Dimensional Boundary Layer Research Articles

Mixed convection boundary-layer stagnation point flow past a vertical stretching/shrinking surface in a nanofluid

A steady two dimensional mixed convection boundary layer flow near a stagnation point on a impermeable vertical stretching/shrinking surface in a nanofluid with controlled nanoparticles volume fraction is investigated numerically. The stretching/shrinking velocity and the ambient fluid velocity are assumed to vary linearly with the distance from the stagnation-point. The governing partial differential equations are first transformed into ordinary differential equations using a similarity transformation, before being solved numerically using a shooting method. The effects of several physical parameters such as mixed convection parameter, stretching/shrinking parameter, Brownian motion parameter, thermophoresis parameter and Lewis number on the skin friction coefficient, heat and mass transfer rates at the surface as well as the velocity, temperature and concentration profiles are analysed and discussed. It was found that increasing the particular values of the mixed convection parameter tends to increase the skin friction coefficient, heat and mass transfer rates at the surface. The results obtained show that the solution is unique for the stretching surface whereas dual (upper and lower branch) solutions exist for the shrinking surface. It should be mentioned that the solutions obtained are only locally similar.

Similarity Solutions for Unsteady Hydromagnetic Free Convection Boundary Layer Flow over Flat Plates with Thermophoresis

This paper is focused on the analysis of thermophoresis on an unsteady two dimensional free convective boundary layer flow along a permeable inclined flat plate in the presence of magnetic field and thermophoresis. Governing time dependent partial differential equations are non- dimensionalzed and transformed into a system of nonlinear ordinary differential equations by applying similarity transformations. These are solved numerically by using the shooting method along with Runge-Kutta sixth order integration scheme. Numerical results for the dimensionless velocity, temperature and concentration profiles have been obtained and displayed graphically. The skin-friction coefficient, wall heat transfer coefficient and wall deposition flux rate have also been obtained and are presented in tabular form. The obtained numerical results also show that increased unsteadiness parameter significantly controls the thermophoretic particle.

BOUNDARY LAYER FLOW AND HEAT TRANSFER WITH VARIABLE VISCOSITY IN THE PRESENCE OF MAGNETIC FIELD

A steady two dimensional boundary layer flow and heat transfer with variable viscosity electrically conducting fluid at T in the presence of magnetic fields and thermal radiation was considered. The governing equations which are partial differential equations were transformed into ordinary differential equations using similarity variables, and the resulting coupled ordinary differential equations were solved using collocation method in MAPLE 18. The velocity and temperature profiles were studied graphically for different physical parameters. The effects of the parameters on velocity and temperature profile were showed.

Global Existence of Weak Solution to the Compressible Prandtl Equations

We study the two dimensional boundary layer problem for compressible Navier–Stokes equations, provided that the density is favourable. We employ the viscous splitting method to establish global existence of weak solutions to the compressible Prandtl equations.

Correlation-Based Transition Transport Modeling for Simulating Crossflow Instabilities

A correlation-based transition model has been developed by Langtry and Menter for modern computational fluid dynamics codes, which is widely used for transition prediction in the field of turbomachinery and aircraft. Langtry’s transition model could simulate bypass, laminar separation and streamwsie Tollmien–Schlichting wave transition. Even so, this model has no ability to predict the transition due to crossflow instabilities in three dimensional boundary layer. In this paper, a new correlation-based transport equation for the transition due to crossflow instabilities has been established based on the experiment data and self-similar equations. The new transport equation is introduced to describe the crosswise displacement thickness Reynolds number growth in boundary layer. This new equation is added to Langtry’s intermittency factor equation to improve the ability of predicting transition. Finally, coupling of these transport equations and Shear Stress Transport (SST) turbulence model completes the new improved transition turbulence model. Comparisons of predictions using the new model with wind tunnel experiments of NLF (2)-0415 infinite swept wing and 6:1 inclined prolate spheroid validate the predictive qualities of the new correlation based transport equation.

Effects of temperature-dependent viscosity and thermal conductivity on mixed convection flow along a magnetized vertical surface

