Flow In Hele-Shaw Cell Research Articles

Energetics and mixing of thermally driven flows in Hele-Shaw cells

Thermally driven flows in fractures play a key role in enhancing the heat transfer and fluid mixing across the Earth's lithosphere. Yet the energy pathways in such confined environments have not been characterised. Building on Letelier et al. (J. Fluid Mech., vol. 864, 2019, pp. 746–767), we introduce novel expressions for energy transfer rates – energetics – of geometrically constrained Rayleigh–Bénard convection in Hele-Shaw cells (HS-RBC) based on two different conceptual frameworks. First, we derived the energetics following the well-established framework introduced by Winters et al. (J. Fluid Mech., vol. 289, 1995, pp. 115–128), in which the gravitational potential energy, $E_{\textit {p}}$ , is decomposed into its available, $E_{\textit {ap}}$ , and background, $E_{\textit {bp}}$ , components. Secondly, we derived the energetics considering a new decomposition for $E_{\textit {p}}$ , named dynamic, $E_{\textit {dp}}$ , and reference, $E_{\textit {rp}}$ , potential energies; $E_{\textit {dp}}$ is defined as the departure of the system's potential energy from the reference state $E_{\textit {rp}}$ , determined by the ‘energy’ of the scalar fluctuations. For HS-RBC, both frameworks lead to the same energy transfer rates at a steady state, satisfying the relationship $\langle E_{\textit {ap}} \rangle _{\tau } = \langle E_{\textit {dp}} \rangle _{\tau } + 1/6$ . Consistent with the work by Hughes et al. (J. Fluid Mech., vol. 729, 2013) on three-dimensional Rayleigh–Bénard convection, we report analytical expressions for the energetics and efficiencies of HS-RBC in terms of the Rayleigh number and the global Nusselt number. Additionally, we performed numerical experiments to illustrate the application of the energetics for the analysis of HS-RBC. Finally, we discuss the impact of the thermal forcing and the geometrical control exerted by Hele-Shaw cells on the development of boundary layers, protoplumes and the self-organisation of large-scale flows.

Stability and hysteresis of Faraday waves in Hele-Shaw cells

The instability of Faraday waves in Hele-Shaw cells is investigated experimentally and theoretically. A novel hydrodynamic model involving capillary action is proposed to capture the variation of the dynamic contact line between two close walls of narrow containers. The amplitude equations are derived from the gap-averaged model. By means of Lyapunov’s first method, a good prediction of the onset threshold of forcing acceleration is obtained, which shows the model’s validity for addressing the stability problem for Faraday waves in Hele-Shaw cells. It is found that the effect of the dynamic contact line is much greater than that of Poiseuille assumption of velocity profile for the cases under investigation. A new dispersion relation is obtained, which agrees well with experimental data. However, we highly recommend the conventional dispersion relation for gravity–capillary waves, which can generally meet common needs. Surface tension is found to be a key factor of interface flows in Hele-Shaw cells. According to our experimental observations, a liquid film is found on the front wall of the Hele-Shaw cell when the wave is falling. As a property of the friction coefficient from molecular kinetics, wet and dry plates show different wetting procedures. Unlike some authors of previous publications, we attribute the hysteresis to the out-of-plane interface shape rather than to detuning, i.e. the difference between natural frequency and response frequency.

Flow-induced phase separation of active particles is controlled by boundary conditions

Active particles, including swimming microorganisms, autophoretic colloids, and droplets, are known to self-organize into ordered structures at fluid-solid boundaries. The entrainment of particles in the attractive parts of their spontaneous flows has been postulated as a possible mechanism underlying this phenomenon. Here, combining experiments, theory, and numerical simulations, we demonstrate the validity of this flow-induced ordering mechanism in a suspension of active emulsion droplets. We show that the mechanism can be controlled, with a variety of resultant ordered structures, by simply altering hydrodynamic boundary conditions. Thus, for flow in Hele-Shaw cells, metastable lines or stable traveling bands can be obtained by varying the cell height. Similarly, for flow bounded by a plane, dynamic crystallites are formed. At a no-slip wall, the crystallites are characterized by a continuous out-of-plane flux of particles that circulate and re-enter at the crystallite edges, thereby stabilizing them. At an interface where the tangential stress vanishes, the crystallites are strictly 2D, with no out-of-plane flux. We rationalize these experimental results by calculating, in each case, the slow viscous flow produced by the droplets and the long-ranged, many-body active forces and torques between them. The results of numerical simulations of motion under the action of the active forces and torques are in excellent agreement with experiments. Our work elucidates the mechanism of flow-induced phase separation in active fluids, particularly active colloidal suspensions, and demonstrates its control by boundaries, suggesting routes to geometric and topological phenomena in an active matter.

