Session T10: Flow Instability: Interfacial and Thin Film II

The role of a deep wavy wall on the natural frequencies of a fluid interface: theory and experiments

The natural frequency of an active fluid whose free surface is exposed to a passive fluid is known to dominantly depend on the density difference of the fluids, the surface tension and the wave number of the disturbances that cause flow oscillations. In addition to these factors, patterned walls are expected to influence the natural frequency. To analyse the role of a patterned substrate with arbitrarily deep indentations a reduced order model is developed in conjunction with a Weighted Residual Integral Boundary Layer (WRIBL) technique. From the analysis, it is learned that the natural frequency generally but not always decreases as the amplitude of the wall pattern increases from a flat wall and that there are exceptional zones of disturbances. These exceptional zones appear when the wave number of the disturbance is twice that of the wall pattern provided the wall has a trigonometric cosine wave pattern and not a trigonometric sine wave. These results are substantiated both by a perturbation series analysis about the flat wall case. The numerical calculations are validated by detailed physical experiments. As a consequence in the shift of the natural frequency, the physical experiments also show a shift in the instability regions when fluid is subject to Faraday excitation.   

On the impact of viscosity ratio on falling two-layer viscous film flow inside a tube

A two-layer falling film consisting of two immiscible viscous fluids with identical density but different viscosity lining the interior of a vertical tube is studied using a long-wave asymptotic model. Linear stability analysis of the model shows an unstable `free-surface mode' consisting largely of perturbations to the free surface, and an `interfacial mode' that can either be stable for all wavenumbers, unstable for a band of long-wave wavenumbers, or unstable only to a band of wavenumbers bounded away from zero. These instabilities grow outside the linear regime and either saturate as a series of waves or continue to grow so that the free surface at a wave crest tends to the center of the tube in finite time, indicating the formation of a plug. Families of traveling wave solutions are found by continuation from Hopf bifurcations that arise due to the instability of one or both modes. The free-surface traveling waves have a turning point that indicates a critical thickness required for plug formation to occur; decreasing the viscosity of the outer layer decreases this critical thickness so that plugs form more readily. The impact on plug formation of fixing the volume flux of each layer, rather than the layer thicknesses, is discussed as well.   

Not Participating

We numerically study the thin liquid film stability via multi-component lattice Boltzmann simulations. The thin liquid film instability triggers a dewetting process, i.e., the reverse process to the spontaneous spreading of a liquid droplet on a solid surface, which takes place when the liquid film is forced to stay in contact with a hydrophobic wall. This situation leads the system to live in a metastable state, whose stability properties depend on the initial height of the film as well as on the surface wetting properties. In this work, we first controlled if our in-house Shan-Chen multi-component lattice Boltzmann model is valid to investigate the break-up of a thin liquid film placed on a flat wall: an infinitesimal perturbation of the initial interface of the liquid film is observed to be sufficient to trigger the film-to-droplet rupture. In this case, we investigated the conditions that trigger this transition in terms of the film surface tension, initial film height, and wetting properties of the flat surface. A further difficulty is then introduced, by analysing how strongly the presence of a solid surface with complex geometry impacts the stability scenario.   

Nonlinear periodic waves on ferrofluid interfaces

We demonstrate that a combination of a radial and azimuthal external static magnetic field causes a linearly unstable circular ferrofluid interface confined in a Hele-Shaw cell to evolve into a stably spinning "gear", driven by interfacial waves. Through weakly nonlinear analysis, we show that the rotation speed can be predicted, which is confirmed by fully nonlinear simulations using a sharp-interface Lagrangian method. To better understand these nonlinear interfacial traveling waves, we derive a long-wave equation for a ferrofluid thin film subject to an angled magnetic field. Interestingly, this new type of generalized Kuramoto--Sivashinsky equation exhibits nonlinear periodic waves as dissipative solitons. A multiple-scale analysis enables the prediction of the nonlinear propagation velocity and further reveals how the linear instability is arrested by the saturating nonlinearity. This new long-wave equation has rich dynamics, such as transitions between different nonlinear periodic states and long-lived multi-periodic wave profiles. These observations shed light on nonlinear interfacial phenomena involving ferrofluids driven by non-uniform magnetic fields, towards their control.   

