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FIG. 8. Static prediction of the existence of asymmetric localised fingers in terms of the relative obstacle size a; and ay The experimental tube had H = 3.07 + 0.01 mm and Bo = 1.058 + 0.008. The solid line shows SE computations for F = 3.06 + 0.01 mm and Bo = 1.050, while the almost indistinguishable dashed line is for H = 3.08 + 0.01 mm and Bx = 1.065. Circles indicate tube geometries where localised fingers are stable when the flow is suddenly stopped (Ca = 0) while squares correspond to tubes where localised fingers are unstable at Ca = 0. The dotted-dashed line is the analytic prediction in the absence of gravity (Bo = 0) corresponding to Eq. (A6).

Figure 8 Static prediction of the existence of asymmetric localised fingers in terms of the relative obstacle size a; and ay The experimental tube had H = 3.07 + 0.01 mm and Bo = 1.058 + 0.008. The solid line shows SE computations for F = 3.06 + 0.01 mm and Bo = 1.050, while the almost indistinguishable dashed line is for H = 3.08 + 0.01 mm and Bx = 1.065. Circles indicate tube geometries where localised fingers are stable when the flow is suddenly stopped (Ca = 0) while squares correspond to tubes where localised fingers are unstable at Ca = 0. The dotted-dashed line is the analytic prediction in the absence of gravity (Bo = 0) corresponding to Eq. (A6).

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FIG. 1. Schematic diagram of the experimental setup. The end view shows the cross-section of the tube, which is rectangular with a centred step-change in depth. The dimensions of the cross-section of the flow tube are: H = 3.07 + 0.01 mm, 3< W <45 mm, w = 4.49 + 0.01 mm, / = 1.09 + 0.01 mm or A = 1.50 + 0.02 mm. The side view illustrates how liquid is withdrawn at constant flow rate Q from one end of the initially liquid-filled tube, while the other end of the tube is open to the atmosphere.
FIG. 1. Schematic diagram of the experimental setup. The end view shows the cross-section of the tube, which is rectangular with a centred step-change in depth. The dimensions of the cross-section of the flow tube are: H = 3.07 + 0.01 mm, 3< W <45 mm, w = 4.49 + 0.01 mm, / = 1.09 + 0.01 mm or A = 1.50 + 0.02 mm. The side view illustrates how liquid is withdrawn at constant flow rate Q from one end of the initially liquid-filled tube, while the other end of the tube is open to the atmosphere.
FIG. 2. Dependence of the finger tip offset from the tube centreline on Ca for a, = 1/4, a, = 0.49, and a = 6.0. The value of the offset 5 was measured 50 mm behind the finger tip, as shown schematically. Inset top-views are shown of the mair finger types: symmetric (solid black symbols), asymmetric non-localised (white, () and (++) symbols), and localised state (solid light grey symbols). Both localised and symmetric fingers are stable for very low values of Ca. The symbols distinguist the perturbations applied to the propagating finger in order to reach the different states: (0) experiments start from rest, (A. experiments start at a high flow rate that is then reduced. A localised finger was formed as a static initial condition (LC) by driving a finger at high flow rate before interrupting the flow, and then restarting the experiment at the desired value of Ca The other perturbations were applied by blocking parts of the cross-section over a short distance at the inlet of the tube (>) one side-channel and region over obstacle blocked, (++) only region over obstacle blocked, or (*) only one side-channe blocked.
FIG. 2. Dependence of the finger tip offset from the tube centreline on Ca for a, = 1/4, a, = 0.49, and a = 6.0. The value of the offset 5 was measured 50 mm behind the finger tip, as shown schematically. Inset top-views are shown of the mair finger types: symmetric (solid black symbols), asymmetric non-localised (white, () and (++) symbols), and localised state (solid light grey symbols). Both localised and symmetric fingers are stable for very low values of Ca. The symbols distinguist the perturbations applied to the propagating finger in order to reach the different states: (0) experiments start from rest, (A. experiments start at a high flow rate that is then reduced. A localised finger was formed as a static initial condition (LC) by driving a finger at high flow rate before interrupting the flow, and then restarting the experiment at the desired value of Ca The other perturbations were applied by blocking parts of the cross-section over a short distance at the inlet of the tube (>) one side-channel and region over obstacle blocked, (++) only region over obstacle blocked, or (*) only one side-channe blocked.
