Local mass conservative Hermite interpolation for the solution of flow problems by a multi-domain boundary element approach

This paper presents a local Hermite radial basis function interpolation scheme for the velocity and pressure fields. The interpolation for velocity satisfies the continuity equation (mass conservative interpolation) while the pressure interpolation obeys the pressure equation. Additionally, the Dual Reciprocity Boundary Element method (DRBEM) is applied to obtain an integral representation of the Navier-Stokes equations. Then, the proposed local interpolation is used to obtain the values of the field variables and their partial derivatives at the boundary of the sub-domains. This interpolation allows one to obtain the boundary values needed for the integral formulas for velocity and pressure at some nodes within the sub-domains. In the proposed approach the boundary elements are merely used to parameterize the geometry, but not for the evaluation of the integrals as it is usually done. The presented multi-domain approach is different from the traditional ones in boundary elements because the resulting integral equations are non singular and the boundary data needed for the boundary integrals are approximated using a local interpolation. Some accurate results for simple Stokes problems and for the Navier-Stokes equations at low Reynolds numbers up to Re = 400 were obtained.     

Properties of Solutions for the Problem of a Joint Slow Motion of a Liquid and a Binary Mixture in a Two-Dimensional Channel

Under study is a conjugate boundary value problemdescribing a joint motion of a binary mixture and a viscous heat-conducting liquid in a two-dimensional channel, where the horizontal component of the velocity vector depends linearly on one of the coordinates. The problemis nonlinear and inverse because the systems of equations contain the unknown time functions—the pressure gradients in the layers. In the case of small Marangoni numbers (the so-called creeping flow) the problem becomes linear. For its solutions the two different integral identities are valid which allow us to obtain a priori estimates of the solution in the uniform metric. It is proved that if the temperature on the channel walls stabilizes with time then, as time increases, the solution of the nonstationary problem tends to a stationary solution by an exponential law.     

A Mixed Analytical/Numerical Method for Velocity and Heat Transfer of Laminar Power-Law Fluids

This paper presents a relatively simple numerical method to investigate the flow and heat transfer of laminar power-law fluids over a semi-infinite plate in the presence of viscous dissipation and anisotropy radiation. On one hand, unlike most classical works, the effects of power-law viscosity on velocity and temperature fields are taken into account when both the dynamic viscosity and the thermal diffusivity vary as a power-law function. On the other hand, boundary layer equations are derived by Taylor expansion, and a mixed analytical/numerical method (a pseudo-similarity method) is proposed to effectively solve the boundary layer equations. This method has been justified by comparing its results with those of the original governing equations obtained by a finite element method. These results agree very well especially when the Reynolds number is large. We also observe that the robustness and accuracy of the algorithm are better when thermal boundary layer is thinner than velocity boundary layer.     

Coupled flow and heat transfer in viscoelastic fluid with Cattaneo–Christov heat flux model

This letter presents a research for coupled flow and heat transfer of an upper-convected Maxwell fluid above a stretching plate with velocity slip boundary. Unlike most classical works, the new heat flux model, which is recently proposed by Christov, is employed. Analytical solutions are obtained by using the homotopy analysis method (HAM). The effects of elasticity number, slip coefficient, the relaxation time of the heat flux and the Prandtl number on velocity and temperature fields are analyzed. A comparison of Fourier’s Law and the Cattaneo–Christov heat flux model is also presented.     

Stagnation-point flow and heat transfer over an exponentially shrinking sheet

An analysis is carried out to investigate the stagnation-point flow and heat transfer over an exponentially shrinking sheet. Using the boundary layer approximation and a similarity transformation in exponential form, the governing mathematical equations are transformed into coupled, nonlinear ordinary differential equations which are then solved numerically by a shooting method with fourth order Runge-Kutta integration scheme. The analysis reveals that a solution exists only when the velocity ratio parameter satisfies the inequality −1.487068 ? c/a. Also, the numerical calculations exhibit the existence of dual solutions for the velocity and the temperature fields; and it is observed that their boundary layers are thinner for the first solution (in comparison with the second). Moreover, the heat transfer from the sheet increases with an increase in c/a for the first solution, while the heat transfer decreases with increasing c/a for the second solution, and ultimately heat absorption occurs.     