Purpose – The purpose of this paper is to report a numerical solution for the problem of steady, two dimensional boundary layer buoyant flow on a vertical magnetized surface, when both the viscosity and thermal conductivity are assumed to be temperature-dependent. In this case, the motion is governed by a coupled set of three nonlinear partial differential equations, which are solved numerically by using the finite difference method (FDM) by introducing the primitive variable formulation. Calculations of the coupled equations are performed to investigate the effects of the different governing parameters on the profiles of velocity, temperature and the transverse component of magnetic field. The effects of the thermal conductivity variation parameter, viscosity variation parameter, magnetic Prandtl number Pmr, magnetic force parameter S, mixed convection parameter Ri and the Prandtl number Pr on the flow structure and heat transfer characteristics are also examined. Design/methodology/approach – FDM. Findings – It is noted that when the Prandtl number Pr is sufficiently large, i.e. Pr=100, the buoyancy force that driven the fluid motion is decreased that decrease the momentum boundary layer and there is no change in thermal boundary layer is noticed. It is also noted that due to slow motion of the fluid the magnetic current generates which increase the magnetic boundary layer thickness at the surface. It is observed that the momentum boundary layer thickness is increased, thermal and magnetic field boundary layers are decreased with the increase of thermal conductivity variation parameter =100. The maximum boundary layer thickness is increased for =100 and there is no change seen in the case of thermal boundary layer thickness but magnetic field boundary layer is deceased. The momentum boundary layer thickness shoot quickly for =40 but is very smooth for =50.There is no change is seen for the case of thermal boundary layer and very clear decay for =40 is noted. Originality/value – This work is original research work.

Influence of convective conditions on three dimensional mixed convective hydromagnetic boundary layer flow of Casson nanofluid

The present work deals with the steady laminar three-dimensional mixed convective magnetohydrodynamic (MHD) boundary layer flow of Casson nanofluid over a bidirectional stretching surface. A uniform magnetic field is applied normal to the flow direction. Similarity variables are implemented to convert the non-linear partial differential equations into ordinary ones. Convective boundary conditions are utilized at surface of the sheet. A numerical technique of Runge–Kutta–Fehlberg (RFK45) is used to obtain the results of velocity, temperature and concentration fields. The physical dimensionless parameters are discussed through tables and graphs.

Hydromagnetic flow of heat absorbing and radiating fluid over exponentially stretching sheet with partial slip and viscous and Joule dissipation

Purpose An investigation of hydromagnetic two dimensional boundary layer flow with heat transfer of a viscous, incompressible, electrically conducting, heat absorbing and optically thick heat radiating fluid over a permeable exponentially stretching sheet considering the effects of viscous and Joule dissipations in the presence of velocity and thermal slip is carried out. Design/methodology/approach Using similarity transform, governing differential equations representing mathematical model of the problem are solved with the help of fourth-order Runge-Kutta method along with shooting technique. Numerical solutions of fluid velocity and fluid temperature are depicted graphically for various values of pertinent flow parameters whereas numerical values of wall velocity gradient and wall temperature gradient are displayed graphically for various values of pertinent flow parameters. Findings Numerical results obtained in this paper are compared with earlier published results and are found to be in excellent agreement. Magnetic field and suction tend to enhance the wall velocity gradient whereas dimensionless co-ordinate, injection and velocity slip factor have reverse effect on it. Suction and heat absorption tend to enhance wall temperature gradient whereas magnetic field, velocity slip factor, injection, thermal radiation, thermal slip factor and viscous dissipation have reverse effect on it. Originality/value The investigation of this problem may have bearing in several engineering processes such as extrusion of plastic sheet, annealing and tinning of copper wire, paper production, crystal growing and glass blowing, continuous casting of metals and spinning of fibers.

Three-dimensional MHD boundary layer flow due to an axisymmetric shrinking sheet with radiation, viscous dissipation and heat source/sink

An analysis is made to study a three dimensional MHD boundary layer flow and heat transfer due to a porous axisymmetric shrinking sheet. The governing partial differential equations of momentum and energy are transformed into self similar non-linear ordinary differential equations by using the suitable similarity transformations. These equations are, then solved by using the variational finite element method. The flow phenomena is characterised by the magnetic parameter <i>M</i>, suction parameter <i>S</i>, porosity parameter <i>Kp</i>, heat source/sink parameter <i>Q</i>, Prandtl number Pr, Eckert number Ec and radiation parameter <i>Rd</i>. The numerical results of the velocity and temperature profiles are obtained and displayed graphically.

3-D Maxwell fluid flow over an exponentially stretching surface using 3-stage Lobatto IIIA formula

The present study looks at the three dimensional boundary layer flow driven by an exponentially stretching surface. An upper-convected Maxwell (UCM) fluid is considered. Characteristics here are characterized by rheological constitutive equations of upper convected Maxwell (UCM) fluid. Involved mathematical modeling constitutes a nonlinear differential system. 3-stage Lobatto IIIA formula is employed to construct the numerical solutions whereas analytic solutions are computed using HAM. Both solutions are compared and found in good agreement. The velocity components are analyzed for the Deborah number and ratio parameters.