The interaction of thin elastomeric substrates with viscous flows: From elastohydrodynamic sedimentation to Hele-Shaw flow

Elastic substrates bounding fluid flows are common in many experimental and industrial settings; their principal purpose is usually to drive or suppress the flow of fluid. The deformation of such elastic layers is becoming increasingly recognised, but the model appropriate for describing its deformation depends on how close to incompressible the coating is. While Poisson's ratios of the most frequently used materials are quoted as being in the range 0.49–0.5, the precise value may change the behaviour of the coating and have knock-on consequences. We will present a model for thin, near-incompressible, elastic foundations, and discuss how its behaviour when coupled to flows at low Reynolds numbers can depend sensitively on how incompressible the coating is. We will demonstrate this by considering two concrete problems: the sedimentation of a cylinder in a viscous fluid above a thin elastic foundation and Hele-Shaw flow bounded by an elastic layer, which has recently been shown to exhibit a flow choking instability.   

Interfacial Instabilities in a Hele-Shaw Cell

A Hele-Shaw cell consists of a thin gap between two parallel plates, along which fluid is forced to flow, providing a constrained geometry suitable for experimental and theoretical study. When one fluid is pumped through the cell to displace a second fluid, the interface between them can, in some cases, develop in an unstable fashion. The Newtonian Saffman-Taylor problem is one such example that has been well studied, predicting an instability driven by the viscosity difference between the two fluids. When the less viscous fluid displaces the more viscous one, the interface is unstable; but if the more viscous fluid displaces the less viscous one, the interface remains stable. The goal of the current study is to explore and analyse two other types of instabilities. The first one is the viscoplastic case of the Saffman-Taylor instability, replacing the displaced Newtonian fluid with a yield stress fluid. The second one is a fracturing instability that occurs when water is displaced by a yield stress fluid. In this case, the theory predicts no Saffman-Taylor instability and thus a stable interface, because the viscoplastic fluid has a higher effective viscosity than water. However, instead of growing asymmetrically, the interface cracks and leads the flow into flower-like shapes.   

Experimental Characterization of Pressure Gradients in the Viscous Fingering Instability

The viscous fingering instability occurs when a less viscous fluid is injected into a more viscous one within a confined geometry such as the thin gap between the plates of a radial Hele-Shaw cell. We determine the mid-plane velocities of our fluids in this cell using alternate injection of dyed and undyed (equal-viscosity) volumes of the inner or outer fluid. While it is apparent that the outer interface at the finger is moving much faster than the inner interface at adjacent valleys, it is observed that this velocity difference is also present significantly away from the interface within the bulk of the inner fluid. By examining velocities as a function of the distance behind the trailing interface, r_inner , we find that the difference in velocities behind a finger and its adjacent valley, △V, depends on the length of the fingers, L_finger, and decays exponentially with distance from the interface as: r*△V = △V₀ exp [-(r_inner – r)/λ]. We also find associated decay length, λ , is  proportional to L_finger. Using this technique we are able to infer an azimuthal pressure gradient.    

The effect of oscillatory translational shear on viscous fingering in miscible fluids

Viscous fingering is an instability that results from the displacement of a more viscous fluid by a less viscous one in a narrow gap, such as between the two plates of a Hele-Shaw cell.  Previous studies showed that the fingering patterns are related to the interfacial structure in the thin dimension traversing the gap.  Here we study if the system can be stabilized against fingering by shearing the two plates with respect to one another.  Shearing the fluid in this way can reduce the viscosity contrast across the interface.  In our experiments, we inject fluid radially from a central hole in one plate while an oscillatory uniaxial translational shear is applied. We find that there is a delay in the instability onset and a suppression in finger length for fingers growing in the direction parallel to the shear as compared to those growing in the perpendicular direction.   

An Experimental and Numerical insight into Diffusive Zones in Radial Miscible Viscous Fingering

Displacement of a high viscous fluid by a less viscous one in a porous medium deforms the interface into finger like patterns, a hydrodynamic instability termed as viscous fingering (VF). Many researchers have carried out research to understand VF due to its application in a variety of fields viz., enhanced oil recovery, CO₂ sequestration, mixing industry, point-of-care devices, to name a few. The growth of VF patterns is known to demonstrate the proportionate growth in humans. A recent study experimentally reported a diffusion dominated zone following the one dominated by convection for radial displacement of miscible fluids. Diffusion is known to stabilize miscible VF and diffusive effects are strong during initial stages of flow when concentration gradient is high. Thus, a diffusion dominated zone is anticipated before convection overtakes and results in VF. However, this initial diffusion driven zone is missing in literature. We provide experimental as well as numerical evidences for this zone for a variety of radial displacements with a point source and a finite source containing both the fluids inside the Hele-Shaw cell.   

