Numerical Analysis of MHD Stagnation Point Flow Towards a Radially Stretching Convectively Heated Disk

The steady stagnation point flow and heat transferof an electrically conducting incompressible viscous fluid isextended to the case where the disk surface is convectivelyheated and radially stretching. The fluid is subjected to anexternal uniform magnetic field perpendicular to the planeof the disk. The governing momentum and energy balanceequations give rise to non-linear boundary value problem.Using a spectral relaxation method with a Chebyshev spectralcollocation method, the numerical solutions are obtained overthe entire range of the physical parameters. Emphasis hasbeen laid to study the effects of viscous dissipation and Jouleheating on the thermal boundary layer. Pertinent results on theeffects of various thermophysical parameters on the velocityand temperature fields as well as local skin friction and localNusselt number are discussed in detail and shown graphicallyand/or in tabular form.

Hydromagnetic Stagnation-Point Flow towards a Radially Stretching Convectively Heated Disk

Mathematical Problems in Engineering ◽

10.1155/2013/616947 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 24

Author(s):

S. Shateyi ◽

O. D. Makinde

Keyword(s):

Stagnation Point ◽

Numerical Solutions ◽

Relaxation Method ◽

Disk Surface ◽

Stagnation Point Flow ◽

Temperature Fields ◽

Physical Parameters ◽

Local Skin ◽

The steady stagnation-point flow and heat transfer of an electrically conducted incompressible viscous fluid are extended to the case where the disk surface is convectively heated and radially stretching. The fluid is subjected to an external uniform magnetic field perpendicular to the plane of the disk. The governing momentum and energy balance equations give rise to nonlinear boundary value problem. Using a spectral relaxation method with a Chebyshev spectral collocation method, the numerical solutions are obtained over the entire range of the physical parameters. Emphasis has been laid to study the effects of viscous dissipation and Joule heating on the thermal boundary layer. Pertinent results on the effects of various thermophysical parameters on the velocity and temperature fields as well as local skin friction and local Nusselt number are discussed in detail and shown graphically and/or in tabular form.

On Spectral Relaxation Approach to Radiating Powell-Eyring Fluid Flow over a Stretching Disk with Newtonian Heating

Defect and Diffusion Forum ◽

10.4028/www.scientific.net/ddf.387.461 ◽

2018 ◽

Vol 387 ◽

pp. 461-473 ◽

Cited By ~ 1

Author(s):

K. Gangadhar ◽

D. Vijaya Kumar ◽

S. Mohammed Ibrahim ◽

Oluwole Daniel Makinde

Keyword(s):

Boundary Layer ◽

Nonlinear Boundary ◽

Numerical Solutions ◽

Relaxation Method ◽

Newtonian Heating ◽

Nonlinear Boundary Value Problems ◽

Local Skin ◽

Boundary Layer Equations ◽

In this study we use a new spectral relaxation method to investigate an axisymmetric law laminar boundary layer flow of a viscous incompressible non-Newtonian Eyring-Powell fluid and heat transfer over a heated disk with thermal radiation and Newtonian heating. The transformed boundary layer equations are solved numerically using the spectral relaxation method that has been proposed for the solution of nonlinear boundary layer equations. Numerical solutions are obtained for the local wall temperature, the local skin friction coefficient, as well as the velocity and temperature profiles. We show that the proposed technique is an efficient numerical algorithm with assured convergence that serves as an alternative to common numerical methods for solving nonlinear boundary value problems. We show that the convergence rate of the spectral relaxation method is significantly improved by using method in conjunction with the successive over-relaxation method. It is observed that CPU time is reduced in SOR method compare with SRM method.

