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Journal articles on the topic 'Hele-Shaw flow; Stokes flow'

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Published: 4 June 2021

Last updated: 14 February 2022

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1

RICHARDSON, S. "Plane Stokes flows with time-dependent free boundaries in which the fluid occupies a doubly-connected region." European Journal of Applied Mathematics 11, no. 3 (June 2000): 249–69. http://dx.doi.org/10.1017/s0956792500004149.

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Consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent region bounded by free surfaces, the motion being driven solely by a constant surface tension acting at the free boundaries. When the fluid region is simply-connected, it is known that this Stokes flow problem is closely related to a Hele-Shaw free boundary problem when the zero-surface-tension model is employed. Specifically, if the initial configuration for the Stokes flow problem can be produced by injection at N points into an empty Hele-Shaw cell, then so can all later configurations. Moreover, there are N invariants; while the N points at which injection must take place move, the amount to be injected at each of these points remains the same. In this paper, we consider the situation when the fluid region is doubly-connected and show that, provided the geometry has an appropriate rotational symmetry, the same results continue to hold and can be exploited to determine the solution of the Stokes flow problem.

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2

Heussler, F. H. C., R. M. Oliveira, M. O. John, and E. Meiburg. "Three-dimensional Navier–Stokes simulations of buoyant, vertical miscible Hele-Shaw displacements." Journal of Fluid Mechanics 752 (July 2, 2014): 157–83. http://dx.doi.org/10.1017/jfm.2014.327.

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AbstractGravitationally and viscously unstable miscible displacements in vertical Hele-Shaw cells are investigated via three-dimensional Navier–Stokes simulations. The velocity of the two-dimensional base-flow displacement fronts generally increases with the unfavourable viscosity contrast and the destabilizing density difference. Displacement fronts moving faster than the maximum velocity of the Poiseuille flow far downstream exhibit a single stagnation point in a moving reference frame, consistent with earlier observations for corresponding capillary tube flows. Gravitationally stable fronts, on the other hand, can move more slowly than the Poiseuille flow, resulting in more complex streamline patterns and the formation of a spike at the tip of the front, in line with earlier findings. A two-dimensional pinch-off governed by dispersion is observed some distance behind the displacement front. Three-dimensional simulations of viscously and gravitationally unstable vertical displacements show a strong vorticity quadrupole along the length of the finger, similar to recent observations for neutrally buoyant flows. This quadrupole results in an inner splitting instability of vertically propagating fingers. Even though the quadrupole’s strength increases for larger destabilizing density differences, the inner splitting is delayed due to the presence of a secondary, outer quadrupole which counteracts the inner one. For large unstable density differences, the formation of a secondary, downward-propagating front is observed, which is also characterized by inner and outer vorticity quadrupoles. This front develops an anchor-like shape as a result of the flow induced by these quadrupoles. Increased spanwise wavelengths of the initial perturbation are seen to result in the formation of the well-known tip-splitting instability. For suitable initial conditions, the inner and tip-splitting instabilities can be seen to develop side by side, affecting different regions of the flow field.

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3

Pozrikidis, C. "The motion of particles in the Hele-Shaw cell." Journal of Fluid Mechanics 261 (February 25, 1994): 199–222. http://dx.doi.org/10.1017/s0022112094000315.

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The force and torque on a particle that translates, rotates, or is held stationary in an incident flow within a channel with parallel-sided walls, are considered in the limit of Stokes flow. Assuming that the particle has an axisymmetric shape with axis perpendicular to the channel walls, the problem is formulated in terms of a boundary integral equation that is capable of describing arbitrary three-dimensional Stokes flow in an axisymmetric domain. The method involves: (a) representing the flow in terms of a single-layer potential that is defined over the physical boundaries of the flow as well as other external surfaces, (b) decomposing the polar cylindrical components of the velocity, boundary surface force, and single-layer potential in complex Fourier series, and (c) collecting same-order Fourier coefficients to obtain a system of one-dimensional Fredholm integral equations of the first kind for the coefficients of the surface force over the traces of the natural boundaries of the flow in an azimuthal plane. In the particular case where the polar cylindrical components of the boundary velocity exhibit a first harmonic dependence on the azimuthal angle, we obtain a reduced system of three real integral equations. A numerical method of solution that is based on a standard boundary element-collocation procedure is developed and tested. For channel flow, the effect of domain truncation on the nature of the far flow is investigated with reference to plane Hagen–Poiseuille flow past a cylindrical post. Numerical results are presented for the force and torque exerted on a family of oblate spheroids located above a single plane wall or within a parallel-sided channel. The effect of particle shape on the structure of the flow is illustrated, and some novel features of the motion are discussed. The numerical computations reveal the range of accuracy of previous asymptotic solutions for small or tightly fitting spherical particles.

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4

RICHARDSON, S. "Two-dimensional Stokes flows with time-dependent free boundaries driven by surface tension." European Journal of Applied Mathematics 8, no. 4 (August 1997): 311–29. http://dx.doi.org/10.1017/s0956792597003057.

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We consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent simply-connected region bounded by a free surface, the motion being driven solely by a constant surface tension acting at the free boundary. Of particular concern here are such flows that start from an initial configuration with the fluid occupying an array of touching circular disks. We show that, when there are N such disks in a general position, the evolution of the fluid region is described by a conformal map involving 2N−1 time-dependent parameters whose variation is governed by N invariants and N−1 first order differential equations. When N=2, or when the problem enjoys some special features of symmetry, the moving boundary of the fluid domain during the motion can be determined by solving purely algebraic equations, the solution of a single differential equation being needed only to link a particular boundary shape to a particular time. The analysis is aided by exploiting a connection with Hele-Shaw free boundary flows when the zero-surface-tension model is employed. If the initial configuration for the Stokes flow problem can be produced by injection (or suction) at N points into an initially empty Hele-Shaw cell, as can the N-disk configuration referred to above, then so can all later configurations; the points where the fluid must be injected move, but the amount to be injected at each of the N points remains invariant. The efficacy of our solution procedure is illustrated by a number of examples, and we exploit the method to show that the free boundary in such a Stokes flow driven by surface tension alone may pass through a cusped state.

