scholarly journals A LARGE PARAMETER SPECTRAL PERTURBATION METHOD FOR NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS THAT MODELS BOUNDARY LAYER FLOW PROBLEMS

2017 ◽  
Vol 9 ◽  
Author(s):  
S.S. Motsa ◽  
T. M. Agbaje ◽  
S. Mondal ◽  
P. Sibanda
Keyword(s):  
Boundary Layer ◽  
Partial Differential Equations ◽  
Nonlinear Systems ◽  
Differential Equations ◽  
Perturbation Method ◽  
Boundary Layer Flow ◽  
Large Parameter ◽  
Flow Problems ◽  
Layer Flow ◽  
Spectral Perturbation

scholarly journals Solving the Nonlinear Boundary Layer Flow Equations with Pressure Gradient and Radiation

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 710
Author(s):  
Michalis A. Xenos ◽  
Eugenia N. Petropoulou ◽  
Anastasios Siokis ◽  
U. S. Mahabaleshwar
Keyword(s):  
Boundary Layer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Pressure Gradient ◽  
Boundary Layer Flow ◽  
Mathematical Formulation ◽  
Dimensionless Temperature ◽  
Partial Differential ◽  
Analytical Results ◽  
Layer Flow

The physical problem under consideration is the boundary layer problem of an incompressible, laminar flow, taking place over a flat plate in the presence of a pressure gradient and radiation. For the mathematical formulation of the problem, the partial differential equations of continuity, energy, and momentum are taken into consideration with the boundary layer simplifications. Using the dimensionless Falkner–Skan transformation, a nonlinear, nonhomogeneous, coupled system of partial differential equations (PDEs) is obtained, which is solved via the homotopy analysis method. The obtained analytical solution describes radiation and pressure gradient effects on the boundary layer flow. These analytical results reveal that the adverse or favorable pressure gradient influences the dimensionless velocity and the dimensionless temperature of the boundary layer. An adverse pressure gradient causes significant changes on the dimensionless wall shear parameter and the dimensionless wall heat-transfer parameter. Thermal radiation influences the thermal boundary layer. The analytical results are in very good agreement with the corresponding numerical ones obtained using a modification of the Keller’s-box method.

scholarly journals Higher Order Compact Finite Difference Schemes for Unsteady Boundary Layer Flow Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
P. G. Dlamini ◽  
S. S. Motsa ◽  
M. Khumalo
Keyword(s):  
Boundary Layer ◽  
Finite Difference ◽  
Differential Equations ◽  
Boundary Layer Flow ◽  
Nonlinear Partial Differential Equations ◽  
Finite Difference Schemes ◽  
Unsteady Boundary Layer ◽  
Compact Finite Difference ◽  
Flow Problems ◽  
Layer Flow

We investigate the applicability of the compact finite difference relaxation method (CFDRM) in solving unsteady boundary layer flow problems modelled by nonlinear partial differential equations. The CFDRM utilizes the Gauss-Seidel approach of decoupling algebraic equations to linearize the governing equations and solve the resulting system of ordinary differential equations using compact finite difference schemes. The CFDRM has only been used to solve ordinary differential equations modelling boundary layer problems. This work extends its applications to nonlinear partial differential equations modelling unsteady boundary layer flows. The CFDRM is validated on two examples and the results are compared to results of the Keller-box method.

Boundary layer flow of a nanofluid past a permeable exponentially shrinking surface with convective boundary condition using Buongiorno’s model

2015 ◽  
Vol 25 (2) ◽  
pp. 299-319 ◽  
Author(s):  
M.M. Rahman ◽  
Alin V. Rosca ◽  
I. Pop
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Boundary Condition ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Skin Friction ◽  
Boundary Layer Flow ◽  
Content Type ◽  
Layer Flow ◽  
Shrinking Surface

Purpose – The purpose of this paper is to numerically solve the problem of steady boundary layer flow of a nanofluid past a permeable exponentially shrinking surface with convective surface condition. The Buongiorno’s mathematical nanofluid model has been used. Design/methodology/approach – Using appropriate similarity transformations, the basic partial differential equations are transformed into ordinary differential equations. These equations have been solved numerically for different values of the governing parameters, stretching/shrinking parameter λ, suction parameter s, Prandtl number Pr, Lewis number Le, Biot number, the Brownian motion parameter Nb and the thermophoresis parameter Nt, using the bvp4c function from Matlab. The effects of these parameters on the reduced skin friction coefficient, heat transfer from the surface of the sheet, Sherwood number, dimensionless velocity, and temperature and nanoparticles volume fraction distributions are presented in tables and graphs, and are in details discussed. Findings – Numerical results are obtained for the reduced skin-friction, heat transfer and for the velocity and temperature profiles. The results indicate that dual solutions exist for the shrinking case (λ<0). A stability analysis has been performed to show that the upper branch solutions are stable and physically realizable, while the lower branch solutions are not stable and, therefore, not physically possible. In addition, it is shown that for a regular fluid (Nb=Nt=0) a very good agreement exists between the present numerical results and those reported in the open literature. Research limitations/implications – The problem is formulated for an incompressible nanofluid with no chemical reactions, dilute mixture, negligible viscous dissipation, negligible radiative heat transfer and a new boundary condition is imposed on nanoparticles and base fluid locally in thermal equilibrium. The analysis reveals that the boundary layer separates from the plate. Beyond the turning point it is not possible to get the solution based on the boundary-layer approximations. To obtain further solutions, the full basic partial differential equations have to be solved. Originality/value – The present results are original and new for the boundary-layer flow and heat transfer past a shrinking sheet in a nanofluid. Therefore, this study would be important for the researchers working in the relatively new area of nanofluids in order to become familiar with the flow behavior and properties of such nanofluids. The results show that in the presence of suction the dual solutions may exist for the flow of a nanofluid over an exponentially shrinking as well as stretching surface.