A Modified Spectral Relaxation Method for Some Emden-Fowler Equations

10.5772/intechopen.96611 ◽

2021 ◽

Author(s):

Gerald Tendayi Marewo

Keyword(s):

Initial Value Problems ◽

Numerical Solutions ◽

Model Problem ◽

Relaxation Method ◽

Initial Instant ◽

Computationally Efficient ◽

Problem Domain ◽

Method Of Solution ◽

In this chapter, we present a modified version of the spectral relaxation method for solving singular initial value problems for some Emden-Fowler equations. This study was motivated by the several applications that these equations have in Science. The first step of the method of solution makes use of linearisation to solve the model problem on a small subinterval of the problem domain. This subinterval contains a singularity at the initial instant. The first step is combined with using the spectral relaxation method to recursively solve the model problem on the rest of the problem domain. We make use of examples to demonstrate that the method is reliable, accurate and computationally efficient. The numerical solutions that are obtained in this chapter are in good agreement with other solutions in the literature.

Spectral Relaxation Method for Powell-Eyring Fluid Flow Past a Radially Stretching Heated Disk Surface in a Porous Medium

Defect and Diffusion Forum ◽

10.4028/www.scientific.net/ddf.387.575 ◽

2018 ◽

Vol 387 ◽

pp. 575-586 ◽

Cited By ~ 4

Author(s):

K. Gangadhar ◽

P.R. Sobhana Babu ◽

Oluwole Daniel Makinde

Keyword(s):

Porous Medium ◽

Nonlinear Boundary ◽

Nonlinear Differential Equations ◽

Slip Flow ◽

Relaxation Method ◽

Disk Surface ◽

Skin Friction Coefficient ◽

Nonlinear Boundary Value Problems ◽

In this study we use a spectral relaxation method to investigate heat transfer in axisymmetric slip flow of a MHD Powell-Eyring fluid over a radially stretching surface embedded in porous medium with viscous dissipation. The transformed governing system of nonlinear differential equations was solved numerically using the spectral relaxation method that has been proposed for the solution of nonlinear boundary layer equations. Results were obtained for the skin friction coefficient, the local Nusselt number as well as the velocity and temperature profiles for the same values of the governing physical and fluid parameters. Validation of the results was reached by the comparison with limiting cases from previous studies in the literature. We show that the proposed technique is an efficient numerical algorithm with assured convergence that serves as an alternative to common numerical methods for solving nonlinear boundary value problems. We show that the convergence rate of the spectral relaxation method is significant improved by using the method in conjunction with the successive over - relaxation method.

Numerical Analysis for the Synthesis of Biodiesel Using Spectral Relaxation Method

Mathematical Problems in Engineering ◽

10.1155/2014/601374 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 2

Author(s):

Z. G. Makukula ◽

S. S. Motsa ◽

S. Shateyi

Keyword(s):

Vegetable Oils ◽

Numerical Solutions ◽

Nonlinear Differential Equations ◽

Relaxation Method ◽

Numerical Results ◽

Reaction Rate Constants ◽

Animal Fats ◽

Alternative Diesel Fuel ◽

Biodiesel is an alternative diesel fuel chemically defined as the mono-alkyl esters of long chain fatty acids derived from vegetable oils or animal fat. It is becoming more attractive as an alternative fuel due to the depleting fossil fuel resources. A mathematical model for the synthesis of biodiesel from vegetable oils and animal fats is presented in this study. Numerical solutions of the model are found using a spectral relaxation method. The method, originally developed for boundary value problems, is an iterative scheme based on the Chebyshev spectral collocation method developed by decoupling systems of equations using Gauss-Seidel type of techniques. The effects of the reaction rate constants and initial concentrations of the reactants on the amount of the final product are being investigated. The accuracy of the numerical results is validated by comparison with known analytical results and numerical results obtained usingode45, an efficient explicit 4th and 5th order Runge-Kutta method used to integrate both linear and nonlinear differential equations.