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5

Oliveira, Rafael M., and Eckart Meiburg. "Miscible displacements in Hele-Shaw cells: three-dimensional Navier–Stokes simulations." Journal of Fluid Mechanics 687 (October 12, 2011): 431–60. http://dx.doi.org/10.1017/jfm.2011.367.

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AbstractThree-dimensional Navier–Stokes simulations of viscously unstable, miscible Hele-Shaw displacements are discussed. Quasisteady fingers are observed whose tip velocity increases with the Péclet number and the unfavourable viscosity ratio. These fingers are widest near the tip, and become progressively narrower towards the root. The film of resident fluid left behind on the wall decreases in thickness towards the finger tip. The simulations reveal the detailed mechanism by which the initial spanwise vorticity of the base flow, when perturbed, gives rise to the cross-gap vorticity that drives the fingering instability in the classical Darcy sense. Cross-sections at constant streamwise locations reveal the existence of a streamwise vorticity quadrupole along the length of the finger. This streamwise vorticity convects resident fluid from the wall towards the centre of the gap in the cross-gap symmetry plane of the finger, while it transports injected fluid laterally away from the finger centre within the mid-gap plane. In this way, it results in the emergence of a longitudinal, inner splitting phenomenon some distance behind the tip that has not been reported previously. This inner splitting mechanism, which leaves the tip largely intact, is fundamentally different from the familiar tip-splitting mechanism. Since the inner splitting owes its existence to the presence of streamwise vorticity and cross-gap velocity, it cannot be captured by gap-averaged equations. It is furthermore observed that the role of the Péclet number in miscible displacements differs in some ways from that of the capillary number in immiscible flows. Specifically, larger Péclet numbers result in wider fingers, while immiscible flows display narrower fingers for larger capillary numbers. Furthermore, while higher capillary numbers are known to promote tip-splitting, inner splitting is delayed for larger Péclet numbers.

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6

Lee, S. Y., R. Teodorescu, and P. Wiegmann. "Viscous shocks in Hele–Shaw flow and Stokes phenomena of the Painlevé I transcendent." Physica D: Nonlinear Phenomena 240, no. 13 (June 2011): 1080–91. http://dx.doi.org/10.1016/j.physd.2010.09.017.

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7

LIN, Y. L. "Large-time rescaling behaviours of Stokes and Hele-Shaw flows driven by injection." European Journal of Applied Mathematics 22, no. 1 (December 2, 2010): 7–19. http://dx.doi.org/10.1017/s0956792510000264.

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In this paper, we give a precise description of the rescaling behaviours of global strong polynomial solutions to the reformulation of zero surface tension Hele-Shaw problem driven by injection, the Polubarinova-Galin equation, in terms of Richardson complex moments. From past results, we know that this set of solutions is large. This method can also be applied to zero surface tension Stokes flow driven by injection and a rescaling behaviour is given in terms of many conserved quantities as well.

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8

Demidov, A. S., J. P. Lohéac, and V. Runge. "Stokes–Leibenson problem for Hele-Shaw flow: a critical set in the space of contours." Russian Journal of Mathematical Physics 23, no. 1 (January 2016): 35–55. http://dx.doi.org/10.1134/s1061920816010039.

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9

Llamoza, Johan, and Desiderio A. Vasquez. "Structures and Instabilities in Reaction Fronts Separating Fluids of Different Densities." Mathematical and Computational Applications 24, no. 2 (May 17, 2019): 51. http://dx.doi.org/10.3390/mca24020051.

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Density gradients across reaction fronts propagating vertically can lead to Rayleigh–Taylor instabilities. Reaction fronts can also become unstable due to diffusive instabilities, regardless the presence of a mass density gradient. In this paper, we study the interaction between density driven convection and fronts with diffusive instabilities. We focus in fluids confined in Hele–Shaw cells or porous media, with the hydrodynamics modeled by Brinkman’s equation. The time evolution of the front is described with a Kuramoto–Sivashinsky (KS) equation coupled to the fluid velocity. A linear stability analysis shows a transition to convection that depends on the density differences between reacted and unreacted fluids. A stabilizing density gradient can surpress the effects of diffusive instabilities. The two-dimensional numerical solutions of the nonlinear equations show an increase of speed due to convection. Brinkman’s equation lead to the same results as Darcy’s laws for narrow gap Hele–Shaw cells. For large gaps, modeling the hydrodynamics using Stokes’ flow lead to the same results.

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10

Zhu, Lailai, and François Gallaire. "A pancake droplet translating in a Hele-Shaw cell: lubrication film and flow field." Journal of Fluid Mechanics 798 (June 15, 2016): 955–69. http://dx.doi.org/10.1017/jfm.2016.357.

