Small Values Of Perturbation Parameter Research Articles

An optimal fitted numerical scheme for solving singularly perturbed parabolic problems with large negative shift and integral boundary condition

This paper deals with numerical solution of singularly perturbed parabolic partial differential equations with large negative shift on the spatial variable and integral boundary condition on the right side of the domain. For small values of perturbation parameter ɛ, solution of the problem exhibits an interior layer besides a parabolic boundary layers. The numerical methods combine the Crank–Nicolson difference scheme for the temporal direction on the uniform mesh and exponentially fitted operator finite difference method in spatial discretization. The integral boundary condition is treated using numerical integration techniques. The rate of convergence of the scheme is optimized by applying Richardson extrapolation technique. The stability and uniform convergence analysis has been carried out. To validate the applicability of the scheme, numerical examples are presented and the method is more accurate and shown to be second order uniformly convergent in both direction, and it also improves the results of the methods (Elango et al., 2021; Gobena and Duressa, 2021) existing in the literature.

Heat and mass transfer of two immiscible flows of Jeffrey fluid in a vertical channel

AbstractIn this article, we examined the effect of heat and mass transfer flow of two immiscible Jeffrey fluids in a vertical channel. The highly nonlinear coupled ordinary differential equations are evaluated using regular perturbation parameters, for small values of perturbation parameter. The effect of Jeffrey's parameter on the flow and the effects of various physical parameters entering into the problem on dimensionless velocity, temperature, and concentration distribution is illustrated graphically. We observe that the Jeffrey parameter, thermal, and mass Grashof number enhance the fluid flow, while the chemical reaction parameter suppresses the fluid flow, also it is established that the Nusselt number is boosted by enhancing the thermal and mass Grashof number. We observed that the results are in very good agreement with the results obtained for a viscous fluid.

An Efficient Superconvergent Spline Collocation Algorithm for Solving Fourth Order Singularly Perturbed Problems

In this work, improvised quintic B-spline collocation method is used to solve a class of fourth order singularly perturbed boundary value problems. The proposed method is an extension of classical collocation technique. Posteriori corrections are made to quintic B-spline interpolant and its derivatives, which leads to formation of improvised quintic B-spline interpolant. The improvisation is done by forcing the spline interpolant to satisfy interpolatory and some special end conditions. These corrections are made to achieve optimal order of convergence, which is not possible by standard formation of splines. Convergence analysis is performed using Green’s function approach. Four test problems are solved to show the effectiveness of the proposed method and its improvement over classical quintic and septic B-spline collocation techniques. The technique is found to be stable even for extremely small values of perturbation parameter.

A domain decomposition method for solving singularly perturbed parabolic reaction‐diffusion problems with time delay

We design and analyse a domain decomposition method for solving singularly perturbed parabolic reaction‐diffusion problems with time delay. Using the asymptotic behavior of the solution, we decompose the original domain of the problem into three overlapping subdomains, two of which are boundary layer subdomains and one is a regular subdomain. On each subdomain, we discretize the problem by the backward Euler scheme in the time direction and the central difference scheme in the spatial direction. The proposed method is shown to be uniformly convergent, having almost second order in space and first order in time. In addition, we prove that the proposed method converges much faster for small values of perturbation parameterAt the end, some numerical results are given in support of theoretical findings.

Effect of viscous dissipation on mixed convection flow in a vertical double passage channel using Robin boundary conditions

The laminar fully developed flow in a vertical double passage channel filled with clear fluid has been discussed using Robin boundary conditions. The thin perfectly conductive baffle is inserted in the channel. The governing equations of the fluid which are coupled and nonlinear are solved analytically by the perturbation method and semi analytically using differential transform method (DTM). The reference temperature of the external fluid is considered to be equal and different. The perturbation method which is valid for small values of perturbation parameter is used to find the combined effects of buoyancy forces and viscous dissipation. The limitation imposed on the perturbation parameter is relaxed by solving the basic equations using differential transform method. The influence of mixed convection parameter, Biot number for symmetric and asymmetric wall temperatures on the velocity, temperature and the Nusselt number is explored at different positions of the baffle. The solutions obtained by differential transform method are justified by comparing with the solutions obtained by perturbation method and the solutions agree very well for small values of the perturbation parameter.Keywords: Baffle, Differential Transform Method, Perturbation Method, Viscous dissipation, Robin Boundary Conditions, Double passage channel.

Fourth-order stable central difference method for self-adjoint singular perturbation problems

In this paper, fourth order stable central difference method is presented for solving self-adjoint singular perturbation problems for small values of perturbation parameter, e . First, the given differential equation was reduced to its conventional form and then it was transformed into linear system of algebraic equations in the form of a three-term recurrence relation, which can easily be solved by using Thomas Algorithm. To validate the applicability of the method, four model examples have been solved for different values of perturbation parameter and mesh sizes. The numerical results are tabulated and compared with some of the previous findings reported in the literature and it is observed that the present method is more efficient. Graphs are also depicted in support of the numerical results. Both theoretical error bounds and numerical rate of convergence have been established for the method. Key Words/Phrases: Singular perturbation; stable central difference method; self-adjoint.

