Showing papers on "Hele-Shaw flow published in 2001"
Book•
[...]
01 Jan 2001
TL;DR: Basic Equations.
Abstract: Basic Equations. Steady Uniform Flow. Control Sections. Gradually Varied Flow. Computation of Gradually Varied Flow. Spatially Varied Flow. Unsteady Flow I. Unsteady Flow II. Artificial Channel Controls. Special Topics. References. Index.
888 citations
TL;DR: In this article, an approximate deconvolution model for large-eddy simulation of incompressible flows is applied to turbulent channel flow and the effect of nonrepresented scales is modeled by a relaxation regularization involving a secondary filter operation.
Abstract: The approximate deconvolution model (ADM) for the large-eddy simulation of incompressible flows is detailed and applied to turbulent channel flow. With this approach an approximation of the unfiltered solution is obtained by repeated filtering. Given a good approximation of the unfiltered solution, the nonlinear terms of the filtered Navier–Stokes equations can be computed directly. The effect of nonrepresented scales is modeled by a relaxation regularization involving a secondary filter operation. Large-eddy simulations are performed for incompressible channel flow at Reynolds numbers based on the friction velocity and the channel half-width of Reτ=180 and Reτ=590. Both simulations compare well with direct numerical simulation (DNS) data and show a significant improvement over results obtained with classical subgrid scale models such as the standard or the dynamic Smagorinsky model. The computational cost of ADM is lower than that of dynamic models or the velocity estimation model.
517 citations
TL;DR: It is demonstrated experimentally that very viscous liquids containing a small amount of high-molecular-weight polymers can be mixed quite efficiently at very low Reynolds numbers, for a simple flow in a curved channel.
Abstract: Mixing in fluids is a rapidly developing area in fluid mechanics, being an important industrial and environmental problem. The mixing of liquids at low Reynolds numbers is usually quite weak in simple flows, and it requires special devices to be efficient. Recently, the problem of mixing was solved analytically for a simple case of random flow, known as the Batchelor regime. Here we demonstrate experimentally that very viscous liquids containing a small amount of high-molecular-weight polymers can be mixed quite efficiently at very low Reynolds numbers, for a simple flow in a curved channel. A polymer concentration of only 0.001% suffices. The presence of the polymers leads to an elastic instability and to irregular flow, with velocity spectra corresponding to the Batchelor regime. Our detailed observations of the mixing in this regime enable us to confirm several important theoretical predictions: the probability distributions of the concentration exhibit exponential tails, moments of the distribution decay exponentially along the flow, and the spatial correlation function of concentration decays logarithmically.
343 citations
TL;DR: This paper presents an analysis of the effects of fluid inertia and pressure on the velocity and vorticity field of electroosmotic flows, using a wall-local, streamline coordinate system.
Abstract: Emerging microfluidic systems have spurred an interest in the study of electrokinetic flow phenomena in complex geometries and a variety of flow conditions. This paper presents an analysis of the effects of fluid inertia and pressure on the velocity and vorticity field of electroosmotic flows. In typical on-chip electrokinetics applications, the flow field can be separated into an inner flow region dominated by viscous and electrostatic forces and an outer flow region dominated by inertial and pressure forces. These two regions are separated by a slip velocity condition determined by the Helmholtz-Smoulochowski equation. The validity of this assumption is investigated by analyzing the velocity field in a pressure-driven, two-dimensional flow channel with an impulsively started electric field. The regime for which the inner/outer flow model is valid is described in terms of nondimensional parameters derived from this example problem. Next, the inertial forces, surface conditions, and pressure-gradient conditions for a full-field similarity between the electric and velocity fields in electroosmotic flows are discussed. A sufficient set of conditions for this similarity to hold in arbitrarily shaped, insulating wall microchannels is the following: uniform surface charge, low Reynolds number, low Reynolds and Strouhal number product, uniform fluid properties, and zero pressure differences between inlets and outlets. Last, simple relations describing the generation of vorticity in electroosmotic flow are derived using a wall-local, streamline coordinate system.
