Microfluidic devices for manipulating fluids are widespread and finding uses in many scientific and industrial contexts. Their design often requires unusual geometries and the interplay of multiple physical effects such as pressure gradients, electrokinetics, and capillarity. These circumstances lead to interesting variants of well-studied fluid dynamical problems and some new fluid responses. We provide an overview of flows in microdevices with focus on electrokinetics, mixing and dispersion, and multiphase flows. We highlight topics important for the description of the fluid dynamics: driving forces, geometry, and the chemical characteristics of surfaces.

Engineering flows in small devices

Macroscopic thin liquid films are entities that are important in biophysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, rheologically complex materials such as polymers solutions or melts, or complex mixtures of phases or components. When the films are subjected to the action of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepening, and development of chaotic responses. Such films can display rupture phenomena creating holes, spreading of fronts, and the development of fingers. In this review a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations. As a result of this long-wave theory, a mathematical system is obtained that does not have the mathematical complexity of the original free-boundary problem but does preserve many of the important features of its physics. The basics of the long-wave theory are explained. If, in addition, the Reynolds number of the flow is not too large, the analogy with Reynolds's theory of lubrication can be drawn. A general nonlinear evolution equation or equations are then derived and various particular cases are considered. Each case contains a discussion of the linear stability properties of the base-state solutions and of the nonlinear spatiotemporal evolution of the interface (and other scalar variables, such as temperature or solute concentration). The cases reducing to a single highly nonlinear evolution equation are first examined. These include: (a) films with constant interfacial shear stress and constant surface tension, (b) films with constant surface tension and gravity only, (c) films with van der Waals (long-range molecular) forces and constant surface tension only, (d) films with thermocapillarity, surface tension, and body force only, (e) films with temperature-dependent physical properties, (f) evaporating/condensing films, (g) films on a thick substrate, (h) films on a horizontal cylinder, and (i) films on a rotating disc. The dynamics of the films with a spatial dependence of the base-state solution are then studied. These include the examples of nonuniform temperature or heat flux at liquid-solid boundaries. Problems which reduce to a set of nonlinear evolution equations are considered next. Those include (a) the dynamics of free liquid films, (b) bounded films with interfacial viscosity, and (c) dynamics of soluble and insoluble surfactants in bounded and free films. The spreading of drops on a solid surface and moving contact lines, including effects of heat and mass transport and van der Waals attractions, are then addressed. Several related topics such as falling films and sheets and Hele-Shaw flows are also briefly discussed. The results discussed give motivation for the development of careful experiments which can be used to test the theories and exhibit new phenomena.

Long-scale evolution of thin liquid films

This article describes the process of formation of droplets and bubbles in microfluidic T-junction geometries. At low capillary numbers break-up is not dominated by shear stresses: experimental results support the assertion that the dominant contribution to the dynamics of break-up arises from the pressure drop across the emerging droplet or bubble. This pressure drop results from the high resistance to flow of the continuous (carrier) fluid in the thin films that separate the droplet from the walls of the microchannel when the droplet fills almost the entire cross-section of the channel. A simple scaling relation, based on this assertion, predicts the size of droplets and bubbles produced in the T-junctions over a range of rates of flow of the two immiscible phases, the viscosity of the continuous phase, the interfacial tension, and the geometrical dimensions of the device.

http://ichf.pong.pl/i/publications/29.pdf

Formation of droplets and bubbles in a microfluidic T-junction-scaling and mechanism of break-up.

