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 Effects of Joule Heating and Viscous Dissipation on Magnetohydrodynamic Boundary Layer Flow of Jeffrey Nanofluid over a Vertically Stretching Cylinder

Coatings ◽  
2021 ◽  
Vol 11 (3) ◽  
pp. 353
Author(s):  
Haroon Ur Rasheed ◽  
Abdou AL-Zubaidi ◽  
Saeed Islam ◽  
Salman Saleem ◽  
Zeeshan Khan ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Mathematical Formulation ◽  
Skin Friction Coefficient ◽  
Deborah Number ◽  
Radiation Absorption ◽  
Partial Differential ◽  
System Parameters ◽  
Jeffrey Nanofluid

This article investigates unsteady magnetohydrodynamic (MHD) mixed convective and thermally radiative Jeffrey nanofluid flow in view of a vertical stretchable cylinder with radiation absorption and heat; the reservoir was addressed. The mathematical formulation of Jeffrey nanofluid is established based on the theory of boundary layer approximations pioneered by Prandtl. The governing model expressions in partial differential equations (PDEs) form was transformed into dimensionless form via similarity transformation technique. The set of nonlinear nondimensional partial differential equations are solved with the help of the homotopic analysis method. For the purpose of accuracy, the optimizing system parameters, convergence, and stability analysis of the analytical algorithm (CSA) were performed graphically. The velocity, temperature, and concentration flow are studied and shown graphically with the effect of system parameters such as Grashof number, Hartman number, Prandtl number, thermal radiation, Schmidt number, Eckert number, Deborah number, Brownian parameter, heat source parameter, thermophoresis parameter, and stretching parameter. Moreover, the consequence of system parameters on skin friction coefficient, Nusselt number, and Sherwood number is also examined graphically and discussed.

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.

Compressibility Effects on the Extended Crocco Relation and the Thermal Recovery Factor in Laminar Boundary Layer Flow

2004 ◽  
Vol 126 (1) ◽  
pp. 32-41 ◽  
Author(s):  
B. W. van Oudheusden
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Pressure Gradient ◽  
Boundary Layer Flow ◽  
Numerical Solutions ◽  
Thermal Recovery ◽  
Recovery Factor ◽  
Perturbation Approach ◽  
Layer Flow ◽  
Compressibility Effects

The relation between velocity and enthalpy in steady boundary layer flow is known as the Crocco relation. It describes that for an adiabatic wall the total enthalpy remains constant throughout the boundary layer, when the Prandtl number (Pr) is one, irrespective of pressure gradient and compressibility. A generalization of the Crocco relation for Pr near one is obtained from a perturbation approach. In the case of constant-property flow an analytic expression is found, representing a first-order extension of the standard Crocco relation and confirming the asymptotic validity of the square-root dependence of the recovery factor on Prandtl number. The particular subject of the present study is the effect of compressibility on the extended Crocco relation and, hence, on the thermal recovery in laminar flows. A perturbation analysis for constant Pr reveals two additional mechanisms of compressibility effects in the extended Crocco relation, which are related to the viscosity law and to the pressure gradient. Numerical solutions for (quasi-)self-similar as well as non-similar boundary layers are presented to evaluate these effects quantitatively.

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.

SEMI-ANALYTICAL SOLUTIONS FOR THE BRUSSELATOR REACTION–DIFFUSION MODEL

2017 ◽  
Vol 59 (2) ◽  
pp. 167-182 ◽  
Author(s):  
H. Y. ALFIFI
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Partial Differential ◽  
Analytical Results ◽  
Transient Solutions ◽  
The Impact

Semi-analytical solutions are derived for the Brusselator system in one- and two-dimensional domains. The Galerkin method is processed to approximate the governing partial differential equations via a system of ordinary differential equations. Both steady-state concentrations and transient solutions are obtained. Semi-analytical results for the stability of the model are presented for the identified critical parameter value at which a Hopf bifurcation occurs. The impact of the diffusion coefficients on the system is also considered. The results show that diffusion acts to stabilize the systems better than the equivalent nondiffusive systems with the increasing critical value of the Hopf bifurcation. Comparison between the semi-analytical and numerical solutions shows an excellent agreement with the steady-state transient solutions and the parameter values at which the Hopf bifurcations occur. Examples of stable and unstable limit cycles are given, and Hopf bifurcation points are shown to confirm the results previously calculated in the Hopf bifurcation map. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with the numerical solutions of partial differential equations.

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