Finite element simulation of magnetohydrodynamic convective nanofluid slip flow in porous media with nonlinear radiation

A numerical investigation of two dimensional steady state laminar boundary layer flow of a viscous electrically-conducting nanofluid in the vicinity of a stretching/shrinking porous flat plate located in a Darcian porous medium is performed. The nonlinear Rosseland radiation effect is taken into account. Velocity slip and thermal slip at the boundary as well as the newly developed zero mass flux boundary conditions are also implemented to achieve physically applicable results. The governing transport equations are reduced to a system of nonlinear ordinary differential equations using appropriate similarity transformations and these are then solved numerically using a variational finite element method (FEM). The influence of the governing parameters (Darcy number, magnetic field, velocity and thermal slip, temperature ratio, transpiration, Brownian motion, thermophoresis, Lewis number and Reynolds number) on the dimensionless velocity, temperature, nanoparticle volume fraction as well as the skin friction, the heat transfer rates and the mass transfer rates are examined and illustrated in detail. The FEM code is validated with earlier studies for non-magnetic non-slip flow demonstrating close correlation. The present study is relevant to high-temperature nano-materials processing operations.

Heat transfer in MHD stagnation point flow of a ferrofluid over a stretchable rotating disk

In this article, heat transfer analysis in three dimensional boundary layer stagnation point flow of incompressible ferrofluid over a stretchable rotating circular disk in the presence of uniform external magnetic field has been investigated. For this purpose, three different types of ferroparticles namely, magnetite (Fe3O4), cobalt ferrite (CoFe2O4) and Mn-Zn ferrite (Mn-ZnFe2O4) are considered with water as a base fluid. The governing partial differential equations of the considered problem are transformed into non-dimensional ordinary differential equations using similarity transformation (Turkyilmazoglu, 2012). The solution of obtained equations is computed numerically using an efficient implicit finite difference scheme. The effects of pertinent parameters namely, volume fraction parameter of magnetic nanoparticle, rotating parameter, magnetic parameter, velocity ratio parameter and Prandtl number on both radial and azimuthal velocity profiles, temperature profile, skin friction coefficients and local Nusselt number are computed and shown through graphs and tables. For magnetite ferroparticle (Fe3O4) radial velocity decreases and azimuthal velocity and temperature increase with increasing value of volume fraction parameter ϕ. The values of skin friction coefficient in radial direction and local Nusselt number of magnetite ferroparticle (Fe3O4) become higher than that of other ferroparticles due to the maximum density and thermal conductivity.

Homotopy analysis solution of MHD slip flow past an exponentially stretching inclined sheet with Soret-Dufour effects

An analysis of steady MHD (magnetohydrodynamic) two dimensional free convective heat and mass transfer boundary layer flow of a viscous fluid towards an exponentially stretching inclined porous sheet in the presence of thermal radiation, Soret and Dufour effects with suction/blowing is presented. The Rossland approximation is used to describe the radiative heat transfer in the limit of optically thick fluids. Velocity slip, thermal slip and concentration slip are considered instead of no-slip condition at the boundary. 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 homotopy analysis method (HAM). Expressions for velocity, temperature and concentration fields are developed in series form. The obtained results are presented through graphs for several sets of values of the parameters and salient features of the solutions are analyzed. A comparison of our HAM results with the available numerical results in the literature (obtained by Runge–Kutta and shooting methods) shows that our results are accurate for wide range of values of the parameters.

Local heat transfer on a finite width surface with laminar boundary layer flow

The effect of a lateral discontinuity in the thermal boundary conditions in two dimensional laminar flow on a flat plate is investigated with numerical and analytical modeling. When the thermal and momentum boundary layers start at the same location, the resulting self-similar two dimensional boundary layer equations were solved numerically. For flow with an unheated starting length, three dimensional numerical simulations were required. For both the two and three dimensional thermal simulations, the Blasius solution for a two dimensional momentum boundary layer was assumed. It is found that all the Nusselt numbers collapse to a single curve when graphed as a function of a spanwise similarity variable. Simple correlations for the local Nusselt number on a rectangular flat plate are presented for a variety of boundary conditions.

Effects of Hydromagnetic and Thermophoresis of Unsteady Forced Convection Boundary Layer Flow over Flat Plates

In this paper, we analyze unsteady two dimensional hydromagnetic forced convection boundary layer flow of a viscous incompressible fluid along flat plates with thermophoresis. The potential flow velocity has been taken as a function of the distance x and time t. The governing partial differential equations are transformed to ordinary differential equation by applying local similarity transformation. The resulting similarity equations are then solved numerically for unsteady case, applying Nachtsheim-Swigert shooting iteration technique with six order Runge-Kutta method. The variations in fluid velocity, fluid temperature and species concentration are displayed graphically and discussed for different material parameters entering into the analysis. The effects of the pertinent parameters on the skin-friction coefficient, wall heat transfer coefficient and wall deposition flux rate are also displayed in tabulated form and discussed them from the physical point of view. An analysis of the obtained results shows that the flow field is influenced appreciably by the magnetic field parameter and the thermophoresis particle deposition.

