Two-dimensional Laminar Boundary Layer Flow Research Articles

Analysis of triple solutions in mixed convection flow and heat transfer characteristics of Ag-water based nanofluid over porous shrinking/stretching sheet

The two-dimensional laminar boundary layer flow and heat characteristics of silver (Ag) water based nanofluid with buoyancy effect over porous shrinking/stretching sheet has been considered. The shooting technique is used for numerical solutions to address the considered physical flow parameters. The numerical investigation shows that the triple solutions exists at the distinct ranges of parameters. Therefore, the analysis of stability is carried out to explore the reliability of solutions. The important results related to the stable solutions indicate that the skin friction declines for λ>0 and enhances for λ<0 when suction, and nanoparticles volume fractions are enhanced. An increase in rate of suction decreases velocity and the temperature profiles. Besides, an increase in silver nanoparticles volume fraction increases velocity and the temperature profiles, respectively. An augmentation in thermal slip factor reduced the temperature. The decrement in heat sink/source constraint resulted the lower temperature.

Numerical Study of Unsteady Boundary Layer Flow of a Biomagnetic Fluid over a Horizontal Stretching Sheet with Magnetic Dipole

This paper focuses on the theoretical and numerical investigation of the unsteady, viscous, incompressible, two-dimensional laminar boundary layer flow of a Newtonian biomagnetic fluid over a stretching sheet under the influence of an applied magnetic field in the presence of heat transfer. The magnetic field is induced by a magnetic dipole placed below the stretching sheet. The magnetic field intensity represents the magneto-thermo-mechanical coupling. This allows exclusion of the biofluid that is distant from the sheet at Curie temperature to avoid further magnetization. The unsteadiness of the flow is discernible in the fluid flow properties. The mathematical model of the problem conforms to the principles of Magnetohydrodynamics (MHD) and Ferrohydrodynamics (FHD). In this work, the study is performed on a specific biofluid, human blood. The modified Stokes principle is used to implement the model under the assumption that along with the three thermodynamic variables P, ρ, and T, the Biomagnetic Fluid Dynamics (BFD) fluid behavior can be characterized as a function of magnetization M. To describe the physical problem, a coupled non-linear system of ordinary differential equations subject to appropriate boundary conditions is derived from Navier-Stokes and thermal energy equations by performing non-dimensionalization of the considered variables. To solve these equations, the dsolve routine in the MAPLE software is used. Numerical results for flow profiles and the local skin friction coefficient (Cfx) and the local Nusselt number (Nux) are discussed for different values of unsteadiness parameter (A), biomagnetic interaction parameter (B) and a rational quantity (ϵ). The achieved results are compared with previously published work for steady state flow, and they seem to be in good agreement. It is found that MHD and FHD interaction parameters affect significantly on the velocity, temperature and pressure field. A successful completion will bring interesting results for better understanding of the biomagnetic fluid flow characteristics and can be beneficial to medical and bioengineering applications; particularly for estimating the characteristics of blood flow in stenosed arteries.

Optimal Homotopy Asymptotic Method for Investigation of Effects of Thermal Radiation, Internal Heat Generation, and Buoyancy on Velocity and Heat Transfer in the Blasius Flow

In this study, analytical examination of effects of internal heat generation, thermal radiation, and buoyancy force on flow and heat transfer in the Blasius flow over flat plate has been presented. The governing nonlinear partial differential equations of the problem are transformed into a set of coupled nonlinear third-order ordinary differential equations by the similarity variable method and have been systematically solved using the optimal homotopy asymptotic method. The main aim of the present study is to inspect the effects of various physical parameters in the flow model on velocity and heat transfer in steady two-dimensional laminar boundary layer flow with convective boundary conditions. The influences of the Grashof number, internal heat generation, the Biot number, radiation parameter, and the Prandtl number on the skin-friction coefficient, the fluid velocity profile, and temperature distribution have been determined and discussed in detail through several plots. The finding revealed that the fluid velocity and temperature delivery upsurge with snowballing in the values of the Biot number and internal heat generation parameters. The temperature profile of the fluid declines contrary to the value of the Grashof number and the Prandtl number but increases with thermal radiation. Moreover, it is found that the skin-friction coefficient and the rate of heat intensify with the Grashof number, internal heat generation, the Biot number, and thermal radiation parameter. The obtained result is likened with the previously published numerical results in a limited case of the problem and shows an excellent agreement.