FIG. 3. Wet fraction m as a function of the capillary number Ca for a, = 0.49 and different values of @,. Black markers: experiments start from rest; grey markers: experiments start at a high flow rate that is then reduced. Markers with a white cross correspond to an oscillatory state: (HI) ay = 2/5, (@) a» = 1/3, (@) ay = 2/7, (V) ay = 1/4, (*&) ay = 1/5, (A) @y = 1/8. The arrow indicates the direction of increasing @,, and illustrative finger shapes are shown as insets. The data for w = 2/7 are explicitly marked because it divides regions of different qualitative behaviour: for a < 2/7, the system does not return to the symmetric state when Ca is reduced from an initially high flow rate. The data for a, = 2/5 anda, = 1/3 were previously presented in Ref. 7.
FIG. 3. Wet fraction m as a function of the capillary number Ca for a, = 0.49 and different values of @,. Black markers: experiments start from rest; grey markers: experiments start at a high flow rate that is then reduced. Markers with a white cross correspond to an oscillatory state: (HI) ay = 2/5, (@) a» = 1/3, (@) ay = 2/7, (V) ay = 1/4, (*&) ay = 1/5, (A) @y = 1/8. The arrow indicates the direction of increasing @,, and illustrative finger shapes are shown as insets. The data for w = 2/7 are explicitly marked because it divides regions of different qualitative behaviour: for a < 2/7, the system does not return to the symmetric state when Ca is reduced from an initially high flow rate. The data for a, = 2/5 anda, = 1/3 were previously presented in Ref. 7.
FIG. 4. Oscillatory pattern “shed” behind a finger propagating under constant flow rate from right to left for a, = 0.49: (a) ay =1/8, Ca = 3.69 x 10-7, (b) a, =1/8, Ca ~ 68 x 10°77, (c) ay = 1/3, Ca = 5.74 x 1073, (d) ay = 1/3, Ca = 7.14 x 1073. The field of view is 400 mm long. Despite the occurrence of oscillations at very different values of Ca, the mechanism underlying their formation is similar in narrow and wide tubes. In particular, the spatial wavelength of the pattern decreases in both cases with increasing Ca, as discussed by Pailha et al.’ in the case of narrow tubes. Figures 4(c) and 4(d) are reprinted with permission from Phys. Fluids 24, 021702 (2012). Copyright 2011 American Institute of Physics.
FIG. 4. Oscillatory pattern “shed” behind a finger propagating under constant flow rate from right to left for a, = 0.49: (a) ay =1/8, Ca = 3.69 x 10-7, (b) a, =1/8, Ca ~ 68 x 10°77, (c) ay = 1/3, Ca = 5.74 x 1073, (d) ay = 1/3, Ca = 7.14 x 1073. The field of view is 400 mm long. Despite the occurrence of oscillations at very different values of Ca, the mechanism underlying their formation is similar in narrow and wide tubes. In particular, the spatial wavelength of the pattern decreases in both cases with increasing Ca, as discussed by Pailha et al.’ in the case of narrow tubes. Figures 4(c) and 4(d) are reprinted with permission from Phys. Fluids 24, 021702 (2012). Copyright 2011 American Institute of Physics.
FIG. 5. Schematic of a three-dimensional air finger propagating from right to left in an axially uniform tube with a partially occluded rectangular cross-section. Behind the finger tip, the finger becomes axially uniform and an illustrative cross-sectiot is shown in white. The remaining thin liquid films are essentially at rest and the majority of the viscous stress on the interfac: occurs near the tip region.
FIG. 5. Schematic of a three-dimensional air finger propagating from right to left in an axially uniform tube with a partially occluded rectangular cross-section. Behind the finger tip, the finger becomes axially uniform and an illustrative cross-sectiot is shown in white. The remaining thin liquid films are essentially at rest and the majority of the viscous stress on the interfac: occurs near the tip region.
