Corner solutions of the Laplace-Young equation (original) (raw)
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Computation of capillary surfaces for the Laplace-Young equation
The Quarterly Journal of Mechanics and Applied Mathematics, 2005
A novel hybrid finite-element/finite-volume numerical method is developed to determine the capillary rise of a liquid with a free surface (under surface tension and gravitational forces). The few known exact analytical solutions are used to verify the numerical computations and establish their accuracy for a range of liquid contact angles. The numerical method is then used to ascertain the limitations of a number of theoretical approximations to solutions for the capillary rise in the linearized limit, for special geometries such as plane walls, concentric cylinders and in a wedge of arbitrary included angle. The existence of a critical wedge angle for a given contact angle is verified. However, the effect of slight practical rounding of wedge corners dramatically reduces the theoretical corner height.
Numerical Solutions for Intermediate Angles of the Laplace-Young Capillary Equations
Capillarity is the phenomena of fluid rise against a solid vertical wall. In this paper, we consider bounded cases of intermediate corner angles (π/2 < α + γ < π/2 + 2γ) , where γ is the angle of contact and 2α is the wedge angle. The Laplace-Young Capillary equations are used to determine the rise of the fluid, especially at corners. While there exist asymptotic expansions for the height rise occurring at the corner of an intermediate angle, not all coefficients are known analytically. Therefore, numerical solutions are necessary, even though only a few numerical methods have been published. We explain our least-squares finite element method used in determining solutions to the Laplace-Young Capillary equations, and then give our numerical results.
The Journal of Chemical Physics, 2014
Many physical problems require explicit knowledge of the equilibrium shape of the interface between two fluid phases. Here, we present a new numerical method which is simply implementable and easily adaptable for a wide range of problems involving capillary deformations of fluid-fluid interfaces. We apply a simulated annealing algorithm to find the interface shape that minimizes the thermodynamic potential of the system. First, for completeness, we provide an analytical proof that minimizing this potential is equivalent to solving the Young-Laplace equation and the Young law. Then, we illustrate our numerical method showing two-dimensional results for fluid-fluid menisci between vertical or inclined walls and curved surfaces, capillary interactions between vertical walls, equilibrium shapes of sessile heavy droplets on a flat horizontal solid surface, and of droplets pending from flat or curved solid surfaces. Finally, we show illustrative three-dimensional results to point out the applicability of the method to micro-or nano-particles adsorbed at a fluid-fluid interface.
Instantaneous viscous flow in a corner bounded by free surfaces
Physics of Fluids, 1996
We study the instantaneous Stokes flow near the apex of a corner of angle 2␣ formed by two plane stress free surfaces. The fluid is under the action of gravity with g ជ parallel to the bisecting plane, and surface tension is neglected. For 2␣Ͼ126.28°, the dominant term of the solution as the distance r to the apex tends to zero does not depend on gravity and has the character of a self-similar solution of the second kind; the exponent of r cannot be obtained on dimensional grounds and the scale of the coefficient depends on the far flow field. Within this angular domain, the instantaneous flow is deeply related to the ͑steady͒ flow in a rigid corner known since Moffatt ͓J. Fluid Mech. 18, 1 ͑1964͔͒ and, as in that case, there may be eddies in the flow. The situation is substantially different for 2␣Ͻ126.28°, as the dominant term is related to gravity and not to the far flow. It has the character of a self-similar solution of the first kind, with the exponent of r being given by dimensional analysis. The solution cannot be continued in time since it leads to the curling of the boundaries. Nevertheless, it provides information on how such a cornered contour may evolve. When 2␣Ͻ180°, the corner angle does not vary as the flow develops; on the other hand, if 2␣Ͼ180°the corner must round or tend to a narrow cusp, depending on the far flow. These predictions are supported by simple experiments.
Computations of spontaneous rise of a rivulet in a corner of a vertical square capillary
Colloids and Surfaces A: Physicochemical and Engineering Aspects, 2018
In this study, the spontaneous rise of a Newtonian liquid in a square capillary with completely or partially wetted walls is investigated using numerical simulations. The flow is modelled using Volume-of-Fluid method with adaptive mesh refinement to resolve the interface for high accuracy. The computations show that for contact angles smaller than 45 • , rivulets appear in the corners of the capillary. At large times the length of the capillary growth approaches the one-third power function of time. The same asymptotic behaviour has been identified in the existing experimental observations for corners of different geometries. The computations predict the dependence of the rate of the rivulet growth on the liquid viscosity, gravity, width of the capillary and the contact angle. The flow in the rivulet is described using a long-wave approximation which considers three regions of the rivulet flow: the flow near the rivulet tip which is described by a similarity solution, intermediate region approaching a static rivulet shape, and the bulk meniscus. Finally, a scaling analysis is proposed which predicts rivulet growth rates for the given parameters.