The proximity effect at superconductor-ferromagnet interfaces produces damped oscillatory behavior of the Cooper pair wave function within the ferromagnetic medium. This is analogous to the inhomogeneous superconductivity, predicted long ago by Fulde and Ferrell (P. Fulde and R. A. Ferrell, 1964, ``Superconductivity in a strong spin-exchange field,'' Phys. Rev. 135, A550--A563), and by Larkin and Ovchinnikov (A. I. Larkin and Y. N. Ovchinnikov, 1964, ``Inhomogeneous state of superconductors,'' Zh. Eksp. Teor. Fiz. 47, 1136--1146 [Sov. Phys. JETP 20, 762--769 (1965)]), and sought by condensed-matter experimentalists ever since. This article offers a qualitative analysis of the proximity effect in the presence of an exchange field and then provides a description of the properties of superconductor-ferromagnet heterostructures. Special attention is paid to the striking nonmonotonic dependence of the critical temperature of multilayers and bilayers on the ferromagnetic layer thickness as well as to the conditions under which ``$\ensuremath{\pi}$'' Josephson junctions are realized. Recent progress in the preparation of high-quality hybrid systems has permitted the observation of many interesting experimental effects, which are also discussed. Finally, the author analyzes the phenomenon of domain-wall superconductivity and the influence of superconductivity on the magnetic structure in superconductor-ferromagnet bilayers.

Proximity effects in superconductor-ferromagnet heterostructures

The dynamics and stability of thin liquid films have fascinated scientists over many decades: the observations of regular wave patterns in film flows down a windowpane or along guttering, the patterning of dewetting droplets, and the fingering of viscous flows down a slope are all examples that are familiar in daily life. Thin film flows occur over a wide range of length scales and are central to numerous areas of engineering, geophysics, and biophysics; these include nanofluidics and microfluidics, coating flows, intensive processing, lava flows, dynamics of continental ice sheets, tear-film rupture, and surfactant replacement therapy. These flows have attracted considerable attention in the literature, which have resulted in many significant developments in experimental, analytical, and numerical research in this area. These include advances in understanding dewetting, thermocapillary- and surfactant-driven films, falling films and films flowing over structured, compliant, and rapidly rotating substrates, and evaporating films as well as those manipulated via use of electric fields to produce nanoscale patterns. These developments are reviewed in this paper and open problems and exciting research avenues in this thriving area of fluid mechanics are also highlighted.

Dynamics and stability of thin liquid films

Understanding the role of thin films in porous media is vital if wettability is to be elucidated at the pore level. The type and thickness of films coating pore walls determines reservoir wettability and whether or not reservoir rock can be altered from its initial state of wettability. Pore shape, especially pore wall curvature, is an important factor in determining wetting-film thicknesses. Yet, pore shape and the physics of thin wetting films are generally neglected in models of flow in porous rocks. This paper incorporates thin-film forces into a collection of star-shaped capillary tubes model to describe the geological development of mixed-wettability in reservoir rock. Here, mixed-wettability refers to continuous and distinct oil and water-wetting surfaces coexisting in the porous medium. The proposed model emphasizes the remarkable role of thin films. New pore-level fluid configurations arise that are quite unexpected. For example, efficient water displacement of oil (i.e, low residual oil saturation) characteristic of mixed-wettability porous media is ascribed to interconnected oil lenses or rivulets which bridge the walls adjacent to a pore corner. Predicted residual oil saturations are approximately 35 % less in mixed-wet rock compared to completely water-wet rock. Calculated capillary pressure curves mimic those of mixed-wet porousmore » media in the primary drainage of water, imbibition of water, and secondary drainage modes. Amott-Harvey indices range from {minus}0.18 to 0.36 also in good agreement with experimental values. (Morrow et al, 1986; Judhunandan and Morrow, 1991).« less

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A pore-level scenario for the development of mixed-wettability in oil reservoirs

