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Volume 6, Issue 1, January 1994

Characteristics of the linearized equations of motion for a compressible fluid
View Description Hide DescriptionThe characteristics and bicharacteristics of the linearized equations of motion for compressible fluid are identical to the wave fronts and rays, respectively, calculated by Fermat’s least‐time principle of geometric acoustics. The bicharacteristics emanating from the middle of a compressible shear layer curve so as to hinder communication in the axial direction.

Nonstirring of an inviscid fluid by a point vortex in a rectangle
View Description Hide DescriptionNumerical evidence is presented that two‐dimensional motion of a large region of incompressible, inviscid fluid due to the motion of a point vortex in a rectangular container leaves does not stir certain finite regions. Such regions are formed by means of the Poincaré mapping in the regular domain for a nonintegrable Hamiltonian system of fluid particle motion.

Singular front formation in a model for quasigeostrophic flow
View Description Hide DescriptionA two‐dimensional model for quasigeostrophic flow which exhibits an analogy with the three‐dimensional incompressible Euler equations is considered. Numerical experiments show that this model develops sharp fronts without the need to explicitly incorporate any ageostrophic effect. Furthermore, these fronts appear to become singular in finite time. The numerical evidence for singular behavior survives the tests of rigorous mathematical criteria.

Apparent dynamic contact angle of an advancing gas–liquid meniscus
View Description Hide DescriptionThe steady motion of an advancing meniscus in a gas‐filled capillary tube involves a delicate balance of capillary, viscous, and intermolecular forces. The limit of small capillary numbers Ca (dimensionless speeds) is analyzed here with a matched asymptotic analysis that links the outer capillary region to the precursor film in front of the meniscus through a lubricating film. The meniscus shape in the outer region is constructed and the apparent dynamic contact angle Θ that the meniscus forms with the solid surface is derived as a function of the capillary number, the capillary radius, and the Hamaker’s constant for intermolecular forces, under conditions of weak gas–solid interaction, which lead to fast spreading of the precursor film and weak intermolecular forces relative to viscous forces within the lubricating film. The dependence on intermolecular forces is very weak and the contact angle expression has a tight upper bound tan Θ=7.48 Ca^{1/3} for thick films, which is independent of the Hamaker constant. This upper bound is in very good agreement with existing experimental data for wetting fluids in any capillary and for partially wetting fluids in a prewetted capillary. Significant correction to the Ca^{1/3} dependence occurs only at very low Ca, where the intermolecular forces become more important and tan Θ diverges slightly from the above asymptotic behavior toward lower values.

Laplace pressure driven drop spreading
View Description Hide DescriptionThis work concerns the spreading of viscousdroplets on a smooth rigid horizontal surface, under the condition of complete wetting (spreading parameter S≳0) with the Laplace pressure as the dominant force. Owing to the self‐similar character foreseeable for this flow, a self‐similar solution is built up by numerical integration from the center of symmetry to the front position to be determined, defined as the point where the free‐surface slope becomes zero. Mass and energy conservation are invoked as the only further conditions to determine the flow. The resulting fluid thickness at the front is a small but finite (≊10^{−7}) fraction of the height at the center. By comparison with experimental results the regime is determined in which the spreading can be described by this solution with good accuracy. Moreover, even within this regime, small but systematic deviations from the predictions of the theory were observed, showing the need to add terms modifying the Laplace pressure force.

Interaction between short‐scale Marangoni convection and long‐scale deformational instability
View Description Hide DescriptionNonlinear evolution of two interacting modes of the Marangoni convection, a long‐scale deformational mode and a short‐scale stationary convective pattern, is considered. It is shown that the interaction between modes stabilizes surface deformation and leads to formation of various convective structures: stationary long‐scale modulated roll patterns, traveling and standing long waves, and can also cause chaotic convection (interfacial turbulence).