Capillary-driven flows along rounded interior corners
Journal of Fluid Mechanics, 2006
The problem of low-gravity isothermal capillary flow along interior corners that are rounded is revisited analytically in this work. By careful selection of geometric length scales and through the introduction of a new geometric scaling parameter T c , the Navier-Stokes equation is reduced to a convenient ∼ O(1) form for both analytic and numeric solutions for all values of corner half-angle α and corner roundedness ratio λ for perfectly wetting fluids. The scaling and analysis of the problem captures much of the intricate geometric dependence of the viscous resistance and significantly reduces the reliance on numerical data compared with several previous solution methods and the numerous subsequent studies that cite them. In general, three asymptotic regimes may be identified from the large second-order nonlinear evolution equation: (I) the 'sharp-corner' regime, (II) the narrow-corner 'rectangular section' regime, and (III) the 'thin film' regime. Flows are observed to undergo transition between regimes, or they may exist essentially in a single regime depending on the system. Perhaps surprisingly, for the case of imbibition in tubes or pores with rounded interior corners similarity solutions are possible to the full equation, which is readily solved numerically. Approximate analytical solutions are also possible under the constraints of the three regimes, which are clearly identified. The general analysis enables analytic solutions to many rounded-corner flows, and example solutions for steady flows, perturbed infinite columns, and imbibing flows over initially dry and prewetted surfaces are provided.
Derivations of the Young-Laplace equation
Capillarity, 2021
The classical Young-Laplace equation relates capillary pressure to surface tension and the principal radii of curvature of the interface between two immiscible fluids. In this paper the required properties of space curves and smooth surfaces are described by differential geometry and linear algebra. The equilibrium condition is formulated by a force balance and minimization of surface energy.
A universal law for capillary rise in corners
Journal of Fluid Mechanics, 2011
We study the capillary rise of wetting liquids in the corners of different geometries and show that the meniscus rises without limit following the universal law: h(t)/a ≈ (γt/ηa)1/3, where γ and η stand for the surface tension and viscosity of the liquid while a,=,sqrtgamma/rhoga\,{=}\,\sqrt{\gamma/\rho g}a,=,sqrtgamma/rhog is the capillary length, based on the liquid density ρ and gravity g. This law is universal in the sense that it does not depend on the geometry of the corner.
Height estimates for exterior problems of capillarity type
Pacific Journal of Mathematics, 1980
This work concerns boundary value problems for a class of nonlinear equations modeled on the physical equations for a capillary free surface in a gravitational field. The results consist principally of estimates for the height of a solution in an exterior domain. Structure conditions reflecting the nonlinearity of the mean curvature operator are imposed on a class of symmetric variational operators in terms of the Legendre transform of the variational integrand. Estimates are found for the boundary height of a rotationally symmetric solution in the exterior of a ball of radius R. These estimates, which are valid for any R 9 are shown to be asymptotically exact as R tends to zero or infinity. This provides a proof of the asymptotic behavior of the boundary height which previously has been derived by a formal perturbation method. An asymptotic characterization of the solution in a neighborhood of the boundary is also given. For a general domain estimates are obtained from a maximum principle due to Finn in which the symmetric solutions serve as comparison functions.
The capillary interaction between two vertical cylinders
Journal of Physics: Condensed Matter, 2012
Particles floating at the surface of a liquid generally deform the liquid surface. Minimizing the energetic cost of these deformations results in an inter-particle force which is usually attractive and causes floating particles to aggregate and form surface clusters. Here we present a numerical method for determining the three-dimensional meniscus around a pair of vertical circular cylinders. This involves the numerical solution of the fully nonlinear Laplace-Young equation using a mesh-free finite difference method. Inter-particle force-separation curves for pairs of vertical cylinders are then calculated for different radii and contact angles. These results are compared with previously published asymptotic and experimental results. For large inter-particle separations and conditions such that the meniscus slope remains small everywhere, good agreement is found between all three approaches (numerical, asymptotic and experimental). This is as expected since the asymptotic results were derived using the linearized Laplace-Young equation. For steeper menisci and smaller inter-particle separations, however, the numerical simulation resolves discrepancies between existing asymptotic and experimental results, demonstrating that this discrepancy was due to the nonlinearity of the Laplace-Young equation.