Equations of magnetohydrodynamic boundary layer with an injection of a dilatable medium

Methods of boundary layer theory are efficient when studying problems of fluid dynamics of multicomponent media with pronounced boundaries between different components. For example, such problems arise when placing screens isolating the main flow of a medium from the surface past which the medium flows, when investigating the mixing layer on the boundary of a submerged jet, when considering thin layers of plasma generated on the surface of a body entering dense layers of the atmosphere at a high velocity, and so on. All such problems are characterized by the presence of an internal boundary on which the rheological and electromagnetic properties of the continuous medium can change. This leads to various diffraction problems for systems of quasilinear partial differential equations. In the present paper, we consider one such problem.     

Reconstruction of Geophysical Layers with Shape Optimization and Level Set Methods

A shape optimization method is used to reconstruct the unknown shape of geophysical layers from boundary heat flux measurements by the use of adjoint fields and level sets. The identification of the shape of the geophysical layers by boundary heat flux measurements is necessary for the efficient use of geothermal energy. The method of speed is used to calculate the shape sensitivities, and the continuous adjoint approach is followed for the computation of the shape derivatives. The unknown shape is described with the help of the level set function; the advantage of the shape function is that no mesh movement or remeshing is necessary, but an additional Hamilton-Jacobi equation has to be solved. This equation is solved in an artificial time, where the velocity represents the movement in the direction of the normal vector of the interface. For large optimization steps, re-initialization of the level set function is also necessary, in order to keep the magnitude of the level set function near unity and also to smooth the level set function. Numerical results are given for measured heat fluxes on the boundary of the domain for different time steps and conductivity ratios. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Natural convection boundary layer flow of fluid with temperature-dependent viscosity from a horizontal elliptical cylinder with constant surface heat flux

This study deals with the temperature-dependent viscosity effects on the natural convection boundary layer on a horizontal elliptical cylinder with constant surface heat flux. The mathematical problem is reduced to a pair of coupled partial differential equations for the temperature and the stream function, and the resulting nonlinear equations are solved numerically by cubic spline collocation method. Results for the heat transfer characteristics are presented as functions of eccentric angle for various values of viscosity variation parameters, Prandtl numbers and aspect ratios. Results show that an increase in the viscosity variation parameter tends to accelerate the fluid flow near the surface and increase the maximum velocity, thus decreasing the velocity boundary layer thickness. As the viscosity variation parameter is increased, the surface temperature tends to decrease, thus increasing the local Nusselt number. Moreover, the local Nusselt number of the elliptical cylinder increases as the Prandtl number of the fluid is increased.     

Investigation of a powerful analytical method into natural convection boundary layer flow

The natural convection boundary layer flow modeled by a system of nonlinear differential equations is considered. By means of similarity transformation, the non-linear partial differential equations are reduced to a system of two coupled ordinary differential equations. The series solutions of coupled system of equations are constructed for velocity and temperature using homotopy analysis method (HAM). Convergence of the obtained series solution is discussed. Finally some figures are illustrated to show the accuracy of the applied method and assessment of various prandtl numbers on the temperature and the velocity is undertaken.     

An Exact Navier-Stokes Solution for Three-Dimensional,Spanwise-Homogeneous Boundary Layers

The plane stagnation flow onto (Hiemenz boundary layer, HBL) and the asymptotic suction boundary layer flow over a flat wall (ASBL) are two boundary layer flows for which the incompressible Navier-Stokes equations are amenable to exact similarity solutions. The Hiemenz solution has been extended to swept Hiemenz flows by superposition of a third, spanwise-homogeneous sweep velocity. This solution becomes singular as the chordwise, tangential base flow component vanishes. In this limit, the homogeneous ASBL solution is valid, which however cannot describe the swept Hiemenz flow, because it does not contain any chordwise velocity. This work presents a generalized three-dimensional similarity solution which describes three-dimensional spanwise homogeneously impinging boundary layers at arbitrary wall-normal suction velocities, using a rescaled similarity coordinate. The HBL and the ASBL are shown to be two limits of this solution. Further extensions consist of oblique impingement or different boundary suction directions, such as slip or stretching walls. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An existence theorem for the Boussinesq equations with non-Dirichlet boundary conditions

The evolution Boussinesq equations describe the evolution of the temperature and velocity fields of viscous incompressible Newtonian fluids. Very often, they are a reasonable model to render relevant phenomena of flows in which the thermal effects play an essential role. In the paper we prescribe non-Dirichlet boundary conditions on a part of the boundary and prove the existence and uniqueness of solutions to the Boussinesq equations on a (short) time interval. The length of the time interval depends only on certain norms of the given data. In the proof we use a fixed point theorem method in Sobolev spaces with non-integer order derivatives. The proof is performed for Lipschitz domains and a wide class of data.     