Small‐scale structures in Boussinesq convection
View Description Hide DescriptionTwo‐dimensional Boussinesq convection is studied numerically using two different methods: a filtered pseudospectral method and a high‐order accurate eno scheme. The issue whether finite time singularity occurs for initially smooth flows is investigated. In contrast to the findings of Pumir and Siggia who reported finite time collapse of the bubble cap, the present numerical results suggest that the strain rate corresponding to the intensification of the density gradient across the front saturates at the bubble cap. Consequently, the thickness of the bubble decreases exponentially. On the other hand, the bubble experiences much stronger straining and intensification of gradients at its side. As the bubble rises, a secondary front also forms from its tail. Together with the primary front, they constitute a pair of tightly bound plus and minus double vortex sheet structure which is highly unstable and vulnerable to viscous dissipation.

Double diffusion in a vertical fluid layer: Onset of the convective regime
View Description Hide DescriptionThis paper considers the onset of double diffusive natural convection in a vertical layer of a binary fluid submitted to horizontal thermal and compositional gradients. The analysis deals with the particular situation where the resulting buoyancy forces (the Grashof numbers corresponding to the thermal and solutal effects) are opposing and of equal intensity. The stability analysis for the infinite layer shows that the purely diffusive (motionless) solution prevails at moderate Grashof numbers, and an analytical expression of the critical Rayleigh number as a function of the Lewis number is obtained. These results are then compared to the critical values which are determined from numerical simulations in an enclosure. Numerical calculations in the transient regime are used to give an interpretation of the stability of the steady state diffusive regime when the buoyancy forces are below the critical value.

Shear flow over a plane wall with an axisymmetric cavity or a circular orifice of finite thickness
View Description Hide DescriptionShear flow over a plane wall that contains an axisymmetric depression or pore is studied using a new boundary integral method which is suitable for computing three‐dimensional Stokes flow within axisymmetric domains. Numerical results are presented for cavities in the shape of a section of a sphere or a circular cylinder of finite length, and for a family of pores or orifices with finite thickness. The results illustrate the distribution of shear stresses over the plane wall and inside the cavities or pores. It is found that in most cases, the distribution of shear stresses over the plane wall, around the depressions, is well approximated with that for flow over an orifice of infinitesimal thickness for which an exact solution is available. The kinematic structure of the flow is discussed with reference to eddy formation and three‐dimensional flow reversal. It is shown that the thickness of a circular orifice or depth of a pore play an important role in determining the kinematical structure of the flow underneath the orifice in the lower half‐space.

Pressure‐driven flow of suspensions of liquid drops
View Description Hide DescriptionThe pressure‐driven flow of a periodic suspension of two‐dimensional viscousdrops in a channel that is bounded by two parallel plane walls is studied numerically using the method of interfacial dynamics, which is an improved version of the boundary integral method. The viscosity of the drops is assumed to be equal to that of the suspending fluid. The effects of capillary number, volume fraction, and number of rows are examined for ordered suspensions, where the drops are initially arranged in several rows on a hexagonal lattice. Results of dynamic simulations for random monodisperse suspensions with up to 12 drops per periodic cell are performed, and the salient features of the motion are discussed. It is found that, in all cases, the drops tend to migrate toward the centerline of the channel, forming either a single row or multiple rows. The effect of the instantaneous suspension microstructure on the effective viscosity is illustrated, and some important differences in the behavior of suspensions in pressure‐driven and shear‐driven flows are identified and discussed. Numerical evidence is presented, suggesting that the behavior of suspensions of high viscositydrops may be significantly different from that of suspensions of drops with small and moderate viscosity.