Foam in porous media exhibits an unusually high apparent viscosity, making it useful in many industrial processes. The rheology of foam, however, is complex and not well understood. Previous pore-level models of foam are based primarily on studies of bubble flow in circular capillaries. A circular capillary, however, lacks the corners that characterize the geometry of the pores. We study the pressure–velocity relation of bubble flow in polygonal capillaries. A long bubble in a polygonal capillary acts as a leaky piston. The ‘piston’ is reluctant to move because of a large drag exerted by the capillary sidewalls. The liquid in the capillary therefore bypasses the bubble through the leaky corners at a speed an order higher than that of the bubble. Consequently, the pressure work is dissipated predominantly by the motion of the fluid and not by the motion of the bubble. This is opposite to the conclusion based on bubble flow in circular capillaries. The discovery of this new flow regime reconciles two groups of contradictory foam-flow experiments.Part 1 of this work studies the fluid films deposited on capillary walls in the limit Ca → 0 (Ca ≡ μU/σ, where μ is the fluid viscosity, U the bubble velocity, and σ the surface tension). Part 2 (Wong et al. 1995) uses the film profile at the back end to calculate the drag of the bubble. Since the bubble length is arbitrary, the film profile is determined here as a general function of the dimensionless downstream distance x. For 1 [Lt ] x [Lt ] Ca−1, the film profile is frozen with a thickness of order Ca2/3 at the centre and order Ca at the sides. For x ∼ Ca−1, surface tension rearranges the film at the centre into a parabolic shape while the film at the sides thins to order Ca4/3. For x [Gt ] Ca−1, the film is still parabolic, but the height decreases as film fluid leaks through the side constrictions. For x ∼ Ca−5/3, the height of the parabola is order Ca2/3. Finally, for x [Gt ] Ca−5/3, the height decreases as Ca1/4x−1/4.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112095001443

The motion of long bubbles in polygonal capillaries. Part 1. Thin films

This work determines the pressure–velocity relation of bubble flow in polygonal capillaries. The liquid pressure drop needed to drive a long bubble at a given velocity U is solved by an integral method. In this method, the pressure drop is shown to balance the drag of the bubble, which is determined by the films at the two ends of the bubble. Using the liquid-film results of Part 1 (Wong, Radke & Morris 1995), we find that the drag scales as Ca2/3 in the limit Ca → 0 (Ca μU/σ, where μ is the liquid viscosity and σ the surface tension). Thus, the pressure drop also scales as Ca2/3. The proportionality constant for six different polygonal capillaries is roughly the same and is about a third that for the circular capillary.The liquid in a polygonal capillary flows by pushing the bubble (plug flow) and by bypassing the bubble through corner channels (corner flow). The resistance to the plug flow comes mainly from the drag of the bubble. Thus, the plug flow obeys the nonlinear pressure–velocity relation of the bubble. Corner flow, however, is chiefly unidirectional because the bubble is long. The ratio of plug to corner flow varies with liquid flow rate Q (made dimensionless by σa2/μ, where a is the radius of the largest inscribed sphere). The two flows are equal at a critical flow rate Qc, whose value depends strongly on capillary geometry and bubble length. For the six polygonal capillaries studied, Qc [Lt ] 10−6. For Qc [Lt ] Q [Lt ] 1, the plug flow dominates, and the gradient in liquid pressure varies with Q2/3. For Q [Lt ] Qc, the corner flow dominates, and the pressure gradient varies linearly with Q. A transition at such low flow rates is unexpected and partly explains the complex rheology of foam flow in porous media.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112095001455

The motion of long bubbles in polygonal capillaries. Part 2. Drag, fluid pressure and fluid flow

