Unsteady motion of a long bubble or droplet in a self-rewetting system

Motivated by the potential use of self-rewetting fluids (i.e., fluids that exhibit a nonmonotonic variation of surface tension with temperature) in various heat-transfer applications, in the present work we formulate and analyze a theoretical model for the unsteady motion of a long bubble or droplet in a self-rewetting system in a nonuniformly heated tube due to a combination of Marangoni effects due to the variation of surface tension with temperature, gravitational effects due to the density difference between the two fluids, and an imposed background flow along the tube. We find that the evolution of the shape (but not of the position) of the bubble or droplet is driven entirely by Marangoni effects and depends on the initial value of its radius in relation to a critical value. In the case in which Marangoni effects are absent, the bubble or droplet always moves with constant velocity without changing shape. In the case in which only Marangoni effects are present, the bubble or droplet either always moves away from or always moves towards the position of minimum surface tension; in the latter case it ultimately fills the entire cross section of the tube at a final stationary position which is closer to the position of minimum surface tension than its original position. In the cases in which either only Marangoni effects and gravitational effects or only Marangoni effects and background-flow effects are present the competition between the two effects can lead to a nonmonotonic evolution of the position of the center of mass of the bubble or droplet. The behavior of a self-rewetting system described in the present work is qualitatively different from that for ordinary fluids, in which case the bubble or droplet always moves with constant velocity without changing shape.

Figure 1

Sketches of surface tension $\sigma$ as a function of temperature $T$ for (a) an ordinary and (b) a self-rewetting fluid.

Figure 2

Sketch of the geometry of the problem: a long axisymmetric bubble or droplet of fluid (fluid 1) of uniform radius $b\left(t\right)$ moving in a second fluid (fluid 2), all contained within a nonuniformly heated vertical tube of constant radius $a$. The linear temperature distribution $T$ imposed on the tube wall and the resulting quadratic variation of surface tension $\sigma$ are shown on the right and the left of the figure, respectively.

Figure 3

Plots of the functions $f_j$ for $j=\mathrm{M}, \mathrm{G}, \mathrm{B}$ defined by Eqs. (38, 39, 40) which arise in the Marangoni (M), gravitational (G), and background-flow (B) contributions to the fluxes appearing in Eqs. (37) as functions of the droplet radius $b$ for $m=0, 1/8, 1/4,...$, 1, $3/2$, 2, 5, 10 and in the limit $m\to \infty$ given by Eqs. (44, 45, 46) (the latter shown with dashed lines).

Figure 4

Plot of the critical droplet radius $b_{\mathrm{c}}$ calculated from Eq. (47) as a function of the viscosity ratio $m$. The asymptotic expressions $b_{\mathrm{c}}\sim exp[-1/\left(4m\right)-1]\to 0$ in the limit $m\to 0^{+}$ and $b_{\mathrm{c}}\sim 1-{(3/2m)}^{1/2}+(5/4m)\to 1^{-}$ in the limit $m\to \infty$ are shown with dashed lines.

Figure 5

Evolution of (a) the radius $b$ and (b) the length $L$ of the droplet calculated numerically from Eq. (48) for various initial radii $b_0$ and initial lengths $L_0=1/b_0^2$ in the case $m=1$ and $M=1$. The critical values $b=b_{\mathrm{c}}\simeq 0.3729$ and $L=L_{\mathrm{c}}=1/b_{\mathrm{c}}^2\simeq 7.1899$ are shown with dashed lines.

Figure 6

Plots of the constant velocity of the droplet $dc/dt$ when Marangoni effects are absent given by Eq. (59) as a function of its initial radius $b_0$ in the case $M=0$ and $\mathrm{\Delta }\rho =100$ when (a) $Q_{\mathrm{b}}=1$, (b) $Q_{\mathrm{b}}=-8$, (c) $Q_{\mathrm{b}}=-25/2$, and (d) $Q_{\mathrm{b}}=-15$ for $m=0, 1/32, 1/16,..., 1/4, 1/2,...$, 2 and in the limit $m\to \infty$ given by Eq. (61) (the latter shown with dashed lines).