Simulation of viscous fingering in miscible displacements with nonmonotonic viscosity profiles
View Description Hide DescriptionThe nonlinear evolution of viscous fingering instabilities in miscible displacement flows in porous media with nonmonotonic viscosity profiles is investigated. The flow is accurately simulated using a Hartley transform based spectral method. A flow with nonmonotonic viscosity profile has an unstable region followed downstream by a stable region. Instabilities first begin in the unstable region and then grow and penetrate the stable region. The striking contrast between viscous fingering in flows with monotonic and nonmonotonic viscosity profiles is the direction of fluid penetration. The nonmonotonicity in the viscosity profile gives rise to a new phenomena of ‘‘reverse’’ fingering in which the displaced fluid fingers through the displacing fluid more readily than vice versa. A forward and a reverse mixing lengths are defined to characterize the growth of the mixing zone in the two directions. At large times, both the forward and reverse mixing lengths grow linearly in time. A model nonmonotonic viscosity profile is used to parametrically study the asymptotic mixing rates. The parametric study shows that for a given end point and maximum viscosities the growth rate of the mixing zone varies nonmonotonically with the length of the stable barrier in the viscosity profile. A physical mechanism is put forth to explain the observed phenomena of reverse fingering and its dependence on the parameters of the problem. Finally, a finger is isolated and its evolution is studied to understand the differences in the mechanisms that control the growth of a finger in flows with monotonic and nonmonotonic viscosity profiles.

Anomalous dispersion in a dipole flow geometry
View Description Hide DescriptionThe dispersion of a passive tracer in fluid flowing between a source and a sink in a Hele–Shaw geometry, characteristic of field scale flows in a layer or fracture, is considered. A combination of analytic and numerical techniques and complementary experimental measurements are employed, leading to a consistent picture. This dispersion process is found to be characterized by a power‐law decay in time of the tracer concentration, with an exponential cutoff at very long times, in strong contrast to the Gaussian behavior associated with the widely used quasi‐one‐dimensional (1‐D) models.

On stationary equivalent modons in an eastward flow
View Description Hide DescriptionModons correspond to isolated dipole vortex solutions of the quasigeostrophic equations. They have been proposed as prototype models for some geophysical (and plasma)vortices. The classical modon solution on a β plane does not permit a Rossby wave field in the exterior or far‐field region of the modon. However, it is qualitatively known that the gravest mode associated with a normal mode decomposition of a stationary modon in a continuously stratified fluid of finite depth necessarily contains a Rossby wave tail in the downstream region if the background flow is eastward. The same effect can be formally recreated in an equivalent‐barotropic model of a stationary modon embedded in a constant eastward zonal flow. An analytical solution to this problem satisfying the correct upstream radiation condition is presented and its dynamical characteristics are discussed.

A low‐dimensional Galerkin method for the three‐dimensional flow around a circular cylinder
View Description Hide DescriptionA low‐dimensional Galerkin method for the three‐dimensional flow around a circular cylinder is constructed. The investigation of the wake solutions for a variety of basic modes, Hilbert spaces, and expansion modes reveals general mathematical and physical aspects which may strongly effect the success of low‐dimensional simulations. Besides the cylinder wake, detailed information about the construction of similar low‐dimensional Galerkin methods for the sphere wake, the boundary‐layer, the flow in a channel or pipe, the Taylor–Couette problem, and a variety of other flows is given.

Hydrodynamic stability of viscous flow between rotating porous cylinders with radial flow
View Description Hide DescriptionA linear stability analysis has been carried out for flow between porous concentric cylinders when radial flow is present. Several radius ratios with corotating and counter‐rotating cylinders were considered. The radial Reynolds number, based on the radial velocity at the inner cylinder and the inner radius, was varied from −30 to 30. The stability equations form an eigenvalue problem that was solved using a numerical technique based on the Runge–Kutta method combined with a shooting procedure. The results reveal that the critical Taylor number at which Taylor vortices first appear decreases and then increases as the radial Reynolds number becomes more positive. The critical Taylor number increases as the radial Reynolds number becomes more negative. Thus, radially inward flow and strong outward flow have a stabilizing effect, while weak outward flow has a destabilizing effect on the Taylor vortexinstability. Profiles of the relative amplitude of the perturbed velocities show that radially inward flow shifts the Taylor vortices toward the inner cylinder, while radially outward flow shifts the Taylor vortices toward the outer cylinder. The shift increases with the magnitude of the radial Reynolds number and as the annular gap widens.