scholarly journals Inclusion of Viscous Dissipation on the Boundary Layer Flow of Cu-TiO2 Hybrid Nanofluid over Stretching/Shrinking Sheet

2021 ◽  
Vol 88 (2) ◽  
pp. 64-79
Author(s):  
Yap Bing Kho ◽  
Rahimah Jusoh ◽  
Mohd Zuki Salleh ◽  
Muhammad Khairul Anuar Mohamed ◽  
Zulkhibri Ismail ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Viscous Dissipation ◽  
Boundary Layer Flow ◽  
Skin Friction Coefficient ◽  
Shrinking Sheet ◽  
Hybrid Nanofluid ◽  
Suction Parameter ◽  
Similarity Transformations ◽  
Layer Flow

The effects of viscous dissipation on the boundary layer flow of hybrid nanofluids have been investigated. This study presents the mathematical modelling of steady two dimensional boundary layer flow of Cu-TiO2 hybrid nanofluid. In this research, the surface of the model is stretched and shrunk at the specific values of stretching/shrinking parameter. The governing partial differential equations of the hybrid nanofluid are reduced to the ordinary differential equations with the employment of the appropriate similarity transformations. Then, Matlab software is used to generate the numerical and graphical results by implementing the bvp4c function. Subsequently, dual solutions are acquired through the exact guessing values. It is observed that the second solution adhere to less stableness than first solution after performing the stability analysis test. The existence of viscous dissipation in this model is dramatically brought down the rate of heat transfer. Besides, the effects of the suction and nanoparticles concentration also have been highlighted. An increment in the suction parameter enhances the magnitude of the reduced skin friction coefficient while the augmentation of concentration of copper and titanium oxide nanoparticles show different modes.

Unsteady Flow and Heat Transfer of a MHD Micropolar Fluid Over a Porous Stretching Sheet in the Presence of Thermal Radiation

2013 ◽  
Vol 29 (3) ◽  
pp. 559-568 ◽  
Author(s):  
G. C. Shit ◽  
R. Haldar ◽  
A. Sinha
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Differential Equations ◽  
Thermal Radiation ◽  
Boundary Layer Flow ◽  
Micropolar Fluid ◽  
Stretching Sheet ◽  
Flow And Heat Transfer ◽  
Non Linear ◽  
Layer Flow

AbstractA non-linear analysis has been made to study the unsteady hydromagnetic boundary layer flow and heat transfer of a micropolar fluid over a stretching sheet embedded in a porous medium. The effects of thermal radiation in the boundary layer flow over a stretching sheet have also been investigated. The system of governing partial differential equations in the boundary layer have reduced to a system of non-linear ordinary differential equations using a suitable similarity transformation. The resulting non-linear coupled ordinary differential equations are solved numerically by using an implicit finite difference scheme. The numerical results concern with the axial velocity, micro-rotation component and temperature profiles as well as local skin-friction coefficient and the rate of heat transfer at the sheet. The study reveals that the unsteady parameter S has an increasing effect on the flow and heat transfer characteristics.

scholarly journals A New Spectral Local Linearization Method for Nonlinear Boundary Layer Flow Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
S. S. Motsa
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Nonlinear Boundary ◽  
Linearization Method ◽  
Variable Boundary ◽  
Flow Problems ◽  
Local Linearization ◽  
Highly Nonlinear ◽  
Layer Flow ◽  
Local Linearization Method

We propose a simple and efficient method for solving highly nonlinear systems of boundary layer flow problems with exponentially decaying profiles. The algorithm of the proposed method is based on an innovative idea of linearizing and decoupling the governing systems of equations and reducing them into a sequence of subsystems of differential equations which are solved using spectral collocation methods. The applicability of the proposed method, hereinafter referred to as the spectral local linearization method (SLLM), is tested on some well-known boundary layer flow equations. The numerical results presented in this investigation indicate that the proposed method, despite being easy to develop and numerically implement, is very robust in that it converges rapidly to yield accurate results and is more efficient in solving very large systems of nonlinear boundary value problems of the similarity variable boundary layer type. The accuracy and numerical stability of the SLLM can further be improved by using successive overrelaxation techniques.