Not Participating

Molten liquid flows that cool as they spread are important in a wide variety of contexts, e.g., lava domes in geophysical flows, reactor core melt in nuclear engineering and molten metal coating flows in chemical and metallurgical engineering. The interplay between the flow and cooling can give rise to a variety of intriguing flow features and fingering instabilities.

Motivated by the above, we consider theoretically a model system of a molten viscous planar liquid dome extruding from a source and spreading over an inclined substrate. The liquid in the dome cools as it spreads, losing its heat to the surrounding colder air and substrate. Lubrication theory is employed to model the spreading flow using coupled nonlinear evolution equations for the dome’s thickness and temperature. The coupling between flow and cooling is via a constitutive relationship for the temperature-dependent viscosity. This model is parameterized by the heat transfer coefficients at both the dome-air and dome-substrate interfaces, the Péclet number, the viscosity-temperature coupling parameter and the substrate inclination angle. A systematic exploration of the parameter space reveals a variety of 1D free surface shapes illustrating the dynamics of a spreading flow undergoing cooling. In particular, we distinguish a new type of solution with the hotter and more mobile liquid piling up behind the dome's colder and less mobile leading edge, forming a distinct hump. A mapping in parameter space is constructed to characterize the existence of the humped solution.

The stability of the 1D hump solution to small-amplitude transverse variations is investigated using linear stability analysis and numerical simulations. The existence of a thermo-viscous fingering instability is revealed; for this instability to occur the presence of the hump is shown to be necessary. Two-dimensional simulations confirm the stability analysis elucidating the underlying thermo-viscous mechanism.   

Network formation as the fingering instability and its inverse problem

Many natural growth processes can be described as a Stefan problem, where the boundary between two phases can move with time. A specific example of such growth is Saffman-Taylor instability, and speaking more generally, Laplacian growth, in which the interface moves with the speed proportional to the gradient of a harmonic field. The growing interface is in general unstable to perturbations that evolve into fingers. The fingers frequently split as they grow, with the daughter branches competing with each other for the available flux. This results in a formation of a ramified, network-like pattern.

In our study, we take a limit of infinitely thin fingers, which grow only at their tips. Such description can be used to understand the formation of blood vessels, river networks, leaf venation or dielectric breakdown patterns, only to mention a few.

We then formulate the inverse problem – can the growth dynamics be inferred from the analysis of the final geometrical structure of the network? We approach this problem numerically, developing tools to simulate the growth and to extract growth laws of the network based on the final pattern. The reliability of the method is validated against synthetic data.   

The dewetting of a thin film sandwiched between a solid substrate and an unbounded liquid bath

We report the dynamics of a thin liquid film of viscosity μ sandwiched between a solid substrate and an unbounded liquid bath of viscosity λμ. In the limit of negligible inertia, the flow depends on two non-dimensional parameters, namely λ and a dimensionless measure of the relative strengths of capillarity and van der Waals forces. We first analyse the linear temporal stability revealing that when the viscosity of the outer bath is much larger than that of the film, λ>>1, the most amplified wavenumber decreases as  k_m∼λ^-1/3, indicating that very slender dewetting structures are expected when λ becomes large. We then march in time the complete Stokes equations to investigate the spatial structure of the flow close to rupture revealing that, for λ<∞, the flow becomes self-similar with the minimum film thickness scaling as  h_min=K(λ) τ^1/3 when τ→0, being τ the time remaining before rupture. The presence of an outer liquid bath affects the self-similar structure obtained by Moreno-Boza et al. 2020 through the prefactor of the film thinning law, K(λ), and the opening angle of the self-similar film shape, θ(λ). For negligible film viscosity, λ→∞, the near-rupture flow is shown to be also self-similar but non-universal, in that the initial conditions dictate the final film shape close to rupture.