Saffman-Taylor fingers with kinetic undercooling.

The Saffman--Taylor finger problem is to predict the shape and, in particular, width of a finger of fluid travelling in a Hele--Shaw cell filled with a different, more viscous fluid. In experiments the width is dependent on the speed of propagation of the finger, tending to half the total cell width as the speed increases. To predict this result mathematically, nonlinear effects on the fluid interface must be considered; usually surface tension is included for this purpose. This makes the mathematical problem sufficiently difficult that asymptotic or numerical methods must be used. We adapt numerical methods used to solve the Saffman--Taylor finger problem with surface tension to instead include the effect of kinetic undercooling, a regularisation effect important in Stefan melting-freezing problems, for which Hele--Shaw flow serves as a leading order approximation when the specific heat of a substance is much smaller than its latent heat. We find the existence of a solution branch where the finger width tends to zero as the propagation speed increases, disagreeing with some aspects of the asymptotic analysis of the same problem. We also find a second solution branch, supporting the idea of a countably infinite number of branches as for the surface tension problem. References P. G. Saffman and G. Taylor. The penetration of a fluid into a porous medium or Hele--Shaw cell containing a more viscous liquid. Proc. R. Soc. London, Ser. A , 245:312--329, 1958. doi:10.1098/rspa.1958.0085 J. W. McLean and P. G. Saffman. The effect of surface tension on the shape of fingers in a Hele--Shaw cell. J. Fluid. Mech. , 102:455--469, 1981. doi:10.1017/S0022112081002735 J. M. Vanden-Broeck. Fingers in a Hele--Shaw cell with surface tension. Phys. Fluids , 26(8):2033--2034, 1983. doi:10.1063/1.864406 S. J. Chapman. On the role of Stokes lines in the selection of Saffman--Taylor fingers with small surface tension. Eur. J. Appl. Math. , 10(06):513--534, 1999. doi:10.1017/S0956792599003848 R. Combescot, T. Dombre, V. Hakim, Y. Pomeau, and A. Pumir. Shape selection of Saffman--Taylor fingers. Phys. Rev. Lett. , 56(19):2036--2039, May 1986. doi:10.1103/PhysRevLett.56.2036 R. Combescot, V. Hakim, T. Dombre, Y. Pomeau, and A. Pumir. Analytic theory of the Saffman--Taylor fingers. Phys. Rev. A , 37(4):1270--1283, February 1988. doi:10.1103/PhysRevA.37.1270 M. Reissig, S. V. Rogosin, and F. Hubner. Analytical and numerical treatment of a complex model for Hele--Shaw moving boundary value problems with kinetic undercooling regularization. Eur. J. Appl. Math. , 10(6):561--579, 1999. doi:10.1017/S0956792599003939 S. J. Chapman and J. R. King. The selection of Saffman--Taylor fingers by kinetic undercooling. J. Eng. Math. , 46(1):1--32, 2003. doi:10.1023/A:1022860705459 D. Bensimon. Stability of viscous fingering. Phys. Rev. A , 33(2):1302--1309, 1986. doi:10.1103/PhysRevA.33.1302 D. A. Kessler and H. Levine. Theory of the Saffman--Taylor ``finger'' pattern. {I}. Phys. Rev. A , 33(4):2621--2633, 1986. doi:10.1103/PhysRevA.33.2621 D. A. Kessler and H. Levine. Theory of the Saffman--Taylor ``finger'' pattern. {II}. Phys. Rev. A , 33(4):2634--2639, 1986. doi:10.1103/PhysRevA.33.2634 A. J. Degregoria and L. W. Schwartz. A boundary integral method for two-phase displacement in Hele--Shaw cells. J. Fluid Mech. , 164:383--400, 1986. doi:10.1017/S0022112086002604 T. Y. Hou, Z. Li, S. Osher, and H. Zhao. A hybrid method for moving interface problems with application to the Hele--Shaw flow. J. Comput. Phys. , 134(2):236--252, 1997. doi:10.1006/jcph.1997.5689 G. Tryggvason and H. Aref. Numerical experiments on Hele--Shaw flow with a sharp interface. J. Fluid Mech. , 136:1--30, 1983. doi:10.1017/S0022112083002037 N. Whitaker. Some numerical methods for the Hele--Shaw equations. J. Comput. Phys. , 111(1):81--88, 1994. doi:10.1006/jcph.1994.1046