Two-Dimensional Laminar Boundary Layer Flow and Heat Transfer over a Moving Wedge Immersed in a Nanofluid

Two-Dimensional Laminar Boundary Layer Flow and Heat Transfer over a Moving Wedge Immersed in a Nanofluid

MHD slip flow and convective heat transfer due to a moving plate with effects of variable viscosity and thermal conductivity

PurposeThe steady two-dimensional laminar boundary layer flow, heat and mass transfer over a flat plate with convective surface heat flux was considered. The governing nonlinear partial differential equations were transformed into a system of nonlinear ordinary differential equations and then solved numerically by Runge–Kutta method with the most efficient shooting technique. Then, the effect of variable viscosity and variable thermal conductivity on the fluid flow with thermal radiation effects and viscous dissipation was studied. Velocity, temperature and concentration profiles respectively were plotted for various values of pertinent parameters. It was found that the momentum slip acts as a boost for enhancement of the velocity profile in the boundary layer region, whereas temperature and concentration profiles decelerate with the momentum slip.Design/methodology/approachNumerical Solution is applied to find the solution of the boundary value problem.FindingsVelocity, heat transfer analysis is done with comparing earlier results for some standard cases.Originality/value100

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.

Unsteady magnetohydrodynamics flow about a stagnation point on a stretching plate embedded in porous medium

This paper presents an unsteady two-dimensional laminar boundary layer flow of an incompressible, viscous and electrically conducting fluid in a vicinity of a stagnation point towards a stretching plate embedded in porous medium with variable free stream velocity. The governing nonlinear partial differential equations of mass, momentum and energy are reduced to a system of nonlinear ordinary differential equations by using suitable similarity variables. The ordinary differential equations are tackled numerically by using Runge-Kutta Fehlberg fourth-fifth order method with shooting technique. The effect of the significant parameters on the fluid temperature and velocity are graphically presented and discussed. The numerically computed results are in good agreement with the available published results in the literature. It is found that the velocity increases with the increasing unsteadiness parameter where the velocity ratio parameter λ = 0.5 but it decreases for λ = 1.5.

Mixed convection flow of jeffrey nanofluid with thermal radiation and double stratification

This article addresses the two-dimensional laminar boundary layer flow of magnetohydrodynamic (MHD) Jeffrey nanofluid with mixed convection. Effects of thermal radiation, thermophoresis, Brownian motion and double stratifications are taken into account. Rosseland's approximation is utilized for the thermal radiation phenomenon. Convergent series solutions of velocity, temperature and nanoparticle concentration are developed. Graphs of dimensionless temperature and nanoparticle concentration are presented to investigate the influences of different emerging parameters. The values of skin-friction coefficient, local Nusselt and Sherwood numbers are computed and discussed for both Jeffrey and viscous fluids cases. We have observed that the temperature profile retarded for the larger values of Deborah number while an enhancement is noticed with the increasing values of ratio of relaxation to retardation times. Increasing values of thermal and nanoparticle concentration stratifications lead to a reduction in the temperature and nanoparticle concentration. The values of local Nusselt and Sherwood numbers are larger for the viscous fluid case when compared with Jeffrey fluid.

Analytical study of Cattaneo–Christov heat flux model for a boundary layer flow of Oldroyd-B fluid**Project supported by the Deanship of Scientific Research (DSR) King Abdulaziz University, Jeddah, Saudi Arabia (Grant No. 32-130-36-HiCi).

