On the Tutte-Krushkal-Renardy polynomial for cell complexes

Recently V. Krushkal and D. Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A. Duval, C. Klivans, and J. Martin. Moreover, after a slight modification, the Tutte-Krushkal-Renardy polynomial evaluated at the origin gives a weighted count of cellular spanning trees, and therefore its free term can be calculated by the cellular matrix-tree theorem of Duval et al. In the case of cell decompositions of a sphere, this modified polynomial satisfies the same duality identity as the original polynomial. We find that evaluating the Tutte-Krushkal-Renardy along a certain line gives the Bott polynomial. Finally we prove skein relations for the Tutte-Krushkal-Renardy polynomial..Comment: Minor revision according to a reviewer comments. To appear in the Journal of Combinatorial Theory, Series

Role of inertia in two-dimensional deformation and breakup of a droplet

We investigate by Lattice Boltzmann methods the effect of inertia on the deformation and break-up of a two-dimensional fluid droplet surrounded by fluid of equal viscosity (in a confined geometry) whose shear rate is increased very slowly. We give evidence that in two dimensions inertia is {\em necessary} for break-up, so that at zero Reynolds number the droplet deforms indefinitely without breaking. We identify two different routes to breakup via two-lobed and three-lobed structures respectively, and give evidence for a sharp transition between these routes as parameters are varied.Comment: 4 pages, 4 figure

Air entrainment through free-surface cusps

In many industrial processes, such as pouring a liquid or coating a rotating cylinder, air bubbles are entrapped inside the liquid. We propose a novel mechanism for this phenomenon, based on the instability of cusp singularities that generically form on free surfaces. The air being drawn into the narrow space inside the cusp destroys its stationary shape when the walls of the cusp come too close. Instead, a sheet emanates from the cusp's tip, through which air is entrained. Our analytical theory of this instability is confirmed by experimental observation and quantitative comparison with numerical simulations of the flow equations

Toroidal drops in viscous flow

Toroidal drops are known since the experiments by Plateau (1854) in rotating fluids. Such shapes and other non-spherical configurations have become of interest in various technological areas, and recently also as potential carriers of drugs (Champion et al., 2007) or building blocks for more complex assemblies (Velev et al., 2000). Such geometry is obtained, for example, when a drop, falling free in a viscous fluid, experiences a finite surface deformation which develops into a toroidal form (Kojima et al., 1984; Baumann et al., 1992; Sostarecz & Belmonte 2003). In this presentation we shall revisit the stable compression of spherical drops in bi-axial viscous extension, within a finite range of the capillary number, Ca, and show that loss of stability can lead to formation of toroidal shapes. We demonstrate numerically that there is a limited range of Ca in which toroidal stationary solutions exist, and that such drops in this flow are inherently unstable (Zabarankin et al., 2013). However, there is a potential of shape stabilization if the drops are comprised of a mild yield stress material. References BAUMANN, N., JOSEPH, D. D., MOHR, P. & RENARDY, Y. 1992 Vortex rings of one fluid in another in free fall. Phys. Fluids A 4 (3), 567–580. CHAMPION, J. A., KATARE, Y. K. & MITRAGOTRI, S. 2007 Particle shape: A new design parameter for micro- and nanoscale drug delivery carriers. J. Contr. Release 121 (1–2), 3–9. KOJIMA, M., HINCH, E. J. & ACRIVOS, A. 1984 The formation and expansion of a toroidal drop moving in a viscous fluid. Phys. Fluids 27 (1), 19–32. PLATEAU, J. 1857 I. Experimental and theoretical researches on the figures of equilibrium of a liquid mass withdrawn from the action of gravity.–Third series. Philosophical Magazine Series 4 14 (90), 1–22. SOSTARECZ, M. C. & BELMONTE, A. 2003 Motion and shape of a viscoelastic drop falling through a viscous fluid. J. Fluid Mech. 497, 235–252. VELEV, O. D., LENHOFF, A. M. & KALER, E. W. 2000 A class of microstructured particles through colloidal crystallization. Science 287 (5461), 2240–2243. ZABARANKIN, M., SMAGIN, I., LAVRENTEVA, O. M. & NIR, A. 2013 Viscous drop in compressional Stokes flow. J. Fluid Mech. 720, 169–191