Nonsimilar boundary layer analysis of double-diffusive convection from a vertical truncated cone in a porous medium with variable viscosity

This work presents a boundary layer analysis about variable viscosity effects on the double-diffusive convection near a vertical truncated cone in a fluid-saturated porous medium with constant wall temperature and concentration. The viscosity of the fluid is assumed to be an inverse linear function of the temperature. A boundary layer analysis is employed to derive the nondimensional nonsimilar governing equations, and the transformed boundary layer governing equations are solved by the cubic spline collocation method to yield computationally efficient numerical solutions. The obtained results are found to be in good agreement with previous papers on special cases of the problem. Results for local Nusselt and Sherwood numbers are presented as functions of viscosity-variation parameter, buoyancy ratio, and Lewis number. For a porous medium saturated with a Newtonian fluid with viscosity proportional to an inverse linear function of temperature, higher value of viscosity-variation parameter leads to the decrease of the viscosity in fluid flow, thus increasing the fluid velocity as well as the local Nusselt number and the local Sherwood number.     

Large Eddy Simulation of a Compressible Boundary Layer

A large eddy simulation of a compressible boundary layer is performed. To generate an appropriate inflow distribution the rescaling technique for compressible flows is discribed. In this method Morkovin's hypothesis in which the total temperature fluctuations are neglected compared with the static temperature fluctuations is applied to rescale and generate the temperature profile at inlet. This new technique is used for various large eddy simulations of subsonic and supersonic three‐dimensional boundary layers of a flat plate. Simulation results for the time‐averaged mean flow and Reynolds stresses are compared with numerical and analytical data to demonstrate the high quality of the method.     

Heat transfer enhancement for boundary layer flow over a wedge by the use of electric fields

The effect of an electric field is investigated for heat transfer properties in a laminar, incompressible, non-isothermal boundary layer gas flow over a wedge. The governing boundary layer equations are reduced to an ordinary differential equation system using similarity transformations. The reduced equations are solved numerically for different values of electric and flow field parameters characterizing the ratio of electric force to fluid inertia force, Joule heating and ion kinetic work. For specific electric field function forms, leading to similarity solutions, velocity boundary layers are observed to become thinner and heat transfer properties are shown to be enhanced near the wall. The level of enhancement is controlled by the electric body force with additional effects of Joule heating and ion kinetic work on the bulk flow. The effects of low and high Prandtl numbers are also demonstrated. Heat transfer enhancement is observed to increase with increasing Prandtl number.     

2D numerical simulation of submerged hydraulic jumps

Hydraulic jumps are usually used to dissipate energy in hydraulic engineering. In this paper, the turbulent submerged hydraulic jumps are simulated by solving the unsteady Reynolds averaged Navier–Stokes equations along with the continuity equation and the standard k–? equations for turbulence modeling. The Lagrangian moving grid method is employed for the simulation of the free surface. In the developed model, kinematic free-surface boundary condition is solved simultaneously with the momentum and continuity equations, so that the water elevation can be obtained along with velocity and pressure fields as part of the solution. Computational results are presented for Froude numbers ranging from 3.2 to 8.2 and submergence factors ranging from 0.24 to 0.85. Comparisons with experimental measurements show that numerical model can simulate the velocity field, variation of free surface, maximum velocity, Reynolds shear and normal stresses at various stations with reasonable accuracy.     

Viscous Flows in a Half Space Caused by Tangential Vibrations on Its Boundary

The paper is devoted to the studies of viscous flows caused by a vibrating boundary. The fluid domain is a half‐space, its boundary is a nondeformable plane that exhibits purely tangential vibrations. Such a simple geometrical setting allows us to study general boundary velocity fields and to obtain general results. From a practical viewpoint, such boundary conditions may be seen as the tangential vibrations of the material points of a stretchable plane membrane. In contrast to the classical boundary layer theory, we aim to build a global solution. To achieve this goal we employ the Vishik–Lyusternik approach, combined with two‐timing and averaging methods. Our main result is: we obtain a uniformly valid in the whole fluid domain approximation to the global solutions. This solution corresponds to general boundary conditions and to three different settings of the main small parameter. Our solution always include the inner part and outer part that both contain oscillating and non‐oscillating components. It is shown that the nonoscillating outer part of the solution is governed either by the full Navier–Stokes equations or the Stokes equations (both with the unit viscosity) and can be interpreted as a steady or unsteady streaming. In contrast to the existing theories of a steady streaming, our solutions do not contain any secular (infinitely growing with the inner normal coordinate) terms. The examples of the spatially periodic vibrations of the boundary and the angular torsional vibrations of an infinite rigid disc are considered. These examples are still brief and illustrative, while the core of the paper is devoted to the adaptation of the Vishik–Lyusternik method to the development of the general theory of vibrational boundary layers.     