We investigate the Cattaneo–Christov heat flux model for a two-dimensional laminar boundary layer flow of an incompressible Oldroyd-B fluid over a linearly stretching sheet. Mathematical formulation of the boundary layer problems is given. The nonlinear partial differential equations are converted into the ordinary differential equations using similarity transformations. The dimensionless velocity and temperature profiles are obtained through optimal homotopy analysis method (OHAM). The influences of the physical parameters on the velocity and the temperature are pointed out. The results show that the temperature and the thermal boundary layer thickness are smaller in the Cattaneo–Christov heat flux model than those in the Fourier’s law of heat conduction.

Thermal Radiation Effects on MHD Boundary Layer Flow over an Exponentially Stretching Surface

The steady two-dimensional laminar boundary layer flow and heat transfer of a viscous incompressible electrically conducting fluid over an exponentially stretching surface in the presence of a uniform magnetic field with thermal radiation are investigated. The governing boundary layer equations are transformed to ordinary differential equations by taking suitable similarity transformation and solved numerically by shooting method. The effects of various parameters such as magnetic parameter, radiation parameter, Prandtl number and Eckert number on local skin-friction coefficient, local Nusselt number, velocity and temperature distributions are computed and represented graphically.

Unsteady hydromagnetic slip flow and heat transfer of nanofluid over a moving surface with prescribed heat and mass fluxes

An unsteady two-dimensional laminar boundary layer flow of viscous, incompressible nanofluid over a vertical stretching surface with hydromagnetic field has been investigated numerically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis in the presence of partial slip on the velocity field. Two types of general heating processes namely, the prescribed surface temperature and concentration and the prescribed wall heat and nanoparticle flux are imposed on the stretching surface. By applying similarity analysis, the governing partial differential equations are transformed into a set of time-dependent nonlinear coupled ordinary differential equations and they are solved by Runge–Kutta–Fehlberg method along with shooting technique. The effects of governing parameters on the dimensionless velocity, temperature and concentration distributions are discussed. Moreover, the numerical results obtained in this study is compared with the existing literature and found that they are in good agreement.

MHD Slip Flow past a Shrinking Sheet

An analysis is made for the steady two-dimensional laminar boundary layer flow of a viscous, incompressible, electrically conducting fluid near a stagnation point past a shrinking sheet with slip in the presence of a magnetic field. The governing boundary layer equations are transformed to ordinary differential equations by taking suitable similarity variables and solved numerically by Shooting method. The effects of the various parameters such as velocity ratio parameter, slip parameter, Prandtl number, Eckert number and magnetic parameter for velocity and temperature distributions have been discussed in detail through graphical representation.

MHD boundary layer flow over a non-linear stretching boundary with suction and injection

In this paper, we give an exact solution to the most celebrated magnetohydrodynamic Falkner–Skan equation. The equation governs the two-dimensional laminar boundary layer flow of a viscous, incompressible and electrically conducting fluid over a semi-infinite flat plate in the presence of magnetic field. Similarity transformations are used to convert the governing coupled non-linear partial differential equations into a highly non-linear ordinary differential equation with boundary conditions. An exact analytical solution is obtained for certain parameters which is then modified and generalized to give an exact solution to all other involved parameters. The results thus obtained are compared with that of direct numerical solutions, which agree well up to desired accuracy. The MHD Falkner–Skan equation exhibits the upper and lower branch solutions that reveal a very interesting velocity profiles for a set of parameters. Results are presented in the form of velocity profiles and skin friction for various values of physical parameters and are discussed in detail.

HOMOTOPY ANALYSIS OF FREE CONVECTION BOUNDARY LAYER FLOW WITH HEAT AND MASS TRANSFER

A steady two-dimensional laminar boundary layer flow of an incompressible viscous fluid over a semi-infinite surface is considered to investigate the accuracy of the homotopy analysis method. The governing coupled nonlinear system of differential equations is solved by means of the HAM approach. Explicit analytical series solutions are obtained and compared with numerical solutions. Good agreement is observed between the numerical results and HAM analytical solutions.