Some geometrical properties of free boundaries in the Hele-Shaw flows

In this paper, we are concerned with certain geometric properties of the moving boundary in the case of two-dimensional viscous fluid flows in Hele-Shaw cells under injection. We study the invariance in time of free boundary for such a bounded flow domain under the assumption of zero surface tension. By applying various results in the theory of univalent functions, we consider the invariance in time of starlikeness of a complex order, almost starlikeness of order α ∈ [0, 1), and almost spirallikeness of type γ∈(−π/2,π/2) and order α ∈ (0, cos γ). This work complements recent work on planar Hele-Shaw flow problems in the case of zero surface tension.

Invariant Geometric Properties in Hele-Shaw Flows

In this paper, we investigate some geometric properties of the moving frontier of a viscous fluid for planar flows in Hele-Shaw cells under injection. We study invariance in time properties of the free boundary for bounded and unbounded domains (with bounded complement) under the assumption of zero surface tension. By applying certain results in the theory of univalent functions we partially solve an open problem of Vasil’ev concerning the invariance in time of starlikeness of order \(\alpha , \alpha \in [0,1)\), for bounded domains. In the case of unbounded domains with bounded complement we analyze the invariance in time of convexity of order \(\alpha , \alpha \in [0,1)\).

On some invariant geometric properties in Hele-Shaw flows with small surface tension

In this paper, by applying methods from complex analysis, we analyse the time evolution of the free boundary of a viscous fluid for planar flows in Hele-Shaw cells under injection in the non-zero surface tension case. We study the invariance in time of α-convexity (for α ∈ [0, 1] this is a geometric property which provides a continuous passage from starlikeness to convexity) for bounded domains. In this case we show that the α-convexity property of the moving boundary in a Hele-Shaw flow problem with small surface tension is preserved in time for α ≤ 0. For unbounded domains (with bounded complement) we prove the invariance in time of convexity.

Image registration algorithm for molecular tagging velocimetry applied to unsteady flow in Hele-Shaw cell

In order to develop velocimetry methods for confined geometries, we propose to combine image registration and volumetric reconstruction from a monocular video of the draining of a Hele-Shaw cell filled with water. The cell’s thickness is small compared to the other two dimensions (e.g. 1×400×800mm3). We use a technique known as molecular tagging which consists in marking by photobleaching a pattern in the fluid and then tracking its deformations. The evolution of the pattern is filmed with a camera whose principal axis coincides with the cell’s gap. The velocity of the fluid along this direction is not constant. Consequently, tracking the pattern cannot be achieved with classical methods because what is observed is the integral of the marked molecules over the entire cell’s gap. The proposed approach is built on top of direct image registration that we extend to specifically model the volumetric image formation. It allows us to accurately measure the motion and the velocity profiles for the entire volume (including the cell’s gap) which is something usually hard to achieve. The results we obtained are consistent with the theoretical hydrodynamic behaviour for this flow which is known as the Poiseuille flow.

Viscous lock-exchange in rectangular channels

In a viscous lock-exchange gravity current, which describes the reciprocal exchange of two fluids of different densities in a horizontal channel, the front between two Newtonian fluids spreads as the square root of time. The resulting diffusion coefficient reflects the competition between the buoyancy-driving effect and the viscous damping, and depends on the geometry of the channel. This lock-exchange diffusion coefficient has already been computed for a porous medium, a two-dimensional (2D) Stokes flow between two parallel horizontal boundaries separated by a vertical height H and, recently, for a cylindrical tube. In the present paper, we calculate it, analytically, for a rectangular channel (horizontal thickness b and vertical height H) of any aspect ratio (H/b) and compare our results with experiments in horizontal rectangular channels for a wide range of aspect ratios (1/10 to 10). We also discuss the 2D Stokes–Darcy model for flows in Hele-Shaw cells and show that it leads to a rather good approximation, when an appropriate Brinkman correction is used.

