Two-dimensional Boundary Layer Flow Research Articles

Analytical Solution to the Boundary Layer Slip Flow and Heat Transfer over a Flat Plate using the Switching Differential Transform Method

In this paper, the two-dimensional steady boundary layer flow and heat transfer over a flat plate with slip velocity and temperature jump conditions at the walls were analyzed. Using similarity transforms, the governing equations were reduced to a system of ordinary differential equations. Semi-analytical solutions to the resulting boundary value problem were obtained using the differential transform method (DTM). In order to cover the asymptotic boundary conditions, a method of switching curves was proposed. In this switching approach, the traditional solution to the DTM, which is valid for finite horizons, was followed by another path that was also an analytical solution to the problem. The main preference of the resulting closed form solution with respect to numerical solution is the possibility of parametric studies. The method was verified using some available numerical data, and the results showed that our proposed method had reasonable efficiency and accuracy.

Magnetohydrodynamic (MHD) Boundary Layer Flow Past a Wedge with Heat Transfer and Viscous Effects of Nanofluid Embedded in Porous Media

The problem of two-dimensional steady laminar MHD boundary layer flow past a wedge with heat and mass transfer of nanofluid embedded in porous media with viscous dissipation, Brownian motion, and thermophoresis effect is considered. Using suitable similarity transformations, the governing partial differential equations have been transformed to nonlinear higher-order ordinary differential equations. The transmuted model is shown to be controlled by a number of thermophysical parameters, viz. the pressure gradient, magnetic, permeability, Prandtl number, Lewis number, Brownian motion, thermophoresis, and Eckert number. The problem is then solved numerically using spectral quasilinearization method (SQLM). The accuracy of the method is checked against the previously published results and an excellent agreement has been obtained. The velocity boundary layer thickness reduces with an increase in pressure gradient, permeability, and magnetic parameters, whereas thermal boundary layer thickness increases with an increase in Eckert number, Brownian motion, and thermophoresis parameters. Greater values of Prandtl number, Lewis number, Brownian motion, and magnetic parameter reduce the nanoparticles concentration boundary layer.

A Study on Non-Newtonian Transport Phenomena in Mhd Fluid Flow From a Vertical Cone With Navier Slip and Convective Heating

Abstract In the current study is to examine numerically the effects of presence of a radial magnetic field, slip and jump conditions on the steady two-dimensional free convective boundary layer flow over an external surface of a vertical cone for an electro-conductive polymer. A proper non-similarity transformation simplifies the system of partial differential equations into a system of ordinary differential equations. The collocation formula in the MATLAB software then solves the system of non-similarity equations. The finding results show that, a weak elevation in temperature is accompanied with the increase in the Carreau fluid parameter, whereas a significant acceleration in the flow is computed near the cone surface. The study is relevant to smart coating transport phenomena.

The study of MHD Nano fluid flow with chemical reaction along with thermophoresis and Brownian motion on boundary layer flow over a linearly stretching sheet

The two-dimensional steady boundary layer flow, of a nanofluid past a stretching sheet with a convective boundary condition in the presence of chemical reaction has been studied. The equation of volume fraction concentration consists of the Brownian motion and the thermophoresis effects. The governing equations are simplified via boundary value approximation and some similarity variables and solved using the Runge-Kutta fourth order method with shooting technique and homotopy perturbation method (HPM) with Pad� approximation (HPM-Pad�). All the salient parameters on the temperature and concentration profiles have been studied and details are given graphically. Thus the results found in this study are being compared with the previously published study, where an excellent agreement is achieved. As presented on tables. It is found that the reduced Nusselt number is a decreasing function of both generative and destructive chemical reaction γ. While the reduced Sherwood number is a decreasing function when chemical reaction parameter, γ0.

Heat transfer analysis for Casson fluid flow over stretching sheet with Newtonian heating and viscous dissipation

A steady two-dimensional boundary layer flow and heat transfer of a Casson fluid over a stretched plate with Newtonian heating boundary condition is investigated. The governing partial differential equations are first reduced to nonlinear ordinary differential equations by using suitable similarity transformations before being solved numerically using an implicit finite difference scheme. The features of the flow and heat transfer characteristics for some embedded parameters, such as Casson fluid parameter β, Eckert number Ec, Prandtl number Pr and conjugate Newtonian heating parameter γ, are analyzed and discussed.