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We adopt a boundary integral method to study the dynamics of a translating droplet confined in a Hele-Shaw cell in the Stokes regime. The droplet is driven by the motion of the ambient fluid with the same viscosity. We characterize the three-dimensional (3D) nature of the droplet interface and of the flow field. The interface develops an arc-shaped ridge near the rear-half rim with a protrusion in the rear and a laterally symmetric pair of higher peaks; this pair of protrusions has been identified by recent experiments (Huerre et al., Phys. Rev. Lett., vol. 115 (6), 2015, 064501) and predicted asymptotically (Burgess & Foster, Phys. Fluids A, vol. 2 (7), 1990, pp. 1105–1117). The mean film thickness is well predicted by the extended Bretherton model (Klaseboer et al., Phys. Fluids, vol. 26 (3), 2014, 032107) with fitting parameters. The flow in the streamwise wall-normal middle plane is featured with recirculating zones, which are partitioned by stagnation points closely resembling those of a two-dimensional droplet in a channel. Recirculation is absent in the wall-parallel, unconfined planes, in sharp contrast to the interior flow inside a moving droplet in free space. The preferred orientation of the recirculation results from the anisotropic confinement of the Hele-Shaw cell. On these planes, we identify a dipolar disturbance flow field induced by the travelling droplet and its $1/r^{2}$ spatial decay is confirmed numerically. We pinpoint counter-rotating streamwise vortex structures near the lateral interface of the droplet, further highlighting the complex 3D flow pattern.

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11

GOYAL, N., H. PICHLER, and E. MEIBURG. "Variable-density miscible displacements in a vertical Hele-Shaw cell: linear stability." Journal of Fluid Mechanics 584 (July 25, 2007): 357–72. http://dx.doi.org/10.1017/s0022112007006428.

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A computational study based on the Stokes equations is conducted to investigate the effects of gravitational forces on miscible displacements in vertical Hele-Shaw cells. Nonlinear simulations provide the quasi-steady displacement fronts in the gap of the cell, whose stability to spanwise perturbations is subsequently examined by means of a linear stability analysis. The two-dimensional simulations indicate a marked thickening (thinning) and slowing down (speeding up) of the displacement front for flows stabilized (destabilized) by gravity. For the range investigated, the tip velocity is found to vary linearly with the gravity parameter. Strongly stable density stratifications lead to the emergence of flow patterns with spreading fronts, and to the emergence of a secondary needle-shaped finger, similar to earlier observations for capillary tube flows. In order to investigate the transition between viscously driven and purely gravitational instabilities, a comparison is presented between displacement flows and gravity-driven flows without net displacements.The linear stability analysis shows that both the growth rate and the dominant wavenumber depend only weakly on the Péclet number. The growth rate varies strongly with the gravity parameter, so that even a moderately stable density stratification can stabilize the displacement. Both the growth rate and the dominant wavelength increase with the viscosity ratio. For unstable density stratifications, the dominant wavelength is nearly independent of the gravity parameter, while it increases strongly for stable density stratifications. Finally, the kinematic wave theory of Lajeunesse et al. (J. Fluid Mech. vol. 398, 1999, p. 299) is seen to capture the stability limit quite accurately, while the Darcy analysis misses important aspects of the instability.

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12

Kulkarni, Venkatesh M., Chu Wee Liang, C. W. Tan, P. A. Aswatha Narayana, and K. N. Seetharamu. "Simulation of Mold Filling for Non-Newtonian Fluids - Part 2." Journal of Microelectronics and Electronic Packaging 3, no. 2 (April 1, 2006): 52–60. http://dx.doi.org/10.4071/1551-4897-3.2.52.

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This paper deals with the flow in the resin transfer molding process commonly used for IC chip encapsulation in the electronic packaging industry. A solution algorithm is presented for modeling the flow of a non-Newtonian fluid obeying a Power-Law model and the algorithm is used to conduct parametric studies in transfer molding. The flow model uses the Hele-Shaw approximation to solve the Navier-Stokes Equations and a pseudo-concentration algorithm for tracking the interface between the resin and the air. The Finite Element Method is employed to reduce the governing partial differential equations to algebraic form. The model is used to study the flow from the transfer ram into the cavity for different dimensions of transfer molding tools. Parametric studies are carried out to obtain balanced filling for transfer molding configuration. Parametric studies could provide a design guideline to optimize the encapsulation process prior to the setting up of an actual manufacturing set.

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13

Demidov, A. S. "Evolution of the Perturbation of a Circle in the Stokes–Leibenson Problem for the Hele-Shaw Flow." Journal of Mathematical Sciences 123, no. 5 (October 2004): 4381–403. http://dx.doi.org/10.1023/b:joth.0000040301.53259.05.

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14

Dalwadi, Mohit P., S. Jonathan Chapman, Sarah L. Waters, and James M. Oliver. "On the boundary layer structure near a highly permeable porous interface." Journal of Fluid Mechanics 798 (May 31, 2016): 88–139. http://dx.doi.org/10.1017/jfm.2016.308.

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The method of matched asymptotic expansions is used to study the canonical problem of steady laminar flow through a narrow two-dimensional channel blocked by a tight-fitting finite-length highly permeable porous obstacle. We investigate the behaviour of the local flow close to the interface between the single-phase and porous regions (governed by the incompressible Navier–Stokes and Darcy flow equations, respectively). We solve for the flow in these inner regions in the limits of low and high Reynolds number, facilitating an understanding of the nature of the transition from Poiseuille to plug to Poiseuille flow in each of these limits. Significant analytical progress is made in the high Reynolds number limit, and we explore in detail the rich boundary layer structure that occurs. We derive general results for the interfacial stress and for the conditions that couple the flow in the outer regions away from the interface. We consider the three-dimensional generalization to unsteady laminar flow through and around a tight-fitting highly permeable cylindrical porous obstacle within a Hele-Shaw cell. For the high Reynolds number limit, we give the coupling conditions and interfacial stress in terms of the outer flow variables, allowing information from a nonlinear three-dimensional problem to be obtained by solving a linear two-dimensional problem. Finally, we illustrate the utility of our analysis by considering the specific example of time-dependent forced far-field flow in a Hele-Shaw cell containing a porous cylinder with a circular cross-section. We determine the internal stress within the porous obstacle, which is key for tissue engineering applications, and the interfacial stress on the boundary of the porous obstacle, which has applications to biofilm erosion. In the high Reynolds number limit, we demonstrate that the fluid inertia can result in the cylinder experiencing a time-independent net force, even when the far-field forcing is periodic with zero mean.