261 citations
TL;DR: In this paper, exact solutions for the following five problems of an Oldroyd-B fluid are obtained: (i) Stokes problem (ii) modified Stokes Problem (iii) the time-periodic Poiseuille flow due to an oscillating pressure gradient (iv) the non-periodical flows between two boundaries, and (v) symmetric flow with an arbitrary initial velocity.
193 citations
TL;DR: In this article, a linear stability analysis of the basic flow becomes unstable at higher Reynolds numbers to four different three-dimensional modes depending on the aspect ratio of the cavity's cross section.
Abstract: The classical rectangular lid-driven-cavity problem is considered in which the motion of an incompressible fluid is induced by a single lid moving tangentially to itself with constant velocity. In a system infinitely extended in the spanwise direction the flow is two-dimensional for small Reynolds numbers. By a linear stability analysis it is shown that this basic flow becomes unstable at higher Reynolds numbers to four different three-dimensional modes depending on the aspect ratio of the cavity’s cross section. For shallow cavities the most dangerous modes are a pair of three-dimensional short waves propagating spanwise in the direction perpendicular to the basic flow. The mode is localized on the strong basic-state eddy that is created at the downstream end of the moving lid when the Reynolds number is increased. In the limit of a vanishing layer depth the critical Reynolds number approaches a finite asymptotic value. When the depth of the cavity is comparable to its width, two different centrifugal-instability modes can appear depending on the exact value of the aspect ratio. One of these modes is stationary, the other one is oscillatory. For unit aspect ratio (square cavity), the critical mode is stationary and has a very short wavelength. Experiments for the square cavity with a large span confirm this instability. It is argued that this three-dimensional mode has not been observed in all previous experiments, because the instability is suppressed by side-wall effects in small-span cavities. For large aspect ratios, i.e., for deep cavities, the critical three-dimensional mode is stationary with a long wavelength. The critical Reynolds number approaches a finite asymptotic value in the limit of an infinitely deep cavity.
158 citations
TL;DR: In this paper, the history of particle deposition in symmetric double-bifurcation airway models is described, assuming spherical noninteracting aerosols that stick to the wall when touching the surface.
Abstract: The flow theory and air flow structures in symmetric double-bifurcation airway models assuming steady laminar, incompressible flow, unaffected by the presence of aerosols, has been described in a companion paper (Part 1). The validated computer simulation results showed highly vortical flow fields, especially around the second bifurcations, indicating potentially complex particle distributions and deposition patterns. In this paper (Part 2), assuming spherical non-interacting aerosols that stick to the wall when touching the surface, the history of depositing particles is described. Specifically, the finite-volume code CFX (AEA Technology) with user-enhanced FORTRAN programs were validated with experimental data of particle deposition efficiencies as a function of the Stokes number for planar single and double bifurcations. The resulting deposition patterns, particle distributions, trajectories and time evolution were analysed in the light of the air flow structures for relatively low (Re D1 = 500) and high (Re D1 = 2000) Reynolds numbers and representative Stokes numbers, i.e. St D1 = 0.04 and St D1 = 0.12. Particle deposition patterns and surface concentrations are largely a function of the local Stokes number, but they also depend on the fluid-particle inlet conditions as well as airway geometry factors. While particles introduced at low inlet Reynolds numbers (e.g. Re D1 = 500) follow the axial air flow, secondary and vortical flows become important at higher Reynolds numbers, causing the formation of particle-free zones near the tube centres and subsequently elevated particle concentrations near the walls. Sharp or mildly rounded carinal ridges have little effect on the deposition efficiencies but may influence local deposition patterns. In contrast, more drastic geometric changes to the basic double-bifurcation model, e.g. the 90°-non-planar configuration, alter both the aerosol wall distributions and surface concentrations considerably.