FIG. 6. Sketches of symmetric static equilibria in tubes of rectangular cross-section in the absence of gravity: (a) unoccluded, (b) low occlusion, and (c) high occlusion. where A* is the dimensional projected area and P* is the dimensional perimeter of the interface within the cross-section. Non-dimensionalising based on the tube half-height H/2 and using Green’s
FIG. 6. Sketches of symmetric static equilibria in tubes of rectangular cross-section in the absence of gravity: (a) unoccluded, (b) low occlusion, and (c) high occlusion. where A* is the dimensional projected area and P* is the dimensional perimeter of the interface within the cross-section. Non-dimensionalising based on the tube half-height H/2 and using Green’s
FIG. 7. Sketches of asymmetric static equilibria in tubes of rectangular cross-section in the absence of gravity: (a) high occlusion, (b) medium occlusion, and (c) low occlusion. In addition to modifying the symmetric solution as described above, the introduction of a partial occlusion allows alternative asymmetric static equilibria for sufficiently high obstacles, see Fig. 7. If the obstacle height is exactly equal to the tube height, then each side branch is an independent rectangular tube with static equilibrium given by Eq. (7), but with a replaced by the half-width of the side branch. These localised asymmetric equilibria may easily be obtained experimentally by introducing air on only one side of the obstacle. If the obstacle height is nearly equal to the channel height the asymmetric state will remain unchanged, see Fig. 7(a), until the gap above the obstacle exceeds the radius of curvature and the arc can no longer meet the side of the obstacle at zero contact angle. Below this height, the equilibrium becomes three quarter circles and an arc of a circle that is pinned on the corner of the obstacle, see Fig. 7(b). As the height of the obstacle is reduced further, the pinned arc evolves smoothly from a quarter circle, meeting the side of the obstacle, to a semi-circle that meets the top of the obstacle when a, = a;,, see Fig. 7(c). In this configuration, the curvature is set by the gap width above the obstacle and there is a degeneracy in the solution because it costs nothing energetically to extend the infinitesimal film above the obstacle. This expansion of the finger over the obstacle allows the smooth transition from localised to non-localised asymmetric fingers. An alternative family of asymmetric solutions is possible in which the interface is pinned
FIG. 7. Sketches of asymmetric static equilibria in tubes of rectangular cross-section in the absence of gravity: (a) high occlusion, (b) medium occlusion, and (c) low occlusion. In addition to modifying the symmetric solution as described above, the introduction of a partial occlusion allows alternative asymmetric static equilibria for sufficiently high obstacles, see Fig. 7. If the obstacle height is exactly equal to the tube height, then each side branch is an independent rectangular tube with static equilibrium given by Eq. (7), but with a replaced by the half-width of the side branch. These localised asymmetric equilibria may easily be obtained experimentally by introducing air on only one side of the obstacle. If the obstacle height is nearly equal to the channel height the asymmetric state will remain unchanged, see Fig. 7(a), until the gap above the obstacle exceeds the radius of curvature and the arc can no longer meet the side of the obstacle at zero contact angle. Below this height, the equilibrium becomes three quarter circles and an arc of a circle that is pinned on the corner of the obstacle, see Fig. 7(b). As the height of the obstacle is reduced further, the pinned arc evolves smoothly from a quarter circle, meeting the side of the obstacle, to a semi-circle that meets the top of the obstacle when a, = a;,, see Fig. 7(c). In this configuration, the curvature is set by the gap width above the obstacle and there is a degeneracy in the solution because it costs nothing energetically to extend the infinitesimal film above the obstacle. This expansion of the finger over the obstacle allows the smooth transition from localised to non-localised asymmetric fingers. An alternative family of asymmetric solutions is possible in which the interface is pinned
TABLE I. Wet fractions at Ca = 0 computed using SE corresponding to the experimental data for a, = 0.49 presented in Fig. 3. The experimental values correspond to straight line extrapolation of the two data points with smallest Ca to the value Ca=0.
TABLE I. Wet fractions at Ca = 0 computed using SE corresponding to the experimental data for a, = 0.49 presented in Fig. 3. The experimental values correspond to straight line extrapolation of the two data points with smallest Ca to the value Ca=0.
FIG. 9. Comparison between two-dimensional SE calculations of cross-sectional finger shapes and experiments for a, = 0.35 and a, = 1/3. Experimental results were obtained with silicone oils of different dynamic viscosities: circles fi = 4.81 x 107-3 kg m~!s~! and squares jz = 9.7 x 10-7 kg m—!s~!. The two-dimensional static solutions, shown as the solid line (green), are in excellent agreement with the experiments, suggesting that the loss of stability of the symmetric finger can be predicted from existence of the non-localised asymmetric state that forms behind the finger tip in the SE calculations. The value of 5 was measured at a distance of 45 mm behind the finger tip.