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15

MARTIN, J., N. RAKOTOMALALA, L. TALON, and D. SALIN. "Viscous lock-exchange in rectangular channels." Journal of Fluid Mechanics 673 (February 14, 2011): 132–46. http://dx.doi.org/10.1017/s0022112010006208.

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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.

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16

Pihler-Puzović, Draga, Anne Juel, Gunnar G. Peng, John R. Lister, and Matthias Heil. "Displacement flows under elastic membranes. Part 1. Experiments and direct numerical simulations." Journal of Fluid Mechanics 784 (November 6, 2015): 487–511. http://dx.doi.org/10.1017/jfm.2015.590.

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The injection of fluid into the narrow liquid-filled gap between a rigid plate and an elastic membrane drives a displacement flow that is controlled by the competition between elastic and viscous forces. We study such flows using the canonical set-up of an elastic-walled Hele-Shaw cell whose upper boundary is formed by an elastic sheet. We investigate both single- and two-phase displacement flows in which the localised injection of fluid at a constant flow rate is accommodated by the inflation of the sheet and the outward propagation of an axisymmetric front beyond which the cell remains approximately undeformed. We perform a direct comparison between quantitative experiments and numerical simulations of two theoretical models. The models couple the Föppl–von Kármán equations, which describe the deformation of the thin elastic membrane, to the equations describing the flow, which we model by (i) the Navier–Stokes equations or (ii) lubrication theory. We identify the dominant physical effects that control the behaviour of the system and critically assess modelling assumptions that were made in previous studies. The insight gained from these studies is then used in Part 2 of this work, where we formulate an improved lubrication model and develop an asymptotic description of the key phenomena.

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17

Li, Huanhao, Chun-Yi Kao, and Chih-Yung Wen. "Labyrinthine and secondary wave instabilities of a miscible magnetic fluid drop in a Hele-Shaw cell." Journal of Fluid Mechanics 836 (December 11, 2017): 374–96. http://dx.doi.org/10.1017/jfm.2017.739.

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A comprehensive experimental study is presented to analyse the instabilities of a magnetic fluid drop surrounded by miscible fluid confined in a Hele-Shaw cell. The experimental conditions include different magnetic fields (by varying the maximum pre-set magnetic field strengths,$H$, and sweep rates,$SR=\text{d}H_{t}/\text{d}t$, where$H_{t}$is the instant magnetic field strength), gap spans,$h$, and magnetic fluid samples, and are further coupled into a modified Péclect number$Pe^{\prime }$to evaluate the instabilities. Two distinct instabilities are induced by the external magnetic fields with different sweep rates: (i) a labyrinthine fingering instability, where small fingerings emerge around the initial circular interface in the early period, and (ii) secondary waves in the later period. Based on 81 sets of experimental conditions, the initial growth rate of the interfacial length,$\unicode[STIX]{x1D6FC}$, of the magnetic drop is found to increase linearly with$Pe^{\prime }$, indicating that$\unicode[STIX]{x1D6FC}$is proportional to the square root of the$SR$and$h^{3/2}$at the onset of the labyrinthine instability. In addition, secondary waves, which are characterised by the dimensionless wavelength$\unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D706}/h$, can only be triggered when the three-dimensional magnetic microconvection is strong enough to make$Pe^{\prime }$exceed a critical value, i.e.$Pe^{\prime }>19\,000$, where$\unicode[STIX]{x1D706}$is the wavelength of the secondary wave. In this flow regime of high$Pe^{\prime }$, the length scale of the secondary wave instability is found to be$\unicode[STIX]{x1D6EC}=7\pm 1$, corresponding to the Stokes regime; meanwhile, in the flow regime of low$Pe^{\prime }$, the flow corresponds to the Hele-Shaw regime introduced by Fernandezet al.(J. Fluid Mech., vol. 451, 2002, pp. 239–260).

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18

Demidov, A. S. "Evolution of the perturbation of a circle in the Stokes-Leibenson problem for the Hele-Shaw flow. Part II." Journal of Mathematical Sciences 139, no. 6 (December 2006): 7064–78. http://dx.doi.org/10.1007/s10958-006-0406-1.

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19

Zhou, Y., P. Y. Lagrée, S. Popinet, P. Ruyer, and P. Aussillous. "Experiments on, and discrete and continuum simulations of, the discharge of granular media from silos with a lateral orifice." Journal of Fluid Mechanics 829 (September 21, 2017): 459–85. http://dx.doi.org/10.1017/jfm.2017.543.

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We compare laboratory experiments, contact dynamics simulations and continuum Navier–Stokes simulations with a $\unicode[STIX]{x1D707}(I)$ visco-plastic rheology, of the discharge of granular media from a silo with a lateral orifice. We consider a rectangular silo with an orifice of height $D$ which spans the silo width $W$, and we observe two regimes. For small enough aperture aspect ratio ${\mathcal{A}}=D/W$, the Hagen–Beverloo relation is obtained. For thin enough silos, ${\mathcal{A}}\gg {\mathcal{A}}_{c}$, we observe a second regime where the outlet velocity varies with $\sqrt{W}$. This new regime is also obtained in the continuum simulations when the friction on side walls is taken into account in a thickness-averaged version of $\unicode[STIX]{x1D707}(I)$ $+$ Navier–Stokes (in the spirit of Hele-Shaw flows). Moreover most of the internal details of the flow field observed experimentally are reproduced when considering this lateral friction. These two regimes are recovered experimentally for a cylindrical silo with a lateral rectangular orifice of height $D$ and arc length $W$. The dependency of the flow rate on the particle diameter is found to be reasonably described experimentally using two geometrical functions that depend respectively on the number of beads through the two aperture dimensions. This is consistent with two-dimensional discrete simulation results: at the outlet, the volume fraction and the velocities depend on the particle diameter and this behaviour is correctly described by those geometrical functions. A similar dependency is observed in the two-dimensional continuum simulations.

