Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global Well-Posedness and Numerical Approximation of a Coupled Darcy-Convection-Diffusion System with Exponential Nonlinearity

Sahil Kundu, Amiya K. Pani, Manoranjan Mishra

TL;DR

The paper analyzes density-driven flow in porous media by coupling Darcy's law with a concentration-transport equation featuring an exponential viscosity and linear adsorption. It establishes global existence and uniqueness of weak solutions in 2D and 3D via a truncation-approximation and Galerkin framework, proves a maximum principle ensuring nonnegativity, and proves exponential decay of the concentration in all $L^p$ norms. The authors validate their theory with pressure-formulation-based numerical simulations (COMSOL), examining how density-contrast, adsorption, and viscosity-contrast affect instability and mixing, revealing nonlinear saturation effects. The results provide a rigorous mathematical foundation for density fingering phenomena and offer insights into how adsorption and viscosity variations influence long-time behavior and mixing efficiency in porous media applications.

Abstract

This paper investigates density driven flow in porous media, focusing on the roles of viscosity contrast, density contrast, and linear adsorption. In this setup, the fluid on top is heavier and more viscous than the fluid below. Under the effect of gravity, this system becomes unstable, and finger-like structures appear. The phenomenon is described mathematically by coupling Darcy's law with a convection-diffusion reaction equation. The nonlinearity in this model arises mainly from the concentration dependence of viscosity and the convective transport term. The existence of a unique pair of weak solutions is shown in both two and three dimensions using the Galerkin approximation method and truncation technique. Moreover, an application of the maximum principle shows non-negativity of the concentration. Additionally, we analyze the long-time behavior of the solution and prove that the concentration converges exponentially to zero in the $L^p$-norm for all $1 \le p \le \infty$ as $t \to \infty.$ To complement the theoretical analysis, we perform numerical simulations based on a pressure formulation. By tracking total kinetic energy and mixing measures over time, we discuss the instability and the mixing efficiency, respectively. The present study reveals that although increasing the density contrast amplifies the total kinetic energy, the marginal impact diminishes with successive increments of density contrast. Similarly, while adsorption acts to suppress mixing, its efficiency in doing so tends to saturate with further increases. These non-linear sensitivities are predicted by our theoretical estimates and confirmed by the numerical simulations.

Global Well-Posedness and Numerical Approximation of a Coupled Darcy-Convection-Diffusion System with Exponential Nonlinearity

TL;DR

The paper analyzes density-driven flow in porous media by coupling Darcy's law with a concentration-transport equation featuring an exponential viscosity and linear adsorption. It establishes global existence and uniqueness of weak solutions in 2D and 3D via a truncation-approximation and Galerkin framework, proves a maximum principle ensuring nonnegativity, and proves exponential decay of the concentration in all norms. The authors validate their theory with pressure-formulation-based numerical simulations (COMSOL), examining how density-contrast, adsorption, and viscosity-contrast affect instability and mixing, revealing nonlinear saturation effects. The results provide a rigorous mathematical foundation for density fingering phenomena and offer insights into how adsorption and viscosity variations influence long-time behavior and mixing efficiency in porous media applications.

Abstract

This paper investigates density driven flow in porous media, focusing on the roles of viscosity contrast, density contrast, and linear adsorption. In this setup, the fluid on top is heavier and more viscous than the fluid below. Under the effect of gravity, this system becomes unstable, and finger-like structures appear. The phenomenon is described mathematically by coupling Darcy's law with a convection-diffusion reaction equation. The nonlinearity in this model arises mainly from the concentration dependence of viscosity and the convective transport term. The existence of a unique pair of weak solutions is shown in both two and three dimensions using the Galerkin approximation method and truncation technique. Moreover, an application of the maximum principle shows non-negativity of the concentration. Additionally, we analyze the long-time behavior of the solution and prove that the concentration converges exponentially to zero in the -norm for all as To complement the theoretical analysis, we perform numerical simulations based on a pressure formulation. By tracking total kinetic energy and mixing measures over time, we discuss the instability and the mixing efficiency, respectively. The present study reveals that although increasing the density contrast amplifies the total kinetic energy, the marginal impact diminishes with successive increments of density contrast. Similarly, while adsorption acts to suppress mixing, its efficiency in doing so tends to saturate with further increases. These non-linear sensitivities are predicted by our theoretical estimates and confirmed by the numerical simulations.
Paper Structure (16 sections, 8 theorems, 93 equations, 9 figures, 1 table)

This paper contains 16 sections, 8 theorems, 93 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical Model
  3. Preliminaries and Concept of Weak Solution
  4. Existence and uniqueness results for the weak solution
  5. Truncation and Bubnov-Galerkin approximation.
  6. A priori bounds
  7. Passage to limit as $n\longrightarrow \infty$.
  8. Uniqueness of Solution
  9. Asymptotic Behavior as $t \to \infty$
  10. Numerical Results and Discussion
  11. Total kinetic energy
  12. Degree of Mixing
  13. Mesh Convergence Study
  14. Instability and Mixing Behavior under Adsorption, Viscosity, and Density Variations
  15. Numerical Study of Exponential Decay in the Reactive Case
  16. ...and 1 more sections

Key Result

Theorem 3.1

If $\Omega \subset \mathbb{R}^d \,(d=2,3)$ is a domain, then for any $\phi \in H^{1}(\Omega)$ there exists a constant $M>0$ depending only on $\Omega$ such that the following inequality holds when $d=2$ and for $d=3$, the inequality

Figures (9)

  • Figure 1: Time evolution of concentration profiles at $t = 150, 300, 450, 600$ (left to right) for $\alpha = 1, 2, 3, 4$ (top to bottom).
  • Figure 2: Energy evolution over time for different values of $\alpha$, with $k = 1$ and $R = 1$.
  • Figure 3: Energy evolution over time for different values of $R$, with $\alpha = 1$ and $k = 1$.
  • Figure 4: Energy evolution over time for different values of $k$, with $\alpha = 1$ and $R = 1$.
  • Figure 5: Variation of the degree of mixing over time for different values of $k$, with $\alpha = 1$ and $R = 1$.
  • ...and 4 more figures

Theorems & Definitions (16)

  • Theorem 3.1: Gagliardo–Nirenberg, cf. migorski2019nonmonotone, garcke2019
  • Definition 3.1: Weak Solution of System \ref{['model']}
  • Theorem 4.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4: Maximum Principle for Concentration
  • ...and 6 more