FIG. 9. Comparison between two-dimensional SE calculations of cross-sectional finger shapes and experiments for a, = 0.35 and a, = 1/3. Experimental results were obtained with silicone oils of different dynamic viscosities: circles fi = 4.81 x 107-3 kg m~!s~! and squares jz = 9.7 x 10-7 kg m—!s~!. The two-dimensional static solutions, shown as the solid line (green), are in excellent agreement with the experiments, suggesting that the loss of stability of the symmetric finger can be predicted from existence of the non-localised asymmetric state that forms behind the finger tip in the SE calculations. The value of 5 was measured at a distance of 45 mm behind the finger tip.
FIG. 10. Comparison between two-dimensional SE calculations of cross-sectional finger shapes and experiments for a; = 0.49 and a, = 1/4. Experimental results (circles) were obtained with silicone oil of dynamic viscosity 4 = 4.81 x 10-3 kg m~!s~!. The localised solution occurs at the point m ~ 0.6, 6 ~ 0.3 and is predicted accurately by the static calculations (solid line and dots, green). The broadening of the finger is qualitatively captured by the static calculations, with good agreement once the finger is detached from both side walls, but the exact details of the evolution are different in the simulations and experiments. The value of 5 was measured at a distance of 45 mm behind the finger tip. we parameterise the two-dimensional solutions and experimental data by the wet fraction, m, and the finger offset, 5, defined in Sec. II and Fig. 2. Results for two different obstacle heights are presented in Figs. 9 and 10. In both cases, the experimental data are in good agreement with the two-dimensional calculations, confirming that the experimental results can be captured by the two-dimensional static model. We reiterate, however, that we can neither determine the capillary numbers nor whether the solutions are experimentally realised without experimental data. Nonetheless, having established the mapping between the experimental data and static solutions, the static calculations reveal the approximate cross-sections of each finger.
FIG. 10. Comparison between two-dimensional SE calculations of cross-sectional finger shapes and experiments for a; = 0.49 and a, = 1/4. Experimental results (circles) were obtained with silicone oil of dynamic viscosity 4 = 4.81 x 10-3 kg m~!s~!. The localised solution occurs at the point m ~ 0.6, 6 ~ 0.3 and is predicted accurately by the static calculations (solid line and dots, green). The broadening of the finger is qualitatively captured by the static calculations, with good agreement once the finger is detached from both side walls, but the exact details of the evolution are different in the simulations and experiments. The value of 5 was measured at a distance of 45 mm behind the finger tip. we parameterise the two-dimensional solutions and experimental data by the wet fraction, m, and the finger offset, 5, defined in Sec. II and Fig. 2. Results for two different obstacle heights are presented in Figs. 9 and 10. In both cases, the experimental data are in good agreement with the two-dimensional calculations, confirming that the experimental results can be captured by the two-dimensional static model. We reiterate, however, that we can neither determine the capillary numbers nor whether the solutions are experimentally realised without experimental data. Nonetheless, having established the mapping between the experimental data and static solutions, the static calculations reveal the approximate cross-sections of each finger.
TABLE II. Wet fractions computed using SE for the proposed limiting steady asymmetric fingers for different tube widths corresponding to the experimental data presented in Fig. 3. The experimental values correspond to the steady (non-oscillatory) asymmetric solution with minimum wet frac- tion. Note that when a@,, = 2/7 the data point with m = 0.33 is not chosen because it is believed to correspond to an oscillatory solution.
TABLE II. Wet fractions computed using SE for the proposed limiting steady asymmetric fingers for different tube widths corresponding to the experimental data presented in Fig. 3. The experimental values correspond to the steady (non-oscillatory) asymmetric solution with minimum wet frac- tion. Note that when a@,, = 2/7 the data point with m = 0.33 is not chosen because it is believed to correspond to an oscillatory solution.
In this configuration the curvature is set by the height of the obstacle and is given bi ky = 1/(1 — a; ). Hence, the total projected perimeter of the finger is FIG. 11. The geometry of an asymmetric static bubble with maximum curvature in a tube with partially occluded cross- section. The half-height of the channel is chosen to be the reference length-scale and is set to 1.
In this configuration the curvature is set by the height of the obstacle and is given bi ky = 1/(1 — a; ). Hence, the total projected perimeter of the finger is FIG. 11. The geometry of an asymmetric static bubble with maximum curvature in a tube with partially occluded cross- section. The half-height of the channel is chosen to be the reference length-scale and is set to 1.
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EngineeringPhysicsMechanicsTwo Phase FlowPipe FlowFlow Instability and TransitionPhysics of FluidsAxial Symmetry
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