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20

MOYERS-GONZÁLEZ, M. A., I. A. FRIGAARD, O. SCHERZER, and T. P. TSAI. "Transient effects in oilfield cementing flows: Qualitative behaviour." European Journal of Applied Mathematics 18, no. 4 (August 2007): 477–512. http://dx.doi.org/10.1017/s0956792507007048.

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We present an unsteady Hele–Shaw model of the fluid–fluid displacements that take place during primary cementing of an oil well, focusing on the case where one Herschel–Bulkley fluid displaces another along a long uniform section of the annulus. Such unsteady models consist of an advection equation for a fluid concentration field coupled to a third-order non-linear PDE (Partial differential equation) for the stream function, with a free boundary at the boundary of regions of stagnant fluid. These models, although complex, are necessary for the study of interfacial instability and the effects of flow pulsation, and remain considerably simpler and more efficient than computationally solving three-dimensional Navier–Stokes type models. Using methods from gradient flows, we demonstrate that our unsteady evolution equation for the stream function has a unique solution. The solution is continuous with respect to variations in the model physical data and will decay exponentially to a steady-state distribution if the data do not change with time. In the event that density differences between the fluids are small and that the fluids have a yield stress, then if the flow rate is decreased suddenly to zero, the stream function (hence velocity) decays to zero in a finite time. We verify these decay properties, using a numerical solution. We then use the numerical solution to study the effects of pulsating the flow rate on a typical displacement.

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21

Tsay, Ruey-Yug, and Sheldon Weinbaum. "Viscous flow in a channel with periodic cross-bridging fibres: exact solutions and Brinkman approximation." Journal of Fluid Mechanics 226 (May 1991): 125–48. http://dx.doi.org/10.1017/s0022112091002318.

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A general solution of the three-dimensional Stokes equations is developed for the viscous flow past a square array of circular cylindrical fibres confined between two parallel walls. This doubly periodic solution, which is an extension of the theory developed by Lee & Fung (1969) for flow around a single fibre, successfully describes the transition in behaviour from the Hele-Shaw potential flow limit (aspect ratio B [Lt ] 1) to the viscous two-dimensional limiting case (B [Gt ] 1, Sangani & Acrivos 1982) for the hydrodynamic interaction between the fibres. These results are also compared with the solution of the Brinkman equation for the flow through a porous medium in a channel. This comparison shows that the Brinkman approximation is very good when B > 5, but breaks down when B [les ] O(1). A new interpolation formula is proposed for this last regime. Numerical results for the detailed velocity profiles, the drag coefficient f, and the Darcy permeability Kp are presented. It is shown that the velocity component perpendicular to the parallel walls is only significant within the viscous layers surrounding the fibres, whose thickness is of the order of half the channel height B′. One finds that when the aspect ratio B > 5, the neglect of the vertical velocity component vz can lead to large errors in the satisfaction of the no-slip boundary conditions on the surfaces of the fibres and large deviations from the approximate solution in Lee (1969), in which vz and the normal pressure field are neglected. The numerical results show that the drag coefficient of the fibrous bed increases dramatically when the open gap between adjacent fibres Δ′ becomes smaller than B′. The predictions of the new theory are used to examine the possibility that a cross-bridging slender fibre matrix can exist in the intercellular cleft of capillary endothelium as proposed by Curry & Michel (1980).

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22

De´nos, R., T. Arts, G. Paniagua, V. Michelassi, and F. Martelli. "Investigation of the Unsteady Rotor Aerodynamics in a Transonic Turbine Stage." Journal of Turbomachinery 123, no. 1 (February 1, 2000): 81–89. http://dx.doi.org/10.1115/1.1314607.

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The paper focuses on the unsteady pressure field measured around the rotor midspan profile of the VKI Brite transonic turbine stage. The understanding of the complex unsteady flow field is supported by a quasi-three-dimensional unsteady Navier–Stokes computation using a k-ω turbulence model and a modified version of the Abu-Ghannam and Shaw correlation for the onset of transition. The agreement between computational and experimental results is satisfactory. They both reveal the dominance of the vane shock in the interaction. For this reason, it is difficult to identify the influence of vane-wake ingestion in the rotor passage from the experimental data. However, the computations allow us to draw some useful conclusions in this respect. The effect of the variation of the rotational speed, the stator–rotor spacing, and the stator trailing edge coolant flow ejection is investigated and the unsteady blade force pattern is analyzed.

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23

Hilgenfeld, L., P. Cardamone, and L. Fottner. "Boundary layer investigations on a highly loaded transonic compressor cascade with shock/laminar boundary layer interactions." Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 217, no. 4 (January 1, 2003): 349–56. http://dx.doi.org/10.1243/095765003322315405.

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Detailed experimental and numerical investigations of the flowfield and boundary layer on a highly loaded transonic compressor cascade were performed at various Mach and Reynolds numbers representative of real turbomachinery conditions. The emerging shock system interacts with the laminar boundary layer, causing shock-induced separation with turbulent reattachment. Steady two-dimensional calculations have been performed using the Navier—Stokes solver TRACE-U. The flow solver employs a modified version of the one-equation Spalart—Allmaras turbulence model coupled with a transition correlation by Abu-Ghannam/Shaw in the formulation by Drela. The computations reproduce well the experimental results with respect to the profile pressure distribution and the location of the shock system. The transitional behaviour of the boundary layer and the profile losses in the wake are properly predicted as well, except for the highest Mach number tested, where large separated regions appear on the suction side.