138 citations
TL;DR: Two microfluidic switches are described based on these two ways for controlling fluid flow through tangential microchannels and theoretical arguments that explain the observed dependence of the flow profiles on the aspect ratio of the channels are presented.
Abstract: This paper describes laminar fluid flow through a three- dimensional elastomeric microstructure formed by two microfluidic channels, fabricated in layers that contact one another face-to-face (typically at a 90 degrees angle), with the fluid flows in tangential contact. There are two ways to control fluid flow through these tangentially connected microchannels. First, the flow profiles through the crossings are sensitive to the aspect ratio of the channels; the flow can be controlled by applying external pressure and changing this aspect ratio. Second, the flow direction of an individual laminar stream in multiphase laminar flow depends on the lateral position of the stream within the channel; this position can be controlled by injecting additional streams of fluid into the channel. We describe two microfluidic switches based on these two ways for controlling fluid flow through tangential microchannels and present theoretical arguments that explain the observed dependence of the flow profiles on the aspect ratio of the channels.
109 citations
TL;DR: In this article, the structure of laminar flow and heat transfer, in a two-dimensional horizontal channel differentially heated, with a built-in triangular prism is investigated from the numerical solutions of complete Navier-Stokes and energy equations.
108 citations
TL;DR: In this paper, it has been shown that the time-discretized lubrication approximation of the one-phase Hele-Shaw flow is the same as the constant limit of the continuous limit of timediscrete gradient flows of the corresponding surface energy functionals with respect to the Wasserstein metric.
Abstract: It has been recently discovered that both the surface tension driven one-phase Hele-Shaw flow and its lubrication approximation can be understood as (continuous limits of time-discretized) gradient flows of the corresponding surface energy functionals with respect to the Wasserstein metric. Here we complete the connection between the two problems, proving that the time-discretized lubrication approximation is the \(\Gamma\)-limit of suitably rescaled time-discretized Hele-Shaw flows in half space.
98 citations
TL;DR: In this article, a spherical drop, placed in a second liquid of the same density and viscosity, is subjected to shear between parallel walls, and the subsequent flow is investigated numerically with a volume-of-fluid continuous-surface-force algorithm.
Abstract: A spherical drop, placed in a second liquid of the same density and viscosity, is subjected to shear between parallel walls. The subsequent flow is investigated numerically with a volume-of-fluid continuous-surface-force algorithm. Inertially driven breakup is examined. The critical Reynolds numbers are examined for capillary numbers in the range where the drop does not break up in Stokes flow. It is found that the effect of inertia is to rotate the drop toward the vertical direction, with a mechanism analogous to aerodynamic lift, and the drop then experiences higher shear, which pulls the drop apart horizontally. The balance of inertial stress with capillary stress shows that the critical Reynolds number scales inversely proportional to the capillary number, and this is confirmed with full numerical simulations. Drops exhibit self-similar damped oscillations towards equilibrium analogous to a one-dimensional mass-spring system. The stationary drop configurations near critical conditions approach an inviscid limit, independent of the microphysical flow- and fluid-parameters.
TL;DR: In this paper, a Fourier-Chebyshev spectral collocation method for Reynolds numbers up to 500 was used to obtain a direct numerical solution for flow and heat transfer past a sphere in a uniform flow.
Abstract: Direct numerical solution for flow and heat transfer past a sphere in a uniform flow is obtained using an accurate and efficient Fourier-Chebyshev spectral collocation method for Reynolds numbers up to 500. We investigate the flow and temperature fields over a range of Reynolds numbers, showing steady and axisymmetric flow when the Reynolds number is less than 210, steady and nonaxisymmetric flow without vortex shedding when the Reynolds number is between 210 and 270, and unsteady three-dimensional flow with vortex shedding when the Reynolds number is above 270. Results from three-dimensional simulation are compared with the corresponding axisymmetric simulations for Re > 210 in order to see the effect of unsteadiness and three-dimensionality on heat transfer past a sphere
TL;DR: The stability of the channel flow of two fluids of different viscosities with a mixed layer in between is demonstrated to be qualitatively different from both interface dominated flows and stratified flows as mentioned in this paper.