Instability of the large Reynolds number flow of a Newtonian fluid over a viscoelastic fluid
View Description Hide DescriptionThe stability of the large Reynolds numberflow of a Newtonian fluid over a much more viscousviscoelastic fluid is studied via a linear analysis. The two fluids are confined within a channel and the flow is driven by the motion of the plate bounding the Newtonian fluid. Matched asymptotic expansions are used to derive the dispersion relation, and the flow is found to be always unstable to an interfacial mode due to the discontinuity in the fluid viscosities. It is shown that even a small amount of elasticity of the viscoelastic fluid can change the stability characteristics considerably.

Note on the Kelvin–Helmholtz instability of stratified fluids
View Description Hide DescriptionIn this paper the conditions under which the Kelvin–Helmholtz instability in stratified fluids is absolute or convective are investigated. It is first shown that at transition from convective to absolute instability two double roots of the dispersion relation coalesce. Based on this property an analytical condition for absolute instability is derived. It is found that the instability is absolute for almost all values of the flow velocity and stratification. Convective instability is found only for a narrow range of flow velocities over the instability threshold when the density ratio parameter, r=ρ_{1}/(ρ_{1}+ρ_{2}), lies in the range 1/3<r<1/2. The different behavior near the instability threshold can be related to the signs of the group velocities of the two waves which coalesce to create the instability: For r<1/3, the group velocities of the two waves have opposite signs, and the resulting instability is absolute, whereas for 1/3<r<1/2, the two waves have group velocities with the same sign, and the instability is convective. This result is also shown to be reflected in the form of the amplitude equation at the instability threshold, which is the linearly unstable Klein–Gordon equation.

Local analysis of the onset of instability in shear flows
View Description Hide DescriptionLocal analysis of the onset of instability in flows that are not exactly parallel is considered. Corrections to the Orr–Sommerfeld equation arising as a consequence of the nonparallelism of the unperturbed flow are studied. The quasiparallel hypothesis is quantified on a model of Gaussian plane wave packets. It appears that the characteristic length scales of the downstream dependence of flow field characteristics must be substantially larger than the inverse of the wave number characterizing the instability. A new eigenvalue problem describing the propagation of these Gaussian wave packets is written. The relation between the marginal and absolute instability analysis for marginal Reynolds numbers is discussed. For flows varying slowly in the downstream direction, closed‐form corrections of the Orr–Sommerfeld equation terms taking account of the x variation of the flow field and of the extension of the propagating wave packets are derived. A first‐order perturbation theory correction of the Orr–Sommerfeld dispersion relation is proposed, allowing the reduction of the calculation of nonparallel corrections of the local instability quantities to quadratures. The proposed theory is applied to two important cases: the Blasius boundary layer and the cylinder wake. For the Blasius boundary layer the basic condition of applicability of the quasiparallel theory is found to be satisfied. However, the nonparallel correction of the critical Reynolds number is found to be non‐negligible and provides a good agreement with experimental results. In the cylinder wake case direct bidimensional simulation results are used to assess the downstream variation of the flow field characteristics. The characteristic length scale of this variation in the near wake is found to be of the order of unity, which is also the magnitude of the wave numbers characterizing the local absolute instabilities in this region. Hence, the Orr–Sommerfeld analysis and any corrections based on the propagation of plane waves in the wake can hardly be expected to provide more than qualitative results.

A three‐dimensional description of solitary waves and their interaction in Marangoni–Bénard layers
View Description Hide DescriptionA dissipation‐modified Boussinesq‐like system of equations governing three‐dimensional long wavelength Marangoni–Bénard oscillatory convection in a shallow layer heated from the air side is presented. Solitary waves and their oblique and head‐on interactions are considered, thus leading to results that compare well with available experimental data.