Variable Viscosity Effects on Time Dependent Magnetic Nanofluid Flow past a Stretchable Rotating Plate

Abstract An attempt has been made to describe the effects of geothermal viscosity with viscous dissipation on the three dimensional time dependent boundary layer flow of magnetic nanofluids due to a stretchable rotating plate in the presence of a porous medium. The modelled governing time dependent equations are transformed a from boundary value problem to an initial value problem, and thereafter solved by a fourth order Runge-Kutta method in MATLAB with a shooting technique for the initial guess. The influences of mixed temperature, depth dependent viscosity, and the rotation strength parameter on the flow field and temperature field generated on the plate surface are investigated. The derived results show direct impact in the problems of heat transfer in high speed computer disks (Herrero et al. [1]) and turbine rotor systems (Owen and Rogers [2]).

Heat transfer analysis of MHD flow due to unsteady bi-directional stretching sheet through porous space

In this article unsteady three dimensional MHD boundary layer flow and heat transfer analysis with constant temperature (CT) and constant heat flux (CH) in a porous medium is considered. The boundary layer flow is governed by a bidirectional stretching sheet. Similarity transformations are used to transform the governing non-linear partial differential equations to ordinary differential equations. Analytical solutions are constructed using homotopy analysis method (HAM). Convergence analysis is also presented through tabular data. The quantities of interest are the velocity, temperature, skin friction coefficient and Nusselt number. The obtained results are validated by comparisons with previously published work in special cases. The results of this parametric study are shown graphically and the physical aspects of the problem are discussed.

Non-similar solutions of mixed convection flow from an exponentially stretching surface

Abstract In this paper we focus on to obtaining non-similar solutions for steady two dimensional double diffusive mixed convection boundary layer flows over an impermeable exponentially stretching sheet in an exponentially moving free stream under the influence of chemically reactive species. The nonlinear partial differential equations governing the flow, temperature and species concentration fields are presented in non-dimensional form with the help of suitable non-similar transformations. The resulting final non-dimensional set of coupled nonlinear partial differential equations is solved by using an implicit finite difference scheme in combination with the Newton’s linearization technique. The effects of various non-dimensional physical parameters on velocity, temperature and species concentration fields are discussed. The results reveal that the streamwise coordinate ξ remarkably influences the flow, thermal and solutal concentration fields which display the existence of non-similar solutions.

Unsteady Magnetohydrodynamic Casson Fluid Flow over a Vertical Cone and Flat Plate with Non-Uniform Heat Source/Sink

The present study focuses the effects of double dispersion, non-uniform heat source/sink and higher order chemical reaction on unsteady, free convective, MHD Casson fluid flow over a vertical cone and flat plate saturated with porous medium. The extensively validated and unconditionally stable numerical solutions are obtained for the governing equations of two dimensional boundary layer flow by using the finite difference scheme of Crank-Nicolson type. The behavior of velocity, temperature and concentration distributions for various controlling parameters of this problem are graphically illustrated and discussed in detail. The average skin friction, Nusselt number and Sherwood number for sundry parameters are presented in tables. Results indicate that an increase in Casson fluid parameter is found to decelerate fluid flow by increasing the plastic dynamic viscosity whereas it enhances the shear stress in the flow regime. The temperature-dependent heat source/sink plays a vital role on controlling the heat transfer however the surface-dependent heat source/sink also has notable influence on the heat transfer characteristics. It is to be noted that higher order chemical reaction has the tendency to dilute the influence of chemical reaction parameter on the species concentration.

Three dimensional boundary layer flow of a viscoelastic nanofluid with Soret and Dufour effects

The present research focuses on the three-dimensional flow of viscoelastic fluid in the presence of Soret and Dufour effects. Effects of thermophoresis and Brownian motion are taken into account. Appropriate similarity transformations lead to nonlinear ordinary differential equations. Solution expressions of velocity, temperature and nanoparticle concentration are computed via homotopy analysis method (HAM). Convergence of obtained solutions is analyzed graphically and numerically. Results are plotted and analyzed for the dimensionless velocities, temperature and nanoparticle concentration. Values of local Nusselt and Sherwood numbers are examined through tabular form. It is observed that Temperature field is enhanced for the larger Brownian motion parameter and an increase in Dufour number gives rise to the temperature and thermal boundary layer thickness.