Chaotic mixing efficiency in different geometries of Hele-Shaw cells

Flows in Hele-Shaw cells are generally laminar, and can be in a first approximation considered as quasi-two-dimensional. Therefore, the only way to mix efficiently inside the cell is to generate a periodic flow and thus create chaos. Here we study the influence of the geometry of the cell on mixing efficiency: to this aim, we use three different geometries of the cell (circle, square and rectangle) and we numerically test them by using two protocols of mixing: the first protocol uses syringes, the second uses pumps. In a view of simplicity, this Hele-Shaw flow is modeled as two-dimensional, allowing us to analytically determine the velocity field corresponding to the different shapes. The results are then analyzed in terms of Poincaré sections (appearance, filling rate, and homogeneity), and the Lyapunov exponents are calculated. We numerically show that the rectangular geometry leads to a better mixing, but also that the aspect ratio of the rectangle plays unexpectedly no important role on mixing.

A201 ホログラフィー再生像の転写によるマイクロスケール流れの可視化

A photochromic method for micro flow visualization has been developed to measure flow fields in micron-scale fluidic devices. The system utilizes the reconstructed images by computer-generated holograms with a UV laser to produce dye streaks in the flow of fluids. Two-dimensional flows in Hele-Shaw cells, such as the flows through a micron-scale aperture and the flow induced by a source and a sink inside a circular micro region, are visualized by using the streak lines generated by a reconstructed image with suitable circle dot patterns. The dotted circle enclosing a source is found to produce useful streak lines for visualization of source and sink flows. The corresponding potential flows are obtained which agree well in flow patterns with the experimental results.

Dynamic blocking of W/O emulsion in a radially expanding flow in Hele-Shaw cell

It is studied the changing of structure of the radial expanding flow of highly concentrated stable reverse water-oil emulsion in a flat capillary channel. The effect of dynamic blocking is discovered, which is that despite the permanent pressure drop between the center and outer forming cells, the emulsion flow with time stands still. Microflows are shown in a state of ”blocking“. The various methods of influencing the system in a state of ”blocking“ are tested. It is found that the movement is resumed in a strong ultrasonic fields.

Planar Elongational Rheometry Using Slit Entry Flow in Hele-Shaw Cell

In our previous study, we observed Hele-Shaw flows of two solutions of a polymer material around two cylinders and through two slits with different lengths. Based on the results, we showed that the potential flow analogy by the Hele-Shaw flow holds when the flow rate is very low in spite of the shear-thinning viscosity and elasticity of the fluids in steady shear flow. It was suggested that the potential flow pattern of the polymer solutions in the Hele-Shaw cell is disturbed by the effect of elongational stress. In this study, a simple method for planar elongational viscosity measurement is devised using the converging flow upstream the slit entry; the elongation rate is estimated from dye streak pattern in the inflow region formed upstream the slit entrance and the elongational stress is obtained from integrated equations of motion. The planar elongational rheometry presented in this work is advantageous due to its applicability to mobile fluids, in addition to its simple experimental set up.

Robin's Modulus in a Hele-Shaw Problem*

Recently, Robin's function and Robin's capacity, generalizations of Green's function and logarithmic capacity respectively, were studied intensively in connection with potential-theoretic problems as well as with physical problems, in particular, their interpretations in terms of minimum energy. Our goal is to give another physical interpretation that comes from free boundary problems for a viscous flow in Hele-Shaw cells with an obstacle inside. We will connect the rate of area change of the phase domain with the rate of change of Robin's reduced modulus of the free boundary in order to get an isoperimetric inequality.

Univalent Functions in Two-Dimensional Free Boundary Problems

The main goal of the paper is to bring together methods of the classical theory of univalent functions and some problems of fluid mechanics. Our interest centers on free boundary problems. We study the time evolution of the free boundary of a viscous fluid in the zero- and nonzero-surface-tension models for planar flows in Hele-Shaw cells either with an extending to infinity free boundary or with a bounded free boundary. We consider special classes of univalent functions that admit an explicit geometric interpretation to characterize the shape of the free interface. Another model is two-dimensional solidification/melting of a nucleus in a forced flow.

