Numerical solution of a moving-boundary problem with variable latent heat

The problem that arises during the movement of the shoreline in a sedimentary ocean basin is a moving-boundary problem with variable latent heat. A numerical method is presented for the solution of this problem. The differential equations governing the above process are converted into initial value problem of vector–matrix form. The time function is approximated by Chebyshev series and the operational matrix of integration is applied. The solution of the problem is then found in terms of Chebyshev polynomials of the second kind. The solution is utilized iteratively in the interface equation to determine time taken to attain a given shoreline position. The numerical results are obtained using Mathematica software and are compared graphically with the values obtained from a published analytical solution.     

A numerical solution for a stochastic beam equation exhibiting purely viscous behavior

The deflection of Euler–Bernoulli beams under stochastic dynamic loading, exhibiting purely viscous behavior, is characterized by partial differential equations of the fourth order. This paper proposes a computational method to determine the approximate solution to such equations. The functions are approximated using two-dimensional shifted Legendre polynomials. An operational matrix of integration and an operational matrix of stochastic integration are derived. The operational matrices assist in breaking down the problem under consideration into a set of algebraic equations that may be solved using any known numerical technique that leads to the solution of the stochastic beam equation. The well-posedness of the problem is studied. The proposed methodology is demonstrated to be practical for addressing the novel stochastic dynamic loading problem by confirming the outcome using a few numerical examples. Thus the effectiveness and applicability of the technique are ensured. The solution quality is explored through diagrams. The accuracy of the method is substantiated by comparing it with the Runge–Kutta method of order 1.5 (R–K 1.5). The absolute error caused by the proposed technique is comparably much less than R–K 1.5. A simulation analysis is carried out with MATLAB, and an algorithm is developed.     

Thermoelastic interaction in the semi-infinite solid medium due to three-phase-lag effect involving memory-dependent derivative

The present study deals with the thermoelastic interaction in a semi-infinite elastic solid with a heat source in the context of three-phase-lag model with memory-dependent derivative. The governing coupled equations, involving time delay and kernel functions are expressed in the vector matrix differential equation form in the Laplace transform domain. The analytical formulations of the problem have been solved by eigenvalue technique. The Honig–Hirdes numerical method is used for the inversion of Laplace transformation. Numerical results are obtained by choosing various types of time delay parameters and kernel functions and graphical representations have been performed accordingly. An extrapolative capability is established by considering the memory-dependent derivative into a three-phase-lag model.     

Effect of Prandtl number on leading edge accretion and ablation: A numerical study of unsteady boundary layer flow over a flat plate

An efficient numerical method, namely, the Runge‐Kutta fourth order integration scheme with shooting technique is employed to give a suitable solution for the unsteady magnetohydrodynamic boundary layer flow of viscous incompressible fluid with accretion or ablation effects over a flat plate under the influence of homogenous first order chemical reaction. When compared to the other numerical techniques such as perturbation methods, this approach provides the accurate numerical results valid uniformly for all nondimensional time. The unsteady behavior of chemically reacting magnetohydrodynamic boundary layer flow is investigated by analyzing the nature of buoyancy and magnetic parameters in the momentum equation. Also, results are extended to the energy and concentration equations by considering the viscous dissipation, Joule heating and chemical reaction effects. With the help of suitable similarity transformations, the highly nonlinear, coupled, time‐dependent partial differential equations are reduced to ordinary differential equations. Furthermore, the numerical solutions in terms of velocity, temperature and concentration profiles within the boundary layer are presented for the various values of control parameters. Also, the impact of physical parameters on the flow, heat and mass transfer characteristics are examined thoroughly. The present investigation reports that, the increasing magnetic parameter increases the temperature field and decreases the velocity field. Also, Eckert number enhance the thermal field whereas, the chemical reaction parameter decays the concentration field. Before concluding the considered problem, present results are validated with the previous results and are found to be in good agreement.     

Effect of Newtonian Heating and Thermal Radiation on Heat and Mass Transfer of Nanofluids over a Stretching Sheet in Porous Media

Laminar boundary layer slip flow from a stretching surface in a nanofluid‐saturated homogenous, isotropic porous medium is studied numerically. A Newtonian heating boundary condition in the presence of thermal radiation is incorporated and a Darcy model utilized for the porous medium. The model used for the nanofluids include the effects of Brownian motion and thermophoresis. A group theoretical analysis is conducted to generate similarity transformations. The governing transport equations are nondimensionalized and rendered into a set of coupled similarity ordinary differential equations using similarity transformations. The transformed equations are then solved using the Runge–Kutta–Fehlberg fourth‐fifth order numerical method with shooting technique. It is shown that the physical quantities of interest depend on a number of parameters. The results are presented in tabular and graphical forms. Comparison of the present numerical solutions with published work shows very good agreement. The study finds applications in high‐temperature nanotechnological materials processing.     

