# Macroscopic thermal properties of quasi-inear cellular medium on example of the liver tissue

### Abstract

There are two main topics of this research: (i) one topic considers overall properties of a nonlinear cellular composite, treated as a model of the liver tissue, and (ii) the other topic concerns the propagation of heat in the nonlinear medium described by the homogenised coefficient of thermal conductivity. For (i) we give a method and find the effective thermal conductivity for the model of the liver tissue, and for the point (ii) we present numerical and analytical treatment of the problem, and indicate the principal difference of heat propagation in linear and nonlinear media. In linear media, as it is well known, the range of the heat field is infinite for all times *t* > 0, and in nonlinear media it is finite. Pennes' equation, which should characterize the heat propagation in the living tissue, is in general a quasi-nonlinear partial differential equation, and consists of three terms, one of which describes Fourier's heat diffusion with conductivity being a function of temperature *T* . This term is just a point of our analysis. We show that a nonlinear character of the medium (heat conductivity dependent on the temperature) changes in qualitative manner the nature of heat transfer. It is proved that for the heat source concentrated initially (*t* = 0) at the space point, the range of heated region (for *t* > 0) is finite. The proof is analytical, and illustrated by a numerical experiment.

### Keywords

heat transport, asymptotic homogenisation, effective heat conductivity,**Computer Assisted Methods in Engineering and Science**, [S.l.], v. 22, n. 4, p. 329-346, jan. 2017. ISSN 2299-3649. Available at: <https://cames.ippt.pan.pl/index.php/cames/article/view/18>. Date accessed: 22 nov. 2019.