first approach reconstructs the interfaces within the CFD solver. The second one represents them as zero level sets, and works instead with surrogate fluid/structure interfaces. For example, the surrogate interfaces obtained simply by joining contiguous segments of the boundary surfaces of the fluid control volumes that are the closest to the zero level sets are explored in this work. All numerical algorithms presented in this paper are applicable with any embedding CFD mesh, whether it is structured or unstructured. Their performance is illustrated by their application to the solution of three‐dimensional fluid–structure interaction problems associated with the fields of aeronautics and underwater implosion. Copyright © 2011 John Wiley & Sons, Ltd.     

Positive solutions to (n-1,1) m-point boundary value problems with dependence on the first order derivative

Using the extension of Krasnoselskii's fixed point theorem in a cone, we prove the existence of at least one positive solution to the nonlinear nth order m-point boundary value problem with dependence on the first order derivative. The associated Green's function for the nth order m-point boundary value problem is given, and growth conditions are imposed on the nonlinear term f which ensures the existence of at least one positive solution. A simple example is presented to illustrate applications of the obtained results.     

MHD Falkner-Skan flow of Maxwell fluid by rational Chebyshev collocation method

The magnetohydrodynamics （MHD） Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite interval. An efficient approach based on the rational Chebyshev collocation method is performed to find the solution to the proposed boundary value problem. The rational Chebyshev collocation method is equipped with the orthogonal rational Chebyshev function which solves the problem on the semi-infinite domain without truncating it to a finite domain. The obtained results are presented through the illustrative graphs and tables which demonstrate the affectivity, stability, and convergence of the rational Chebyshev collocation method. To check the accuracy of the obtained results, a numerical method is applied for solving the problem. The variations of various embedded parameters into the problem are examined.     

Weakly-imposed Dirichlet boundary conditions for non-Newtonian fluid flow

We propose a new formulation for weakly imposing Dirichlet boundary conditions in non-Newtonian fluid flow. It is based on the Gerstenberger–Wall formulation for Newtonian fluids [1], but extended to non-Newtonian fluids. It uses a stabilization term in the weak form that is independent from the actual fluid model used, except for an adjustable parameter κ, having the physical dimension of a viscosity. The new formulation is tested, combined with an extended finite element method, for the flow past a cylinder between two walls using both a generalized Newtonian and a viscoelastic fluid. It is shown that the convergence is optimal for the generalized Newtonian fluid by comparing with a converged boundary-fitted solution using traditional strong boundary conditions. Also the solution of the viscoelastic fluid compares very well with a traditional solution using a boundary-fitted mesh and strong Dirichlet boundary conditions. For both fluid models we also test various values of the κ parameter and it turns out that a value equal to the zero-shear-viscosity gives good results. But, it is also shown that a wide range of κ values can be chosen without sacrificing accuracy.     

Some Exactly Solvable Problems of the Radiation of Three–Dimensional Periodic Internal Waves

An exact solution of the problem of the generation of three–dimensional periodic internal waves in an exponentially stratified, viscous fluid is constructed in a linear approximation. The wave source is an arbitrary part of the surface of a vertical circular cylinder which moves in radial, azimuthal, and vertical directions. Solutions satisfying exact boundary conditions, describe both the beam of outgoing waves and wave boundary layers of two types: internal boundary layers, whose thickness depends on the buoyancy frequency and the geometry of the problem, and viscous boundary layers, which, as in a homogeneous fluid, are determined by kinematic viscosity and frequency. Asymptotic solutions are derived in explicit form for cylinders of large, intermediate, and small dimensions relative to the natural scales of the problem.     

WELL-POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INVISCID COMPRESSIBLE ADIABATIC FLUID ＊

Introduction Itisreasonabletoconsiderarealfluidasanidealoneinfluidmechanicsundermany conditions.Forinstance,ingeneral,forthedistributionofthefluidfieldaroundtheaerocraft,mostpartofthefluidfieldmayberegardedasidealfluidexceptforasmallpartwhereeffects ofviscousandheatconductioninthethinlayernearthesurfacemustbeconsidered.Evenif thefluidiscompletelysupposedasidealonethroughoutthefluidfield,thequitereasonable resultsarealsogained,thereforestudyingidealfluidhasnotonlytheoreticalsignificancebut also…