Barycentric rational interpolation method for numerical investigation of magnetohydrodynamics nanofluid flow and heat transfer in nonparallel plates with thermal radiation

In this study, the problem of heat transfer in the steady two‐dimensional flow of an incompressible viscous magnetohydrodynamics nanofluid from a sink or source between two shrinkable or stretchable plates under the effect of thermal radiation has been studied. The governing differential equations have been solved numerically using a collocation method based on the barycentric rational basis functions. This method employs the derivative operational matrix of the barycentric rational bases and the weights that were introduced by Floater and Hormann. The influence of some embedding parameters, such as the solid volume fraction , the Reynolds number , the Hartmann number , the Prandtl number , the radiation parameter , the stretching‐shrinking parameter, , and the angle of the channel on the temperature distribution and velocity profile has been illustrated by graphs and tables. Numerical results reveal the efficiency and high accuracy of the proposed scheme compared to the previously existing solutions. Furthermore, the implementation of the proposed method is fast and the run time is short.     

A Comparison of ODE Solution Methods for Spacecraft Thermal Problems

Abstract

In spacecraft thermal design and analysis practice, the lumped-parameter network formulation is used extensively to construct mathematical models. The models lake the form of a system of coupled, nonlinear, first-order ordinary differential equations (ODEs). The number of equations may vary from a few tens to a few thousands. It is necessary to solve these equations in an efficient and economical way. This article reviews various methods available for the numerical solution of ODEs. General-purpose codes available for this are discussed. Numerical experiments are conducted to investigate the efficiency of various methods. The results indicate that the Crank-Nicholson method with provision for automatic selection of step size to control the local truncation error is a very good choice for the solution of spacecraft thermal problems.     

A novel analytical solution of mixed convection about an inclined flat plate embedded in a porous medium using the DTM-Padé

In this study, the problem of mixed convection about an inclined flat plate embedded in a porous medium is performed. The similarity transformations are applied to reduce governing partial differential equations (PDEs) to a set of nonlinear coupled ordinary differential equations (ODEs) in dimensionless form. An efficient mathematical technique, called the differential transform method (DTM), is used to solve the nonlinear differential equations governing the problem in the form of series with easily computable terms. Then, Padé approximant is applied to the solutions to increase the convergence of given series. It has been attempted to show the reliability and performance of the DTM in comparison with the numerical method (fourth-order Runge–Kutta) in solving this problem. The obtained solutions, in comparison with the numerical solutions admit a remarkable accuracy.     

Analytical and numerical solutions of heat and mass transfer of boundary layer flow in the presence of a transverse magnetic field

The present investigation aims to study the effect of a transverse magnetic field with the presence of an adverse pressure gradient on the two‐dimensional laminar incompressible boundary layer flow over a flat plate. Using appropriated similarity transformations, the partial differential equations governing the studied problems are transformed into the ordinary nonlinear differential equations. Thereafter, these equations are solved numerically and analytically using the fourth‐order Runge‐Kutta method featuring shooting technique and the Adomian decomposition method, respectively. Obtained results reveal an excellent agreement between analytical and numerical data for temperature and concentration profiles.     

A Precise Integration Boundary-Element Method for Solving Transient Heat Conduction Problems with Variable Thermal Conductivity

In this article, a combined approach of the radial integration boundary element method (RIBEM) and the precise integration method is presented for solving transient heat conduction problems with variable thermal conductivity. First, the system of ordinary differential equations on the boundary integral equation can be obtained by the RIBEM. Then, the precise integration method is adopted to solve the system of ordinary differential equations. Finally, three numerical examples are presented to demonstrate the performance of the present method. The results show that the present approach can obtain satisfactory performance even for very large time-step size.     

Numerical modelling of second‐grade fluid flow past a stretching sheet

The present numerical study reports the chemically reacting boundary layer flow of a magnetohydrodynamic second‐grade fluid past a stretching sheet under the influence of internal heat generation or absorption with work done due to deformation in the presence of a porous medium. To distinguish the non‐Newtonian behaviour of the second‐grade fluid with those of Newtonian fluids, a very popularly known second‐grade fluid flow model is used. The fourth order momentum equation with four appropriate boundary conditions along with temperature and concentration equations governing the second‐grade fluid flow are coupled and highly nonlinear in nature. Well‐established similarity transformations are efficiently used to reduce the dimensional flow equations into a set of nondimensional ordinary differential equations with the necessary conditions. The standard bvp4c MATLAB solver is effectively used to solve the fluid flow equations to get the numerical solutions in terms of velocity, temperature, and concentration fields. Numerical results are obtained for a different set of physical parameters and their behaviour is described through graphs and tables. The viscoelastic parameter enhances the velocity field whereas the magnetic and porous parameters suppress the velocity field in the flow region. The temperature field is magnified for increasing values of the heat source/sink parameter. However, from the present numerical study, it is noticed that the flow of heat occurs from sheet to the surrounding ambient fluid. Before concluding the considered problem, our results are validated with previous results and are found to be in good agreement.     