Abstract: The stability of the channel flow of two fluids of different viscosities with a mixed layer in between is demonstrated to be qualitatively different from both interface dominated flows and stratified flows. More important, this flow displays unexpected changes in stability when the mixed layer overlaps the critical layer of the disturbance: this feature can be exploited for flow control. When these layers are distinct, the flow is mildly destabilized when the less viscous fluid is in the outer region. When the layers overlap, however, there is an order of magnitude stabilization of the flow. The reverse occurs when the more viscous fluid is in the outer region. This behavior may be explained by the balance of stresses in the critical layer.
TL;DR: In this paper, the effects of different inlet Reynolds and Stokes numbers in a triple bifurcation of the human respiratory system were simulated for several combinations of relatively high and low inlet numbers, and it was shown that preferential concentration of particles can be induced by the secondary vortical flow in the tubes when the inlet number is high enough.
Abstract: Considering steady laminar incompressible flow in a triple bifurcation, which represents generations three to six of the human respiratory system, air flow fields and micron-particle transport have been simulated for several combinations of relatively high and low inlet Reynolds and Stokes numbers. While the upstream bifurcations are hardly affected by the third bifurcation, complex air and particle flow fields occur in the daughter tubes leading to the third dividers. Variations in Reynolds number, 500≤Re≤2000, and Stokes number, 0.04≤St≤0.12, cause locally changing vortical air flows as well as irregular particle motions. Preferential concentration of particles can be induced by the secondary vortical flow in the tubes when the inlet Reynolds number is high enough. The air and particle velocity profiles in the third daughter tubes are still quite different from those in the upstream tubes, which indicates that additional downstream effects are possible. This work may contribute to respiratory dose estimation in health risk assessment studies, as well as the analyses of drug aerosol delivery.
TL;DR: In this paper, a finite volume method is used to calculate the steady incompressible flow in rectangular cavities, where the flow is driven by two opposing cavity side walls which move with constant velocities tangentially to themselves.
Abstract: The two-dimensional steady incompressible flow in rectangular cavities is calculated numerically by a finite volume method. The flow is driven by two opposing cavity side walls which move with constant velocities tangentially to themselves. Depending on the cavity aspect ratio and the two side-wall Reynolds numbers different flow states exist. Their range of existence and the bifurcations between different states are investigated by a continuation method accurately locating the bifurcation points. When both side walls move in opposite directions up to seven solutions are found to exist for the same set of parameters. Three of these are point-symmetric and four are asymmetric with respect to the center of the cavity, if the side-wall Reynolds numbers have the same magnitude. When the walls move in the same direction, up to five different flow states are found. In this case only a single mirror symmetric solution exists for equal Reynolds numbers.
TL;DR: In this paper, a two-phase flow analysis is used to analyze sediment concentration profiles in uniform open-channel flows over flat, sediment-starved beds that have high concentrations of single-sized sediment.
Abstract: Two-phase flow analysis is used to analyze sediment concentration profiles in uniform open-channel flows over flat, sediment-starved beds that have high concentrations of single-sized sediment. Two-phase flow analysis can explicitly incorporate the effects of particle-particle interactions and particle inertia. Conventional convection-diffusion modeling cannot directly represent these phenomena and are thus limited. Both the two-phase flow formulation and the convection-diffusion modeling are compared against experimental data collected in sediment-starved sediment-laden flows. The two-phase flow model is shown to simulate the effect of both particle-particle interactions and particle inertia in these experimental flows. Simple criteria are given to determine when particle-particle interactions and particle inertia are important in sediment-laden open-channel flows over a flat bed. The current two-phase approach requires empirical formulas of the turbulence quantities and further experimental and analytical work is necessary to develop improved models for the velocity distribution and turbulence quantities.