Multiple physical aspects during the flow and heat transfer analysis of Carreau fluid with nanoparticles

The current work is concerned with the two-dimensional boundary layer flow of a non-Newtonian fluid in the presence of nanoparticles. The heat and mass transfer mechanism for Carreau nanofluid flow due to a radially stretching/shrinking sheet is further investigated in this article. The governing physical situation is modelled in the form of partial differential equations and are simplified to a system of non-linear ordinary differential equations by employing dimensionless variables. Numerical simulations for non-dimensional velocity, temperature and concentration fields has been performed with the assistance of built-in Matlab solver bvp4c routine. One significant computational outcome of this study is the existence of multiple numerical solutions for the flow fields. The impacts of various developing parameters, for instance, Weissenberg number, power-law index, shrinking parameter, suction parameter, Prandtl number, Schmidt number, Brownian motion and thermophoresis parameter on the velocity, temperature and nanoparticles concentration are visualized through tables and graphical experiment. The numerical results demonstrate that the rates of heat and mass transfer are raised by higher Weissenberg number for first solution and an inverse is seen for second solution. Moreover, an increasing trend is seen in nanofluids temperature for both solutions with greater values of thermophoresis parameter. In addition, the numerical results obtained by the applied technique are validated with existing literature and found to be in an excellent agreement.

Numerical investigations on magnetic field modeling for Carreau non-Newtonian fluid flow past an isothermal sphere

In the current study, the effects of radial magnetic field, slip and jump conditions on the steady two-dimensional free convective boundary layer flow over an external surface of an isothermal sphere for an electro-conductive polymer are numerically studied. It is assumed that the studied fluid has a non-Newtonian rheological behavior and follows the Carreau fluid model. In this investigation, the formulation of the Carreau fluid model has been used first time for describing the present boundary layer problem, and then the resulting partial differential equations are transformed to ordinary differential equations by using non-similarity transformations. The obtained ordinary differential equations are solved numerically by a well-known method named as Keller-Box method. The finding results show that a weak elevation in temperature is accompanied with the increase in the Carreau fluid parameter, whereas a significant acceleration in the flow is computed near the sphere surface. It is shown also that an increase in the thermal slip parameter allows to strongly decrease both the skin friction coefficient and the local Nusselt number. The skin friction coefficient is also depressed with increasing magnetic body force parameter. Moreover, it is observed that an increase in the momentum slip parameter allows to decrease the skin friction coefficient, whereas the local Nusselt number is reduced with the increase in the Carreau fluid parameter. It is found also that the skin friction coefficient is increased with greater stream-wise coordinate, whereas the local Nusselt number is reduced with the increase in this parameter.

Finite element analysis of magnetohydrodynamic flow over flat surface moving in parallel free stream with viscous dissipation and Joule heating

PurposeThe purpose of this paper is to investigate two-dimensional viscous incompressible magnetohydrodynamic boundary layer flow and heat transfer of an electrically conducting fluid over a continuous moving flat surface considering the viscous dissipation and Joule heating.Design/methodology/approachSuitable similarity variables are introduced to reduce the governing nonlinear boundary layer partial differential equations to ordinary differential equations. A numerical solution of the resulting two-point boundary value problem is carried out by using the finite element method with the help of Gauss elimination technique.FindingsA comparison of obtained results is made with the previous work under the limiting cases. Behavior of flow and thermal fields against various governing parameters like mass transfer parameter, moving flat surface parameter, magnetic parameter, Prandtl number and Eckert number are analyzed and demonstrated graphically. Moreover, shear stress and heat flux at the moving surface for various values of the physical parameters are presented numerically in tabular form and discussed in detail.Originality/valueThe work is relatively original, as very little work has been reported on magnetohydrodynamic flow and heat transfer over a continuous moving flat surface. Viscous dissipation and Joule heating are neglected in most of the previous studies. The numerical method applied to solve governing equations is finite element method which is new and efficient.

