scholarly journals Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularisation

2014 ◽  
Vol 25 (6) ◽  
pp. 707-727 ◽  
Author(s):  
MICHAEL C. DALLASTON ◽  
SCOTT W. McCUE
Keyword(s):  
Viscous Fluid ◽  
Finite Time ◽  
Analytic Solutions ◽  
Flow Rule ◽  
Geometric Flow ◽  
Numerical Solution Method ◽  
Kinetic Undercooling ◽  
Finger Width ◽  
Loss Of Regularity ◽  
Discrete Family

We examine the effect of a kinetic undercooling condition on the evolution of a free boundary in Hele-Shaw flow, in both bubble and channel geometries. We present analytical and numerical evidence that the bubble boundary is unstable and may develop one or more corners in finite time, for both expansion and contraction cases. This loss of regularity is interesting because it occurs regardless of whether the less viscous fluid is displacing the more viscous fluid, or vice versa. We show that small contracting bubbles are described to leading order by a well-studied geometric flow rule. Exact solutions to this asymptotic problem continue past the corner formation until the bubble contracts to a point as a slit in the limit. Lastly, we consider the evolving boundary with kinetic undercooling in a Saffman-Taylor channel geometry. The boundary may either form corners in finite time, or evolve to a single long finger travelling at constant speed, depending on the strength of kinetic undercooling. We demonstrate these two different behaviours numerically. For the travelling finger, we present results of a numerical solution method similar to that used to demonstrate the selection of discrete fingers by surface tension. With kinetic undercooling, a continuum of corner-free travelling fingers exists for any finger width above a critical value, which goes to zero as the kinetic undercooling vanishes. We have not been able to compute the discrete family of analytic solutions, predicted by previous asymptotic analysis, because the numerical scheme cannot distinguish between solutions characterised by analytic fingers and those which are corner-free but non-analytic.

scholarly journals Rigorous results in existence and selection of Saffman–Taylor fingers by kinetic undercooling

2018 ◽  
Vol 30 (1) ◽  
pp. 63-116
Author(s):  
XUMING XIE
Keyword(s):  
Surface Tension ◽  
Rigorous Results ◽  
Engineering Mathematics ◽  
Stokes Lines ◽  
Kinetic Undercooling ◽  
Finger Width ◽  
Linear Differential ◽  
Finger Tip ◽  
Selection Of ◽  
Hele Shaw Cell

The selection of Saffman–Taylor fingers by surface tension has been extensively investigated. In this paper, we are concerned with the existence and selection of steadily translating symmetric finger solutions in a Hele–Shaw cell by small but non-zero kinetic undercooling (ε2). We rigorously conclude that for relative finger width λ near one half, symmetric finger solutions exist in the asymptotic limit of undercooling ε2 → 0 if the Stokes multiplier for a relatively simple non-linear differential equation is zero. This Stokes multiplier S depends on the parameter $\alpha \equiv \frac{2 \lambda -1}{(1-\lambda)}\epsilon^{-\frac{4}{3}}$ and earlier calculations have shown this to be zero for a discrete set of values of α. While this result is similar to that obtained previously for Saffman–Taylor fingers by surface tension, the analysis for the problem with kinetic undercooling exhibits a number of subtleties as pointed out by Chapman and King (2003, The selection of Saffman–Taylor fingers by kinetic undercooling, Journal of Engineering Mathematics, 46, 1–32). The main subtlety is the behaviour of the Stokes lines at the finger tip, where the analysis is complicated by non-analyticity of coefficients in the governing equation.

Slow steady rotation of axially symmetric bodies in a viscous fluid

1961 ◽  
Vol 10 (1) ◽  
pp. 17-24 ◽  
Author(s):  
R. P. Kanwal
Keyword(s):  
Angular Velocity ◽  
Flow Field ◽  
Viscous Fluid ◽  
Stokes Flow ◽  
Flow Problem ◽  
Analytic Solutions ◽  
The Body ◽  
Axially Symmetric ◽  
Steady Rotation ◽  
Axis Of Symmetry

The Stokes flow problem is considered here for the case in which an axially symmetric body is uniformly rotating about its axis of symmetry. Analytic solutions are presented for the heretofore unsolved cases of a spindle, a torus, a lens, and various special configurations of a lens. Formulas are derived for the angular velocity of the flow field and for the couple experienced by the body in each case.