Finite Element Analysis of Radiative Mixed Convection Magneto‐Micropolar Flow in a Darcian Porous Medium with Variable Viscosity and Convective Surface Condition

A mathematical study is presented for the collective influence of the buoyancy parameter, convective boundary parameter and temperature dependent viscosity on the steady mixed convective laminar boundary flow of a radiative magneto‐micropolar fluid adjacent to a vertical porous stretching sheet embedded in a Darcian porous medium. The fluid viscosity is assumed to vary as an inverse linear function of temperature. Using appropriate transformations, the governing equations of the problem under consideration are transformed into a system of dimensionless nonlinear ordinary differential equations, which are then solved with the well‐tested, efficient finite element method. The results obtained are depicted graphically to illustrate the effect of the various important controlling parameters on velocity, microrotation, and temperature functions. The skin friction coefficient, wall couple stress, and the rate of heat transfer have also been computed and presented in tabular form. Comparison of the present numerical results with earlier published data has been performed and the results are found to be in good agreement, thus validating the accuracy of the present numerical code. The study finds applications in conducting polymer flows in filtration systems, trickle bed magnetohydrodynamics in chemical engineering, electro‐conductive materials processing, and so on.     

Comparison study of differential transform method with finite difference method for magnetoconvection in a vertical channel

In this paper, coupled nonlinear equations governing the flow for magnetoconvection in a vertical channel for open and short circuits are solved. The calculations are carried out by using differential transformation method (DTM) which is a semi‐numerical–analytical solution technique. By using DTM, the nonlinear constrained governing equations are reduced to recurrence relations and related initial conditions are transformed into a set of algebraic equations. The principle of differential transformation is briefly introduced, and then applied for the aforementioned problems. The current results are then compared with those derived from the finite difference method (FDM) and perturbation method (PM) in order to verify the accuracy of the proposed method. The findings reveal that the DTM can achieve more suitable results in predicting the solution of such problems. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21035     

Steady incompressible magnetohydrodynamics Casson boundary layer flow past a permeable vertical and exponentially shrinking sheet: A stability analysis

Steady incompressible magnetohydrodynamic mixed convection boundary layer flow of a Casson fluid on an exponentially vertical shrinking sheet using the non‐Newtonian heating equation is investigated in this paper. There are three main objectives of this study, namely, to develop a new mathematical model, to obtain multiple solutions, and to perform stability analysis. The governing partial differential equations have been changed into nonlinear ordinary differential equations. The resultant equations of boundary value problems are then converted into the equivalent initial value problems using the shooting method before they can be solved using Runge‐Kutta of order four. The numerical results are obtained and found to be in good agreement with the published literature. The results also indicate that the velocity boundary layer becomes thinner as the magnetic, slip, and Casson parameters increase. Dual solutions for temperature and velocity distributions are obtained. Furthermore, the results suggest that the presence of the force of buoyancy (opposing flow case) would cause the occurrence of dual solutions. However, based on the stability analysis, only the first solution is stable.     

Analytical modeling of heat convection in magnetized micropolar fluid by using modified differential transform method

In this study, a new analytical method (DTM‐Padé) and the numerical method (by using a fourth‐order RungeBKutta and shooting method) were compared to solve convective heat transfer for a micropolar fluid in the presence of uniform magnetic field. It was shown that the differential transform method (DTM) solutions are only valid for small values of independent variables; therefore the DTM is not applicable for solving magnetohydrodynamic (MHD) boundary‐layer equations. The new method (DTM‐Padé) has removed this problem. Numerical comparisons between the DTM‐Padé and the numerical method revealed that the new method is a powerful method for solving MHD boundary‐layer equations. Finally, the analytical and numerical solutions of the problem for different values of the dimensionless parameters are shown simultaneously. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com ). DOI 10.1002/htj.20337     

Approximate Analytical Solution of Stagnation Point Flow and Heat Transfer over an Exponential Stretching Sheet with Convective Boundary Condition