Falkner–Skan boundary layer flow of a power-law fluid past a stretching wedge

The steady two-dimensional laminar boundary layer flow of a power-law fluid past a permeable stretching wedge beneath a variable free stream is studied in this paper. Using appropriate similarity variables, the governing equations are reduced to a single third order highly nonlinear ordinary differential equation in the dimensionless stream function, which is solved numerically using the Runge–Kutta scheme coupled with a conventional shooting procedure. The flow is governed by the wedge velocity parameter λ, the transpiration parameter f 0, the fluid power-law index n, and the computed wall shear stress is f″(0). It is found that dual solutions exist for each value of f 0, m and n considered in λ − f″(0) parameter space. A stability analysis for this self-similar flow reveals that for each value of f 0, m and n, lower solution branches are unstable while upper solution branches are stable. Very good agreements are found between the results of the present paper and that of Weidman et al. [28] for n = 1 (Newtonian fluid) and m = 0 (Blasius problem [31]).

Non-uniform double slot suction (injection) into water boundary layer flows over a cylinder

An analysis is performed to study the effect of non-uniform double slot suction (injection) into a steady two-dimensional laminar boundary layer flow when fluid properties such as viscosity and Prandtl number are inverse linear functions of temperature. Non-similar solutions have been obtained from the starting point of the streamwise co-ordinate to the exact point of separation. By applying an implicit finite difference scheme in combination with the quasi-linearization technique and an appropriate selection of the finer step sizes along the streamwise direction, the difficulties arising at the starting point of the streamwise co-ordinate, at the edges of the slot and at the point of separation have been overcome. The results indicate that the separation can be delayed by non-uniform double slot suction and also by moving the slot downstream. However, the effect of non-uniform double slot injection is just the opposite.

Falkner-Skan equation for flow past a moving wedge with suction or injection

The characteristics of steady two-dimensional laminar boundary layer flow of a viscous and incompressible fluid past a moving wedge with suction or injection are theoretically investigated. The transformed boundary layer equations are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The effects of Falkner-Skan power-law parameter (m), suction/injection parameter (f0) and the ratio of free stream velocity to boundary velocity parameter (λ) are discussed in detail. The numerical results for velocity distribution and skin friction coefficient are given for several values of these parameters. Comparisons with the existing results obtained by other researchers under certain conditions are made. The critical values off 0,m and λ are obtained numerically and their significance on the skin friction and velocity profiles is discussed. The numerical evidence would seem to indicate the onset of reverse flow as it has been found by Riley and Weidman in 1989 for the Falkner-Skan equation for flow past an impermeable stretching boundary.

An analysis of second-order effects on laminar boundary-layer flow

Treated in the present paper are the second-order effects, i.e. the effect of displacement thickness, longitudinal curvature, external shear and slip, on the two-dimensional laminar boundary-layer flow of incompressible fluid. The analysis is developed in terms of the stream-function co-ordinates proposed for the analysis of the rotational flow by the senior author previously. An inverse problem is set in the co-ordinates plane, and, within the framework of the second-order approximation, the problem is reduced to solving a parabolic partial differential equation for the total head. An implicit finite-difference scheme is then devised to solve the equation, and the practical computations are carried out for the flow along a flat plate or the one around a parabolic body, by making use of an electronic computer.

Similarity conditions for heat transfer in incompressible two-dimensional laminar boundary layer flow

Similarity conditions are presented for the solution of some problems of heat transfer in incompressible two-dimensional boundary layer flow. The treatment holds for forced convection as well as for free convection. For free convection no a priori restriction is made with respect to geometry or temperature distribution of the solid surface. For forced convection the treatment is restricted to uniform bulk flow parallel to a flat surface of non-uniform temperature or heat flux. The results are summarized in some tables that facilitate comparison with older work.