Mixed convection flow of an electrically conducting fluid in a vertical channel using Robin boundary conditions with heat source/sink

A numerical study of steady, mixed convection of electrically conducting fluid in a vertical channel with heat source/sink is analyzed using Robin boundary conditions. The plate exchanges heat with an external fluid. Both conditions of equal and different reference temperature of the external fluid are considered. The governing equations are solved analytically using regular perturbation method and numerically by Runge–Kutta fourth order method with shooting technique. The graphs illustrating the effects of various parameters involved in the problem on the flow as well as average velocity and Nusselt number are presented and discussed. It is found that the effect of negative electric field load parameter is to aid the flow while the effect of positive electric field load parameter is to oppose the flow as compared to the case of short circuit. The analytical and numerical solutions agree very well for small values of perturbation parameter.

Free Convective Flow of Electrically Conducting and Viscous Immiscible Fluid Flow in a Vertical Channel in the Presence of First‐Order Chemical Reaction

In this paper, the effects of chemical reaction on free convective flow of electrically conducting and viscous incompressible immiscible fluids are analyzed. The coupled nonlinear equations governing the heat and mass transfer are solved analytically and numerically with appropriate boundary conditions for each fluid and the solutions have been matched at the interface. The analytical solutions are solved by using regular perturbation method valid for small values of perturbation parameter and numerically by using finite difference method. The numerical results for various values of thermal Grashof number, mass Grashof number, Hartman number, viscosity ratio, width ratio, conductivity ratio, and chemical reaction parameter have been presented graphically in the presence and in the absence of electric field load parameter. In addition, the closed form expression for volumetric flow rate, Nusselt number, species concentration, and total heat rate added to the flow is also analyzed. The solutions obtained by finite difference method and perturbation method agree very well to the order of 10−4 for small values of perturbation parameter.

Free and Forced Convective Flow in a Vertical Channel Filled with Composite Porous Medium Using Robin Boundary Conditions

Mixed convection flow and heat transfer in a vertical channel filled with composite porous medium using Robin boundary conditions is analyzed. The flow is modeled using the Darcy-Lapwood-Brinkman model. The viscous and Darcy dissipation terms are included in energy equation. The plate exchanges heat with an external fluid. Both the conditions of equal and different reference temperature of the external fluid are considered. The governing equations are coupled and non-linear because of inclusion of dissipation terms and buoyancy forces. The equations are solved using perturbation method valid for small values of perturbation parameter. However, the restriction on the perturbation parameter is relaxed by finding the solutions of governing equations by using Differential Transform Method. The effects of various parameters such as mixed convection parameter, porous parameter, viscosity ratio, width ratio, conductivity ratio and the Biot numbers on the flow are discussed. The percentage of error between perturbation method and differential transformation method increases as the perturbation parameter increases for both equal and unequal Biot numbers.

Mixed convection of a permeable fluid in a vertical channel with boundary conditions of a third kind

AbstractNumerical investigation of a steady mixed convective flow through a fluid‐saturated porous media in a vertical channel with boundary conditions of the third kind including the effects of viscous dissipation and Darcy dissipation has been studied. The plates exchange heat with an external fluid. Both conditions of equal and of different reference temperatures of the external fluid are considered. First, the simpler cases of either negligible Brinkman number or negligible Grashof number are addressed with the help of analytical solutions. The combined effects of buoyancy forces and viscous dissipation are analyzed by a perturbation series method valid for small values of perturbation parameter. To relax the conditions on the perturbation parameter, the governing equations are also evaluated numerically by a shooting technique that uses the classical explicit Runge–Kutta method of four slopes as an integration scheme and the Newton–Raphson method as a correction scheme. The problem is analyzed for different values of mixed convection parameters, porous parameter for equal and unequal Biot numbers, keeping the wall temperatures symmetric or asymmetric. The graphical results illustrating the effects of various parameters on the flow as well as average velocity and Nusselt numbers are presented. Further the analytical and numerical solutions agree very well for small values of the perturbation parameter. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21019

Effect of viscous dissipation on mixed convection flow in a vertical double passage channel using Robin boundary conditions

The laminar fully developed flow in a vertical double passage channel filled with clear fluid has been discussed using Robin boundary conditions. The thin perfectly conductive baffle is inserted in the channel. The governing equations of the fluid which are coupled and nonlinear are solved analytically by the perturbation method and semi analytically using differential transform method (DTM). The reference temperature of the external fluid is considered to be equal and different. The perturbation method which is valid for small values of perturbation parameter is used to find the combined effects of buoyancy forces and viscous dissipation. The limitation imposed on the perturbation parameter is relaxed by solving the basic equations using differential transform method. The influence of mixed convection parameter, Biot number for symmetric and asymmetric wall temperatures on the velocity, temperature and the Nusselt number is explored at different positions of the baffle. The solutions obtained by differential transform method are justified by comparing with the solutions obtained by perturbation method and the solutions agree very well for small values of the perturbation parameter.Keywords: Baffle, Differential Transform Method, Perturbation Method, Viscous dissipation, Robin Boundary Conditions, Double passage channel.