Fracture of a viscous liquid

When a viscous liquid hits a pool of liquid of same nature, the impact region is hollowed by the shock. Its bottom becomes extremely sharp if increasing the impact velocity, and we report that the curvature at that place increases exponentially with the flow velocity, in agreement with a theory by Jeong and Moffatt. Such a law defines a characteristic velocity for the collapse of the tip, which explains both the cusp-like shape of this region, and the instability of the cusp if increasing (slightly) the impact velocity. Then, a film of the upper phase is entrained inside the pool. We characterize the critical velocity of entrainment of this phase and compare our results with recent predictions by Eggers

A discrete systems approach to cardinal spline Hermite interpolation

AbstractA cardinal spline Hermite interpolation problem is posed by specifying values, and m−1 derivatives, m⩾1, at uniformly spaced knots tk; it may be solved by means of a generalized spline function w(t) (a standard spline function when m=1), piecewise a polynomial of degree n−1=2m+p−1, p⩾0, with w(j)(t) continuous across the knots for j=0,1,2,…,m+p−1. The problem is studied here for p>0 in the context of an (m+p)-dimensional system of linear recursion equations satisfied by the values of the m-th through m+p−1-st derivatives of w(t) at the knots, whose homogeneous term involves a p×p matrix A . In the case m=1 we relate the characteristic polynomial of A and certain controllability notions to the standard B-spline and we proceed to show how systems-theoretic ideas can be used to generate systems of basis splines for higher values of m

Hydrodynamic Singularities

We give a brief overview of the physical significance of singularities in fluid mechanics

Analysis of the String Structure Near Break-up of A Slender Jet of An Upper Convected Maxwell Liquid

In this paper, we analytically study the string structure near the break-up of a slender jet of a viscoelastic liquid surrounded by air. The governing equations are derived from the conservation laws of mass and momentum, and the rheological equation of the jet. The rheological equation of the jet is assumed to satisfy an Upper Convected Maxwell (UCM) model. Introducing a stretch variable and then applying a transformation, we obtain a coupled system of nonlinear differential equations. Via these equations, we then show that the UCM jet does not break up in finite time, which physically means that it has sufficient time to exhibit the string structure before it breaks up due to the dominant surface force

Macroscopic Model for Head-On Binary Droplet Collisions in a Gaseous Medium

In this Letter, coalescence-bouncing transitions of head-on binary droplet collisions are predicted by a novel macroscopic model based entirely on fundamental laws of physics. By making use of the lubrication theory of Zhang and Law [Phys. Fluids 23, 042102 (2011)], we have modified the Navier-Stokes equations to accurately account for the rarefied nature of the interdroplet gas film. Through the disjoint pressure model, we have incorporated the intermolecular van der Waals forces. Our model does not use any adjustable (empirical) parameters. It therefore encompasses an extreme range of length scales (more than 5 orders of magnitude): from those of the external flow in excess of the droplet size (a few hundred μm) to the effective range of the van der Waals force around 10 nm. A state of the art moving adaptive mesh method, capable of resolving all the relevant length scales, has been employed. Our numerical simulations are able to capture the coalescence-bouncing and bouncing-coalescence transitions that are observed as the collision intensity increases. The predicted transition Weber numbers for tetradecane and water droplet collisions at different pressures show good agreement with published experimental values. Our study also sheds new light on the roles of gas density, droplet size, and mean free path in the rupture of the gas film

Stability of plane Poiseuille flow of a fluid with pressure-dependent viscosity

We study the linear stability of a plane Poiseuille flow of an incompressible fluid whose viscosity depends linearly on the pressure. It is shown that the local critical Reynolds number is a sensitive function of the applied pressure gradient and that it decreases along the channel. While in the limit of small pressure gradients conventional results for a pressure-independent Newtonian fluid are recovered, a significant stabilisation of the flow and an elongation of the critical disturbance wavelength are observed when the longitudinal pressure gradient is increased. These features drastically distinguish the stability characteristics of a piezo-viscous flow from its pressure-independent Newtonian counterpart