The effect of surface tension and kinetic undercooling on a radially-symmetric melting problem

Saffman-Taylor fingers with anisotropic surface tension

A model for the evolution of phase boundaries reminiscent of the phase-field model is considered. The equation related to conservation of thermal energy is diffusive and coupled to an equation for the order parameter, which contains a nonlinear convolution operator, related to the limit of an interacting particle model with Kac-potential. Under diffusive rescaling the solutions converge to solutions of the Stefan problem with kinetic undercooling and surface tension.

We study analytically the selection, due to surface tension, of Saffman-Taylor fingers in the sector geometry recently introduced experimentally by Thom\'e et al. [Phys. Fluids A 1, 224 (1989)]. The finger shape for zero surface tension is written as the solution of an integral equation. This leads to an explicit solution in the case where the sector angle is small. The selection due to surface tension is controlled, as in similar problems, by the singular points of a differential equation. For divergent fingers, we find for these singularities a geometric situation which is qualitatively new. Indeed we obtain one ``axis'' singularity and two ``side'' singularities. When the axis singularity is ``dominant,'' the finger disappears, which explains the branch-merging phenomenon found numerically by Ben Amar [Phys. Rev. A 44, 3673 (1991)]. Our analysis is carried out completely explicitly in the case of small-angle sector, and we indeed find branch merging, either by a WKB analysis or by a numerical solution of our differential equation. For convergent fingers, we find either an ``axis'' singularity or two ``side'' singularities, and the results are similar to those for the standard Saffman-Taylor fingers in the linear geometry.

New numerical solutions to the so-called selection problem for one and two steadily translating bubbles in an unbounded Hele-Shaw cell are presented. Our approach relies on conformal mapping which, for the two-bubble problem, involves the Schottky-Klein prime function associated with an annulus. We show that a countably infinite number of solutions exist for each fixed value of dimensionless surface tension, with the bubble shapes becoming more exotic as the solution branch number increases. Our numerical results suggest that a single solution is selected in the limit that surface tension vanishes, with the scaling between the bubble velocity and surface tension being different to the well-studied problems for a bubble or a finger propagating in a channel geometry.

I In part I of this thesis similarity solutions to the equations of three phase flow through porous media are examined. The three phases are water, steam, and a noncondensing phase, most likely oil. The main purpose of analyzing such flows is to gain understanding of the steam flooding of oil fields. Provided steam is being injected at a higher pressure than the initial field pressure, it is shown that there will always be at least two saturation shocks. As one increases the pressure of the injected steam several regimes are encountered; first the flow develops a region where all the steam is completely condensed, then the position of two of the shocks are interchanged, and finally one of the shocks grows weaker and is eventually replaced by an expansion fan. In sections 12 and 13 the stability of steadily moving condensation fronts in porous media is analyzed. For one special problem it is found that the sign of the jump in pressure gradient at the interface determines whether the interfaces are stable or unstable. This result is applied with some caution to the similarity solutions found in the earlier sections. II Recently McLean analyzed the shapes of fingers in a Hele-Shaw cell, including the effects of surface tension. His work resolved the question of the uniqueness of the shapes first brought up by Saffman and Taylor in their analysis that did not include surface tension. It is however felt that there are still unresolved problems. In determining the pressure jump across an interface there are two principal radii of curvature. McLean only took into account the effect of the larger of these, assuming that the other was constant along the outline of the finger. Unless the smaller radius is very nearly constant, it should in fact give a larger contribution to the jump in pressure. In this thesis the effect of this smaller radius of curvature is modelled by assuming that it is a function of the normal velocity of the mean two dimensional surface of the finger. It is found that if one only takes into account the smaller radius of curvature, the problem is not uniquely determined, as in the case with no surface tension at all. When both radii of curvature are taken into account, the effect of the smaller radius of curvature is to modify the finger shapes in a way that is qualitatively in agreement with experimental data. Also, McLean's results are checked by an independent numerical scheme, and the results are found to be in excellent agreement. Using both methods of solution a second solution branch other than that described by McLean was also found.

