Mathematical modeling of near-wall flows of two-phase mixture with evaporating droplets

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Analytical investigation of boundary layer growth and swirl intensity decay rate in a pipe

In this research, the developing turbulent swirling flow in the entrance region of a pipe is investigated analytically by using the boundary layer integral method. The governing equations are integrated through the boundary layer and obtained differential equations are solved with forth-order Adams predictor-corrector method. The general tangential velocity is applied at the inlet region to consider both free and forced vortex velocity profiles. The comparison between present model and available experimental data demonstrates the capability of the model in predicting boundary layer parameters (e.g. boundary layer growth, shear rate and swirl intensity decay rate). Analytical results showed that the free vortex velocity profile can better predict the boundary layer parameters in the entrance region than in the forced one. Also, effects of pressure gradient inside the boundary layer is investigated and showed that if pressure gradient is ignored inside the boundary layer, results deviate greatly from the experimental data.     

Comment on the paper “Magnetohydrodynamic effects on natural convection flow of a nanofluid in the presence of heat source due to solar energy,N. Anbuchezhian,K. Srinivasan,K. Chandrasekaran,R. Kandasamy,Meccanica (2013) 48:307–321”

In the above paper a theoretical investigation of MHD convective flow and heat transfer of an incompressible viscous nanofluid past a porous vertical stretching sheet in the presence of variable stream condition is presented. The governing boundary layer equations are transformed by a Lie symmetry group transformation and the ordinary differential equations are solved numerically using Runge–Kutta Gill method.     

Magnetoaerodynamic boundary layer flow with pressure gradient and separation

This study considers the large interaction parameter magnetoaerodynamic boundary layer associated with the free stream flow of a conducting fluid over an infinitely long circular insulator cylinder with the applied magnetic field normal to the distant free stream flow. The investigation is conducted in two parts; a theoretical solution of the associated boundary layer equations and a qualitative experimental investigation to allow visualization of flow separation caused by the magnetic field. The general integral formulation of Galerkin-Kantorovich-Dorodnitsyn is used to determine the boundary layer thickness, momentum thickness, displacement thickness, approximate separation point, and velocity profiles.     

Turbulent flow in converging nozzles,part one: boundary layer solution

The boundary layer integral method is used to investigate the development of the turbulent swirling flow at the entrance region of a conical nozzle. The governing equations in the spherical coordinate system are simplified with the boundary layer assumptions and integrated through the boundary layer. The resulting sets of differential equations are then solved by the fourth-order Adams predictor-corrector method. The free vortex and uniform velocity profiles are applied for the tangential and axial velocities at the inlet region, respectively. Due to the lack of experimental data for swirling flows in converging nozzles, the developed model is validated against the numerical simulations. The results of numerical simulations demonstrate the capability of the analytical model in predicting boundary layer parameters such as the boundary layer growth, the shear rate, the boundary layer thickness, and the swirl intensity decay rate for different cone angles. The proposed method introduces a simple and robust procedure to investigate the boundary layer parameters inside the converging geometries.     

The new solution for the axially symmetrical laminar boundary layer equations between two parallel sherical surfaces

The transformations, which are similar to Mangler’s transformations, are given in this paper. They change the entrance region flow of axially symmetrical laminar boundary layer between two parallel spherical surfaces into the flow of two-dimensional boundary layer, and simplify the problems. The simplified equations can be solved by the two-dimensional boundary layer theory and numerical methods. Therefore, a new way is opened up to solve the diffusive laminar flow in the entrance region between two parallel spherical surfaces.     

THE NEW SOLUTION FOR THE AXIALLY SYMMETRICAL LAMINAR BOUNDARY LAYER EQUATIONS BETWEEN TWO PARALLEL SPHERICAL SURFACES

The transformations, which are similar to Mangler's transformations, are given in this paper. They change the entrance region flow of axially symmetrical laminar boundary layer between two parallel spherical surfaces into the flow of two-dimensional boundary layer, and simplify the problems. The simplified equattons can be solved by the two-dimensional boundary layer theory and numerical methods. Therefore, a new way is opened up to solve the diffusive laminar fiow in the entrance region between two parallel spherical surfaces.     

The new solutions for two kinds of axially symmetrical laminar boundary layer equations

The transformations, which are similar to Mangier's transformation, are given in this paper, and make the two kinds of entrance region flow of axially symmetrical laminar boundary layer in internal way into the flow of two-dimensional boundary layer, and simplify the proboems. The simplified equations can be solved by the 2-D boundary layer theory. Therefore, a new way is opened up to solve the axially symmetrical flow in the entrance region of internal way.Project Supported by the Scientific Fund of Chinese Academy of Science.     