The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid

1958 ◽  
Vol 245 (1242) ◽  
pp. 312-329 ◽  
Keyword(s):  
Porous Medium ◽  
Viscous Fluid ◽  
Normal Modes ◽  
Oil Fields ◽  
Viscous Oil ◽  
Dense Fluid ◽  
Two Dimensional Flow ◽  
Finger Width ◽  
Infinite Set ◽  
Hele Shaw Cell

When a viscous fluid filling the voids in a porous medium is driven forwards by the pressure of another driving fluid, the interface between them is liable to be unstable if the driving fluid is the less viscous of the two. This condition occurs in oil fields. To describe the normal modes of small disturbances from a plane interface and their rate of growth, it is necessary to know, or to assume one knows, the conditions which must be satisfied at the interface. The simplest assumption, that the fluids remain completely separated along a definite interface, leads to formulae which are analogous to known expressions developed by scientists working in the oil industry, and also analogous to expressions representing the instability of accelerated interfaces between fluids of different densities. In the latter case the instability develops into round-ended fingers of less dense fluid penetrating into the more dense one. Experiments in which a viscous fluid confined between closely spaced parallel sheets of glass, a Hele-Shaw cell, is driven out by a less viscous one reveal a similar state. The motion in a Hele-Shaw cell is mathematically analogous to two-dimensional flow in a porous medium. Analysis which assumes continuity of pressure through the interface shows that a flow is possible in which equally spaced fingers advance steadily. The ratio λ = (width of finger)/(spacing of fingers) appears as the parameter in a singly infinite set of such motions, all of which appear equally possible. Experiments in which various fluids were forced into a narrow Hele-Shaw cell showed that single fingers can be produced, and that unless the flow is very slow λ = (width of finger)/(width of channel) is close to ½, so that behind the tips of the advancing fingers the widths of the two columns of fluid are equal. When λ = ½ the calculated form of the fingers is very close to that which is registered photographically in the Hele-Shaw cell, but at very slow speeds where the measured value of λ increased from ½ to the limit 1.0 as the speed decreased to zero, there were considerable differences. Assuming that these might be due to surface tension, experiments were made in which a fluid of small viscosity, air or water, displaced a much more viscous oil. It is to be expected in that case that λ would be a function of μ U/T only, where μ is the viscosity, U the speed of advance and T the interfacial tension. This was verified using air as the less viscous fluid penetrating two oils of viscosities 0.30 and 4.5 poises.

scholarly journals Analytical Study of an Isotropic Viscoplastic Sea Ice Model in Idealized Configurations

2015 ◽  
Vol 45 (2) ◽  
pp. 331-354 ◽  
Author(s):  
Jérôme Sirven ◽  
Bruno Tremblay
Keyword(s):  
Sea Ice ◽  
Compressive Stress ◽  
Large Scale ◽  
Yield Curve ◽  
Initial Conditions ◽  
Analytic Solutions ◽  
Flow Rule ◽  
Large Area ◽  
Plastic Regime ◽  
Ice Model

