Power generalization of the linear constitutive equations of heat and mass transfer and the variants of writing the equations of momentum transfer, heat and diffusion arising from them

V. A. Pavlovsky

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 18 (2022) no. 4, pp. 527-534

Résumé

Currently, when solving problems of heat and mass transfer, linear constitutive equations are used — in hydrodynamics, the viscous stress tensor is proportional to the strain rate tensor (Newton's rheological ratio), in heat transfer, the heat flux density is linearly related to the temperature gradient (Fourier's heat conduction law), in mass transfer, the diffusion flux density proportional to the concentration gradient (Fick's law). When writing these linear governing equations, proportionality coefficients are used, which are called the viscosity coefficient, thermal conductivity coefficient and diffusion coefficient, respectively. Such constitutive equations are widely used to describe the processes of heat and mass transfer in a laminar flow regime. For turbulent flows, these equations are unsuitable, it is necessary to introduce into consideration the empirical turbulent coefficients of viscosity $\mu _t$, thermal conductivity $\lambda_t$ and diffusion $D_t$. However, to describe turbulent flows, it is possible to go in another way — to modify the linear constitutive relations by giving them a nonlinear power-law form. Two-parameter power-law generalizations of Newton's, Fourier's and Fick's formulas for shear stress, heat flux density and diffusion, which, depending on the value of the exponents, can be used to describe the processes of heat and mass transfer both in laminar and turbulent fluid flow. Also, this generalization can be used to describe the behavior of power-law fluids and flows of polymer solutions exhibiting the Toms effect.

Détail

Keywords: hydrodynamics, heat transfer, Newton's, Fick's formulas, power generalizations
Mots-clés : diffusion, Fourier's, turbulence.