Time‐domain BEM solution of convection–diffusion‐type MHD equations

The two‐dimensional convection–diffusion‐type equations are solved by using the boundary element method (BEM) based on the time‐dependent fundamental solution. The emphasis is given on the solution of magnetohydrodynamic (MHD) duct flow problems with arbitrary wall conductivity. The boundary and time integrals in the BEM formulation are computed numerically assuming constant variations of the unknowns on both the boundary elements and the time intervals. Then, the solution is advanced to the steady‐state iteratively. Thus, it is possible to use quite large time increments and stability problems are not encountered. The time‐domain BEM solution procedure is tested on some convection–diffusion problems and the MHD duct flow problem with insulated walls to establish the validity of the approach. The numerical results for these sample problems compare very well to analytical results. Then, the BEM formulation of the MHD duct flow problem with arbitrary wall conductivity is obtained for the first time in such a way that the equations are solved together with the coupled boundary conditions. The use of time‐dependent fundamental solution enables us to obtain numerical solutions for this problem for the Hartmann number values up to 300 and for several values of conductivity parameter. Copyright © 2007 John Wiley & Sons, Ltd.     

Combined effects of rotation and translation on a MHD flow due to compressible viscous fluid

Summary The modification of an axi-symmetric viscous flow due to a relative rotation of a disk or fluid by a translation of the boundary are studied. The fluid is taken to be compressible and electrically conducting. The equations governing the motion are solved iteratively through a central-difference scheme. The effect of an axial magnetic field and disk temperature on the flow and heat transfer are included in the present analysis. The translation of the disk or fluid generates a velocity field at each plane parallel to the disk (secondary flow). The cartesian components of the velocity due to the secondary flow are oscillatory in nature when a rigid body rotation of the free stream along with a translation of the disk is considered. The magnetic field damps out the velocity field, and reduces the thickness of the boundary layer. The cross component of wall shear due to secondary flow acts in a direction opposite to the rotation of the disk or fluid for all cases of the motion. The rise in disk temperature produces an increment in the magnitude of the wall shear associated with the secondary flow.     

Diffusion and Electrical Processes in the Turbulent Boundary Layer and in the Neighborhood of the Stagnation Point

The problem of electric current (engine current) formation in aircraft jet engine ducts as a result of the development of electrical diffusion boundary layers on the surfaces of the duct and internal engine components is investigated. It is assumed that the outer flow containing electrons and positive ions is quasi-neutral and that the electrical quasi-neutrality is violated (and the electric engine current develops) in the wall flow zone as a result of the difference between the electron and ion diffusion coefficients. The problem of the development of an electrical diffusion boundary layer inside the turbulent gasdynamic boundary layer on a plane surface is formulated and solved. The engine current distribution along the duct is found for various values of a turbulent viscosity on the boundary of the gasdynamic boundary layer which affect the laminar-turbulent transition point.The electrical diffusion processes that occurs when an electrically quasi-neutral hydrodynamic stream impinges on a plane surface (simulation of the flow in the neighborhood of a stagnation point) is studied. In this case the Navier-Stokes equations have a self-similar solution. It is shown that the system of electrohydrodynamic equations also has a self-similar solution. The electrical parameter fields are determined and the engine current is found on the basis of this solution.     

MHD stagnation-point flow and heat transfer towards stretching sheet with induced magnetic field

The problem of the steady magnetohydrodynamic (MHD) stagnation-point flow of an incompressible viscous fluid over a stretching sheet is studied. The effect of an induced magnetic field is taken into account. The nonlinear partial differential equations are transformed into ordinary differential equations via the similarity transformation. The transformed boundary layer equations are solved numerically using the shooting method. Numerical results are obtained for various magnetic parameters and Prandtl numbers. The effects of the induced magnetic field on the skin friction coefficient, the local Nusselt number, the velocity, and the temperature profiles are presented graphically and discussed in detail.     

Effect of Hall current on MHD natural convection flow from vertical permeable flat plate with uniform surface heat flux

The effect of the Hall current on the magnetohydrodynamic (MHD) natural convection flow from a vertical permeable flat plate with a uniform heat flux is analyzed in the presence of a transverse magnetic field.It is assumed that the induced magnetic field is negligible compared with the imposed magnetic field.The boundary layer equations are reduced to a suitable form by employing the free variable formulation (FVF) and the stream function formulation (SFF).The parabolic equations obtained from FVF are numer...     

