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Finite-Difference Operators for 2D problems

Sobczyk T.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2020

Vol. 68, nr 6

1535--1541

This paper presents the concept of using algorithms for reducing the dimensions of finite-difference equations of two-dimensional (2D) problems, for second-order partial differential equations. Solutions are predicted as two-variable functions over the rectangular domain, which are periodic with respect to each variable and which repeat outside the domain. Novel finite-difference operators, of both the first and second orders, are developed for such functions. These operators relate the value of derivatives at each point to the values of the function at all points distributed uniformly over the function domain. A specific feature of the novel operators follows from the arrangement of the function values as well as the values of derivatives, which are rectangular matrices instead of vectors. This significantly reduces the dimensions of the finite-difference operators to the numbers of points in each direction of the 2D area. The finite-difference equations are created exemplary elliptic equations. An original iterative algorithm is proposed for reducing the process of solving finite-difference equations to the multiplication of matrices.

Effects of heat and mass transfer on a moving vertical cylinder with chemically reactive species diffusion

Ganesan P., Loganathan P.

International Journal of Applied Mechanics and Engineering

2003

Vol. 8, no 2

195-205

A numerical analysis is carried out to examine the combined heat and mass transfer along with chemical reaction characteristics in a free convective flow over a moving semi-infinite vertical cylinder. The governing equations are solved using an implicit finite difference scheme of Crank-Nicolson type. The effects of the chemical reaction in transient velocity, temperature, concentration, local and average skin-friction, Nusselt number and Sherwood number profiles are illustrated and discussed for various physical parametric values. It is observed that with an increase in the chemical reaction parameter K or the Schmidt number Sc, velocity decreases, but velocity increases with an increasing thermal Grashof number Gr or mass Grashof number Gc. The surface mass transfer strongly depends on the Schmidt number and the chemical reaction and it decreases with their increasing values.