Saleh Tanveer

Surprises in viscous fingering

In this paper, we review some aspects of viscous fingering in a Hele-Shaw cell that at first sight appear to defy intuition. These include singular effects of surface tension relative to the corresponding zero-surface-tension problem both for the steady and unsteady problem. They also include a disproportionately large influence of small effects like local inhomogeneity of the flow field near the finger tip, or of the leakage term in boundary conditions that incorporate realistic thin-film effects. Through simple explicit model problems, we demonstrate how such properties are not unexpected for a system approaching structural instability or ill-posedness.

Evolution of Hele-Shaw Interface for Small Surface Tension

We consider the time-evolving displacement of a viscous fluid by another fluid of negligible viscosity in a Hele-Shaw cell, either in a channel or a radial geometry, for idealized boundary conditions developed by McLean & Saffman. The interfacial evolution is conveniently described by a time-dependent conformal map z(£, t) that maps a unit circle (or a semicircle) in the £ plane into the viscous fluid flow region in the physical z-plane. Our paper is concerned with the singularities of the analytically continued z((,,t) in |£| > 1, which, on approaching |£| = 1, correspond to localized distortions of the actual interface. For zero surface tension, we extend earlier results to show that for any initial condition, each singularity, initially present in |£| > 1, continually approaches |£| = 1, the boundary of the physical domain, without any change in the singularity form. However, depending on the singularity type, it may or may not impinge on |£| = 1 in finite time. Under some assumptions, we give analytical evidence to suggest that the ill-posed initial value problem in the physical domain |£| ≤ 1 can be imbedded in a well-posed problem in |£| ≤ 1. We present a numerical scheme to calculate such solutions. For each initial singularity of a certain type, which in the absence of surface tension would have merely moved to a new location £s(t) at time t from an initial £s(0), we find an instantaneous transformation of the singularity structure for non-zero surface tension B; however, for 0 < B << 1, surface tension effects are limited to a small ‘inner’ neighbourhood of £s(t) when t << B-1 Outside the inner region, but for ( — £s(t)1, the singular behaviour of the zero surface tension solution z0 is reflected in On the other hand, for each initial zero of z£, which for B = 0 remains a zero of z0£ at a location £0(t) that is generally different from £0(0), surface tension spawns new singularities that move away from £0(t) and approach the physical domain |£| = 1. We find that even for 0 < B << 1, it is possible for z — z0 — O(1) or larger in some neighbourhood where z0£ is neither singular nor zero. Our findings imply that for a small enough B, the evolution of a Hele-Shaw interface is very sensitive to prescribed initial conditions in the physical domain.

On the formation of Moore curvature singularities in vortex sheets

Moore (1979) demonstrated that the cumulative influence of small nonlinear effects on the evolution of a slightly perturbed vortex sheet is such that a curvature singularity can develop at a large, but finite, time. By means of an analytical continuation of the problem into the complex spatial plane, we find a consistent asymptotic solution to the problem posed by Moore. Our solution includes the shape of the vortex sheet as the curvature singularity forms. Analytic results are confirmed by comparison with numerical solutions. Further, for a wide class of initial conditions (including perturbations of finite amplitude), we demonstrate that 3/2-power singularities can spontaneously form at t =0+ in the complex plane. We show that these singularities propagate around the complex plane. If two singularities collide on the real axis, then a point of infinite curvature develops on the vortex sheet. For such an occurrence we give an asymptotic description of the vortex-sheet shape at times close to singularity formation.

Multiple scales in streamer discharges, with an emphasis on moving boundary approximations

Streamer discharges determine the very first stage of sparks or lightning, and they govern the evolution of huge sprite discharges above thunderclouds as well as the operation of corona reactors in plasma technology. Streamers are nonlinear structures with multiple inner scales. After briefly reviewing basic observations, experiments and the microphysics, we start from density models for streamers, i.e. from reaction–drift–diffusion equations for charged-particle densities coupled to the Poisson equation of electrostatics, and focus on derivation and solution of moving boundary approximations for the density models. We recall that so-called negative streamers are linearly stable against branching (and we conjecture this for positive streamers as well), and that streamer groups in two dimensions are well approximated by the classical Saffman–Taylor finger of two fluid flow. We draw conclusions on streamer physics, and we identify open problems in the moving boundary approximations.

