A comprehensive model of functionals for three-dimensional non-isothermal steady flow with molecular and convective diffusion and a chemical reaction of arbitrary order

Abstract

A mathematical model of diffusion–reaction, velocity and temperature functionals is developed using the calculus of variation and the concept of “local potential”. The diffusion–reaction functional, which involves both molecular and convective diffusion, describes the general case of three-dimensional steady-state diffusion of a single component with a chemical reaction of arbitrary order and complexity while the velocity and temperature functionals cover viscosity, pressure, convection and heat generation terms. The functionals have the feature of being minimum at the stationary state and can, therefore, be minimized and solved for velocity, temperature and concentration. An important feature of the present formulation for the diffusion–reaction problem is that the reaction term is linearized during the course of solution. Such linearization is inherent in the present formulation and makes the functional applicable to reactions of any type and complexity. Furthermore, the convective terms for both diffusion and heat transfer and the viscous and reaction generation terms in the energy equation are also linear and can be handled with relative ease in any numerical solution. The functionals are verified by yielding, upon minimization, the appropriate diffusion–reaction, momentum, continuity and energy equations as their Euler–Lagrange equations. Finally, the developed functionals are independent of the coordinate system and can be applied in any appropriate system of coordinates chosen for convenience.