Dynamic van der Waals theory

We present a dynamic van der Waals theory starting with entropy and energy functional with gradient contributions. The resultant hydrodynamic equations contain the stress arising from the density gradient. It provides a general scheme of two-phase hydrodynamics involving the gas-liquid transition in nonuniform temperature. Some complex hydrodynamic processes with evaporation and condensation are examined numerically. They are (i) adiabatically induced spinodal decomposition, (ii) piston effect with a bubble in liquid, (iii) temperature and velocity profiles around a droplet in heat flow, (iv) efficient latent heat transport at small liquid densities (the mechanism of heat pipes), (v) boiling in gravity with continuous bubble formation and rising, and (vi) spreading and evaporation of liquid on a heated boundary wall.

Figure 1

Density profiles in phase separation after lowering of the boundary temperature from $1.1T_c$ to $0.91T_c$ at $t=0$. Here the liquid layer (in black) at the boundary acts as a piston adiabatically expanding the interior.

Figure 2

Adiabatic time evolution of the temperature $T$ and the density $n$ at the cell center (lower solid lines) in the early stage after lowering of the boundary temperature from $1.1T_c$ to $0.91T_c$. Also shown is the inverse isothermal compressibility $1∕K_T$ multiplied by $K_c=n_c{k}_B{T}_c$ (dotted line). The entropy $s$ per particle (upper line) remains nearly unchanged, indicating that the process is adiabatic. The $s_c$ is the critical value. All the quantities are those at the cell center still without domains in this early stage.

Figure 3

Early stage time evolution due to the piston effect after a change of $T_b∕T_c$ from 0.875 to 0.895 without gravity. A bubble with radius $50\ell =L∕4$ is at the center. Upper plate: $T∕T_c$ vs $t∕t_0$ at the center $y=L∕2$ and in the liquid at $y=L∕5$ and $4L∕5$, where $x=L∕2$. Lower plate: $\mathit{v}$ at $t∕t_0=55$, where the gas region is strongly perturbed.

Figure 4

$(T-T_t)∕(T_b-T_t)$, $p∕p_c$, and $v_{0}n$ vs $y∕L$ at $t∕t_0=200$ and at $x=L∕2$, with $T_b=0.895T_c$ and $T_t=0.875T_c$, in the same run as in Fig. 3. The thermal diffusion layers are reaching at the bubble due to the small system size.

Figure 5

$T$ (upper plate) and $\mathit{v}$ and $n$ (lower plate) in a steady heat-conducting state at $T_b=0.945T_c$ and $T_t=0.875T_c$ under zero gravity. A bubble is attracted to the warmer boundary wall. Temperature homogeneity inside the bubble is attained by latent heat convection. The horizontal direction is along the $x$ axis in the text.

Figure 6

$T$ (upper plate) and $\mathit{v}$ and $n$ (lower plate) around a cool liquid droplet at $t=4565t_0$ with $T_b=0.91T_c$ and $T_t=0.875T_c$ under zero gravity. Condensation is taking place at the interface and velocity field is induced.

Figure 7

Density profiles after that in Fig. 6. Illustrated are slow migration and rapid spreading after collision with the upper boundary.

Figure 8

$T$ (upper plate) and $\mathit{v}$ and $n$ (lower plate) in a steady heat-conducting state at $T_b=0.89T_c$ and $T_t=0.875T_c$ under zero gravity.

Figure 9

$T$ (upper plate) and $\mathit{v}$ and $n$ (lower plate) in a steady heat-conducting state at $T_b=0.91T_c$ and $T_t=0.875T_c$ under $g^{*}={10}^{-4}$.

Figure 10

Evaporation of a liquid layer at $g^{*}={10}^{-4}$ after increasing the bottom temperature from $0.91T_c$ to $1.1T_c$, where the initial state is given in Fig. 9.

Figure 11

Changeovers of boiling with increasing $T_b$ at $T_t=0.775T_c$ and $g^{*}={10}^{-4}$. Upper plates: nonboiling state at $T_b=0.91T_c$ (left), and weakly boiling state with a single bubble at $T_b=0.945T_c$ (middle and right). Middle plates: transition from weak to strong boiling after a change of $T_b$ from $0.945T_c$ to $1.0T_c$ at $t=34245t_0$. Lower plates: dynamical steady state with continuous bubble formation and rising.

Figure 12

Temperature (upper plate) and velocity field (lower plate) in a fully developed boiling state at $T_b=T_c$ and $T_t=0.775T_c$ under $g^{*}={10}^{-4}$ at $t=63047t_0$.

Figure 13

Normalized heat flux $Q_b\left(x\right)∕Q_0$ at the bottom in boiling at $t=34600t_0$ in (a) and $t=63047t_0$ in (b), corresponding to the first panel in Fig. 11 for (a) and to Fig. 12 for (b). Here $Q_b\left(x\right)$ is defined by Eq. (3.18) and $Q_0=ϵ\ell ∕v_0{t}_0$.

Figure 14

Spreading and evaporation of a liquid region on the bottom with $T_b∕T_c=0.875$ in (a) (top), 0.915 in (b) (middle), and 0.975 in (c) (bottom), with the initial temperature being at $0.875T_c$.

Figure 15

Velocity $\mathit{v}$ (upper panel) and normalized heat flux $Q_b\left(x\right)∕Q_0$ at the bottom (lower panel) at $t=500t_0$ with $T_b∕T_c=0.875$ obtained from the run giving the series (a) in Fig. 14. Here $Q_0=ϵ\ell ∕v_0{t}_0$. Absorption is maximum at the contact points, where evaporation is taking place.