Dynamics of a thermally driven film climbing the outside of a vertical cylinder

The dynamics of a film climbing the outside of a vertical cylinder under the competing effects of a thermally driven surface tension gradient and gravity is examined through numerical simulations of a thin-film model for the film height. The model, including boundary conditions, depends on three parameters, the scaled cylinder radius $\widehat{R},$ the upstream film height $h_{\infty },$ and the downstream precursor film thickness $b,$ and reduces to the model for Marangoni driven film climbing a vertical plate in the limit $\widehat{R}\to \infty .$ The axisymmetric advancing front displays dynamics similar to that found along a vertical plate where, depending on $h_{\infty },$ the film forms a single Lax shock, an undercompressive double shock, or a rarefaction-undercompressive shock. A linear stability analysis of the Lax shock reveals the number of fingers that form along the contact line increases linearly with cylinder circumference while no fingers form for sufficiently small cylinders (below $\widehat{R}\approx 1.15$ when $b=0.1$). The substrate curvature controls the height of the Lax shock, bounds on $h_{\infty }$ that define the three distinct solutions, and the maximum growth rate of contact line perturbations to the Lax shock when $\widehat{R}=O\left(1\right),$ whereas the three solutions and the stability of the Lax shock converge to the behavior one observes on a vertical plate when $\widehat{R}\ge O\left(10\right).$ An energy analysis reveals that the azimuthal curvatures of the base state and perturbation, which arise from the annular geometry of the film, promote instability of the advancing contact line.

Figure 1

Schematic of a thermally driven thin film climbing a vertical cylinder of radius, $R$.

Figure 2

Lax shock solutions connecting $h_{\infty }=0.3$ to $b=0.1$ along cylinders with dimensionless radius $\widehat{R}=1.0$ (thick solid line), 1.45 (thick dashed line), and 5.0 (thin dotted-dashed line) and along a plate (thin solid line). The thin dashed line represents the initial data.

Figure 3

Difference in capillary ridge heights $▵h_{\max }=h_{\max }^{\mathrm{cylinder}}-h_{\max }^{\mathrm{plate}}$ versus scaled cylinder radius for $h_{\infty }=0.3,b=0.1(\circ$); $h_{\infty }=0.3,b=0.05(▵$); and $h_{\infty }=0.3323,b=0.1$ ($\blacksquare$). For sufficiently large $\widehat{R},▵h_{max}\sim {\widehat{R}}^{-2.04}$ (solid line). Inset shows the asymptotic approach in $\widehat{R}$ of the capillary ridge height along the cylinder to the height along the plate (dashed lines).

Figure 4

Computed values of the threshold of the upstream film thickness $h_2$ as a function of the scaled cylinder radius with $b=0.1.$ The advancing contact line forms a single Lax shock when $h_{\infty }<h_2.$

Figure 5

Undercompressive double shock structure connecting $h_{\infty }=0.4$ to $b=0.1$ at computational time 4800 for a plate (solid) and cylinders with $\widehat{R}=3.0$ (dashed) and $\widehat{R}=2.0$ (dotted-dashed).

Figure 6

Nonconvex flux function $f\left(h\right)=h^2-h^3.$ The slope of the chord connecting $h_{\infty }$ to $b$ is the speed of a single Lax shock. The slopes of the chords connecting $h_{\mathrm{uc}}$ to $b$ and $h_{\infty }$ to $h_{\mathrm{uc}}$ are the speeds of the leading undercompressive shock and trailing Lax shock of an undercompressive double shock. The points signify the following: A, $b=0.1$; B, $h_{\infty }=0.4$; C, $h_{\mathrm{uc}}^{\mathrm{plate}}\left(0.1\right)$; D, $h_{\mathrm{uc}}^{\mathrm{cylinder}}\left(0.1\right)$ with $\widehat{R}=3.0$; and E, $h_{\mathrm{uc}}^{\mathrm{cylinder}}\left(0.1\right)$ with $\widehat{R}=2.0$, where $h_{\mathrm{uc}}=h_{\mathrm{uc}}\left(b\right).$

Figure 7

Asymptotic approach of the undercompressive shock height along a cylinder for sufficiently large $\widehat{R}$ with $h_{\infty }=0.4$ for $b=0.1$ ($\circ$) and $b=0.05$ ($▵$) to the height along a plate $h_{\mathrm{uc}}^{\mathrm{plate}}\left(0.1\right)=0.5679$ and $h_{\mathrm{uc}}^{\mathrm{plate}}\left(0.05\right)=0.6528$ (dashed lines).

