A Finite-Element Singular-Perturbation Technique for Convection-Diffusion Problems—Part 1: The One-Dimensional Case

1981 ◽  
Vol 48 (2) ◽  
pp. 265-271 ◽  
Author(s):  
Rafael F. Diaz-Munio ◽  
L. Carter Wellford
Keyword(s):  
Finite Element ◽  
Singular Perturbation ◽  
Perturbation Technique ◽  
Diffusion Equations ◽  
Shape Functions ◽  
Singular Perturbation Technique ◽  
One Dimensional ◽  
Convection Diffusion ◽  
The One ◽  
Standard Galerkin Method

Approximation procedures for the solution of convection-diffusion equations, occurring in various physical problems, are considered. Several finite-element algorithms based on singular-perturbation methods are proposed for the solution of these equations. A method of variational matched asymptotic expansions is employed to develop shape functions which are particularly useful when convection effects dominate diffusion effects in these problems. When these shape functions are used, in conjunction with the standard Galerkin method, to solve convection-diffusion equations, increased solution accuracy is obtained. Numerical results for various one-dimensional problems are presented to establish the workability of the developed methods.

A Finite-Element Singular-Perturbation Technique for Convection-Diffusion Problems—Part 2: Two-Dimensional Problems

1981 ◽  
Vol 48 (2) ◽  
pp. 272-275 ◽  
Author(s):  
Rafael F. Diaz-Munio ◽  
L. Carter Wellford
Keyword(s):  
Finite Element ◽  
Variational Method ◽  
Perturbation Technique ◽  
Diffusion Equations ◽  
Galerkin Methods ◽  
Two Dimensional ◽  
Singular Perturbation Technique ◽  
Convection Diffusion ◽  
Solution Algorithms ◽  
Diffusion Problems

Approximation procedures for the solution of two-dimensional convection-diffusion problems are introduced. In these procedures finite-element techniques are utilized. The developed solution algorithms are based on a variational method of matched asymptotic expansions. When these techniques are used in conjunction with standard Galerkin methods, to solve convection-diffusion equations, highly accurate solutions are obtained. Numerical results for certain two-dimensional problems are presented to establish the accuracy of the proposed procedures.

scholarly journals Laminar condensation on a moving drop. Part 1. Singular perturbation technique

1984 ◽  
Vol 139 ◽  
pp. 105-130 ◽  
Author(s):  
J. N. Chung ◽  
P. S. Ayyaswamy ◽  
S. S. Sadhal
Keyword(s):  
Singular Perturbation ◽  
Perturbation Technique ◽  
Continuous Phase ◽  
Transport Processes ◽  
Forced Flow ◽  
Saturated Steam ◽  
Singular Perturbation Technique ◽  
Spherical Drop ◽  
Coupled Equations ◽  
Induced Velocity

In this paper, laminar condensation on a spherical drop in a forced flow is investigated. The drop experiences a strong, radial, condensation-induced velocity while undergoing slow translation. In view of the high condensation velocity, the flow field, although the drop experiences slow translation, is not in the Stokes-flow regime. The drop environment is assumed to consist of a mixture of saturated steam (condensable) and air (non-condensable). The study has been carried out in two different ways. In Part 1 the continuous phase is treated as quasi-steady and the governing equations for this phase are solved through a singular perturbation technique. The transient heat-up of the drop interior is solved by the series-truncation numerical method. The solution for the total problem is obtained by matching the results for the continuous and dispersed phases. In Part 2 both the phases are treated as fully transient and the entire set of coupled equations are solved by numerical means. Validity of the quasi-steady assumption of Part 1 is discussed. Effects due to the presence of the non-condensable component and of the drop surface temperature on transport processes are discussed in both parts. A significant contribution of the present study is the inclusion of the roles played by both the viscous and the inertial effects in the problem treatment.

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