Deposition and Thinning of the Human Tear Film

Introduction Understanding the role of thin films in porous media is vital if wettability is to be elucidated at the pore level. The type and thickness of films coating pore walls determines reservoir wettability and whether or not reservoir rock can be altered from its inlial state of wettability. Pore shape. especially pore wall curvature, is an important factor in determining wetting film thickness. Yet, pore shape and the physics of thin wetting films are generally neglected in models of flow in porous rocks. This paper incorporates thin wetting film forces into a collection of capillary tubes model to describe the geological development of so-called mixed-wettability in reservoir rock. Our model emphasizes the remarkable role of thin films. New pore-level fluid configurations arise that are quite unexpected. For example, efficient water displacement of oil (i.e., low residual oil saturation) characteristic of mixed-wettability porous media comes about due to interconnected oil lenses or rivulets which bridge the walls adjacent to a pore corner. Predicted residual oil saturations are found to be approximately 35 percent less in mixed-wet rock when compared to completely water-wet rock. Calculated capillary pressure curves mimic those of mixed-wet porous media in the primary drainage, imbibition, and secondary drainage modes. Amott-Harvey indices range from -0.18 to 0.36 also in good agreement with experimental values1**. References and illustrations at end of paper The wettability of reservoir rock is a critical factor in determining the displacement effectiveness and ultimate oil recovery by drive fluids, such as water. Since the most wetting fluid tends to occupy the smallest and, hence, most hydrodynamically resistive pore channels while the least wetting fluid distributes to the largest and least resistive pore channels, wettability is a prime factor controlling multiphase flow and phase trapping. Therefore, understanding how wettability is established at the pore level is crucial if predictive flow models are to be developed. Wettability in porous media is generally classified as either homogeneous or heterogeneous. For the homogeneous case the entire rock surface has a uniform molecular affinity for either water or oil. Conversely, heterogeneous wettability indicates distinct surface regions that exhibit different affinities for oil or water. Three broad classifications of homogeneous wetting exist: strongly water-wet, strongly oil-wet, and intermediate wet. If smooth representative rock surfaces can be prepared, then contact angles for water-wet surfaces, measured through the water phase, are near zero. Whereas, for oil-wet surfaces they are near 180". In the case of intermediatewetting, the rock has neither a strong affinity for water nor oil, and contact angles range roughly from 45" to 135" 3. Two types of heterogeneous wettability are generally recognized. Mixed-wettability refers to distinct and separate water-wet and oil-wet surfaces which coexist and span a porous medium. Dalmation, also speckled or spotted, wettability refers to continuous water-wet surfaces enclosing A PORE-LEVEL SCENARIO FOR THE DEVELOPMENT OF 2 MIXED-WETTABILITY IN OIL RESERVOIRS SPE 24880 macroscopic regions of discontinuous oil-wet surfaces or vice-versa 4. For many years, it was common petroleumengineering practice to assume that oil reservoirs are strongly water-wet. Because most reservoir rock is highly siliceous and because oil reservoirs evolve by oil migrating into initially brine-occupied pore space, it was thought that the rock surface maintains a strong affinity for water even in the presence of oil5. In 1973, however, Salathie16 established that reservoirs with mixed-wettability display low residual oil (i.e., oil trapped as isolated globules) and consequently high displacement efficiency, as gauged by the ratio of oil recovered after waterflooding to the original oil in place. More recently, Fassi-Fihri et a17 used cryoscanning electron microscopy to confirm that wettability is heterogeneous on the pore scale in actual reservoir media and that mineralogy greatly affects the type of wettability. The purpose of this paper is to develop a porelevel picture of how mixed-wettability might form and evolve in reservoir rock initially filled with brine. Before embarking on this task we briefly explore the relationship between thin films and wettability. Thin Films and Wettability A succinct description of mixed-wettability follows when the effects of thin films which coat and adhere to solid surfaces are considered. Thin-film forces in fact control wettability. Figure 1 displays a schematic diagram of an apparatus used to quantify thin-film forces in liquid films adjacent to a solid surface8. An annular porous disk, which allows flow of wetting fluid, is sealed against a solid substrate. A nonwetting fluid meniscus enters from below the solid substrate disk assembly. Far away from the solid substrate, the meniscus is hemispherical. When the central portion of the meniscus is pressed against the wall, a thin film forms. As the pressure of the nonwetting fluid rises compared to that in the wetting fluid, the film