The temporal evolution of an interface separating two phases is studied using renormalization group and scaling theory and exact calculation of a sharp interface model incorporating surface tension and kinetic undercooling. Under conditions favoring rapid solidification the characteristic length, R(t), varies as t(1/2) while the total surface area of the interface, S(t), varies as t((d-1)/2). This complements the results of Jasnow and Vinals who found R(t) approximately t in the quasistatic regime. The transition in exponents from R(t) approximately t to R(t) approximately t(1/2), as solidification proceeds from the quasistatic to the rapid solidification regime, suggests a complex evolution toward a self-similar late stage growth.

The influence of fluctuations due to thermal excitations on the interactions between membranes is reviewed. The complex interplay between the bare interactions (for example Van der Waals, hydration or electrostatic forces) and the membranes’ thermal fluctuations is analyzed within a linear functional renormalization group (FRG), for membranes with a significant surface tension and for membranes with a vanishing surface tension and non-zero curvature rigidity.In all cases, the fluctuations erase the details of the interactions which characterize each particular system and produce universal behaviour.KeywordsHarmonic ApproximationSteric InteractionLamellar PhasisHydration Free EnergyFunctional Renormalization GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Viscous fingering in a wedge-shaped Hele–Shaw cell is analyzed. The shape of a self-dilating viscous finger is shown to obey a time-independent nonliner integrodifferential equation that is solved numerically. The results show that there is a continuum of zero-surface-tension solutions for any wedge angle between 0 and π/2. When surface tension is taken into account only a discrete set of solutions exist. In contrast to the classical case, different branches merge for finite values of the surface tension parameter.

Classical nucleation theory was used as the basis for investigating the effects of dilute oxygen solute on the nucleation behavior of pure zirconium. Undercooling distributions were obtained for zirconium samples via electrostatic levitation experiments and kinetic parameters were determined using statistical analysis. Oxygen and residual impurity content were determined using glow discharge mass spectrometry. The work of formation of a critical nucleus, ΔG*, and the kinetic prefactor, Kv, increased with decreasing oxygen content. To examine the effects of oxygen on ΔG*, a sharp interface model was assumed to allow separation of volume free energy and interfacial free energy components. Dilute oxygen has a minimal effect on the volume free energy. A temperature-dependent reduced interfacial free energy, αm, was used to examine the effect of oxygen on interfacial free energy. Experimentally derived values for αm and the corresponding interface configurational entropy values varied significantly from theoretical predictions for “pure” zirconium. Oxygen compositional effects increased the configurational entropy of the interface atoms, resulting in a decrease in the reduced interfacial free energy.

The transient convective burning of n-octane droplets interacting within single-layer arrays in a hot gas flow perpendicular to the layer is studied numerically, with considerations of droplet surface regression, deceleration due to the drag of the droplets, internal liquid motion, variable properties, non-uniform liquid temperature and surface tension. Infinite periodic arrays, semi-infinite periodic arrays with one row of droplets (linear array) or two rows of droplets, and finite arrays with nine droplets with centers in a plane are investigated. All arrays are aligned orthogonal to the free stream direction. This paper compares the behavior of semi-infinite periodic arrays and finite arrays with the behavior of previously studied infinite periodic arrays. Furthermore, it identifies the critical values of the initial Damköhler number for bifurcations in flame behavior at various initial droplet spacing for all these arrays. The initial flame shape is either an envelope flame or a wake flame as determined by the initial Damköhler number, the array configuration and the initial droplet spacing. The critical initial Damköhler number separating initial wake flames from initial envelope flames decreases with increasing interaction amongst droplets at intermediate droplet spacing (when the number of rows in the array increases or the initial droplet spacing decreases for a specific number of rows in the array). In the transient process, an initial wake flame has a tendency to develop from a wake flame to an envelope flame, with the moment of wake-to-envelope transition advanced for the increasing interaction amongst droplets at intermediate droplet spacing. For the array with nine droplets with centers in a plane, the droplets at different types of positions have different critical initial Damköhler number and different wake-to-envelope transition time for initial wake flame.