TL;DR: In this article, a quasi-two-dimensional magnetohydrodynamics (MHD) model was proposed to study the behavior of two-dimensional turbulent flows, where a strong external magnetic field B perpendicular to the flow direction of an electrically conducting fluid will suppress velocity gradients in the direction of B. The resulting approximation is known as quasi-2D MHD.
Abstract: Magnetohydrodynamics (MHD) offers a unique opportunity to study the behavior of two-dimensional turbulent flows. A strong external magnetic field B perpendicular to the flow direction of an electrically conducting fluid will suppress velocity gradients in the direction of B. The resulting approximation is known as quasi-two-dimensional MHD. An experimental configuration is presented which meets this requirement, along with a spatially extended probe used to visualize the two-dimensional flow kinematics inside the opaque liquid metal flow. As a prototypical example, the wake behind a circular cylinder is investigated for Reynolds numbers up to R=10 000. New and unexpected vortex patterns are observed that deviate significantly from usual hydrodynamic flows. Also, stability limits for the transition from stationary to nonstationary flow patterns are experimentally determined for the cylinder wake and another type of shear flow profile. These results confirm existing theoretical predictions and thus validate...
TL;DR: In this paper, the stability of a multi-layer Poiseuille flow of two Bingham fluids is investigated and it is shown that this two-fluid flow is frequently more stable than the equivalent flow of either fluid alone.
Abstract: We consider the stability of a multi-layer plane Poiseuille flow of two Bingham fluids. It is shown that this two-fluid flow is frequently more stable than the equivalent flow of either fluid alone. This phenomenon of super-stability results only when the yield stress of the fluid next to the channel wall is larger than that of the fluid in the centre of the channel, which need not have a yield stress. Our result is in direct contrast to the stability of analogous flows of purely viscous generalised Newtonian fluids, for which short wavelength interfacial instabilities can be found at relatively low Reynolds numbers. The results imply the existence of parameter regimes where visco-plastic lubrication is possible, permitting transport of an inelastic generalised Newtonian fluid in the centre of a channel, lubricated at the walls by a visco-plastic fluid, travelling in a stable laminar flow at higher flow rates than would be possible for the single fluid alone.
TL;DR: In this article, an unsteady Navier-Stokes solver for incompressible fluid is coupled with a level set approach to describe free surface motions, where the two-phase flow of air and water is approximated by the flow of a single fluid whose properties, such as density and viscosity, change across the interface.
Abstract: An unsteady Navier–Stokes solver for incompressible fluid is coupled with a level set approach to describe free surface motions. The two-phase flow of air and water is approximated by the flow of a single fluid whose properties, such as density and viscosity, change across the interface. The free surface location is captured as the zero level of a distance function convected by the flow field. To validate the numerical procedure, two classical two-dimensional free surface problems in hydrodynamics, namely the oscillating flow in a tank and the waves generated by the flow over a bottom bump, are studied in non-breaking conditions, and the results are compared with those obtained with other numerical approaches. To check the capability of the method in dealing with complex free surface configurations, the breaking regime produced by the flow over a high bump is analyzed. The analysis covers the successive stages of the breaking phenomenon: the steep wave evolution, the falling jet, the splash-up and the air entrainment. In all phases, numerical results qualitatively agree with the experimental observations. Finally, to investigate a flow in which viscous effects are relevant, the numerical scheme is applied to study the wavy flow past a submerged hydrofoil. Copyright © 2001 John Wiley & Sons, Ltd.
TL;DR: In this article, the Saffman-Taylor instability of an air bubble expanding into a non-Newtonian fluid in a Hele-Shaw cell was studied theoretically, with the motivation of understanding suppression of tip-splitting and the formation of dendritic structures observed in the flow of complex fluids, such as polymeric liquids or liquid crystals.