This study is concerned with the stagnation point flow and heat transfer over an exponential stretching sheet via an approximate analytical method known as optimal homotopy asymptotic method (OHAM). The governing partial differential equations are converted into ordinary nonlinear differential equations using similarity transformations available in the literature. The heat transfer problem is modeled using two‐point convective boundary condition. These equations are then solved using the OHAM approach. The effects of controlling parameters on the dimensionless velocity, temperature, friction factor, and heat transfer rate are analyzed and discussed through graphs and tables. It is found that the OHAM results match well with numerical results obtained by Runge–Kutta Fehlberg fourth‐fifth order method for different assigned values of parameters. The rate of heat transfer increases with the stretching parameter. It is also found that the stretching parameter reduces the hydrodynamic boundary layer thickness whereas the Prandtl number reduces the thermal boundary layer thickness.     

MHD boundary‐layer flow over a stretching surface with internal heat generation or absorption

This article is concerned with the steady laminar magnetohydrodynamic boundary‐layer flow past a stretching surface with uniform free stream and internal heat generation or absorption in an electrically conducting fluid. A constant magnetic field is applied in the transverse direction. A uniform free stream of constant velocity and temperature is passed over the sheet. The effects of free convection and internal heat generation or absorption are also considered. The governing boundary layer and temperature equations for this problem are first transformed into a system of ordinary differential equations using similarity variables, and then solved by a new analytical method and numerical method, by using a fourth‐order Runge–Kutta and shooting method. Velocity and temperature profiles are shown graphically. It is shown that the differential transform method solutions are only valid for small values of independent variables but the results obtained by the DTM‐Padé are valid for the entire solution domain with high accuracy. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21054     

Influence of variable viscosity and thermal conductivity,hydrodynamic, and thermal slips on magnetohydrodynamic micropolar flow: A numerical study

Thermophysical and wall‐slip effects arise in many areas of nuclear technology. Motivated by such applications, in this article, the collective influence of variable‐viscosity, thermal conductivity, velocity and thermal slip effects on a steady two‐dimensional magnetohydrodynamic micropolar fluid over a stretching sheet is analyzed numerically. The governing nonlinear partial differential equations have been converted into a system of nonlinear ordinary differential equations using suitable coordinate transformations. The numerical solutions of the problem are expressed in the form of nondimensional velocity and temperature profiles and discussed from their graphical representations. The Nachtsheim‐Swigert shooting iteration technique together with the sixth‐order Runge‐Kutta integration scheme has been applied for the numerical solution. A comparison with the existing results has been done, and an excellent agreement is found. Further validation with the Adomian decomposition method is included for the general model. Interesting features in the heat and momentum characteristics are explored. It is found that a greater thermal slip and thermal conductivity elevate thermal boundary layer thickness. Increasing Prandtl number enhances the Nusselt number at the wall but reduces wall couple stress (microrotation gradient). Temperatures are enhanced with both the magnetic field and viscosity parameter. Increasing momentum (hydrodynamic) slip is found to accelerate the flow and elevate temperatures.     

H -adaptive RBF-FD method for the high-dimensional convection-diffusion equation

Radial basis function-generated finite difference method (RBF-FD) has been a popular method for simulating the derivatives of a function and has been successfully applied for the partial differential equations (PDEs). In this paper we introduce an effective h-adaptive RBF-FD method to the convection-diffusion equation in high-dimension space including two dimensions and three dimensions. The derivative of the solution is represented on overlapping a new influence domain through RBF-FD by using Thin Plane Spline Radial Basis Functions (TPS) augmented with additional polynomial functions. The number of the nodes added in the domain by the h-adaptive RBF-FD method is triggered by an error indicator, which very simply depends on the local residual norm. Several numerical examples are given to demonstrate the validity of h-adaptive RBF-FD method for the convection-diffusion equation in high-dimension space.     

Non‐Darcian Unsteady Flow of a Micropolar Fluid over a Porous Stretching Sheet with Thermal Radiation and Chemical Reaction

An analysis is presented to investigate the effects of a chemical reaction on an unsteady flow of a micropolar fluid over a stretching sheet embedded in a non‐Darcian porous medium. The governing partial differential equations are transformed into a system of ordinary differential equations by using similarity transformation. The resulting nonlinear coupled differential equations are solved numerically by using a fourth‐order Runge–Kutta scheme together with shooting method. The influence of pertinent parameters on velocity, angular velocity (microrotation), temperature, concentration, skin friction coefficient, Nusselt number, and Sherwood number has been studied and numerical results are presented graphically and in tabular form. Comparisons with previously published work are performed and the results are found to be in excellent agreement. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21090