THREE-DIMENSIONAL EXTRUDATE SWELL: FORMULATION WITH THE STREAM TUBE METHOD AND NUMERICAL RESULTS FOR A NEWTONIAN FLUID

Stream tube analysis, already applied to two-dimensional extrudate swell problems involving rate and integral constitutive equations for incompressible fluids, is now considered in the problem of free surface determination in a three-dimensional flow situation. The method allows computation of the unknown free surface by considering only a ‘peripheral stream tube’ limited by the wall and the jet surface and an inner stream surface. Those boundary surfaces are determined by considering the conservation equations together with boundary condition equations, solved by the Levenberg/Marquatdt optimization algorithm. The method leads to a considerable reduction in the number of degrees of freedom and the storage area. As in a previous study in the two-dimensional case, singularity problems in the vicinity of the junction points between the wall and the free surface are avoided. However, the numerical method still allows evaluati on of stress peaks due to the singularity at the exit, as may be observed for results obtained with a Newtonian fluid in a duct of square cross-section.     

Laminar boundary layer in a swirling flow

The spatial non-self-similar boundary layer in a compressible gas in a swirling flow is studied. Boundary-layer equations are written in variables ensuring constancy of the coefficients of first derivatives and are solved by the finite-difference method. Boundary-layer peculiarities in the presence of a return circulation region in the channel are clarified.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 43–49, January–February, 1976.     

Viscous dissipation and thermal radiation effects on the magnetohydrodynamic (MHD) flow and heat transfer over a stretching slender cylinder

An axisymmetric magnetohydrodynamic (MHD) boundary layer flow and heat transfer of a fluid over a slender cylinder are investigated numerically. The effects of viscous dissipation, thermal radiation, and surface transverse curvature are taken into account in the simulations. For this purpose, the governing partial differential equations are transformed to ordinary differential equations by using appropriate similarity transformations. The resultant ordinary differential equations along with appropriate boundary conditions are solved by the fourth-order Runge–Kutta method combined with the shooting technique. The effects of various parameters on the velocity and temperature profiles, local skin friction coefficient, and Nusselt number are analyzed.     

MHD stagnation flow of a micropolar fluid through a porous medium

The unsteady MHD boundary layer flow of a micropolar fluid near the forward stagnation point of a two dimensional plane surface is investigated by using similarity transformations. The transformed nonlinear differential equations are solved by an analytic method, namely homotopy analysis method (HAM). The solution is valid for all values of time. The effect of MHD and porous medium, non dimensional velocity and the microrotation are presented graphically and discussed. The coefficient of skin friction is also presented graphically.     

Computation of the viscous shock layer on blunted cones

A numerical method is described for solving the equations of the compressible viscous shock layer on smooth spherically blunted axisymmetric cones at zero angle of attack and flow of a perfect gas. Effective use is made of the scheme of separating the original system of equations into parabolic (second order) and inviscid (first order) subsystems, which are solved by intrinsic methods. The results of the computations are presented. The method is capable of natural generalization to the case of nonequilibrium physical and chemical processes and diffusion. In most published papers dealing with computation of the compressible shock layer, the authors examine either the vicinity of the stagnation point or a certain region of spherical blunting [1–5]. In all the papers except [4, 5], a number of simplified assumptions have been made regarding the flow picture. Very few papers [6–8] have calculated the viscous shock layer on the forward surface of blunted bodies. In [6, 7] an approximate examination was made only of hyperboloids and paraboloids of revolution, which have very favorable geometry. Reference [8] used a approximate Karman—Polhausen integral method for a very simple system of equations. The method proposed here is essentially an accurate numerical method for solution of the viscous shock layer equations.     

MHD stagnation point flow towards heated shrinking surface subjected to heat generation/absorption

The magnetohydrodynamic(MHD) stagnation point flow of micropolar fluids towards a heated shrinking surface is analyzed.The effects of viscous dissipation and internal heat generation/absorption are taken into account.Two explicit cases,i.e.,the prescribed surface temperature(PST) and the prescribed heat flux(PHF),are discussed.The boundary layer flow and energy equations are solved by employing the homotopy analysis method.The quantities of physical interest are examined through the presentation of plots/tabulated values.It is noticed that the existence of the solutions for high shrinking parameters is associated closely with the applied magnetic field.     

MHD boundary-layer flow of an upper-convected Maxwell fluid in a porous channel

Two-dimensional magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell fluid is investigated in a channel. The walls of the channel are taken as porous. Using the similarity transformations and boundary layer approximations, the nonlinear partial differential equations are reduced to an ordinary differential equation. The developed nonlinear equation is solved analytically using the homotopy analysis method. An expression for the analytic solution is derived in the form of a series. The convergence of the obtained series is shown. The effects of the Reynolds number Re, Deborah number De and Hartman number M are shown through graphs and discussed for both the suction and injection cases.