Non-linear radiation effects on magnetic/non-magnetic nanoparticles with different base fluids over a flat plate

This work investigates the two-dimensional steady convective boundary layer flow and heat transfer of Newtonian/non-Newtonian base fluids with magnetic/non-magnetic nanoparticles over a flat plate which incorporates non-linear thermal radiation and slip effects. We considered magnetite and aluminium oxide as magnetic and non-magnetic nanoparticles suspending inside the two sorts of base fluids specifically Water and Sodium Alginate. For physical significance we analyzed the behavior on non-Newtonian profiles by employing Casson model individually. The particular intrigue lies in looking the impacts of non-linear thermal radiation on the behavior of the flow. The solution of wide class of boundary value problems are facilitated by the change of the partial differential equations administering the flow utilizing similarity transformations into ordinary differential equations. The ODE’s are numerically handled by applying fourth order Runge-Kutta integration scheme in association with shooting procedure. The novel results for the dimensionless velocity and temperature inside the boundary layer are exhibited graphically for various parameters that describe the flow. A graphical demonstration is given for the skin friction coefficient and the local Nusselt number.

On the fluid dynamics of unsteady separated stagnation-point flow of a power-law fluid on the surface of a moving flat plate

The transformation group theoretic approach is applied to perform an analysis of the unsteady separated stagnation-point (USSP) flow over a moving flat plate immersed in a non-Newtonian power-law fluid. This unsteady two-dimensional laminar boundary layer flow of a power-law fluid is governed by the classical Ostwald–de Waele power-law model. The system of governing partial differential equations reduces to an autonomous third order nonlinear ordinary differential equation via two-parameter group theoretic method. By using an efficient shooting method, we have established the existence of two different kinds of solutions, namely: (i) attached flow solution (AFS) and (ii) reverse flow solution (RFS) of this USSP flow problem under a definite negative value of the unsteadiness parameter β with a given value of the acceleration parameter a and for all values of the power-law index n considered in the present study. We have also found a unique AFS solution that is irrespective of n when β≥0. Dual solutions of the governing boundary layer equation are found for all kinds of fluids with the fixed values of a=1 and β=−1. The role of the rheological index n on the displacement thickness, skin-friction coefficient and velocity boundary layer are explored numerically. The results of this analysis reveal that the wall shear stress is always positive for AFS flow and negative for RFS flow for the above values of a and β with any given value of n. A novel result that emerges from this analysis is that the magnitude of the wall shear stress decreases and finally it approaches asymptotically to a constant value with the increase in the non-Newtonian behavior (n>1) of the fluids for both the cases of AFS and RFS.

Effects of Heat Source/Sink and Chemical Reaction on MHD Maxwell Nanofluid Flow Over a Convectively Heated Exponentially Stretching Sheet Using Homotopy Analysis Method

The aim of this paper is to study the effects of chemical reaction and heat source/sink on a steady MHD (magnetohydrodynamic) two-dimensional mixed convective boundary layer flow of a Maxwell nanofluid over a porous exponentially stretching sheet in the presence of suction/blowing. Convective boundary conditions of temperature and nanoparticle concentration are employed in the formulation. Similarity transformations are used to convert the governing partial differential equations into non-linear ordinary differential equations. The resulting non-linear system has been solved analytically using an efficient technique, namely: the homotopy analysis method (HAM). Expressions for velocity, temperature and nanoparticle concentration fields are developed in series form. Convergence of the constructed solution is verified. A comparison is made with the available results in the literature and our results are in very good agreement with the known results. The obtained results are presented through graphs for several sets of values of the parameters and salient features of the solutions are analyzed. Numerical values of the local skin-friction, Nusselt number and nanoparticle Sherwood number are computed and analyzed.

Effects of heat and mass transfer on unsteady boundary layer flow of a chemical reacting Casson fluid

Abstract In this study, an endeavor is to observe the unsteady two-dimensional boundary layer flow with heat and mass transfer behavior of Casson fluid past a stretching sheet in presence of wall mass transfer by ignoring the effects of viscous dissipation. Chemical reaction of linear order is also invoked here. Similarity transformation have been applied to reduce the governing equations of momentum, energy and mass into non-linear ordinary differential equations; then Homotopy analysis method (HAM) is applied to solve these equations. Numerical work is done carefully with a well-known software MATHEMATICA for the examination of non-dimensional velocity, temperature, and concentration profiles, and then results are presented graphically. The skin friction (viscous drag), local Nusselt number (rate of heat transfer) and Sherwood number (rate of mass transfer) are discussed and presented in tabular form for several factors which are monitoring the flow model.

