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Free Surface Deformation and Cusp Formation During the Drainage of a Very Viscous Fluid

We report an experimental study of the deformation of the free surface of a very viscous fluid when drained out of a container through a circular orifice. At some critical height of the fluid, the free surface deforms and forms a dimple. At later times, this dimple becomes a cusp. We found that the height of the dimple measured from the bottom of the tank as well as the curvature of the dimple before the cusp formation, follows power laws behaviors. We found that this behavior was due to an interplay between viscous pressure and surface tension

the form of the deformed free surface are no longer the same. The dimple appears far away from the drainage hole in the vertical direction, and it evolves into a sharp dimple at early stages of the draining. In this paper, we report such effect where the free surface of a very viscous fluid deforms into a dimple when drained from a cylindrical container. At later times this dimple evolves into a cusp. Such free surface deformations are also found in selective withdrawal [2,3,4,5,6,7,8], where the flow is steady whereas our flow is gravity driven and non-steady. We, however, fitted the profile of the dimple to a power law of the z ∝ x α and found that the power α is a linear function of time. This might be an effect of the non-steadiness of the interface motion.
A scheme of the set up and the axis notation is displayed in figure 1. We used viscous liquids (from the PDMS family: called also silicon oils) which viscosity ranges from 10 to where the thread of air bubbles appear will be called t f and will be associated to h f . In other words, at h f the interface undergoes a topological transition in a manner analogous to the pinch-off of a viscous thread. At the transition, the curvature κ of the interface at the dimple becomes infinite, i.e. a cusp forms.
We measured the height of the dimple h f and its corresponding time t f with an error of 1 mm and 1 sec respectively. This is checked by looking at the video frame just before some air bubbles start nucleating at the dimple's apex. We use standard video with a frame frequency of 30 frames per second. We define the non dimensional numberh = D is the hole diameter as shown in figure 1. The flow rate is varied by varying the hole diameter. We measured the dependence ofh versus the dimensionless time scale defined in our case as τ = In this expression ν is the kinematic viscosity µ/ρ. The result is displayed in figure 3. The Log-Log plot shows a power law dependence and the line is best fit by a 2/3 power law:h ∼ τ 2/3 . We monitored, h ∞ ,which is the height of the liquid far away from the cusp region as shown in Fig. 1, versus time and found the flow rate which gives the velocity at the draining hole. The Reynolds number defined as ρU R µ is equal to 3.3 × 10 −3 for the liquid of viscosity 10 Pa. s. and is equal to 2.7 × 10 −6 for a liquid of viscosity 100 Pa.s. Where ρ is the density of the liquid, U is the flow velocity of the liquid at the hole and R is the exit-hole radius This cusp formation is analogous to the selective withdrawal and to drop pinch off. Here the critical dimple height h f is analogous to the point z 0 for thread pinch-off [7]. The distance between the current dimple height and h f , dh = h − h f , is analogous to the minimal thread radius for thread pinch-off [5,6]. As the interface's dimple sharpens to a cusp, the capillary stress γκ diverges. To check this ideas we measured the radius of curvature R of the dimple, by fitting it to a polynomial, while it is still smooth before the appearance of the bubbles, If we suppose that, before the cusp is developed, the only forces present in the system are the pressure drop at the hole and the pressure difference across the interface, then the scaling can be self consistently derived as the following: The divergence in capillary stress is balanced by a divergence in the viscous stress at the exit hole and which scales as βµQ/(dh) 3 [9], where Q is the flux of the fluid out of the drain hole, β is a number of the order of unity and µ is the viscosity. Here we are assuming that the flow associated with the dimple formation is a sink flow which is verified experimentally by monitoring the flow far and close to the hole by monitoring the flow using PIV (Particle Image Velocimetry) and we used passive tracers such as small bubbles to monitor the displacement of a volume element of the liquid. The air bubble we followed is around 200 µm in diameter with a rise velocity smaller than 10 µm/sec measured by tracking the bubble path. Since the flow at the hole has a velocity of 1 -2 mm/sec, we suppose that the ascending velocity of the tracer bubble is negligible over the time scale of the observation and the bubble follows essentially the flow. From this equation [11] we know that the only quantity on which v can depend, other than z and t is ν. From these three quantities only one dimensionless combination can be found which is η = z/(νt) 2 . From this scaling we note that the vertical distance z goes like t 1/2 . This explain the behavior r ∝ (t * − t) 1/2 .
We interpret the second result with a simple (heuristic) argument as the following: When the tracer is far from the singularity, the velocity field is the due to a dipole, since the singularity has an image across the horizontal plate, due to the no-slip condition. We know from low reynolds number hydrodynamics [9] that the dipolar contribution to the velocity field, due to a sink flow, is such that v ∼ 1/r 2 , and since the tracer is passive as we pointed it out earlier in the text, the velocity is simply dr/dt, solving in t, we find that r ∼ (t * − t) 1/3 , which is the behavior found experimentally. This proves experimentally that the flow in the tank is indeed a sink flow. This property will allow us to use the results of sink flow mainly the pressure drop at the sink which is given by βµQ/(dh) 3 . We will suppose our process as quasi-static and we can balance capillary forces and viscous forces due to the sink flow at every step of the drainage. From the balance γκ ≈ βµQ/(dh) 3 we see that close to 4 the formation of the cusp, κ ≈ 1/(dh) 3 . Measurements of κ and dh show that they have the following power law scaling with the time remaining before the formation of the cusp: κ ∝ (t f − t) −2 and dh ∝ (t f − t) 2/3 . Although the scaling is explained self-consistently, it is unclear at present what determines the values of these exponents. These exponents and the curvature drop in the inset of Fig. 4, are a signature of finite time singularity.
In summary, we have examined the motion of an air-liquid interface due to drainage at very low reynolds numbers where viscosity is dominant. We found that the prior to cusp formation, the interface height and the dimple curvature follow scaling laws dictated by the interplay between surface tension and viscous pressure at the drainage hole.  The crossover between these two tendencies lies at the hole diameter. These data points were obtained for the same run.