Viscous froth lens

Microscale models of foam structure traditionally incorporate a balance between bubble pressures and surface tension forces associated with curvature of bubble films. In particular, models for flowing foam microrheology have assumed this balance is maintained under the action of some externally imposed motion. Recently, however, a dynamic model for foam structure has been proposed, the viscous froth model, which balances the net effect of bubble pressures and surface tension to viscous dissipation forces: this permits the description of fast-flowing foam. This contribution examines the behavior of the viscous froth model when applied to a paradigm problem with a particularly simple geometry: namely, a two-dimensional bubble “lens.” The lens consists of a channel partly filled by a bubble (known as the “lens bubble”) which contacts one channel wall. An additional film (known as the “spanning film”) connects to this bubble spanning the distance from the opposite channel wall. This simple structure can be set in motion and deformed out of equilibrium by applying a pressure across the spanning film: a rich dynamical behavior results. Solutions for the lens structure steadily propagating along the channel can be computed by the viscous froth model. Perturbation solutions are obtained in the limit of a lens structure with weak applied pressures, while numerical solutions are available for higher pressures. These steadily propagating solutions suggest that small lenses move faster than large ones, while both small and large lens bubbles are quite resistant to deformation, at least for weak applied back pressures. As the applied back pressure grows, the structure with the small lens bubble remains relatively stiff, while that with the large lens bubble becomes much more compliant. However, with even further increases in the applied back pressure, a critical pressure appears to exist for which the steady-state structure loses stability and unsteady-state numerical simulations show it breaks up by route of a topological transformation.

Figure 1

The equilibrium lens shape assuming a unit width channel. The parameter $l$ is the relative distance that the equilibrium lens extends across the channel (measured in terms of the channel width, which we take to be unity here) and is related to the equilibrium lens area, pressure, and surface energy [see Eqs. (2, 4, 5)]. The parameter $l$ is also directly related to the length of each curved arm of the lens bubble, ${\mathcal{L}}^{\circ }$, via ${\mathcal{L}}^{\circ }=2\pi l∕\left(3\sqrt{3}\right)$ [Eq. (3)].

Figure 2

A distorted lens driven by an imposed back pressure $p_b$ and traveling at some (unknown) migration velocity $v$. The so-called spanning film turns through an (unknown) angle $\delta \varphi$, but all films still meet at $\frac{2\pi }{3}$ angles at the vertex (see the close-up view of the vertex on the right of the figure). The lens pressure $p_l$ is also affected.

Figure 3

A local element of film which has turned through an angle $\varphi$ relative to its (vertical) orientation at the channel wall. In a time $dt$, the film displaces in the normal direction by an amount $v_{\perp }dt$, while the apparent horizontal displacement of the structure is $v_{\perp }dt∕\cos \varphi$.

Figure 4

Steady-state lens structures for a medium-size lens with $l=0.5$ and various back pressures driving the system towards the right: equilibrium lens $p_b=0$ (solid line), $p_b=2$ (long-dashed line), $p_b=4$ (short-dashed line), the end of the original steady-solution branch $p_b\approx 4.64$ (denser dotted line), and a lens on the point of detachment, with $p_b\approx 3.40$ on a new solution branch (fainter dotted line).

Figure 5

The vertical vertex position (denoted $y_v$) across the channel as a function of the back pressure $p_b$ for small $(l=0.1)$, medium-size $(l=0.5)$, and large $(l=0.9$) lenses. Solid lines indicate the original branch of steady solutions (the vertex displaces upwards as $p_b$ grows along this branch), while dashed lines (with higher $y_v$ values) indicate a new steady-solution branch ($y_v$ displaces downwards as $p_b$ grows): the latter branch is believed unstable: see, e.g., the discussion in Sec. 7B.

Figure 6

The steady-state turning angle along the spanning film $\delta \varphi$ as a function of back pressure $p_b$ for small $(l=0.1)$, medium-size $(l=0.5)$, and large $(l=0.9)$ lenses. Solid lines indicate the original branch of steady solutions, while dashed lines (with higher $\delta \varphi$ values) indicate a new steady-solution branch: the latter branch is believed unstable. The dotted lines indicate the asymptotic formula for the weakly driven lens for each $l$ value.

Figure 7

The steady-state migration velocity $v$ of the lens structure as a function of back pressure $p_b$. Solid line: $l=0.1$. Long-dashed line: $l=0.5$. Short-dashed line: $l=0.9$. For the $l=0.9$ data both the original steady-solution branch (lower part of the short-dashed curve) and the new branch (upper part of the short-dashed curve which loops backward) are shown. Only the original branch is shown for other $l$ values, as the branches virtually overlie each other in these cases. For clarity the inset shows a close-up view of the region $0\le p_b\le 5$, $0\le v\le 5$. Returning to the main plot, the asymptotics for the weakly driven lens are also shown (dotted lines, one for each $l$ value), giving velocity values slightly less than the computed ones. In all cases, however, the computed velocities satisfy $v<p_b$.

Figure 8

The steady-state pressure of the lens bubble $p_l$ normalized by equilibrium pressure $p_l^{\circ }$ as a function of (normalized) back pressure $p_b∕p_l^{\circ }$. Various lens sizes are shown, $l=0.1$, $l=0.5$, and $l=0.9$. On the original solution branch, solid lines show regions where $p_l>p_b+v$ [as a result the film mobility parameter $a_b$ defined by Eq. (13) satisfies $0<a_b<1$]. Long-dashed lines show regions where $p_b<p_l<p_b+v$ (as a result $a_b>1$). The short-dashed line (which only applies for $l=0.9$) shows regions where $p_b>p_l$ (and hence $a_b$ is negative). The dense dotted lines show the new solution branch (Sec. 6D). The faint dotted lines show $p_l=p_l^{\circ }+\frac{1}{2}{p}_b$ (which is the asymptotic behavior for a weakly driven lens) and $p_l=p_l^{\circ }+p_b$. All data values lie between these two limits.

