P. Constantin, T.F. Dupont, R.E. Goldstein, L.P. Kadanoff, M.J. Shelley and S.-M. Zhou, Droplet breakup in a model of the Hele-Shaw cell, Phys. Rev. E 47 (1993), pp. 4169.  Home/Search   Document Not in Database   Summary   APS   Related Articles   Check

.... of multi component fluid flows and are still poorly understood as the collision of material surfaces produces a singularity in the fluid equations (e.g. 39] The relative simplicity of the governing equations makes Hele Shaw flows natural test cases in which to study this phenomena (e.g. see [12, 15, 3, 30, 29, 28, 31, 38]) Indeed, one of the goals of our work is to develop insight that can be used in the study of viscous fluids in more general geometries using diffuse interface models. There are several ways in which interface transitions may be modelled. In interface tracking algorithms (see e.g. 61, 40, 77, ....

P. Constantin, T.F. Dupont, R.E. Goldstein, L.P. Kadanoff, M.J. Shelley and S.-M. Zhou, Droplet breakup in a model of the Hele-Shaw cell, Phys. Rev. E 47 (1993), pp. 4169.

.... with permeability b 2 =12, see for example [96, 86, 48] In addition, the relative simplicity of the equations of motion also makes Hele Shaw flows ideal test cases in which to develop mathematical theory and numerical methods to study singularities and topological transitions, see for instance [15, 22, 3, 40, 39, 38, 41, 50, 52, 34]. The latter is our point of view here. Topological transitions such as pinchoff and reconnection of interfaces are fundamental features of multi component fluid flows. The details of such transitions are important in many physical processes as mixing and reaction rates may sensitively depend on ....

P. Constantin, T.F. Dupont, R.E. Goldstein, L.P. Kadanoff, M.J. Shelley and S.-M. Zhou, Droplet breakup in a model of the Hele-Shaw cell, Phys. Rev. E 47 (1993), pp. 4169.

.... fs 0g : 7) The prime still denotes differentiation w. r. t. the spatial variable x; the factor of 2 in (6) makes some subsequent formulas more convenient. Heuristic and systematic derivations of (6) from the surface tension driven single phase Hele Shaw free boundary problem can be found in [11, 18, 1]. 6) is a fourth order parabolic equation with a degenerate mobility. We observe ffl The fourth order parabolic equation (6) on Q : fs 0g for s is supplemented by Dirichlet boundary conditions for s and s 0 on the boundary Q. ffl The normal velocity of the free boundary fs(t) 0g is ....

....of fs 0g in finite time. More precisely: There is a time t 1 and a point x s. t. lim t t s(t; x ) 0 and x 2 (lim t t a(t) lim t t b(t) where fs(t) 0g = a(t) b(t) In fact, it is believed, but not proved in the present situation, that such an event occurs, see [18, 11, 12, 1]; there is a rigorous result in a slightly different setting [4] This behavior suggests that the only continuation beyond t is s. t. fs(t) 0g splits in two components: fs(t) 0g = a(t) x Gamma (t) x (t) b(t) for t t . Hence the evolution (6, 7) may favor a change of ....

P. Constantin, T. F. Dupont, R. E. Goldstein, L. P. Kadanoff, M. J. Shelley, S.--M. Zhou, Droplet breakup in a model of the Hele--Shaw cell, Phys. Rev. E, 47 (6) (1993), 4169 -- 4181.

....and spatial scales of the singularities, we would implement a code that preserves positivity and has an adaptive spatial mesh. Finally, we are not using a fully implicit timestepping scheme; a basic such scheme would only be O( t) but could be made O( t 2 ) using Richardson extrapolation [12]. It would likely be very stable, but slow. For speed, we chose a partially implicit scheme and then checked for numerical instabilities when we held the timestep xed. We observed none. Also, it seems unlikely that our adaptive timestepper was taking small timesteps in order to control numerical ....

P. Constantin, T. F. Dupont, R. E. Goldstein, L. P. Kadano, M. J. Shelley and S.-M. Zhou. Droplet breakup in a model of the Hele{Shaw cell. Phys Rev E, 47(6):4169-4181, 1993.

....by this mathematical model. Equation (1.1) can be derived from a lubrication approximation of the Navier Stokes equations for flow of thin viscous films with surface tension. Use of this model to study flow in Hele Shaw cells has been extensively considered by Constantin, Kadanoff, Bertozzi et al. [10,11,12]. The lubrication equation with various 1 n 3 has been used to describe flows in many applications [10] Hele Shaw cells, polymeric liquids, thin viscous films, and contact line motion with slip conditions. Analysis of the solutions of (1.1) has been studied by Bernis et al. 2,3,4,5] Bertozzi ....