AbstractAnalytic solutions of a mechanical sea ice model are computed in idealized configurations. They are then used to study the properties of this model. It classically assumes that the ice behaves at large scale as an isotropic viscoplastic medium. The plastic regime is characterized by a Mohr–Coulomb yield curve. The flow rule corresponds to the one used in granular mediums and depends on a parameter δ that characterizes the expansion properties of the medium. Using simple model configurations, this study first shows that a sliding of the ice along the coast must be permitted; otherwise, the model generally has no solution when the plastic regime is active. This study then shows that the viscous regime is reached only if the stress remains nearly uniform over a large area. For a stress having no particular properties, the plastic regime acts everywhere. In this case, the compressive stress may reach the maximum value allowed by the model close to the coastline. The extension of the domain where the compressive stress is at its maximum depends on δ and the direction of the forcing field. Over this domain, the ice behaves as a fluid material with a small negative viscosity. Last, the authors found that neither the existence of the solution nor its unicity are guaranteed in this stationary model. This result does not imply that the unicity is lost in the transient problem; it suggests that the evolution of sea ice depends not only on the forcing, but also on the initial conditions or history of the system.

On the unsteady squeezing of a viscous fluid from a tube

1979 ◽  
Vol 21 (1) ◽  
pp. 65-74 ◽  
Author(s):  
F. M. Skalak ◽  
C. Y. Wang
Keyword(s):  
Viscous Fluid ◽  
Numerical Scheme ◽  
Stokes Equations ◽  
Analytic Solutions ◽  
Linear Ordinary Differential Equation ◽  
Navier Stokes ◽  
Relative Importance ◽  
Dimensional Parameter ◽  
Navier Stokes Equations ◽  
Flow Reversals

AbstractViscous fluid is squeezed out from a shrinking (or expanding) tube whose radius varies with time as (1 – βt)½. The full Navier–Stokes equations reduce to a non-linear ordinary differential equation governed by a non-dimensional parameter S representing the relative importance of unsteadiness to viscosity. This paper studies the analytic solutions for large | S | through the method of matched asymptotic expansions. A simple numerical scheme for integration is presented. It is found that boundary layers exist near the walls for large | S |. In addition, flow reversals and oscillations of the velocity profile occur for large negative S (fast expansion of the tube).

Contact in a viscous fluid. Part 2. A compressible fluid and an elastic solid

2010 ◽  
Vol 646 ◽  
pp. 339-361 ◽  
Author(s):  
N. J. BALMFORTH ◽  
C. J. CAWTHORN ◽  
R. V. CRASTER
Keyword(s):  
Viscous Fluid ◽  
Half Space ◽  
Compressible Fluid ◽  
Finite Time ◽  
Asymptotic Methods ◽  
Elastic Solid ◽  
Two Dimensional ◽  
Infinite Time ◽  
Elastic Wall ◽  
Final Approach

A lubrication theory is presented for the effect of fluid compressibility and solid elasticity on the descent of a two-dimensional smooth object falling under gravity towards a plane wall through a viscous fluid. The final approach to contact, which takes infinite time in the absence of both effects, is determined by numerical and asymptotic methods. Compressibility can lead to contact in finite time either during inertially generated oscillations or if the viscosity decreases sufficiently quickly with increasing pressure. The approach to contact is invariably slowed by allowing the solids to deform elastically; specific results are presented for an underlying elastic wall modelled as a foundation, half-space, membrane or beam.

Highly accelerated, free-surface flows

1993 ◽  
Vol 248 ◽  
pp. 449-475 ◽  
Author(s):  
Michael S. Longuet-Higgins
Keyword(s):  
Free Surface ◽  
Finite Time ◽  
Initial Conditions ◽  
Analytic Solutions ◽  
Free Surface Flows ◽  
Parametric Form ◽  
Surface Flows ◽  
Pass Through ◽  
Accelerated Flow ◽  
Special Case

Accelerations exceeding 20g in surface waves have been observed both in experiments and in numerically computed flows with a free surface. The present paper describes a family of analytic solutions which display such behaviour. They are expressible in parametric form as z = F sinh ω + iG cosh ω + γω + iH, where F, G and H are functions of the time t only, and γ is linear in t. ω is a complex parameter which is real at the free surface. The functions F(t) and G(t) satisfy two nonlinear, coupled ODEs, which can be solved numerically. Typically the solutions pass through an ‘inertial shock’, or singularity in the time, where the displacements vary as t2/3, the velocities as t1/3 and the accelerations as t-4/3. In this class of solution the free surface develops a cusp as t → ∞. In a special case, F and G vary as t4/7 and the cusp is reached in finite time. Gravity is neglected, but plays a part in setting up the initial conditions for the highly accelerated flow.In future papers it will be shown that more general solutions exist in which the acceleration is momentarily large but bounded.