Abstract: We study theoretically the Saffman–Taylor instability of an air bubble expanding into a non-Newtonian fluid in a Hele–Shaw cell, with the motivation of understanding suppression of tip-splitting and the formation of dendritic structures observed in the flow of complex fluids, such as polymeric liquids or liquid crystals. A standard visco-elastic flow model is simplified in the case of flow in a thin gap, and it is found that there is a distinguished limit where shear thinning and normal stress differences are apparent, but elastic response is negligible. This observation allows formulation of a generalized Darcy’s law, where the pressure satisfies a nonlinear elliptic boundary value problem. Numerical simulation shows that shear-thinning alone modifies considerably the pattern formation and can produce fingers whose tip-splitting is suppressed, in agreement with experimental results. These fingers grow in an oscillating fashion, shedding “side-branches” from their tips, closely resembling solidification patterns. A careful analysis of the parametric dependencies of the system provides an understanding of the conditions required to suppress tip-splitting, and an interpretation of experimental observations, such as emerging length-scales.
TL;DR: In this paper, a closed-form expression was derived to estimate the pressure drop through sinusoidally constricted tubes, and good agreement was found in the creeping flow regime for the pressure-drop versus flow rate relationship.
Abstract: Creeping flow of a Newtonian fluid through tubes of varying radius is studied. Using an asymptotic series solution for low Reynolds number flow, velocity profiles and streamlines are obtained for constricted tubes, for various values of constriction wavelength and amplitude. A closed-form expression is derived to estimate the pressure drop through this type of tube. The results obtained with this new expression are compared to data from previous experimental and numerical studies for sinusoidally constricted tubes. Good agreement is found in the creeping flow regime for the pressure drop versus flow rate relationship. Our method offers an improvement over the integrated form of the Hagen–Poiseuille equation (i.e., lubrication approximation), which does not account for the wavelength of the constrictions.
TL;DR: In this article, an experimental study was conducted of incompressible, moderate Reynolds number flow of air over heated rectangular blocks in a two-dimensional, horizontal channel using holographic interferometry combined with high-speed cinematography was used to visualize the unsteady temperature fields in self- sustained oscillatory flow.
Abstract: An experimental study was conducted of incompressible, moderate Reynolds number flow of air over heated rectangular blocks in a two-dimensional, horizontal channel. Holographic interferometry combined with high-speed cinematography was used to visualize the unsteady temperature fields in self- sustained oscillatory flow. Experiments were conducted in the laminar, transitional and turbulent flow regimes for Reynolds numbers in the range from Re = 520 to Re = 6600. Interferometric measurements were obtained in the thermally and fluiddynamically periodically fully developed flow region on the ninth heated block. Flow oscillations were first observed between Re = 1054 and Re = 1318. The period of oscillations, wavelength and propagation speed of the Tollmien–Schlichting waves in the main channel were measured at two characteristic flow velocities, Re = 1580 and Re = 2370. For these Reynolds numbers it was observed that two to three waves span one geometric periodicity length. At Re = 1580 the dominant oscillation frequency was found to be around 26 Hz and at Re = 2370 the frequency distribution formed a band around 125 Hz. Results regarding heat transfer and pressure drop are presented as a function of the Reynolds number, in terms of the block-average Nusselt number and the local Nusselt number as well as the friction factor. Measurements of the local Nusselt number together with visual observations indicate that the lateral mixing caused by flow instabilities is most pronounced along the upstream vertical wall of the heated block in the groove region, and it is accompanied by high heat transfer coefficients. At Reynolds numbers beyond the onset of oscillations the heat transfer in the grooved channel exceeds the performance of the reference geometry, the asymmetrically heated parallel plate channel.
TL;DR: In this article, a simulation of turbulent flow between a rotating and a stationary disk is performed by integrating the time-dependent Navier-Stokes equations until a statistically steady state is reached and with the aim of providing both long-time statistics and an exposition of coherent structures obtained by conditional sampling.