....a positive region of support, c must be positive and we must restrict 1=2 ff 1. Then, the corresponding interface velocity is positive and (2.6) represents spreading similarity solutions for 3=2 n 3 [5,8] The similarity solution (2. 6) satisfies the Dirichlet or time dependent pressure [9,11,12] boundary conditions h(0; t) c 1=ff ; h xx (0; t) 1 Gamma ff ff 2 p t c 1=ff Gamma2 : 2:9) Different boundary conditions, imposed at finite H (away from the interface) can be satisfied by assuming more general forms for the ansatz (2.7) with additional higher order terms in H . For ....

P. Constantin, T. F. Dupont, L. P. Kadanoff, and M. J. Shelley, Droplet breakup in a model of the Hele-Shaw cell, Physical Review E 47 (1993), 4169--4181.

....to force together two pieces of interface. Second, we consider the well known lubrication approximation [17] accurate for thin necks. Goldstein, Pesci, and Shelley [18] have derived this model for two fluid flows, and demonstrated formation of a singularity driven by gravity. Constantin et al. [19] have studied the single fluid model without gravity and demonstrated the existence of singularities in infinite time; Dupont et al. 20] demonstrated the existence of singularities in finite time, driven by the boundary conditions. From this work, we know that the local dynamics does not ....

....of singularity formation in the lubrication model. establish the validity of the lubrication model, and support our case for finitetime singularity formation in the full system. 3. 1 Thin Neck Asymptotics The lubrication model is usually constructed using rather heuristic arguments (see, e.g. [19]) we are aware of only one systematic derivation [18] based on a boundary integral formulation. We shall present a different derivation based on a standard multiple scales expansion. Formally, this expansion is based on assumptions about derivatives of the solution which cease to be valid in the ....

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Peter Constantin, Todd F. Dupont, Raymond E. Goldstein, Leo P. Kadanoff, Michael Shelley, and Su-Min Zhou. Droplet breakup in a model of the Hele-Shaw cell. Phys. Rev. E, 47:4169--4181, 1993.

....for the thickness h of the film of the form h t r Delta (h n r Deltah) 0; x 2 R d h(x; 0) h 0 (x) n 0 (1) where n depends on the geometry of the problem and any boundary conditions relevant for the fluid. Examples include a thin neck in the Hele Shaw cell for which n = 1 and d = 1 [11] and a thin film on a solid surface with no slip on the fluid solid interface for which n = 3 and d = 2 [14] The problem that I consider here is the evolution from strictly positive initial data. For a discussion of evolution from nonnegative initial data see [7, 2, 3] Topological transitions in ....

....= 0:8 Gamma cos( x) 0:25 cos(2 x) 14) Since the initial data is symmetric about x = 0 and the equation preserves the symmetry, I compute the solution on [0; 1] and use reflection symmetry at the boundary. The numerical method used is an adaptation of a code used in [6, 8] and earlier papers [11, 12]. It is a conventional finite difference method using an implicit, two level scheme based on central differences. The implicit time step is crucial to remove stiffness associated with the parabolicity of the equation. On each time step, the fully nonlinear difference equations are solved via ....

Peter Constantin, Todd F. Dupont, Raymond E. Goldstein, Leo P. Kadanoff, Michael J. Shelley, and Su-Min Zhou. Droplet breakup in a model of the Hele-Shaw cell. Physical Review E, 47(6):4169--4181, June 1993.

....h t = Gammar Delta (f(h)r Deltah) in one space dimension. This equation, derived from a lubrication approximation , models surface tension dominated motion of thin viscous films and spreading droplets [14] The equation with f(h) jhj also models a thin neck of fluid in the Hele Shaw cell [9, 10, 22]. In such problems h(x; t) is the local thickness of the the film or neck. This paper will consider the properties of weak solutions which are more relevant to the droplet problem than to Hele Shaw. For simplicity we consider periodic boundary conditions with the interpretation of modeling a ....

....directly extended to the case where f(h) is a sum of such terms. This equation, derived from a lubrication approximation , models surface tension dominated motion of thin viscous films and spreading droplets [14] The equation with f(h) jhj also models a thin neck of fluid in the Hele Shaw cell [9, 10, 22]. In such problems h is the thickness of the the film or neck. This paper considers weak solutions that are zero on a set of non zero measure, hence are much more relevant to the droplet problem than to Hele Shaw. We briefly compare this fourth order problem to the well known second order ....

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Peter Constantin, Todd F. Dupont, Raymond E. Goldstein, Leo P. Kadanoff, Michael J. Shelley, and Su-Min Zhou. Droplet breakup in a model of the Hele-Shaw cell. Physical Review E, 47(6):4169--4181, June 1993. THE LUBRICATION APPROXIMATION FOR THIN FILMS 37

....for the thickness h of the film of the form h t r Delta (h n r Deltah) 0; x 2 R d h(x; 0) h 0 (x) 1) where n depends on the geometry of the problem and boundary conditions on any liquid solid interfaces. Examples include a thin neck in the Hele Shaw cell for which n = 1 and d = 1 [24] and a thin film on a solid surface with no slip on the fluid solid interface for which n = 3 and d = 2 [34] Periodic boundary conditions have been proposed for the Hele Shaw problem [33, 44] to describe an infinite periodic neck and for the thin film problem in 2 D [15, 18] to describe a ....