Contact in a viscous fluid. Part 1. A falling wedge

2010 ◽  
Vol 646 ◽  
pp. 327-338 ◽  
Author(s):  
C. J. CAWTHORN ◽  
N. J. BALMFORTH
Keyword(s):  
Viscous Fluid ◽  
Stokes Flow ◽  
Finite Time ◽  
Approximate Analytical Solution ◽  
Plane Surface ◽  
Two Dimensional ◽  
Infinite Time ◽  
Logarithmic Divergence ◽  
Sharp Corners ◽  
Made In

Computations are presented of the upward force on a two-dimensional wedge descending towards a plane surface due to the Stokes flow of an intervening viscous fluid. The predictions are compared with those of lubrication theory and an approximate analytical solution; all three predict a logarithmic divergence of the force with the minimum separation. An object falling vertically under gravity will therefore make contact with an underlying plane surface in finite time if roughened by asperities with sharp corners (with smooth surfaces, contact is made only after infinite time). Contact is still made in finite time if the roughened object also moves horizontally or rotates as it falls.

On classical solvability for the Hele-Shaw moving boundary problems with kinetic undercooling regularization

1995 ◽  
Vol 6 (5) ◽  
pp. 421-439 ◽  
Author(s):  
Yu. E. Hohlov ◽  
M. Reissig
Keyword(s):  
Harmonic Functions ◽  
Moving Boundary ◽  
Local Existence ◽  
Analytic Solutions ◽  
Moving Boundary Problems ◽  
Boundary Problems ◽  
Classical Solvability ◽  
Kinetic Undercooling

In this paper, Hele-Shaw moving boundary problems with kinetic undercooling regularization are studied. By application of the nonlinear abstract Cauchy–Kovalevskaya theorem the local existence of analytic solutions is shown. Therefore we have to study the behaviour of the gradient of harmonic functions up to the boundary of the domain.

scholarly journals Atmospheric Predictability: Revisiting the Inherent Finite-Time Barrier

2019 ◽  
Vol 76 (12) ◽  
pp. 3883-3892 ◽  
Author(s):  
Tsz Yan Leung ◽  
Martin Leutbecher ◽  
Sebastian Reich ◽  
Theodore G. Shepherd
Keyword(s):  
Finite Time ◽  
Lipschitz Continuity ◽  
Initial Conditions ◽  
Analytic Solutions ◽  
Navier Stokes ◽  
Model Resolution ◽  
Spectral Slope ◽  
Potential Conflict ◽  
Apparent Contradiction ◽  
Atmospheric Flows

Abstract The accepted idea that there exists an inherent finite-time barrier in deterministically predicting atmospheric flows originates from Edward N. Lorenz’s 1969 work based on two-dimensional (2D) turbulence. Yet, known analytic results on the 2D Navier–Stokes (N-S) equations suggest that one can skillfully predict the 2D N-S system indefinitely far ahead should the initial-condition error become sufficiently small, thereby presenting a potential conflict with Lorenz’s theory. Aided by numerical simulations, the present work reexamines Lorenz’s model and reviews both sides of the argument, paying particular attention to the roles played by the slope of the kinetic energy spectrum. It is found that when this slope is shallower than −3, the Lipschitz continuity of analytic solutions (with respect to initial conditions) breaks down as the model resolution increases, unless the viscous range of the real system is resolved—which remains practically impossible. This breakdown leads to the inherent finite-time limit. If, on the other hand, the spectral slope is steeper than −3, then the breakdown does not occur. In this way, the apparent contradiction between the analytic results and Lorenz’s theory is reconciled.

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