Mixed Convection Boundary Layer Flow Near the Lower Stagnation Point of a Cylinder Embedded in a Porous Medium Using a Thermal Nonequilibrium Model

2016 ◽  
Vol 138 (8) ◽  
Author(s):  
Haliza Rosali ◽  
Anuar Ishak ◽  
Ioan Pop
Keyword(s):  
Boundary Layer ◽  
Porous Medium ◽  
Differential Equations ◽  
Mixed Convection ◽  
Stagnation Point ◽  
Boundary Layer Flow ◽  
Thermal Conductivity Ratio ◽  
Convection Boundary Layer ◽  
Convection Boundary ◽  
Layer Flow

The present paper analyzes the problem of two-dimensional mixed convection boundary layer flow near the lower stagnation point of a cylinder embedded in a porous medium. It is assumed that the Darcy's law holds and that the solid and fluid phases of the medium are not in thermal equilibrium. Using an appropriate similarity transformation, the governing system of partial differential equations are transformed into a system of ordinary differential equations, before being solved numerically by a finite-difference method. We investigate the dependence of the Nusselt number on the solid–fluid parameters, thermal conductivity ratio and the mixed convection parameter. The results indicate that dual solutions exist for buoyancy opposing flow, while for the assisting flow, the solution is unique.

Mixed convection boundary layer flow past a vertical flat plate embedded in a non-Darcy porous medium saturated by a nanofluid

2014 ◽  
Vol 24 (5) ◽  
pp. 970-987 ◽  
Author(s):  
Natalia C. Roşca ◽  
Alin V. Roşca ◽  
Teodor Groşan ◽  
Ioan Pop
Keyword(s):  
Boundary Layer ◽  
Porous Medium ◽  
Differential Equations ◽  
Mixed Convection ◽  
Flat Plate ◽  
Boundary Layer Flow ◽  
Convection Boundary Layer ◽  
Content Type ◽  
Layer Flow ◽  
Vertical Flat Plate

Purpose – The purpose of this paper is to numerically solve the problem of steady mixed convection boundary layer flow past a vertical flat plate embedded in a fluid-saturated porous medium filled by a nanofluid. The non-Darcy equation model along with the mathematical nanofluid model proposed by Tiwari and Das (2007) has been used. Design/methodology/approach – Using appropriate similarity transformations, the basic partial differential equations are transformed into ordinary differential equations. These equations have been solved numerically for different values of the nanoparticle volume fraction, the mixed convection and the non-Darcy parameters using the bvp4c function from Matlab. A stability analysis has been also performed. Findings – Numerical results are obtained for the reduced skin-friction, heat transfer and for the velocity and temperature profiles. The results indicate that dual solutions exist for the opposing flow case (λ<0). The stability analysis indicates that for the opposing flow case, the lower solution branch is unstable, while the upper solution branch is stable. In addition, it is shown that for a regular fluid (φ=0) a very good agreement exists between the present numerical results and those reported in the open literature. Research limitations/implications – The problem is formulated for three types of nanoparticles, namely, copper (Cu), alumina (Al2O3) and titania (TiO2). However, the paper present results here only for the Cu nanoparticles. The analysis reveals that the boundary layer separates from the plate. Beyond the turning point it is not possible to get the solution based on the boundary-layer approximations. To obtain further solutions, the full basic partial differential equations have to be solved. Practical implications – Nanofluids have many practical applications, for example, the production of nanostructured materials, engineering of complex fluids, for cleaning oil from surfaces due to their excellent wetting and spreading behavior, etc. Social implications – Nanofluids could be applied to almost any disease treatment techniques by reengineering the nanoparticle properties. Originality/value – The present results are original and new for the boundary-layer flow and heat transfer past a vertical flat plate embedded in a porous medium saturated by a nanofluid. Therefore, this study would be important for the researchers working in porous media in order to become familiar with the flow behavior and properties of such nanofluids.

scholarly journals Slip Effects on Boundary Layer Flow and Heat Transfer Along a Stretching Cylinder

2013 ◽  
Vol 18 (2) ◽  
pp. 447-459 ◽  
Author(s):  
S. Mukhopadhyay ◽  
R.S.R Gorla
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Differential Equations ◽  
Boundary Layer Flow ◽  
Numerical Solutions ◽  
Velocity Slip ◽  
Heat Equations ◽  
Stretching Cylinder ◽  
Linear Ordinary Differential Equations ◽  
Layer Flow

An axi-symmetric laminar boundary layer flow of a viscous incompressible fluid and heat transfer towards a stretching cylinder is presented. Velocity slip is considered instead of the no-slip condition at the boundary. Similarity transformations are used to convert the partial differential equations corresponding to the momentum and heat equations into non-linear ordinary differential equations. Numerical solutions of these equations are obtained by the shooting method. It is found that the velocity decreases with increasing the slip parameter. The skin friction as well as the heat transfer rate at the surface is larger for a cylinder compared to those for a flat plate.

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