....periodic neck and for the thin film problem in 2 D [15, 18] to describe a periodic array of droplets. In y University of Chicago, Department of Mathematics, Chicago, IL 60637 z The work was supported by the National Science Foundation and the Department of Energy 2 A. L. Bertozzi addition, [24, 25] considered the pressure boundary conditions for the 1 D problem on the interval [ Gamma1; 1] PBC) h( Sigma1) 1; h xx ( Sigma1) p to model a Hele Shaw experiment in which a constant external pressure was applied to a thin neck contained between two fixed boundaries. Furthermore, the ....

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P. Constantin, T. F. Dupont, R. E. Goldstein, L. P. Kadanoff, M. J. Shelley, and S.- M. Zhou, Droplet breakup in a model of the Hele-Shaw cell, Physical Review E, 47 (1993), pp. 4169--4181.

....droplets. For such problems, the power n depends on the boundary condition on the liquid solid interface: no slip gives n = 3 while various Navier slip conditions can yield n 3. The same equation in one space dimension with f(h) h was also shown to model a thin neck in the Hele Shaw cell [2]. Other applications include Cahn Hilliard models with degenerate mobility [3] population dynamics [4] and problems in plasticity [5] In all examples, in order to have a physical solution, h must be nonnegative. Equation (1.1) describes fourth order degenerate diffusion. Like the second order ....

Peter Constantin, Todd F. Dupont, Raymond E. Goldstein, Leo P. Kadanoff, Michael J. Shelley, and Su-Min Zhou. Droplet breakup in a model of the Hele-Shaw cell. Physical Review E, 47(6):4169--4181, June 1993.

....there exist mechanisms that allow topology changes. For breakup of a two dimensional fluid droplet in a Hele Shaw cell forced by external pressures or imposed flow, several different types of local similarity solutions (with different scaling properties) can occur: an infinite time singularity [11]; a finite time singularity in which the pinch point moves with constant velocity [12] and a finite time singularity in which the pinch point is stationary [13, 14] Varying the initial and boundary conditions leads to the different singular behaviors. Moreover, the same array of singularity ....

....In the region of the thin neck, a lubrication approximation reduces the two dimensional Hele Shaw system to a one dimensional model. We denote by h(x; t) the half width of the neck; then assuming that jh x j 1, and that h 1 so that the pressure p p(x) yields the lubrication equation [11] h t (hh xxx ) x = 0: 1) This equation also follows from a systematic asymptotic expansion in h [18, 15] We are interested in the rupture of thin necks, when h(x; t) 0 in (1) at a finite time. In the lubrication model (2) the fourth order term is degenerate and plays an interesting role ....

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Peter Constantin, Todd F. Dupont, Raymond E. Goldstein, Leo P. Kadanoff, Michael Shelley, and Su-Min Zhou. Droplet breakup in a model of the HeleShaw cell. Phys. Rev. E, 47:4169--4181, 1993.

....Hele Shaw flows provide a simple, but nontrivial, situation in which to study them. The majority of studies concern analytical and numerical studies of simplified lubrication , or long wave, descriptions of Hele Shaw flows, which yield nonlinear, but local, PDEs for the layer thickness (e.g. see [175, 39, 58, 66, 24, 67, 3, 25, 68, 69]. Two studies that have sought to study topological singularities in the full Hele Shaw problem are Shelley, Goldstein, Pesci [175] and Almgren [2] who both applied boundary integral methods. Shelley et al. 175] give an initial study of topological singularity formation in Hele Shaw flows, ....

P. Constantin, T.F. Dupont, R.E. Goldstein, L.P. Kadano#, M.J. Shelley, and S.M. Zhou. Droplet breakup in a model of the Hele--Shaw cell. Physical Review E, 47(6):4169--4181, 1993. 38

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Peter Constantin, Todd F. Dupont, Raymond E. Goldstein, Leo P. Kadanoff, Michael J. Shelley, and Su-Min Zhou. Droplet breakup in a model of the Hele-Shaw cell. Physical Review E, 47(6):4169--4181, June 1993.

No context found.

Peter Constantin, Todd F. Dupont, Raymond E. Goldstein, Leo P. Kadanoff, Michael Shelley, and Su-Min Zhou. Droplet breakup in a model of the Hele-Shaw cell. Phys. Rev. E, 47:4169--4181, 1993.

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