Abstract: Turbulent flow between a rotating and a stationary disk is studied. Besides its fundamental importance as a three-dimensional prototype flow, such flow fields are frequently encountered in rotor–stator configurations in turbomachinery applications. A direct numerical simulation is therefore performed by integrating the time-dependent Navier–Stokes equations until a statistically steady state is reached and with the aim of providing both long-time statistics and an exposition of coherent structures obtained by conditional sampling. The simulated flow has local Reynolds number r2ω/v = 4 × 105 and local gap ratio s/r = 0.02, where ω is the angular velocity of the rotating disk, r the radial distance from the axis of rotation, v the kinematic viscosity of the fluid, and s the gap width.The three components of the mean velocity vector and the six independent Reynolds stresses are compared with experimental measurements in a rotor–stator flow configuration. In the numerically generated flow field, the structural parameter a1 (i.e. the ratio of the magnitude of the shear stress vector to twice the mean turbulent kinetic energy) is lower near the two disks than in two-dimensional boundary layers. This characteristic feature is typical for three-dimensional boundary layers, and so are the misalignment between the shear stress vector and the mean velocity gradient vector, although the degree of misalignment turns out to be smaller in the present flow than in unsteady three-dimensional boundary layer flow. It is also observed that the wall friction at the rotating disk is substantially higher than at the stationary disk.Coherent structures near the disks are identified by means of the λ2 vortex criterion in order to provide sufficient information to resolve a controversy regarding the roles played by sweeps and ejections in shear stress production. An ensemble average of the detected structures reveals that the coherent structures in the rotor–stator flow are similar to the ones found in two-dimensional flows. It is shown, however, that the three-dimensionality of the mean flow reduces the inter-vortical alignment and the tendency of structures of opposite sense of rotation to overlap. The coherent structures near the disks generate weaker sweeps (i.e. quadrant 4 events) than structures in conventional two-dimensional boundary layers. This reduction in the quadrant 4 contribution from the coherent structures is believed to explain the reduced efficiency of the mean flow in producing Reynolds shear stress.
TL;DR: In this article, a true-to-mechanism model is proposed, which considers steady-state two-phase flow in porous media as a composition of two prototype flows, namely ganglion dynamics (GD) and connected-oil pathway flow (CPF).
TL;DR: In this article, the authors consider the flow within a rotating horizontal cylinder containing a small amount of a very viscous liquid which completely coats the cylinder surface and show that, under creeping flow conditions, the addition of the hydrostatic pressure term to the standard lubrication equation leads to film thickness profiles which, over a broad range of parameters, are in close agreement with those obtained experimentally, as well as via the solution to the full Stokes equations.
Abstract: We consider the flow within a rotating horizontal cylinder containing a small amount of a very viscous liquid which completely coats the cylinder surface. We show that, under creeping flow conditions, the addition of the hydrostatic pressure term to the standard lubrication equation leads to film thickness profiles which, over a broad range of parameters, are in close agreement with those obtained experimentally, as well as via the solution to the full Stokes equations.
TL;DR: In this article, the shape of a viscous drop deforming in several time-dependent flow fields, including that due to a potential vortex, has been studied, and it is found that the drop deformation is non-monotonic with time in its approach to an equilibrium value.
Abstract: The shape of a two-dimensional viscous drop deforming in several time-dependent flow fields, including that due to a potential vortex, has been studied. Vortex flow was approximated by linearizing the induced velocity field at the drop centre, giving rise to an extensional flow with rotating axes of stretching. A generalization of the potential vortex, a flow we have called rotating extensional flow, occurs when the frequency of revolution of the flow is varied independently of the shear rate. Drops subjected to this forcing flow exhibit an interesting resonance phenomenon. Finally we have studied drop deformation in an oscillatory extensional flow.Calculations were performed at small but non-zero Reynolds numbers using an ADI front-tracking/finite difference method. We investigate the effects of interfacial tension, periodicity, viscosity ratio, and Reynolds number on the drop dynamics. The simulation reveals interesting behaviour for steady stretching flows, as well as time-dependent flows. For a steady extensional flow, the drop deformation is found to be non-monotonic with time in its approach to an equilibrium value. At sufficiently high Reynolds numbers, the drop experiences multiple growth–collapse cycles, with possible axes reversal, before reaching a final shape. For a vortex flow, the long-time deformation reaches a steady value, and the drop attains a revolving steady elliptic shape. For rotating extensional flows as well as oscillatory extensional flows, the maximum value of deformation displays resonance with variation in parameters, first increasing and then decreasing with increasing interfacial tension or forcing frequency. A simple ODE model with proper forcing is offered to explain the observed phenomena.