A note on third-order structure functions in turbulence

Starting from the Navier–Stokes equation, we rigorously prove that a modified third–order structure function, Stilde3 (r) asymptotically equals –4 over 3 ∈r (∈ is the dissipation rate) in an inertial regime. From this result, we rigorously confirm the Kolmogorov four–fifths law, without the Kolmogorov assumption on isotropy. Our definition of the structure function involves a solid angle averaging over all possible orientations of the displacement vector y, besides space–time averaging. Direct numerical simulation for a highly symmetric flow for a Taylor Reynolds number of up to 155, shows that the flow remains significantly anisotropic and that, without solid angle averaging, the resulting structure functions approximately satisfy these scaling relations over some range of r = lyl for some orientation of y, but not for another.

Singular effects of surface tension in evolving Hele-Shaw flows

In this paper, we present evidence to show that a smoothly evolving zero-surface tension solution of the Hele-Shaw equations can be singularly perturbed by the presence of arbitrarily small non-zero surface tension in order-one time. These effects are explained by the impact of ‘daughter singularities’ on the physical interface, whose formation was suggested in a prior paper (Tanveer 1993). For the case of finger motion in a channel, it is seen that the daughter singularity effect is strong enough to produce the transition from a finger of arbitrary width to one with the selected steady-state width in O( 1) time.

Existence and Uniqueness for a Class of Nonlinear Higher-Order Partial Differential Equations in the Complex Plane

We prove existence and uniqueness results for nonlinear third-order partial differential equations of the form

Short Time Existence and Borel Summability in the Navier–Stokes Equation in ℝ3

We consider the Navier–Stokes initial value problem, where 𝒫 is the Hodge-Projection to divergence free vector fields in the assumption that ‖f‖μ, β < ∞ and ‖ v 0 ‖μ+2, β < ∞ for β ≥ 0,μ > 3, where and [fcirc] (k) = ℱ [f (·)] (k) is the Fourier transform in x. By Borel summation methods we show that there exists a classical solution in the form t ∈ ℂ, , and we estimate α in terms of ‖ [vcirc]0 ‖μ+2, β and ‖ [fcirc] ‖μ, β. We show that ‖ [vcirc] (·; t) ‖μ+2, β < ∞. Existence and t-analyticity results are analogous to Sobolev spaces ones. An important feature of the present approach is that continuation of v beyond t = α−1 becomes a growth rate question of U(·, p) as p → ∞, U being is a known function. For now, our estimate is likely suboptimal. A second result is that we show Borel summability of v for v 0 and f analytic. In particular, Borel summability implies a the Gevrey-1 asymptotics result: , where , with A 0 and B 0 are given in terms of to v 0 and f and for small t, with ,

Convection effects on radial segregation and crystal–melt interface in vertical Bridgman growth

The influence of convection caused by horizontal heat transfer through the sides of a vertical Bridgman apparatus is studied analytically. The case when the heat transfer across the side walls is small is considered, so that the resulting interfacial deformation and fluid velocities are also small. This allows one to linearize the Navier–Stokes equations and express the interfacial conditions about a planar interface through a Taylor expansion. Using a no‐tangential stress condition on the side walls, asymptotic expressions for both the interfacial slope and radial segregation at the crystal–melt interface are obtained in closed form in the limit of large thermal Rayleigh number. It is suggested that these can be reduced by appropriately controlling a specific heat transfer property at the edge of the insulation zone on the solid side.

Global Existence for a Translating Near-Circular Hele–Shaw Bubble with Surface Tension

This paper concerns global existence for arbitrary nonzero surface tension of bubbles in a Hele–Shaw cell that translate in the presence of a pressure gradient. When the cell width to bubble size is sufficiently large, we show that a unique steady translating near-circular bubble symmetric about the channel centerline exists, where the bubble translation speed in the laboratory frame is found as part of the solution. We prove global existence for symmetric sufficiently smooth initial conditions close to this shape and show that the steady translating bubble solution is an attractor within this class of disturbances. In the absence of side walls, we prove stability of the steady translating circular bubble without restriction on symmetry of initial conditions. These results hold for any nonzero surface tension despite the fact that a local planar approximation for the steady shape near the front of the bubble would suggest Saffman–Taylor instability. We exploit a boundary integral approach that is particul...