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24

FERNANDEZ, J., P. KUROWSKI, P. PETITJEANS, and E. MEIBURG. "Density-driven unstable flows of miscible fluids in a Hele-Shaw cell." Journal of Fluid Mechanics 451 (January 25, 2002): 239–60. http://dx.doi.org/10.1017/s0022112001006504.

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Density-driven instabilities between miscible fluids in a vertical Hele-Shaw cell are investigated by means of experimental measurements, as well as two- and three-dimensional numerical simulations. The experiments focus on the early stages of the instability growth, and they provide detailed information regarding the growth rates and most amplified wavenumbers as a function of the governing Rayleigh number Ra. They identify two clearly distinct parameter regimes: a low-Ra, ‘Hele-Shaw’ regime in which the dominant wavelength scales as Ra−1, and a high-Ra ‘gap’ regime in which the length scale of the instability is 5±1 times the gap width. The experiments are compared to a recent linear stability analysis based on the Brinkman equation. The analytical dispersion relationship for a step-like density profile reproduces the experimentally observed trend across the entire Ra range. Nonlinear simulations based on the two- and three-dimensional Stokes equations indicate that the high-Ra regime is characterized by an instability across the gap, wheras in the low-Ra regime a spanwise Hele-Shaw mode dominates.

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CUMMINGS, L. J., S. D. HOWISON, and J. R. KING. "Two-dimensional Stokes and Hele-Shaw flows with free surfaces." European Journal of Applied Mathematics 10, no. 6 (December 1999): 635–80. http://dx.doi.org/10.1017/s0956792599003964.

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26

Hedenmalm, Håkan, and Sergei Shimorin. "Hele–Shaw flow on hyperbolic surfaces." Journal de Mathématiques Pures et Appliquées 81, no. 3 (2002): 187–222. http://dx.doi.org/10.1016/s0021-7824(01)01222-3.

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27

Zeybek, M., and Y. C. Yortsos. "Parallel flow in Hele-Shaw cells." Journal of Fluid Mechanics 241 (August 1992): 421–42. http://dx.doi.org/10.1017/s0022112092002106.

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We consider the parallel flow of two immiscible fluids in a Hele-Shaw cell. The evolution of disturbances on the fluid interfaces is studied both theoretically and experimentally in the large-capillary-number limit. It is shown that such interfaces support wave motion, the amplitude of which for long waves is governed by a set of KdV and Airy equations. The waves are dispersive provided that the fluids have unequal viscosities and that the space occupied by the inner fluid does not pertain to the Saffman-Taylor conditions (symmetric interfaces with half-width spacing). Experiments conducted in a long and narrow Hele-Shaw cell appear to validate the theory in both the symmetric and the non-symmetric cases.

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28

Morris, S. J. S. "Stability of thermoviscous Hele-Shaw flow." Journal of Fluid Mechanics 308 (February 10, 1996): 111–28. http://dx.doi.org/10.1017/s0022112096001413.

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Viscous fingering can occur as a three-dimensional disturbance to plane flow of a hot thermoviscous liquid in a Hele-Shaw cell with cold isothermal walls. This work assumes the principle of exchange of stabilities, and uses a temporal stability analysis to find the critical viscosity ratio and finger spacing as functions of channel length, Lc. Viscous heating is taken as negligible, so the liquid cools with distance (x) downstream. Because the base flow is spatially developing, the disturbance equations are not fully separable. They admit, however, an exact solution for a liquid whose viscosity and specific heats are arbitrary functions of temperature. This solution describes the neutral disturbances in terms of the base flow and an amplitude, A(x). The stability of a given (computed) base flow is determined by solving an eigenvalue problem for A(x), and the critical finger spacing. The theory is illustrated by using it to map the instability for variable-viscosity flow with constant specific heat. Two fingering modes are predicted, one being a turning-point instability. The preferred mode depends on Lc. Finger spacing is comparable with the thermal entry length in a long channel, and is even larger in short channels. When applied to magmatic systems, the results suggest that fingering will occur on geological scales only if the system is about freeze.

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29

Nizamova, A. D., and Valiev A. A. Valiev. "Mathematical model of oil displacement by water in a plane channel." Multiphase Systems 15, no. 3-4 (2020): 208–11. http://dx.doi.org/10.21662/mfs2020.3.131.

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Unstable displacement of immiscible liquids in a plane channel is a topical research in both theoretical and practical applications. In this paper, we consider a plane channel filled with an incompressible fluid. Over time, another fluid is injected into the channel. The fluids are immiscible. The paper builds a mathematical model of the process of oil displacement by water in a plane channel, which allows further numerical studies and comparison of the results with the obtained experimental data using the example of the Hele-Show cell. The mathematical model for a multiphase, multicomponent flow consists of the Navier-Stokes equations, the equations of conservation of mass, momentum and energy. Modern methods for modeling the dynamics of "viscous fingers“ are based mainly on numerical methods for solving systems of differential equations using the pressure gradient, viscosity and capillary forces as parameters. The influence of these parameters must be determined experimentally. To solve the problem, a quasi-hydrodynamic approach is used, based on the addition of a certain small parameter and allowing one to describe stable schemes with central differences. The complexity of solving such problems lies in the size of the considered models, which in practice have a wide range of applications from micro-scale to orders of one centimeter. A comprehensive study will allow us to evaluate and analyze the entire process as a whole, as well as to establish flow parameters to improve the efficiency of displacement and increase oil recovery, since in the numerical modeling of the process it is easier to create many independent experiments with the same initial data, in contrast to the experimental study.