Figure 7

Evolution of the position of the center of mass of the droplet $c$ when only Marangoni effects are present given by Eq. (62) for initial positions $c_0=-2, -3/2, -1, ...,$ 2 in the case $m=1, M=1, \mathrm{\Delta }\rho =0$, and $Q_{\mathrm{b}}=0$ when (a) $b_0=1/4$ ($<b_{\mathrm{c}}\simeq 0.3729$) and (b) $b_0=1/2$ ($>b_{\mathrm{c}}$).

Figure 8

Examples of the four qualitatively different forms of the evolution of a droplet when only Marangoni effects are present when $c_0>0$ in the case $m=1, M=1, \mathrm{\Delta }\rho =0, Q_{\mathrm{b}}=0$, for which $b_{\mathrm{c}}\simeq 0.3729$. In (a) and (b), $L_0=9, b_0=1/3$ ($<b_{\mathrm{c}}$), with (a) $c_0=10$, (b) $c_0=2$, and the snapshots are shown at $t=0$, 15, 30, 45, whereas in (c) and (d), $L_0=4, b_0=1/2$ ($>b_{\mathrm{c}}$), with (c) $c_0=5/2$, (d) $c_0=1$, and the snapshots are shown at $t=0$, 20, 40, 60. In each snapshot the center of mass of the droplet is denoted by a dot ($•$). Note that the corresponding evolutions when $c_0<0$ are simply the mirror images of those when $c_0>0$ in the plane $z=0$, and the evolutions when $c_0=0$ have $c\equiv 0$ for all $t$.

Figure 9

Evolution of the position of the center of mass of the droplet $c$ when only Marangoni and gravitational effects are present given by Eq. (57) in the case $m=1, M=1, \mathrm{\Delta }\rho =10$, and $Q_{\mathrm{b}}=0$ for initial positions (a) $c_0=-14, -12, -10,...$, 12, 14 when $b_0=1/4(<b_{\mathrm{c}}\simeq 0.3729)$ and (b) $c_0=-15, -12, -9,...$, 12, 15 when $b_0=1/2(>b_{\mathrm{c}})$. The evolutions corresponding to the critical values (a) $c_0=c_1^{\mathrm{G}}\simeq -6.7523$ and $c_0=c_2^{\mathrm{G}}\simeq -3.3930$, and (b) $c_0=c_1^{\mathrm{G}}\simeq 8.5571$ and $c_0=c_2^{\mathrm{G}}\simeq -6.8558$, which separate the three different behaviors, are shown with dashed lines.

Figure 10

Evolution of the position of the center of mass of the droplet $c$ when only Marangoni and background-flow effects are present given by Eq. (57) in the case $m=1, M=1, \mathrm{\Delta }\rho =0$, and $Q_{\mathrm{b}}=1/10$ for initial positions (a) $c_0=-10, -9, -8,...$, 3, 4 when $b_0=1/4(<b_{\mathrm{c}}\simeq 0.3729)$ and (b) $c_0=-5, -5/2,0,..., 25/2$, 15 when $b_0=1/2(>b_{\mathrm{c}})$. The evolutions corresponding to the critical values (a) $c_0=c_1^{\mathrm{B}}\simeq -5.9993$ and $c_0=c_2^{\mathrm{B}}\simeq -4.4817$, and (b) $c_0=c_1^{\mathrm{B}}\simeq 6.7484$ and $c_0=c_2^{\mathrm{B}}\simeq 4.0561$, which separate the three different behaviors, are shown with dashed lines.

Figure 11

Plots of (a) the radial velocity $u_1$ and (b) the normalized axial velocity $w_1/z$ at the interface $r=b$ given by Eqs. (A2) and (A3), respectively, when only Marangoni effects are present as functions of $b$ for $m=0, 1/4, 1/2, 3/4$, 1, 2, 5, and 10 and in the limit $m\to \infty$ (the latter shown with dashed lines) in the case $M=1, Q_{\mathrm{b}}=0$, and $\mathrm{\Delta }\rho =0$.

Figure 12

Plots of (a) the transverse velocity $u_1$ and (b) the normalized axial velocity $w_1/z$ at the interface $x=b$ given by Eqs. (B9) and (B10), respectively, when only Marangoni effects are present as functions of $b$ for $m=0, 1/4, 1/2, 3/4$, 1, 2, 5, and 10 and in the limit $m\to \infty$ (the latter shown with dashed lines) in the case $M=1, Q_{\mathrm{b}}=0$, and $\mathrm{\Delta }\rho =0$.