Effects of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference

This paper concerns with studying the steady and unsteady MHD micropolar flow and mass transfers flow with constant heat source in a rotating frame of reference in the presence chemical reaction of the first-order, taking an oscillatory plate velocity and a constant suction velocity at the plate. The plate velocity is assumed to oscillate in time with a constant frequency; it is thus assumed that the solutions of the boundary layer are the same oscillatory type. The governing dimensionless equations are solved analytically after using small perturbation approximation. The effects of the various flow parameters and thermophysical properties on the velocity and temperature fields across the boundary layer are investigated. Numerical results of velocity profiles of micropolar fluids are compared with the corresponding flow problems for a Newtonian fluid. The results show that there exists completely oscillating behavior in the velocity distribution.     

Nonisothermic flow of gas mixture in a channel at intermediate Knudsen numbers

Solution of the problem of gas mixture flow in a plane channel at intermediate Knudsen numbers is considered on the basis of the 20-moment approximation as a function of distribution. The applied method consists of averaging moment equations valid throughout the flow region (including the Knudsen layers) with the determination of boundary values of macroscopic parameters on the wall using the approximate Loyalka method /1,2/. Expressions are obtained for a binary mixture for the mean molar velocity averaged over the channel cross section, difference of component velocities, and the relative heat flux in the presence of longitudinal gradients of partial pressures, and for the temperature gradients. Respective kinetic coefficients of the Onsager matrix are calculated. Dependence of these coefficients on the Knudsen number, and the properties of molecule scatter on the channel wall are analyzed in detail in the case of one-component gas and of a binary mixture with small relative difference of mass and diameters of molecule scatter.     

Examples of boundary layers associated with the incompressible Navier-Stokes equations

The author surveys a few examples of boundary layers for which the Prandtl boundary layer theory can be rigorously validated. All of them are associated with the incompressible Navier-Stokes equations for Newtonian fluids equipped with various Dirichlet boundary conditions (specified velocity). These examples include a family of (nonlinear 3D) plane parallel flows, a family of (nonlinear) parallel pipe flows, as well as flows with uniform injection and suction at the boundary. We also identify a key ingredient in establishing the validity of the Prandtl type theory, i.e., a spectral constraint on the approximate solution to the Navier-Stokes system constructed by combining the inviscid solution and the solution to the Prandtl type system. This is an additional difficulty besides the wellknown issue related to the well-posedness of the Prandtl type system. It seems that the main obstruction to the verification of the spectral constraint condition is the possible separation of boundary layers. A common theme of these examples is the inhibition of separation of boundary layers either via suppressing the velocity normal to the boundary or by injection and suction at the boundary so that the spectral constraint can be verified. A meta theorem is then presented which covers all the cases considered here.     

Velocity and Concentration Boundary Layers on Spinning Fibers

In this paper a mathematical model for the viscose wet spinning process is presented: We consider a single fibre, which is produced by pressing a basic solution of viscose into a bath containing sulphuric acid. H₂SO₄ diffuses into the viscose solution and reacts with the natrium hydroxide so that a solidifying fibre is formed which is pulled through the bath by drives. Due to the movement of the fibre and of diffusive transport of sulphuric acid into the fibre velocity and concentration boundary layers develop. Starting from the laminar boundary layer equations we investigate the flow and concentration fields in the bath induced by the fibre utilizing the Local Non‐Similarity method. Mass transfer in the fibre is modelled by transport equations for sulphuric acid and natrium hydroxide taking into account the neutralization reaction. The model of the fibre is coupled to the bath phase model by appropriate boundary conditions for the mass flow density and the chemical potential of sulphuric acid. The non‐constant diameter of the fibre is taken into account by a perturbation approach.