THERMOPHORESIS AND HEAT GENERATION/ABSORPTION EFFECTS ON MAGNETOHYDRODYNAMIC FLOW OF JEFFREY FLUID OVER POROUS OSCILLATORY STRETCHING SURFACE WITH CONVECTIVE BOUNDARY CONDITIONS

In the current work, the effects of thermophoresis, heat generation/absorption, chemical reaction, and thermal radiation on two-dimensional boundary layer flow of a Jeffrey fluid over an oscillatory stretching surface embedded in a porous medium are investigated. Unlike typical studies, the idea of convective boundary conditions is used to investigate heat and mass transfer phenomenon. With the help of appropriate dimensionless variables, the number of independent variables is reduced in the governing equations, which are then solved analytically by using the homotopy analysis method. The effects of involved physical parameters such as Deborah number, ratio of relaxation to retardation time, Hartmann number, Prandtl number, heat absorption/generation parameter, thermal and concentration and Biot numbers, chemical reaction parameter, Schmidt number and thermophoresis parameter on dimensionless velocity, temperature, and concentration distributions are investigated and discussed quantitatively with the help of various graphs. It is observed that amplitude of oscillations in velocity increases with increasing Deborah number while it follows an opposite trend with increasing porosity parameter. Moreover, the heat transfer increases with increasing porosity parameter and ratio of relaxation to retardation time while it decreases with increasing Deborah number.

Experimental study of the influence of the Prandtl number on the convective heat transfer from a square cylinder

A detailed convective heat transfer study was conducted for a square cylinder subjected to uniform air and water streams at various inclination angles using a wind tunnel and a towing tank approach, respectively. Since the same test object was considered for both fluids, the comparison of the air and water heat transfer data enabled an assessment of the influence of the Prandtl number on the convective heat transfer. The reliability of the test apparatus was checked by means of mean heat transfer results obtained for air for which accurate literature data were available. The new results of the present experimental study demonstrated that the classical assumption m = 1/3 for the Prandtl number exponent in heat transfer correlations was only valid in cases where a two-dimensional boundary layer flow governed the convective heat transfer. The classical Prandtl number exponent assumption failed in the case of flow regimes dominated by separation and reattachment. The new experimental data indicated that for such three-dimensional flows virtually higher values should be used for the Prandtl number exponent.

Mixed convection boundary layer flow past a horizontal circular cylinder in viscoelastic nanofluid with constant wall temperature

The steady of two-dimensional convection boundary layer flow of viscoelastic nanofluid over a circular cylinder with constant wall temperature is investigated in this paper. Carboxymethyl cellulose solution-water (CMC-water) is chosen as the base fluid and copper as a nanoparticle with the Prandtl number Pr = 6.2. The governing boundary layer partial differential equations are transformed into dimensionless forms. Then the obtained equations are solved numerically by using the Keller-Box method. This paper focus on the effect of selected parameter on the flow and heat transfer characteristics and be presented in graphs. The results show that, the velocity profiles are increased while the temperature profiles are decreased by increasing the values of nanoparticles volume fraction and viscoelastic parameter, respectively. Also, the values of reduced skin friction are increased by increasing mixed convection parameter, but the values of heat transfer coefficient produce an opposite behavior with an increasing in mixed convection parameter.

Magnetohydrodynamic (MHD) Jeffrey fluid over a stretching vertical surface in a porous medium

This paper presents the study of steady two-dimensional mixed convection boundary layer flow and heat transfer of a Jeffrey fluid over a stretched sheet immersed in a porous medium in the presence of a transverse magnetic field. The governing partial differential equations are reduced to nonlinear ordinary differential equations with the aid of similarity transformation, which are then solved numerically using an implicit finite difference scheme. The effects of some of the embedded parameters, such as Deborah number β, magnetic parameter M, mixed convection parameter λ, porosity parameter γ and Prandtl number Pr, on the flow and heat transfer characteristics, are given in forms of tables and graphs.

