Radhey S. Gupta

Isotherm Migration Method in two dimensions

Abstract The Isotherm Migration Method is extended to two dimensions. The equations are formulated and a convenient finite-difference method of solution is described for a variety of initial and boundary conditions. Particular attention is devoted to Stefan problems in which phase changes occur on a moving interface. As an example the solidification of a square prism of fluid is solved in detail and the numerical results are compared with those obtained by earlier authors.

Variable time step methods for one-dimensional stefan problem with mixed boundary condition

Abstract The variable time step method introduced by Douglas and Gallie for solving a one-dimensional Stefan problem with constant heat flux at the fixed end is extended to cover a more general boundary condition. The numerical results are obtained for solidification of a liquid initially at its fusion temperature. A method due to Goodling and Khader is discussed in detail and some practical aspects of its implementation are investigated. The same problem is solved by the “modified variable time step” method earlier suggested by the present authors. The results from all the methods are almost identical. An approximate analytical solution is obtained by the heat-balance integral method.

A modified variable time step method for the one-dimensional stefan problem

Abstract Variable time step methods have been suggested by some authors for solving one-dimensional Stefan problems. These methods are found to suffer from certain drawbacks. They may not converge to a solution under more rigorous conditions or may not be applicable to problems with general boundary conditions. In the present paper a modified method is proposed. The results computed from various methods clearly show the superiority of the present method over the others. The numerical results are also compared with those obtained from the integral method and are found to be in very good agreement.

Moving grid method without interpolations

Abstract In their method of solving a one-dimensional moving boundary problem Crank and Gupta suggest a grid system which moves with the interface. The method requires some interpolations to be carried out which they perform by using a cubic spline or an ordinary polynomial. In the present paper these interpolations are avoided by employing a Taylors expansion in space and time dimensions. A practical diffusion problem is solved and the results are compared with those obtained from other methods.

Treatment of multi-dimensional moving boundary problems by coordinate transformation

Abstract A method based on coordinate transformation which transforms the time-varying domain into an invariant one, is given for solving multi-dimensional solidification/melting problems. The present method differs from its predecessors in that it transforms one space variable only while others remain unchanged. This results in tremendous amounts of saving in computations in comparison to methods proposed by earlier authors. Two sample problems are treated by the method in two dimensions one concerned with solidification of the square prism and the other with melting in a rectangular prism. The numerical results obtained by the present method are found to be in good agreement with those due to earlier authors. The method is further extended to three dimensions through a problem of cuboid solidification. Obtained in the present paper are perhaps the only available results for the complete solidification of the cuboid.

Complete numerical solution of the oxygen diffusion problem involving a moving boundary

Abstract A problem of oxygen diffusion in an absorbing tissue, first discussed by Crank and Gupta [1], is solved by a variable time step method. This problem has been attempted by many other authors using various techniques. Different authors have traced the moving boundary to different stages and the total absorption time has been calculated by some kind of interpolation/extrapolation at the end. In the present method, the time for complete absorption emerges as part of the computing procedure itself. In addition, the method described here requires, in general, a smaller number of time steps than do the other methods. Yet, the numerical results compare extremely well with those due to earlier authors.

Variable time-step method with coordinate transformation

Abstract The variable time-step methods for solving moving boundary problems are presented by transforming the variable space domain. This results in dissociating the mode of advancement of the boundary from the size of the space mesh. That is, a small movement of the moving boundary may be chosen for computing the time interval while the space domain is subdivided into larger space meshes. As a consequence, an enormous amount of saving in computer time may be achieved by using the proposed method. Two sample problems are selected for the illustration of the method.

Constrained integral method for solving moving boundary problems

Abstract In solving a moving boundary problem by the conventional integral method, the various parameters in the choice of a temperature/concentration profile are expressed in terms of the position of the moving boundary only, while in the present method they are expressed as functions of the position of the moving boundary plus an additional parameter at the fixed surface. This new parameter is taken to be the space derivative, when a Dirichlet condition is prescribed at the fixed end, or to be the function value when a Neumann-type condition is prescribed there. By doing so a control is provided at both ends, i.e. the moving boundary as well as the fixed end. Finally two simultaneous first-order differential equations are obtained which give the position of the moving boundary and the value of the unknown additional parameter in an implicit manner. Two sample problems with different types of boundary conditions at the fixed end are considered for testing the suggested method. The results seem to be in very good agreement with those due to earlier authors who have solved the problem using other techniques.

Approximate analytical methods for multi-dimensional Stefan problems

Abstract Two approximate analytical methods are presented for solving two-dimensional Stefan problems. One of them selects one-dimensional quadratic temperature profiles in one variable at the preselected values of the other variable. The second one is based on piecewise linear profiles. The first method is extended for solving a three-dimensional problem also. The numerical results for some selected problems are obtained by using the present methods as well as some methods due to other authors. The merits of the proposed methods are discussed.

An efficient approach to isotherm migration method in two dimensions

Analyse numerique par deux methodes lune implicite et lautre explicite du mouvement de linterface liquide-solide au cours de la solidification dun liquide dans un prisme rectangulaire de longueur definie