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Phase segregation of liquid-vapor systems with a gravitational field

A. Lamura

TL;DR

This paper investigates how gravity influences the phase-separation dynamics of a symmetric liquid–vapor system described by a van der Waals equation of state, using a two-dimensional isothermal lattice Boltzmann model. Gravity enhances coarsening along the vertical direction, increasing the effective growth exponent from the gravity-free inertial value of $\alpha=\tfrac{2}{3}$ up to about $\alpha\approx \tfrac{5}{3}$ as the external force strengthens, and drives the system toward a final stratified state with liquid at the bottom and vapor at the top. The authors confirm persistent sharp interfaces via a Porod law along the gravity direction, and they reveal that the wall-adjacent layer thickness grows from $L(t)\sim t^{2/3}$ to $L(t)\sim g\,t^{5/3}$ before saturating, with the steady density profile in agreement with a one-dimensional theory. Overall, the work provides a quantitative framework for gravity-driven coarsening in nonideal fluids and establishes a scalable LB approach for exploring hydro-dynamically governed phase separation under external fields.

Abstract

Phase separation in the presence of external forces has attracted considerable attention since the initial works for solid mixtures. Despite this, only very few studies are available which address the segregation process of liquid-vapor systems under gravity. We present here an extensive study which takes into account both hydrodynamic and gravitational effects on the coarsening dynamics. An isothermal formulation of a lattice Boltzmann model for a liquid-vapor system with the van der Waals equation of state is adopted. In the absence of gravity, the growth of domains follows a power law with the exponent $2/3$ of the inertial regime. The external force deeply affects the observed morphology accelerating the coarsening of domains and favoring the liquid accumulation at the bottom of the system. Along the force direction, the growth exponent is found to increase with the gravity strength still preserving sharp interfaces since the Porod's law is found to be verified. The time evolution of the average thickness $L$ of the layers of accumulated material at confining walls shows a transition from an initial regime where $L \simeq t^{2/3}$ ($t$: time) to a late-time regime $L \simeq g t^{5/3}$ with $g$ the gravitational acceleration. The final steady state, made of two overlapped layers of liquid and vapor, shows a density profile in agreement with theoretical predictions.

Phase segregation of liquid-vapor systems with a gravitational field

TL;DR

This paper investigates how gravity influences the phase-separation dynamics of a symmetric liquid–vapor system described by a van der Waals equation of state, using a two-dimensional isothermal lattice Boltzmann model. Gravity enhances coarsening along the vertical direction, increasing the effective growth exponent from the gravity-free inertial value of up to about as the external force strengthens, and drives the system toward a final stratified state with liquid at the bottom and vapor at the top. The authors confirm persistent sharp interfaces via a Porod law along the gravity direction, and they reveal that the wall-adjacent layer thickness grows from to before saturating, with the steady density profile in agreement with a one-dimensional theory. Overall, the work provides a quantitative framework for gravity-driven coarsening in nonideal fluids and establishes a scalable LB approach for exploring hydro-dynamically governed phase separation under external fields.

Abstract

Phase separation in the presence of external forces has attracted considerable attention since the initial works for solid mixtures. Despite this, only very few studies are available which address the segregation process of liquid-vapor systems under gravity. We present here an extensive study which takes into account both hydrodynamic and gravitational effects on the coarsening dynamics. An isothermal formulation of a lattice Boltzmann model for a liquid-vapor system with the van der Waals equation of state is adopted. In the absence of gravity, the growth of domains follows a power law with the exponent of the inertial regime. The external force deeply affects the observed morphology accelerating the coarsening of domains and favoring the liquid accumulation at the bottom of the system. Along the force direction, the growth exponent is found to increase with the gravity strength still preserving sharp interfaces since the Porod's law is found to be verified. The time evolution of the average thickness of the layers of accumulated material at confining walls shows a transition from an initial regime where (: time) to a late-time regime with the gravitational acceleration. The final steady state, made of two overlapped layers of liquid and vapor, shows a density profile in agreement with theoretical predictions.

Paper Structure

This paper contains 6 sections, 20 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Numerical model
  3. Results
  4. Phase separation dynamics
  5. Final state
  6. Conclusions

Figures (7)

  • Figure 1: Contour plots of the density $n_r$ in the cases with $E_r=0$ (top), $0.41 \times 10^{-2}$ (middle), $1.84 \times 10^{-2}$ (bottom) on a portion of size $(511 \Delta s \times 2047 \Delta s)$ of the whole system at times $t/(5.2 \times 10^3 \Delta t)= 2$ (a), $3$ (b), $4$ (c), $5$ (d), $6$ (e), $7$ (f).
  • Figure 2: Time evolution of the size of domains $R_y(t)$ along the vertical direction in the cases with $E_r= 0$ (black line), $0.12 \times 10^{-2}$ (cyan line), $0.20 \times 10^{-2}$ (yellow line), $0.41 \times 10^{-2}$ (red line), $0.82 \times 10^{-2}$ (purple line), $1.23 \times 10^{-2}$ (green line), $1.84 \times 10^{-2}$ (blue line), and of the spherically-averaged domain size $R_{sf}(t)$ for the fully periodic system without gravity (black dots). The black and blue dashed lines are guides to the eye and have slopes $0.66$ and $1.66$, respectively. Inset: Time evolution of the size of domains $R_x(t)$ along the horizontal direction in the same cases of the main panel. The black dashed line is a guide to the eye and has slope $0.66$.
  • Figure 3: Scaled structure factor $G(k_y,t)/R_y(t)$, averaged along the $k_x$-direction at time $t/(5.2 \times 10^3 \Delta t)= 3$, as a function of $k_y R_y(t)$ in the cases with $E_r= 0$ (black symbols), $0.41 \times 10^{-2}$ (red symbols), $1.23 \times 10^{-2}$ (green symbols), $1.84 \times 10^{-2}$ (blue symbols). The dashed line has slope $-2$.
  • Figure 4: Scaled thickness of layers at walls $L(t) E_r^{2/3}/\Delta s$ as a function of the scaled time $t E_r /\Delta t$ in the cases with $E_r= 0.12 \times 10^{-2}$ (cyan symbols), $0.20 \times 10^{-2}$ (yellow symbols), $0.41 \times 10^{-2}$ (red symbols), $0.82 \times 10^{-2}$ (purple symbols), $1.23 \times 10^{-2}$ (green symbols), $1.84 \times 10^{-2}$ (blue symbols). The black straight lines have slopes $0.66$ and $1.66$. Upper inset: Thickness of layers at walls $L(t)/\Delta s$ as a function of time $t/\Delta t$ for the same cases of the main panel. Lower inset: Scaled size of domains along the vertical direction $R_y(t) E_r^{2/3}/\Delta s$ as a function of the scaled time $t E_r /\Delta t$ in the same cases of the main panel. The black straight line has slope $1.66$.
  • Figure 5: Separation depth $S(t)$ in the cases with $E_r= 0$ (black line), $0.41 \times 10^{-2}$ (red line), $1.23 \times 10^{-2}$ (green line), $1.84 \times 10^{-2}$ (blue line). Inset: Profile of the density $n_r$, averaged along the horizontal direction, as a function of the vertical coordinate $y/H$ in the case with $E_r=1.84 \times 10^{-2}$ at times $t/(5.2 \times 10^3 \Delta t)= 4$ (blue line), $5$ (red line), $6$ (cyan line), $7$ (purple line), denoted by the corresponding colored arrows in the main panel.
  • ...and 2 more figures