Numerical study of magnetohydrodynamics (MHD) boundary layer slip flow of a Maxwell nanofluid over an exponentially stretching surface with convective boundary condition

This paper focuses on a theoretical analysis of a steady two-dimensional magnetohydrodynamic boundary layer flow of a Maxwell fluid over an exponentially stretching surface in the presence of velocity slip and convective boundary condition. This model is used for a nanofluid, which incorporates the effects of Brownian motion and thermophoresis. The resulting non-linear partial differential equations of the governing flow field are converted into a system of coupled non-linear ordinary differential equations by using suitable similarity transformations, and the resultant equations are then solved numerically by using Runge-Kutta fourth order method along with shooting technique. A parametric study is conducted to illustrate the behavior of the velocity, temperature and concentration. The influence of significant parameters on velocity, temperature, concentration, skin friction coefficient and Nusselt number has been studied and numerical results are presented graphically and in tabular form. The reported numerical results are compared with previously published works on various special cases and are found to be an in excellent agreement. It is found that momentum boundary layer thickness decreases with the increase of magnetic parameter. It can also be found that the thermal boundary layer thickness increases with Brownian motion and thermophoresis parameters.

Finite Element Analysis of Eyring–Powell Nano Fluid Over an Exponential Stretching Sheet

An analysis is made to study the two-dimensional boundary layer flow of Eyring–Powell nanofluid over an exponentially stretching sheet. The nanofluid model considered in the paper incorporates the effect of Brownian motion and thermophoresis. The condition for zero normal flux of nanoparticles at the stretching sheet is defined to impulse the particles away from surface. The governing partial differential equations are transformed into a set of coupled non-linear ordinary differential equations with appropriate similarity transformations. The transformed boundary layer equations are solved numerically, using variational finite element method. The flow features and the heat transfer characteristics and nanoparticle volume fraction are analyzed and discussed in detail for several sets of values of the governing flow parameters. Also the effects of the various flow parameters on the skin friction and the rate of heat transfer are obtained and illustrated graphically. A comparison of the present results with the existing published results shows excellent agreement, which confirms the validity of the present methodology.

Impact of Cattaneo-Christov Heat Flux Model on the Flow of Maxwell Ferromagnetic Liquid Along a Cold Flat Plate Embedded with Two Equal Magnetic Dipoles

The target of the current study is to inspect theoretically two-dimensional boundary layer flow of a Maxwell ferromagnetic fluid toward a flat plate. An external magnetic field due to two equal line dipole which are equidistant from the wall and perpendicular to the flow plane is applied. Cattaneo-Christov heat flux model is utilized in modified form of Fourier’s Law to disclose the heat transfer characteristic. Governing flow problem is normalized into ordinary differential equation by adopting similarity transform procedure. The solution of resulting non-linear ODE’s are solved by shooting technique based on Runge-Kutta algorithm with the help of MATLAB. Characteristic of sundry parameter like magneto-thermomechanical (ferrohydrodynamic) interaction parameter, dimensionless thermal relaxation, Prandtl number and Debora number on velocity and temperature profile are displayed via graphs and in tabular form. It is also pointed out that temperature profile suppresses by varying values of the thermal relaxation time and Prandtl number and increasing behaviour is seen against ferrohydrodynamic interaction. Present numerical results are compared with those published previously in the literature for the case of Newtonian fluid (α₁ → 0) and found an excellent agreement.

Mixed convection boundary layer flow of viscoelastic nanofluid past a horizontal circular cylinder: Case of constant heat flux

The steady of two-dimensional convection boundary layer flow of a viscoelastic nanofluid over a circular cylinder is investigated in this paper. Carboxymethyl cellulose solution (CMC) is chosen as the base fluid and copper as a nanoparticle with the Prandtl number Pr = 6.2. The governing boundary layer partial differential equations are transformed into dimensionless forms. Then they are solved numerically by using the Keller-Box method. This paper focus on the effects of selected parameter on the flow and heat transfer characteristics and be presented in graphs. The results show that, the velocity profiles and the temperature profiles are increased by increasing the values of nanoparticles volume fraction. While velocity profile decreases when viscoelastic parameter is increase. The reverse trend is observed for the temperature profiles. Also, the values of reduced skin friction are increased by increasing mixed convection parameter, but the values of heat transfer coefficient produce an opposite behaviour with an increasing in mixed convection parameter.