The problem of interface pattern selection in nonlinear dissipative systems is critical in many fields of science, occurring in physical, chemical and biological systems. One of the simplest pattern formations is the Saffman-Taylor finger pattern that forms when a viscous fluid is displaced by a less viscous fluid. Such finger-shaped patterns have been observed in distinctly different fields of science (hydrodynamics, combustion and crystal growth) and this has led to a search for a unified concept of pattern formation, as first proposed by the classic work of D'arcy Thomson. Two-dimensional finger-shaped patterns, observed in flame fronts and the ensembled average shape of the diffusion-limited aggregation pattern, have been shown to be similar to Saffman-Taylor finger shapes. Here we present experimental studies that establish that the cell shapes formed during directional solidification of alloys can be described by the form of the Saffman-Taylor finger shape equation when a second phase is present in the intercellular region.

We treat two related moving boundary problems. The first is the ill-posed Stefan problem for melting a superheated solid in one Cartesian coordinate. Mathematically, this is the same problem as that for freezing a supercooled liquid, with applications to crystal growth. By applying a front-fixing technique with finite differences, we reproduce existing numerical results, concentrating on solutions that break down in finite time. This sort of finite time blow-up is characterised by the speed of the moving boundary becoming unbounded in the blow-up limit. The second problem, which is an extension of the first, is proposed to simulate aspects of a particular two phase Stefan problem with surface tension. We study this novel moving boundary problem numerically, and provide results that support the hypothesis that it exhibits a similar type of finite time blow-up as the more complicated two phase problem. The results are unusual in the sense that it appears the addition of surface tension transforms a well-posed problem into an ill-posed one. References S. D. Howison, J. R. Ockendon, and A. A. Lacey. Singularity development in moving-boundary problems. Q. J. Mech. Appl. Math., 38(3):343--360, 1985. doi:10.1093/qjmam/38.3.343 A. A. Lacey and J. R. Ockendon. Ill-posed free boundary problems. Control and Cybernetics, 14:275--296, 1985. A. Fasano and M. Primicerio. General free-boundary problems for the heat equation I. J. Math. Anal. Appl, 57:694--723, 1977. A. Fasano, M. Primicerio, and A. A. Lacey. New results on some classical parabolic free-boundary problems. Quart. Appl. Math., 38:439--460, 1981. M. A. Herrero and J. J. L. Velazquez. Singularity formation in the one-dimensional supercooled Stefan problem. Euro. J. Appl. Math., 7:119--150, 1996. doi:10.1137/04060528X B. Sherman. A general one-phase Stefan problem. Quart. Appl. Math., 28:377--382, 1970. J. R. King and J. D. Evans. Regularization by kinetic undercooling of blow-up in the ill-posed Stefan problem. SIAM J. Appl. Math., 65(5):1677--1707, 2005. doi:10.1137/04060528X J. Crank. Free and moving boundary problems. Clarendon Press, Oxford, 1984. T. C. Illingworth and I. O. Golosnoy. Numerical solutions of diffusion-controlled moving boundary problems which conserve solute. J. Comp. Phys., 209:207--225, 2005. doi:10.1016/j.jcp.2005.02.031 F. Liu and D. L. S. McElwain. A computationally efficient solution technique for moving-boundary problems in finite media. IMA J. Appl. Math, 59:71--84, 1997. doi:10.1093/imamat/59.1.71 S. W. McCue, B. Wu, and J. M. Hill. Classical two-phase Stefan problem for spheres. Proc. R. Soc. A, 464:2055--2076, 2008. doi:10.1098/rspa.2007.0315 S. W. McCue, B. Wu, and J. M. Hill. Micro/nanoparticle melting with spherical symmetry and surface tension. IMA J. Appl. Math., 74:439--457, 2009. doi:10.1093/imamat/hxn038 B. Wu, S. W. McCue, P. Tillman, and J. M. Hill. Single phase limit for melting nanoparticles. Appl. Math. Model., 33:2349--2367, 2009. doi:10.1016/j.apm.2008.07.009 H. G. Landau. Heat conduction in a melting solid. Quart. J. Appl. Math., 8:81--94, 1950. L. F. Shampine and M. W. Reichelt. The MATLAB ODE Suite. SIAM Journal on Scientific Computing, 18:1--22, 1997. S. C. Gupta. The classical Stefan problem: Basic concepts, modelling and analysis. Elsevier, Amsterdam, 2003.

Domain walls at finite temperature