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30

GRAF, F., E. MEIBURG, and C. HÄRTEL. "Density-driven instabilities of miscible fluids in a Hele-Shaw cell: linear stability analysis of the three-dimensional Stokes equations." Journal of Fluid Mechanics 451 (January 25, 2002): 261–82. http://dx.doi.org/10.1017/s0022112001006516.

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We consider the situation of a heavier fluid placed above a lighter one in a vertically arranged Hele-Shaw cell. The two fluids are miscible in all proportions. For this configuration, experiments and nonlinear simulations recently reported by Fernandez et al. (2002) indicate the existence of a low-Rayleigh-number (Ra) ‘Hele-Shaw’ instability mode, along with a high-Ra ‘gap’ mode whose dominant wavelength is on the order of five times the gap width. These findings are in disagreement with linear stability results based on the gap-averaged Hele-Shaw approach, which predict much smaller wavelengths. Similar observations have been made for immiscible flows as well (Maxworthy 1989).In order to resolve the above discrepancy, we perform a linear stability analysis based on the full three-dimensional Stokes equations. A generalized eigenvalue problem is formulated, whose numerical solution yields both the growth rate and the two-dimensional eigenfunctions in the cross-gap plane as functions of the spanwise wavenumber, an ‘interface’ thickness parameter, and Ra. For large Ra, the dispersion relations confirm that the optimally amplified wavelength is about five times the gap width, with the exact value depending on the interface thickness. The corresponding growth rate is in very good agreement with the experimental data as well. The eigenfunctions indicate that the predominant fluid motion occurs within the plane of the Hele-Shaw cell. However, for large Ra purely two-dimensional modes are also amplified, for which there is no motion in the spanwise direction. Scaling laws are provided for the dependence of the maximum growth rate, the corresponding wavenumber, and the cutoff wavenumber on Ra and the interface thickness. Furthermore, the present results are compared both with experimental data, as well as with linear stability results obtained from the Hele-Shaw equations and a modified Brinkman equation.

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31

BOOS, W., and A. THESS. "Thermocapillary flow in a Hele-Shaw cell." Journal of Fluid Mechanics 352 (December 10, 1997): 305–30. http://dx.doi.org/10.1017/s0022112097007477.

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We formulate a simple theoretical model that permits one to investigate surface-tension-driven flows with complex interface geometry. The model consists of a Hele-Shaw cell filled with two different fluids and subjected to a unidirectional temperature gradient. The shape of the interface that separates the fluids can be arbitrarily complex. If the contact line is pinned, i.e. unable to move, the problem of calculating the flow in both fluids is governed by a linear set of equations containing the characteristic aspect ratio and the viscosity ratio as the only input parameters. Analytical solutions, derived for a linear interface and for a circular drop, demonstrate that for large aspect ratio the flow field splits into a potential core flow and a thermocapillary boundary layer which acts as a source for the core. An asymptotic theory is developed for this limit which reduces the mathematical problem to a Laplace equation with Dirichlet boundary conditions. This problem can be efficiently solved utilizing a boundary element method. It is found that the thermocapillary flow in non-circular drops has a highly non-trivial streamline topology. After releasing the assumption of a pinned interface, a linear stability analysis is carried out for the interface under both transverse and longitudinal temperature gradients. For a semi-infinite fluid bounded by a freely movable surface long-wavelength instability due to the temperature gradient across the surface is predicted. The mechanism of this instability is closely related to the long-wave instability in surface-tension-driven Bénard convection. A linear interface heated from the side is found to be linearly stable. The possibility of experimental verification of the predictions is briefly discussed.

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32

BALSA, THOMAS F. "Secondary flow in a Hele-Shaw cell." Journal of Fluid Mechanics 372 (October 10, 1998): 25–44. http://dx.doi.org/10.1017/s0022112098002171.

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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.

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33

Almgren, Robert, Wei-Shen Dai, and Vincent Hakim. "Scaling behavior in anisotropic Hele-Shaw flow." Physical Review Letters 71, no. 21 (November 22, 1993): 3461–64. http://dx.doi.org/10.1103/physrevlett.71.3461.

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34

Ceniceros †, Hector D., and José M. Villalobos. "Topological reconfiguration in expanding Hele—Shaw flow." Journal of Turbulence 3 (January 2002): N37. http://dx.doi.org/10.1088/1468-5248/3/1/037.

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35

Kondic, Ljubinko, Peter Palffy-Muhoray, and Michael J. Shelley. "Models of non-Newtonian Hele-Shaw flow." Physical Review E 54, no. 5 (November 1, 1996): R4536—R4539. http://dx.doi.org/10.1103/physreve.54.r4536.

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36

Mishuris, Gennady, Sergei Rogosin, and Michal Wrobel. "MOVING STONE IN THE HELE‐SHAW FLOW." Mathematika 61, no. 2 (April 8, 2015): 457–74. http://dx.doi.org/10.1112/s0025579314000461.

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37

Hedenmalm, Haakan, and Anders Olofsson. "Hele-Shaw flow on weakly hyperbolic surfaces." Indiana University Mathematics Journal 54, no. 4 (2005): 1161–80. http://dx.doi.org/10.1512/iumj.2005.54.2651.

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38

Goldstein, Raymond E., Adriana I. Pesci, and Michael J. Shelley. "Instabilities and singularities in Hele–Shaw flow." Physics of Fluids 10, no. 11 (November 1998): 2701–23. http://dx.doi.org/10.1063/1.869795.

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39

Mishuris, Gennady, Sergei Rogosin, and Michal Wrobel. "Hele-Shaw flow with a small obstacle." Meccanica 49, no. 9 (May 15, 2014): 2037–47. http://dx.doi.org/10.1007/s11012-014-9919-8.

