•I have revised the manuscript “Spectral methods for capillary surfaces described by bounded generating curves” and submit these changes for consideration for publication.•I have substantially ...replaces the code snippets with a flowchart and I have added clear and detailed summaries of the goals of each table, as indicated was desired by the referee.
We consider capillary surfaces that are constructed by bounded generating curves. This class of surfaces includes radially symmetric and lower dimensional fluid-fluid interfaces. We use the arc-length representation of the differential equations for these surfaces to allow for vertical points and inflection points along the generating curve. These considerations admit fluids in capillary tubes, sessile drops, and fluids in annular tubes as well as other examples.
We present a pseudo-spectral method for approximating solutions of the associated boundary value problems based on interpolation by Chebyshev polynomials. This method is observably more stable than the traditional shooting method and it is computationally lean and fast. The algorithm is also adaptive, but does not use the adaptive automation in Chebfun.
We consider radially symmetric capillary surfaces that are described by bounded generating curves. We use the arc-length representation of the differential equations for these surfaces to allow for ...vertical points and inflection points along the generating curve. These considerations admit capillary tubes, sessile drops, and fluids in annular tubes as well as other examples. We present a multi-scale pseudo-spectral method for approximating solutions of the associated boundary value problems based on interpolation by Chebyshev polynomials. The multi-scale approach is based on a domain decomposition with adaptive refinements within each subdomain. Key words. capillarity, spectral methods AMS subject classifications. 76B45, 65N35, 35Q35, 34B60
The main contribution of this paper is the precise numerical identification of a model set of parameters for a floating object/container system which admits three distinct equilibrium configurations, ...two of which are local energy minimizers among pseudo-equilibrium configurations. This numerical result strongly suggests the existence of a physical system in which a circular object can be observed to float in a centrally symmetric position in two geometrically distinct configurations, i.e., at two different heights. Thus, the general dependence of observable stable equilibria on the physical parameters of the problem is both shown to be much more complicated than originally anticipated and likely to depend on additional information, e.g., the initial positioning of the floating object. We show the existence of at least one equilibrium configuration in any situation which the density of the floating object is less than that of the liquid bath. We also give a collection of conditions under which all equilibrium fluid interfaces can be shown to project simply onto the base of the container.
We consider the physical configuration of a container which holds a finite number of movable solid objects and three immiscible fluids. Each fluid volume is prescribed, and the container is ...completely filled. The configuration is modeled using sets of finite perimeter, and the energy of the configuration is given using the theory of functions of bounded variation. We show that there exists an admissible configuration that attains the minimum energy.
The symmetry of floating drops is considered. Under conditions that the free boundary is contained in a horizontal plane it is shown that all three component interfaces are symmetric about a vertical ...line. PUBLICATION ABSTRACT
The height of the surface of a fluid in an annular tube is explored using a shooting method to solve a boundary value problem where the radii and the contact angles are given. The contact angles on ...the inner and outer tube surface need not be the same. These surfaces are then extended so that they are no longer graphs. The extended surfaces are shown to solve a boundary value problem over an annular base domain where given inclination angles are achieved at given radii.
An existence theorem for floating drops due to Elcrat, Neel, and Siegel is generalized. The theorem applies to all radially symmetric domains, and to both light and heavy floating drops, and utilizes ...new results in annular capillary theory.
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