thins and the meniscus or Plateau border recedes into the corner. An experiment proceeds by applying a capillary pressure difference (i.e., PC = Pnw Pw) across the Plateau-border meniscus and measuring the resulting thickness of the film by interferometry or ellipsometry. At equilibrium, the pressure of the wetting fluid in the thin film and the Plateau border are identical. The film is flat, yet the capillary pressure is nonzero. Consequently, the standard Young-Laplace equation of capillarity does not describe the thin film portion of Fig. 1. Rather, thin-film forces must be incorporated into the Young-Laplace equation by adding a term. n(h), referred to as disjoining pressure9-l2 : In the augmented Young-Laplace equation, o refers to the bulk interfacial tension between the wetting and nonwetting phases for an interface with principal radii of curvature rl and r2. Disjoining pressure is a function of film thickness, h. If disjoining pressure is positive, the two interfaces are repelled, whereas if disjoining pressure is negative the two interfaces are attracted. When films are thin (e.g., 100 nm), the disjoining pressure is significant compared to the other terms in Eqn. (1). For the flat film in Fig. 1, the curvature is zero so that at equilibrium PC = ll(h). Equation 1 may be cast in dimensionless form by a characteristic force, o, and a characteristic dimension, am. Representative values of o and am are 50 mN1m and 150 pm respectively. With this scaling, the first term on the right side of Eqn. (1) varies from approximately 113 to 10 whereas a disjoining pressure of 500 Pa makes the second term of order 1. A schematic disjoining pressure isotherm is drawn in Fig. 2. For certain values of disjoining pressure, for example l l1 , there are three possible equilibrium film thicknesses. The outermost films have thicknesses of order 10 to 100 nm. The central portion of the n versus h curve ( i.e., where dnldh is positive) is an unstable region13. A film in this thickness region spontaneously thickens or thins to a new stable configuration. The innermost films are quite thin, possibly on the order of one or several monolayers of solvent molecules. The effective range of the disjoining pressure is denoted by hp. Many factors contribute to the shape of a disjoining pressure isotherm. Three major force components are electrostatic interactions, van der Waals interactions, and hydration forces14. The first, electrostatic forces, originate from the overlap of ionic clouds known as electrical double layers present at each interface. For symmetric charged films, these are repulsive stabilizing forces15. M e l r ~ s e ' ~ and Hallq7 give thorough discussions of double-layer repulsive forces and their effects on film stability. Dispersive van der Waals forces are usually attractive and destabilize thin-aqueous films14. Lastly, strong repulsive hydration forces are important at film thicknesses approaching molecular dimensions1' . Clearly, the mineral content of pore walls has a large effect on thin-film interaction forces. For example, when a drop of crude oil is contacted wlh a brine-covered surface, crude oil adheres more readily (i.e., thick a ueous films break more easily) on calcite than Because calcite has a more positive surface charge, under identical conditions there is lesser stabilizing repulsive action from the electrostatic overlap forces. Oil-drop adhesion tests provide a simple approach to characterize the wetting behavior of crude oils against solid surface^^*'^^^^. When adhesion occurs, it is found that the solidloillwater boundary is fixed or "pinned" at the three-phase contact linela. Accordingly, the contact line of a retracting oil drop does not recede along the solid substrate. In subsequent sections, the importance of contact-angle pinning in mixed-wettability systems will become evident. SPE 214880 A. R. KOVSCEK , H. WONG, AND C. J. RADKE Thin-film forces determine the contact angle. Direct integration of the augmented Young-Laplace equation yields the following relationship between contact angle, 8, and disjoining pressure2' In Eqn, (2), h is the equilibrium film thickness of interest. Implicit in Eqn. (2) is the constraint that hphm cc 1. Integration over the positive purely repulsive portions of Fig. 2 (i.e., the thick outer films) results in zero contact angle. Conversely, nonzero contact angles up to 90° are possible when Eqn. (2) is integratedz1 over the attractive negative portions of Fig. 2. The above equation is derived for a meniscus attached to a flat solid surface. Nevertheless, it still applies for solid curved surfaces as long as the film thickness is much smaller than the radius of curvature of the surface. Consider now films on significantly curved surfaces. Figure 3 displays one 

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Harris Wong

Papers

A pore-level scenario for the development of mixed-wettability in oil reservoirs

The motion of long bubbles in polygonal capillaries. Part 1. Thin films

The motion of long bubbles in polygonal capillaries. Part 2. Drag, fluid pressure and fluid flow

Deposition and Thinning of the Human Tear Film

A Pore-Level Scenario for the Development of Mixed Wettability in Oil Reservoirs