Michael Renardy

Hyperbolicity and Change of Type in the Flow of Viscoelastic Fluids.

The equations governing the flow of viscoelastic liquids are classified according to the symbol of their differential operators. Propagation of singularities is discussed and conditions for a change of type are investigated. The vorticity equation for steady flow can change type when a critical condition involving speed and stresses is satisfied. This leads to a partitioning of the field of flow into subcritical and supercritical regions, as in the problem of transonic flow.

Numerical simulation of breakup of a viscous drop in simple shear flow through a volume-of-fluid method

A spherical drop, placed in a second liquid of the same density, is subjected to shearing between parallel plates. The subsequent flow is investigated numerically with a volume-of-fluid (VOF) method. The scheme incorporates a semi-implicit Stokes solver to enable computations at low Reynolds number. Our simulations compare well with previous theoretical, numerical, and experimental results. For capillary numbers greater than the critical value, the drop deforms to a dumbbell shape and daughter drops detach via an end-pinching mechanism. The number of daughter drops increases with the capillary number. The breakup can also be initiated by increasing the Reynolds number.

Linear stability of plane couette flow of an upper convected maxwell fluid

Abstract We investigate the linear stability of plane Couette flow of an upper convected Maxwell fluid using a spectral method to compute the eigenvalues. No instabilities are found. This is in agreement with the results of Ho and Denn [1] and Lee and Finlayson [2], but contradicts “proofs” of instability by Gorodtsov and Leonov [3] and Akbay and Frischmann [4,5]. The errors in those arguments are pointed out. We also find that the numerical discretization can generate artificial instabilities (see also [1,6]). The qualitative behavior of the eigenvalue spectrum as well as the artificial instabilities is discussed.

Pyramidal and toroidal water drops after impact on a solid surface

Superhydrophobic surfaces generate very high contact angles as a result of their microstructure. The impact of a water drop on such a surface shows unusual features, such as total rebound at low impact speed. We report experimental and numerical investigations of the impact of approximately spherical water drops. The axisymmetric free surface problem, governed by the Navier–Stokes equations, is solved numerically with a front-tracking marker-chain method on a square grid. Experimental observations at moderate velocities and capillary wavelength much less than the initial drop radius show that the drop evolves to a staircase pyramid and eventually to a torus. Our numerical simulations reproduce this effect. The maximal radius obtained in numerical simulations precisely matches the experimental value. However, the large velocity limit has not been reached experimentally or numerically. We discuss several complications that arise at large velocity: swirling motions observed in the cross-section of the toroidal drop and the appearance of a thin film in the centre of the toroidal drop. The numerical results predict the dry-out of this film for sufficiently high Reynolds and Weber numbers. When the drop rebounds, it has a top-heavy shape. In this final stage, the kinetic energy is a small fraction of its initial value.

A numerical study of the asymptotic evolution and breakup of Newtonian and viscoelastic jets

Numerical and asymptotic methods are used to study the surface-tension driven instability of the jet. For the Newtonian case, we establish a relationship between the initial shape of the jet and the asymptotic approach to breakup. For the Oldroyd-B fluid, no breakup occurs, and we study the evolution of the jet for large time. On the other hand, the numerical solutions show finite time breakup for the Giesekus model. For the upper convected Maxwell model, nonunique discontinuous solutions exist, and the choice of the physically appropriate solution depends on regularizing terms in the equations. We discuss one such regularization involving higher order derivatives resulting from taking into account the axial curvature of the free surface.

A note on the equations of a thermoelastic plate

Abstract It is proved that the equations of a linear thermoelastic plate are associated with an analytic semigroup.

A new volume-of-fluid formulation for surfactants and simulations of drop deformation under shear at a low viscosity ratio

Abstract A numerical algorithm for the linear equation of state is developed for the volume-of-fluid interface-tracking code SURFER++, using the continuous surface stress formulation for the description of interfacial tension. This is applied to deformation under simple shear for a liquid drop in a much more viscous matrix liquid. We choose a Reynolds number and capillary number at which the drop settles to an ellipsoidal steady state, when there is no surfactant. The viscosity ratio is selected in a range where experiments have shown tip streaming when surfactants are added. Our calculations show that surfactant is advected by the flow and moves to the tips of the drop. There is a threshhold surfactant level, above which the drop develops pointed tips, which are due to surfactant accumulating at the ends of the drop. Fragments emitted from these tips are on the scale of the mesh size, pointing to a shortcoming of the linear equation of state, namely that it does not provide a lower bound on interfacial tension. One outcome is the possibility of an unphysical negative surface tension on the emitted drops.

An existence theorem for model equations resulting from kinetic theories of polymer solutions

A local existence and uniqueness theorem is proved for a set of partial differential equations modelling the flow of polymer solutions. The constitutive relations considered here are motivated by kinetic theory. The stress tensor is given by an integral which involves the solution of a linear diffusion equation. The coefficients of this diffusion equation depend on the velocity gradient.

Structure of the spectrum in zero Reynolds number shear flow of the UCM and Oldroyd-B liquids

Abstract We provide a mathematical analysis of the spectrum of the linear stability problem for one and two layer channel flows of the upper-convected Maxwell (UCM) and Oldroyd-B fluids at zero Reynolds number. For plane Couette flow of the UCM fluid, it has long been known (Gorodstov and Leonov, J. Appl. Math. Mech. (PMM) 31 (1967) 310) that, for any given streamwise wave number, there are two eigenvalues in addition to a continuous spectrum. In the presence of an interface, there are seven discrete eigenvalues. In this paper, we investigate how this structure of the spectrum changes when the flow is changed to include a Poiseuille component, and as the model is changed from the UCM to the more general Oldroyd-B. For a single layer UCM fluid, we find that the number of discrete eigenvalues changes from two in Couette flow to six in Poiseuille flow. The six modes are given in closed form in the long wave limit. For plane Couette flow of the Oldroyd-B fluid, we solve the differential equations in closed form. There is an additional continuous spectrum and a family of discrete modes. The number of these discrete modes increases indefinitely as the retardation time approaches zero. We analyze the behavior of the eigenvalues in this limit.

Two-Dimensional cusped interfaces

Two-dimensional cusped interfaces are line singularities of curvature. We create such cusps by rotating a cylinder half immersed in liquid. A liquid film is dragged out of the reservoir on one side and is plunged in at the other, where it forms a cusp at finite speeds, if the conditions are right. Both Newtonian and non-Newtonian fluids form cusps, but the transition from a rounded interface to a cusp is gradual in Newtonian liquids and sudden in non-Newtonian liquids. We present an asymptotic analysis near the cusp tip for the case of zero surface tension, and we make some remarks about the effects of a small surface tension. We also present the results of numerical simulations showing the development of a cusp. In those simulations, the fluid is filling an initially rectangular domain with a free surface on top. The fluid enters from both sides and is sucked out through a hole in the bottom.