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40

Pareschi, Lorenzo, Giovanni Russo, and Giuseppe Toscani. "A kinetic approximation of Hele–Shaw flow." Comptes Rendus Mathematique 338, no. 2 (January 2004): 177–82. http://dx.doi.org/10.1016/j.crma.2003.11.006.

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41

Perrin, Charlotte. "A remark on memory effects in constrained fluid systems." ESAIM: Proceedings and Surveys 69 (2020): 56–69. http://dx.doi.org/10.1051/proc/202069056.

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The goal of this note is to put into perspective the recent results obtained on memory effects in partially congested fluid systems of Euler or Navier-Stokes type with former studies on free boundary obstacle problems and Hele-Shaw equations. In particular, we relate the notion of adhesion potential initially introduced in the context of dense suspension flows with the one of Baiocchi variable used in the analysis of free boundary problems.

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42

Aronsson, Gunnar, and Ulf Janfalk. "On Hele–Shaw flow of power-law fluids." European Journal of Applied Mathematics 3, no. 4 (December 1992): 343–66. http://dx.doi.org/10.1017/s0956792500000905.

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This paper reviews the governing equations for a plane Hele–Shaw flow of a power-law fluid. We find two closely related partial differential equations, one for the pressure and one for the stream function. Some mathematical results for these equations are presented, in particular some exact solutions and a representation theorem. The results are applied to Hele–Shaw flow. It is then possible to determine the flow near an arbitrary corner for any power-law fluid. Other examples are also given.

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43

ENTOV, V. M., and P. ETINGOF. "On a generalized two-fluid Hele-Shaw flow." European Journal of Applied Mathematics 18, no. 1 (February 2007): 103–28. http://dx.doi.org/10.1017/s0956792507006869.

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Generalized two-phase fluid flows in a Hele-Shaw cell are considered. It is assumed that the flow is driven by the fluid pressure gradient and an external potential field, for example, an electric field. Both the pressure field and the external field may have singularities in the flow domain. Therefore, combined action of these two fields brings into existence some new features, such as non-trivial equilibrium shapes of boundaries between the two fluids, which can be studied analytically. Some examples are presented. It is argued, that the approach and results may find some applications in the theory of fluids flow through porous media and microfluidic devices controlled by electric field.

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44

McDONALD, N. R. "Generalised Hele-Shaw flow: A Schwarz function approach." European Journal of Applied Mathematics 22, no. 6 (May 16, 2011): 517–32. http://dx.doi.org/10.1017/s0956792511000210.

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An equation governing the evolution of a Hele-Shaw free boundary flow in the presence of an arbitrary external potential – generalised Hele-Shaw flow – is derived in terms of the Schwarz functiong(z,t) of the free boundary. This generalises the well-known equation ∂g/∂t= 2∂w/∂z, wherewis the complex potential, which has been successfully employed in constructing many exact solutions in the absence of external potentials. The new equation is used to re-derive some known explicit solutions for equilibrium and time-dependent free boundary flows in the presence of external potentials, including those with singular potential fields, uniform gravity and centrifugal forces. Some new solutions are also constructed that variously describe equilibrium flows with higher order hydrodynamic singularities in the presence of electric point sources and an unsteady solution describing bubbles under the combined influence of strain and centrifugal potential.

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45

KHALID, A. H., N. R. McDONALD, and J. M. VANDEN-BROECK. "Hele-Shaw flow driven by an electric field." European Journal of Applied Mathematics 25, no. 4 (October 10, 2013): 425–47. http://dx.doi.org/10.1017/s0956792513000351.

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The behaviour of two-dimensional finite blobs of conducting viscous fluid in a Hele-Shaw cell subject to an electric field is considered. The time-dependent free boundary problem is studied both analytically using the Schwarz function of the free boundary and numerically using a boundary integral method. Various problems are considered, including (i) the behaviour of an initially circular blob of conducting fluid subject to an electric point charge located arbitrarily within the blob, (ii) the delay in cusp formation on the free boundary in sink-driven flow due to a strategically placed electric charge and (iii) the stability of exact steady solutions having both hydrodynamic and electric forcing.

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46

Tryggvason, Grétar, and Hassan Aref. "Finger-interaction mechanisms in stratified Hele-Shaw flow." Journal of Fluid Mechanics 154 (May 1985): 287–301. http://dx.doi.org/10.1017/s0022112085001537.

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Interactions between a few fingers in sharply stratified Hele-Shaw flow are investigated by numerical integration of the initial-value problem. It is shown that fingers evolving from an initial perturbation of an unstable interface consisting of a single wave are rather insensitive to variations of the control parameters governing the flow. Initial perturbations with at least two waves, on the other hand, lead to important finger-interaction and selection mechanisms at finite amplitude. On the basis of the results reported here many features of an earlier numerical study of the ‘statistical-fingering’ regime can be rationalized.

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Hansen, E. B., and H. Rasmussen. "A numerical study of unstable Hele-Shaw flow." Computers & Mathematics with Applications 38, no. 5-6 (September 1999): 217–30. http://dx.doi.org/10.1016/s0898-1221(99)00228-x.

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48

Miranda, José A., and Michael Widom. "Parallel flow in Hele-Shaw cells with ferrofluids." Physical Review E 61, no. 2 (February 1, 2000): 2114–17. http://dx.doi.org/10.1103/physreve.61.2114.

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49

Glasner, Karl. "A diffuse interface approach to Hele Shaw flow." Nonlinearity 16, no. 1 (October 28, 2002): 49–66. http://dx.doi.org/10.1088/0951-7715/16/1/304.

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50

Takaki, Ryuji. "Hele Shaw Flow between Flexible and Rigid Walls." Journal of the Physical Society of Japan 54, no. 1 (January 15, 1985): 8–10. http://dx.doi.org/10.1143/jpsj.54.8.

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