TL;DR: In this article, a local analytic representation valid near the particle and matching it to an external field calculated by a standard finite-difference (or finite-element) method is presented.
TL;DR: In this paper, Barenblatt's and George and Castillo's power laws are used to estimate the friction velocities for flows over a smooth surface, but only the skin friction law has good prediction for the rough wall data, and the results indicate that the power law formulations can model a wider extent of the flow than the classical logarithmic profile.
Abstract: Scaling laws for the overlap region of near-wall turbulent flows are of particular interest to turbulence researchers and engineers. For the mean flow at sufficiently high Reynolds numbers, the classical boundary layer theory proposes a logarithmic law for the overlap region. On the other hand, at low Reynolds numbers, refined measurements and direct numerical simulation results indicate that the log law region becomes negligibly small. Instead, power laws have received increasing attention as an alternative formulation for the overlap region at low Reynolds numbers. In the present study, we use open channel flow measurements to assess the ability of the power laws proposed by Barenblatt [J. Fluid Mech. 248, 513 (1993)] and George and Castillo [Appl. Mech. Rev. 50, 689 (1997)] to describe the overlap region in low Reynolds number boundary layers on smooth and rough surfaces. The skin friction laws derived from the power laws are also used to estimate the friction velocity, which values are then compared to measurements obtained by other reliable techniques. The results indicate that at low Reynolds numbers the power law formulations can model a wider extent of the flow than the classical logarithmic profile. Both Barenblatt’s and George and Castillo’s power laws give an excellent prediction of the friction velocities for flows over a smooth surface, but only the skin friction law proposed by George and Castillo gives good prediction for the rough wall data.
TL;DR: In this paper, the effect of steady base suction and blowing on the stability and dynamics of the cylinder wake was investigated at low Reynolds numbers, using numerical simulation and stability analysis.
Abstract: The effect of steady base suction and blowing on the stability and dynamics of the cylinder wake is investigated at low Reynolds numbers, using numerical simulation and stability analysis. Simulation results show that, in the supercritical Reynolds number regime (Re>47), slight blowing or high enough suction stabilizes the wake; in the subcritical regime, suction destabilizes the wake and results in vortex shedding, whereas blowing has no detectable effect on the flow stability. At supercritical Reynolds numbers, the transition from unsteady to steady flow at a critical suction flow rate is accompanied by simultaneous symmetry breaking, resulting in strongly asymmetric steady flow. For finite flow domain, the flow undergoes another transition from steady asymmetric to steady symmetric flow at even higher suction flow rates. The dynamics of vortex shedding in the controlled flow can be strongly modified, in comparison to the uncontrolled flow. Global stability analysis confirms the results of numerical sim...
TL;DR: In this paper, the authors derived the appropriate inertial corrections to the Darcy law in a Hele-Shaw cell based on a perturbative method and a polynomial approximation to the velocity field.
Abstract: This note presents a derivation of the appropriate inertial corrections to the Darcy law in a Hele–Shaw cell based on a perturbative method and a polynomial approximation to the velocity field. The obtained equation is optimal in the sense that every method of weighted residuals will converge to it as the number of test functions is increased. A good agreement with the study of the shear instability in a Hele–Shaw cell at low Reynolds number is found.