Transient cavities near boundaries. Part 1. Rigid boundary

Journal of Fluid Mechanics ◽

10.1017/s0022112086000988 ◽

1986 ◽

Vol 170 ◽

pp. 479-497 ◽

Cited By ~ 310

Author(s):

J. R. Blake ◽

B. B. Taib ◽

G. Doherty

Keyword(s):

Stagnation Point ◽

Numerical Solutions ◽

Relative Magnitude ◽

Boundary Integral ◽

Stagnation Point Flow ◽

Boundary Integral Method ◽

Physical Parameters ◽

Jet Formation ◽

Rigid Boundary ◽

Parameter Ranges

The growth and collapse of transient vapour cavities near a rigid boundary in the presence of buoyancy forces and an incident stagnation-point flow are modelled via a boundary-integral method. Bubble shapes, particle pathlines and pressure contours are used to illustrate the results of the numerical solutions. Migration of the collapsing bubble, and subsequent jet formation, may be directed either towards or away from the rigid boundary, depending on the relative magnitude of the physical parameters. For appropriate parameter ranges in stagnation-point flow, unusual ‘hour-glass’ shaped bubbles are formed towards the end of the collapse of the bubble. It is postulated that the final collapsed state of the bubble may be two toroidal bubbles/ring vortices of opposite circulation. For buoyant vapour cavities the Kelvin impulse is used to obtain criteria which determine the direction of migration and subsequent jet formation in the collapsing bubble.

Slip Velocity and Temperature Jump on Dissipative CASSON Fluid with CATTANEO-CHRISTOV Heat Flux Model: Spectral Relaxation Method

International Journal of Engineering & Technology ◽

10.14419/ijet.v7i4.10.20905 ◽

2018 ◽

Vol 7 (4.10) ◽

pp. 240

Author(s):

K. Gangadhar ◽

K. V. Ramana ◽

T. Kannan ◽

B. Rushi Kumar

Keyword(s):

Heat Flux ◽

Differential Equations ◽

Numerical Solutions ◽

Fluid Velocity ◽

Thermal Relaxation ◽

Relaxation Method ◽

Casson Fluid ◽

Fluid Temperature ◽

A numerical analysis is performed for investigating the slip flow of a viscous dissipative Casson fluid towards a stretching sheet with Cattaneo-Christov heat flux and variable viscosity. The nonlinear partial differential equations are transformed with appropriate similarity variables into a system of nonlinear ordinary differential equations. Numerical solutions are carried out by using efficient Spectral relaxation method. Notable accuracy of the present results has been obtained with previous results in a limiting sense from the literature. It is found that thermal relaxation time has an inverse relationship with the fluid temperature. Interestingly, the fluid velocity is gradually decreasing with higher values of slip factor.

Numerical solutions for the problem of the boundary layer flow of a Powell–Eyring fluid over an exponential sheet using the spectral relaxation method

Indian Journal of Physics ◽

10.1007/s12648-019-01583-8 ◽

2019 ◽

Vol 94 (9) ◽

pp. 1369-1374

Author(s):

M M Khader

Keyword(s):

Boundary Layer ◽

Boundary Layer Flow ◽

Numerical Solutions ◽

Relaxation Method ◽

Layer Flow ◽

MHD stagnation point transient flow of a nanofluid past a stretching sheet: SRM approach

Latin American Applied Research - An international journal ◽

10.52292/j.laar.2019.77 ◽

2019 ◽

Vol 49 (3) ◽

pp. 205-211

Author(s):

Bhuvaneshvar Kumar ◽

G. S. Seth

Keyword(s):

Stagnation Point ◽

Stretching Sheet ◽

Transient Flow ◽

Fluid Velocity ◽

Velocity Slip ◽

Relaxation Method ◽

Opposite Case ◽

Boundary Layer Equations ◽

Stagnation point nanofluid flow over a stretching sheet embedded in a porous medium is investigated in the present model by taking Navier’s velocity sip into account. The spectral relaxation method (SRM) is utilized to solve boundary layer equations. The variation of nanofluid velocity, concentration and temperature corresponding to some dominant flow parameters is displayed via graphs. The findings reveal that when stretching sheet is moving faster than free stream then porous permeability, unsteadiness, velocity slip and magnetic parameters have tendency to reduce fluid velocity but in opposite case, they behave as an assisting parameters for flow field.