On the geometry of Hele-Shaw flows with small surface tension

We study the time evolution of the free boundary of a viscous fluid in the non-zero-surface-tension models for planar flows in Hele-Shaw cells under injection. Applying methods of conformal map we prove that certain geometric properties, such as starlikeness and directional convexity, are preserved in time.

Univalent Functions in the Dynamics of Viscous Flows

The main goal of the paper is to apply methods of the theory of univalent functions to some problems of fluid mechanics. Our interest centers on free boundary problems. We study the time evolution of the free boundary of a viscous fluid in the zero- and non-zero-surface-tension models for planar flows in Hele-Shaw cells either with an extending to infinity free boundary or with a bounded free boundary. We consider special classes of univalent functions which admit an explicit geometric interpretation to characterize the shape of the free interface. The paper contains a survey part as well as new results. We also set up some new problems.

Parallel flow in hele-shaw cells with ferrofluids

Parallel flow in a Hele-Shaw cell occurs when two immiscible liquids flow with relative velocity parallel to the interface between them. The interface is unstable due to a Kelvin-Helmholtz type of instability in which fluid flow couples with inertial effects to cause an initial small perturbation to grow. Large amplitude disturbances form stable solitons. We consider the effects of applied magnetic fields when one of the two fluids is a ferrofluid. The dispersion relation governing mode growth is modified so that the magnetic field can destabilize the interface even in the absence of inertial effects. However, the magnetic field does not affect the speed of wave propogation for a given wave number. We note that the magnetic field creates an effective interaction between the solitons.

Saffman–Taylor instability in yield-stress fluids

When a fluid is pushed by a less viscous one the well-known Saffman–Taylor instability phenomenon arises, which takes the form of fingering. Since this phenomenon is important in a wide variety of applications involving strongly non-Newtonian fluids – in other words, fluids that exhibit yield stress – we undertake a full theoretical examination of Saffman–Taylor instability in this type of fluid, in both longitudinal and radial flows in Hele-Shaw cells. In particular, we establish the detailed form of Darcy's law for yield-stress fluids. Basically the dispersion equation for both flows is similar to equations obtained for ordinary viscous fluids but the viscous terms in the dimensionless numbers conditioning the instability contain the yield stress. As a consequence the wavelength of maximum growth can be extremely small even at vanishing velocities. Additionally an approximate analysis shows that the fingers which are left behind at the beginning of destabilization should tend to stop completely. Fingering of yield-stress fluids therefore has some peculiar characteristics which nevertheless are not sufficient to explain the fractal pattern observed with colloidal systems.

Secondary flow in a Hele-Shaw cell

We examine the flow in a horizontal Hele-Shaw cell in which the undisturbed unidirectional flow at infinity is required to stream around a vertical cylinder spanning the gap between the two (horizontal) plates of the cell. A combination of matched asymptotic expansions and numerical methods is employed to elucidate the structure of the boundary layer near the surface of the cylinder. The two length scales of the problem are the gap, h, and the length of the body, l; it is assumed that h/l<<1. The characteristic Reynolds number based on l is O(1). The length scales associated with the boundary layer and the classical Hele-Shaw flow pattern are O(h) and O(l), respectively.It is found that the boundary layer contains streamwise vorticity. This vorticity is generated at the three no-slip surfaces (the two plates and the cylinder wall) as a result of the cross-flow induced by the streamwise acceleration/deceleration of the flow around the curved cylinder. The strength of the secondary flow, hence the associated streamwise vorticity, is proportional to changes in body curvature. The validity of the classical Hele-Shaw flow is examined systematically, and higher-order corrections are worked out. This results in a displacement thickness that is roughly 30% of the gap. In other words, the lowest-order correction to the classical Hele-Shaw flow may be obtained by requiring the outer flow (on the scale O(l)) to satisfy the no-penetration boundary condition on a displaced cylinder surface. The boundary layer contains ‘corner’ vortices at the intersections of the horizontal plates and the vertical cylinder surface.

Fracture-like patterns and dendrites from perturbed saffman-taylor flows

Abstract We have studied the effect that two different perturbations have on the flow in Hele-Shaw cells. The introduction of geometrical anisotropy in the cell adds a rich variety of new morphologies whose dynamics reveal interesting regularities. The use of viscoelastic materials, on the other hand, has allowed for the observation of a dynamical transition from viscous fingering into fracture-like behaviour, with many intriguing regularities and with several open questions.