Figure 8

Rarefaction-undercompressive shock solution formed along a cylinder with $h_{\infty }=0.8,b=0.1,s=s(h_{\infty },b)=0.17$, and $\widehat{R}=5.0.$ The undercompressive shock height is $h_{\mathrm{uc}}=0.5717.$ The initial data are represented by the dashed line. The numerical solution (solid line) and nonclassical inviscid rarefaction-undercompressive shock (16) ($\circ$) are at time 1400.

Figure 9

Dispersion relation for Lax shock solutions plotted on a linear-logarithmic scale with $h_{\infty }=0.3$ and $b=0.1$ for the plate (thick solid line) and for cylinders with scaled radius $\widehat{R}=$ 1.0 (thin solid line), 1.45 (thin dashed line), 5.0 (dotted-dashed line), and 30.0 (thick dashed line).

Figure 10

Cutoff mode (filled symbols) and most unstable mode (open symbols) versus scaled cylinder radius derived from linear stability theory with $h_{\infty }=0.3$ and for $b=0.1$ ($\circ$) and $b=0.05$ ($\square$). For $b=0.1$ and $\widehat{R}\ge 1.75, q_{\mathrm{cutoff}}\sim {\widehat{R}}^{0.97}$, and $q^{*}\sim {\widehat{R}}^{0.97}$ (solid lines). For $b=0.05$ and $\widehat{R}\ge 2.0, q_{\mathrm{cutoff}}\sim {\widehat{R}}^{1.00}$, and $q^{*}\sim {\widehat{R}}^{1.00}$ (dashed lines).

Figure 11

Difference in maximum growth rate $▵{\beta }^{*}={\beta }_{\mathrm{cylinder}}^{*}-{\beta }_{\mathrm{plate}}^{*}$ versus scaled cylinder radius derived from linear stability theory with $h_{\infty }=0.3$ and for $b=0.1$ ($\circ$) and $b=0.05$ ($\square$). For $b=0.1$ and $\widehat{R}\ge 1.75, ▵{\beta }^{*}\sim {\widehat{R}}^{-2.05}$ (solid line) and for $b=0.05$ and $\widehat{R}\ge 2.0, ▵{\beta }^{*}\sim {\widehat{R}}^{-1.88}$ (dashed line). Inset shows the asymptotic approach in $\widehat{R}$ of the maximum growth rate along the cylinder to the value along the plate.

Figure 12

Eigenfunction $G$ corresponding to the largest eigenvalue plotted for several eigenmodes for a cylinder with (a) $\widehat{R}=1.45,h_{\infty }=0.3,b=0.1$ and (b) $\widehat{R}=5.0,h_{\infty }=0.3,b=0.05.$

Figure 13

Energy results plotted as a function of wave number for a cylinder with (a) $\widehat{R}=1.45,h_{\infty }=0.3,b=0.1;$ (b) $\widehat{R}=5.0,h_{\infty }=0.3,b=0.1;$ and (c) $\widehat{R}=5.0,h_{\infty }=0.3,b=0.05.$ Terms 1–11 for the cylinder model are listed in Table 1. To aid comparison, terms with the same or analogous physical basis as the plate model are indicated in the legend (labelled I–VIII). The insets show behavior near $\dot{E}=0.$

Figure 14

Energy results plotted as a function of wave number for a plate with $h_{\infty }=0.3$ and $b=0.1.$ Terms I–VIII for the plate model are listed in Table 2. The insets show behavior near $\dot{E}=0.$