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Hydrodynamic Poroelasticity with Thermal Effects

Michael Eden, Meraj Alam, Prakash Kumar, G P Raja Sekhar

TL;DR

The paper develops a linear hydrodynamic thermo-elastic model for a two-phase tissue mixture (fluid and solid) and proves the existence of a unique weak solution within an implicit evolution framework. It presents a thorough nondimensionalization and variational formulation, then establishes well-posedness by first solving the hydrodynamics subproblem and then addressing the coupled thermo-mechanical system via implicit parabolic theory. A one-dimensional reduced model on short time scales yields semi-analytical solutions, revealing how fluid flow, solid deformation, and heat transfer interact under varying permeability, heat exchange, and conductivity. The findings highlight the delicate balance between hydraulic coupling and thermal effects in tissue-like media and set the stage for incorporating tumor growth and nonlinear material behavior in future work.

Abstract

This study proposes and explores a linear hydrodynamic thermo-elasticity system within mixture models, comprising fluid and solid phases, with a focus on biological tissues, particularly tumor-related phenomena. Although tumor growth is not yet incorporated, this work aims to comprehend the interaction between thermal effects and hydrodynamics on short-time scales where the tumor size typically remains stable. We establish the existence of a unique weak solution within the framework of implicit evolution equations, overcoming challenges posed by intricate coupling mechanisms within the system. To further investigate the model, we then study the one-dimensional model and explore in detail the complex interplay between fluid flow, solid deformation, and heat transfer. This complex coupled system of equations is reduced for the short time scale to obtain the semi-analytical solution.

Hydrodynamic Poroelasticity with Thermal Effects

TL;DR

The paper develops a linear hydrodynamic thermo-elastic model for a two-phase tissue mixture (fluid and solid) and proves the existence of a unique weak solution within an implicit evolution framework. It presents a thorough nondimensionalization and variational formulation, then establishes well-posedness by first solving the hydrodynamics subproblem and then addressing the coupled thermo-mechanical system via implicit parabolic theory. A one-dimensional reduced model on short time scales yields semi-analytical solutions, revealing how fluid flow, solid deformation, and heat transfer interact under varying permeability, heat exchange, and conductivity. The findings highlight the delicate balance between hydraulic coupling and thermal effects in tissue-like media and set the stage for incorporating tumor growth and nonlinear material behavior in future work.

Abstract

This study proposes and explores a linear hydrodynamic thermo-elasticity system within mixture models, comprising fluid and solid phases, with a focus on biological tissues, particularly tumor-related phenomena. Although tumor growth is not yet incorporated, this work aims to comprehend the interaction between thermal effects and hydrodynamics on short-time scales where the tumor size typically remains stable. We establish the existence of a unique weak solution within the framework of implicit evolution equations, overcoming challenges posed by intricate coupling mechanisms within the system. To further investigate the model, we then study the one-dimensional model and explore in detail the complex interplay between fluid flow, solid deformation, and heat transfer. This complex coupled system of equations is reduced for the short time scale to obtain the semi-analytical solution.
Paper Structure (11 sections, 6 theorems, 77 equations, 8 figures, 3 tables)

This paper contains 11 sections, 6 theorems, 77 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mathematical Model and Setting
  3. Nondimensionalization
  4. Variational formulation
  5. Well-posedness
  6. Hydrodynamics
  7. Heat problem
  8. Reduced model
  9. Solution procedure
  10. Results and Discussion
  11. Conclusions

Key Result

Lemma 1

Let $f_f,f_s\in L^2(\Omega)$ and $f_N\in L^2(\Gamma_N)$. For every $\vartheta_s\in L^2(\Omega)$, there is a unique solution $(V_f,U_s)\in \mathbf{H}^1_{\Gamma}(\Omega)\times \mathbf{H}^1_{0}(\Omega)$ to the problem given by a:w4a:w5. The corresponding pressure function $P\in L^2(\Omega)$ is then giv where $C>0$ is independent of $\vartheta$.

Figures (8)

  • Figure 1: Schematic representation of the 1D model.
  • Figure 2: For high porosity $\phi_f=0.9$, permeability $Da=0.0583$, and $\kappa_f=5$, $\kappa_s=5$, $N=2$, $\hbox{Pe}_f=0.7596$.
  • Figure 3: For low porosity $\phi_f=0.5$ permeability, $Da=0.0002$, and $\kappa_f=5$, $\kappa_s=5$, $N=2$, $\hbox{Pe}_f=0.7596$.
  • Figure 4: For high porosity $\phi_f=0.9$, permeability $Da=0.0583$, and $\kappa_f=5$, $\kappa_s=5$, $N=0.2$, $\hbox{Pe}_f=0.7596$.
  • Figure 5: For high porosity $\phi_f=0.9$, permeability $Da=0.0583$, and $\kappa_f=5$, $\kappa_s=1$, $N=0.2$, $\hbox{Pe}_f=0.7596$.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Remark
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 3 more