Figure 9

The surface energy of the lens structure (denoted $E$) normalized by the equilibrium energy $E^{\circ }$ as a function of back pressure $p_b$ for various lens sizes $l$. Only a second-order increase in energy is seen for small $p_b$. On the original solution branch, the solid lines indicate regions where the back film mobility parameter $a_b$ satisfies $0<a_b<1$, the long-dashed lines indicate regions where $a_b>1$, and the short-dashed line (in the case of $l=0.9$) regions where $a_b$ is negative. Meanwhile, the dotted lines show the new solution branch (see Sec. 6D), which is of higher energy and believed unstable.

Figure 10

The (critical) pressure $p_b$ (solid line with $+$) and velocity $v$ (dashed line with $\times$) at which steady solutions break down as functions of the lens size parameter $l$.

Figure 11

The breakdown of the lens structure following a topological transformation. The former spanning film is flat and propagates with velocity equal to back pressure $v=p_b$. The former lens bubble (area $A_l$) remains behind and is semicircular in shape with radius $(2A_l∕\pi {)}^{1∕2}$, the relation between $A_l$ and the lens size parameter $l$ being given by Eq. (2).

Figure 12

The velocity of a lens on the point of detachment as a function of the back pressure corresponding to detachment (solid line). The long-dashed extension of the line towards lower back pressures and velocities corresponds to parameter values which cannot be realized starting from an equilibrium lens (regardless of the stability or otherwise of the nearly detached lens structure), because the resulting nearly detached lens would have an area greater than an equilibrium lens with $l=1$. The short-dashed extension towards even lower back pressures and velocities corresponds to parameter values that cannot be realized for any lens structure as the nearly detached lens bubble itself would span more than one channel width. The zero-velocity point corresponds to a back pressure for which the spanning film is a circular arc subtending $\frac{\pi }{3}$. The approximate velocity formula, Eq. (45) (dotted line), asymptotes to the data for large back pressure.

Figure 13

The sliding matchbox system. A foam (e.g., the lens structure) is confined between glass plates that slide over an impenetrable barrier. As drawn, the plates move to the left at speed $v_{box}$ relative to some laboratory frame, while the barrier and the lens structure (if and when it reaches steady state) do not move in the laboratory frame.

Figure 14

Dimensionless time for the occurrence of the topological transformation (lens bubble detaching from the spanning film) as a function of the driving velocity $v_{box}$ for rapidly driven systems that have no steady state. The initial condition upon start up of the driving velocity was the equilibrium lens structure. Various lens sizes are shown: Solid line with $+:l=0.9$. Long-dashed line with $\times :l=0.7$. Short-dashed line with $\circ :l=0.5$. Dotted line with $◻:l=0.3$. The vertical lines to the left of each curve show the velocities at which the time to topological transformation should diverge to infinity (since steadily propagating lenses are possible below these velocities): Solid line: $l=0.9$. Long-dashed line: $l=0.7$. Short-dashed line: $l=0.5$. Dotted line: $l=0.3$.

Figure 15

Distance along the channel (in units of one channel width) for the occurrence of the topological transformation (lens bubble detaching from the spanning film) as a function of the driving velocity $v_{box}$ for rapidly driven systems that have no steady state. The initial condition upon start up of the driving velocity was the equilibrium lens structure. Various lens sizes are shown: solid line with $+:l=0.9$. Long-dashed line with $\times :l=0.7$. Short-dashed line with $\circ :l=0.5$. Dotted line with $◻:l=0.3$. The vertical lines to the left of each curve show the velocities at which the distance to topological transformation should diverge to infinity (since steadily propagating lenses are possible below these velocities): Solid line: $l=0.9$. Long-dashed line: $l=0.7$. Short-dashed line: $l=0.5$. Dotted line: $l=0.3$.

Figure 16

A segment of length $ds$ of moving foam film. The film is confined between plates separated by a distance $H$, with the plate separation being much smaller than the total film length. For clarity the plates are not actually shown in the figure. The film has a curvature $K$ in the direction across the plates and a curvature $\kappa$ along them. These curvatures are associated with spatial changes in film tangents $\mathit{T}$ and $\mathit{t}$, respectively, the changes in $\mathit{T}$ being between upper and lower plates (see ${\mathit{T}}_{upper}$ and ${\mathit{T}}_{lower}$ in the figure) and those in $\mathit{t}$ being between positions along the film $s$ and $s+ds$. These spatial changes in the tangents, coupled with surface tension, lead to forces on the film. A pressure difference $\mathrm{\Delta }p$ across the film also produces force. At the plates, the film edges swell into Plateau border channels (indicated as bold curves in the figure). These channels incur viscous drag forces, according to their rate of motion over the plates $v_{\perp }$ (a projection of their velocity in the direction of the film normal $\mathit{n}$, which is a vector in the plane of the plates and at right angles to $\mathit{t}$). Consideration of separate force balances on the upper plate, on the lower plate and on the film segment excluding the plate regions, can be used to derive the two-dimensional viscous froth model.