A pressure impulse theory for hemispherical liquid impact problems

Liquid impact problems for hemispherical fluid domain are considered. By using the concept of pressure impulse we show that the solution of the flow induced by the impact is reduced to the derivation of Laplace's equation in spherical coordinates with Dirichlet and Neumann boundary conditions. The structure of the flow at the impact moment is deduced from the spherical harmonics representation of the solution. In particular we show that the slip velocity has a logarithmic singularity at the contact line. The theoretical predictions are in very good agreement both qualitatively and quantitatively with the first time step of a numerical simulation with a Navier-Stokes solver named Gerris.Comment: 11 pages, 14 figures, Accepted for publication in European Journal of Mechanics - B/Fluid

Three-dimensional solutions for the geostrophic flow in the Earth's core

In his seminal work, Taylor (1963) argued that the geophysically relevant limit for dynamo action within the outer core is one of negligibly small inertia and viscosity in the magnetohydrodynamic equations. Within this limit, he showed the existence of a necessary condition, now well known as Taylor's constraint, which requires that the cylindrically-averaged Lorentz torque must everywhere vanish; magnetic fields that satisfy this condition are termed Taylor states. Taylor further showed that the requirement of this constraint being continuously satisfied through time prescribes the evolution of the geostrophic flow, the cylindrically-averaged azimuthal flow. We show that Taylor's original prescription for the geostrophic flow, as satisfying a given second order ordinary differential equation, is only valid for a small subset of Taylor states. An incomplete treatment of the boundary conditions renders his equation generally incorrect. Here, by taking proper account of the boundaries, we describe a generalisation of Taylor's method that enables correct evaluation of the instantaneous geostrophic flow for any 3D Taylor state. We present the first full-sphere examples of geostrophic flows driven by non-axisymmetric Taylor states. Although in axisymmetry the geostrophic flow admits a mild logarithmic singularity on the rotation axis, in the fully 3D case we show that this is absent and indeed the geostrophic flow appears to be everywhere regular.Comment: 29 Pages, 8 figure

Drag force on a sphere moving towards an anisotropic super-hydrophobic plane

We analyze theoretically a high-speed drainage of liquid films squeezed between a hydrophilic sphere and a textured super-hydrophobic plane, that contains trapped gas bubbles. A super-hydrophobic wall is characterized by parameters $L$ (texture characteristic length), $b_1$ and $b_2$ (local slip lengths at solid and gas areas), and $\phi_1$ and $\phi_2$ (fractions of solid and gas areas). Hydrodynamic properties of the plane are fully expressed in terms of the effective slip-length tensor with eigenvalues that depend on texture parameters and $H$ (local separation). The effect of effective slip is predicted to decrease the force as compared with expected for two hydrophilic surfaces and described by the Taylor equation. The presence of additional length scales, $L$, $b_1$ and $b_2$, implies that a film drainage can be much richer than in case of a sphere moving towards a hydrophilic plane. For a large (compared to $L$) gap the reduction of the force is small, and for all textures the force is similar to expected when a sphere is moving towards a smooth hydrophilic plane that is shifted down from the super-hydrophobic wall. The value of this shift is equal to the average of the eigenvalues of the slip-length tensor. By analyzing striped super-hydrophobic surfaces, we then compute the correction to the Taylor equation for an arbitrary gap. We show that at thinner gap the force reduction becomes more pronounced, and that it depends strongly on the fraction of the gas area and local slip lengths. For small separations we derive an exact equation, which relates a correction for effective slip to texture parameters. Our analysis provides a framework for interpreting recent force measurements in the presence of super-hydrophobic surface.Comment: 9 pages, 5 figures, submitted to PRE; EPAPS file include

Axisymmetric multiphase lattice Boltzmann method

A lattice Boltzmann method for axisymmetric multiphase flows is presented and validated. The method is capable of accurately modeling flows with variable density. We develop the classic Shan-Chen multiphase model [ Phys. Rev. E 47 1815 (1993)] for axisymmetric flows. The model can be used to efficiently simulate single and multiphase flows. The convergence to the axisymmetric Navier-Stokes equations is demonstrated analytically by means of a Chapmann-Enskog expansion and numerically through several test cases. In particular, the model is benchmarked for its accuracy in reproducing the dynamics of the oscillations of an axially symmetric droplet and on the capillary breakup of a viscous liquid thread. Very good quantitative agreement between the numerical solutions and the analytical results is observed

Unsteady axisymmetric flow and heat transfer over time-dependent radially stretching sheet

AbstractThis article address the boundary layer flow and heat transfer of unsteady and incompressible viscous fluid over an unsteady stretching permeable surface. First of all modeled nonlinear partial differential equations are transformed to a system of ordinary differential equations by using similarity transformations. Analytic solution of the reduced problem is constructed by using homotopy analysis method (HAM). To validate the constructed series solution a numerical counterpart is developed using shooting algorithm based on Runge-Kutta method. Both schemes are in an excellent agreement. The effects of the pertinent parameters on the velocity and energy profile are shown graphically and examined in detail