L. E. Scriven

Hydrodynamic Model of Steady Movement of a Solid / Liquid / Fluid Contact Line

Abstract Movement of the contact line over a solid surface violates the adherence, or “no slip,” boundary condition which is otherwise obeyed by flowing liquids. A flat fluid interface moving steadily over a flat solid is modeled with the creeping flow approximation, which turns out to be self-consistent. Adherence is required except at the contact line itself. Though the velocity field appears to be realistic, stresses and viscous dissipation are found to increase without bound at the contact line. The way the hydrodynamic model breaks down suggests that in reality there may be steep gradients, rheological anomalies, and discontinuous processes around the contact line. Slip and the role of long-range forces are explored with the aid of the lubrication flow approximation.

On cellular convection driven by surface-tension gradients: effects of mean surface tension and surface viscosity

The onset of steady, cellular convection driven by surface tension gradients on a thin layer of liquid is examined in an extension of Pearsons (1958) stability analysis. By accounting for the possibility of shape deformations of the free surface it is found that there is no critical Marangoni number for the onset of stationary instability and that the limiting case of ‘zero wave-number’ is always unstable. Surface viscosity of a Newtonian interface is found to inhibit stationary instability. A simple criterion is found for distinguishing visually the dominant force, buoyancy or surface tension, in cellular convection in liquid pools.

Pendular rings between solids: meniscus properties and capillary force

The Laplace–Young equation is solved for axisymmetric menisci, analytically in terms of elliptic integrals for all possible types of pendular rings and liquid bridges when the effect of gravity is negligible, numerically for selected other cases in order to assess gravitys effect. Meniscus shapes, mean curvatures, areas and enclosed volumes are reported, as are capillary forces. It is shown that capillary attraction may become capillary repulsion when wetting is imperfect. The special configurations of vanishing capillary force and of zero mean curvature are treated. The range of utility of the convenient ‘circle approximation’ is evaluated.

An Integral Constitutive Equation for Mixed Flows: Viscoelastic Characterization

A viscoelasticconstitutive equation of the single‐integral form has been designed. Its memory function is factored into a time‐dependent part and a strain‐dependent part. The time function is the usual series of exponential relaxations. Its relaxation times and weighting coefficients are determined by nonlinear regression on linear viscoelastic data: stress relaxation after small‐strain and small‐amplitude sinusoidal oscillations. The relative accuracy of linear and nonlinear regression fitting is compared. The strain‐dependent function is new. It is of a simple sigmoidal form with only two parameters: one determined from shear and the other from extensional data. Its sigmoidal form provides a finite linear viscoelastic region, a steady viscosity in uniaxial extension, and a well‐behaved power‐law shear viscosity at high shear rate. An efficient strategy for collecting sufficient data to determine the parameters of the equation is described. Predictions of the equation are tested against shear and extension data collected on the Rheometrics System Four for two polydimethylsiloxanes and against data for other polymer melts from the literature. Both uniaxial and biaxial extension as well as shear data are described. Transient shear normal stresses are somewhat underpredicted. The constitutive equation has the potential for modeling mixed shear and extensional flows as encountered in processing operations and is simple enough to be attractive for efficient computer‐aided analysis by modern finite element methods.

The oscillations of a fluid droplet immersed in another fluid

From an analysis of small oscillations of a viscous fluid droplet immersed in another viscous fluid a general dispersion equation is derived by which frequency and rate of damping of oscillations can be calculated for arbitrary values of droplet size, physical properties of the fluids, and interfacial viscosity and elasticity coefficients. The equation is studied for two distinct extremes of interfacial characteristics: (i) a free interface between the two fluids in which only a constant, uniform interfacial tension acts; (ii) an ‘inextensible’ interface between the two fluids, that is, a highly condensed film or membrane which, to first order, cannot be locally expanded or contracted. Results obtained are compared with those previously published for various special cases.When the viscosities of both fluids are low, the primary contribution to the rate of damping of oscillations is generally the viscous dissipation in a boundary layer near the interface, in both the free and inextensible interface situations. For this reason inviscid velocity profiles, which do not account for the boundarylayer flow, do not lead to good approximations to the damping rate. The two exceptions in which the approximation based on inviscid profiles is adequate occur when the interface is free and either the interior or exterior fluid is a gas of negligible density and viscosity.

Monte Carlo simulation of model amphiphile‐oil–water systems

A square‐lattice model of amphiphile‐oil–water systems is developed in which oil and water molecules occupy single sites and amphiphiles occupy chains of sites. Energies and free energies estimated by Monte Carlo sampling of configuration space show that when the head, or water‐loving portion, of the amphiphile has no tendency to hydrate or surround itself with water, as opposed to surrounding itself with other heads, the capability of even long amphiphiles to solubilize repellant oil and water into a single phase is weak. Although the Monte Carlo free energies deviate markedly from those given by quasichemical theory, the deviation of the phase behavior is modest. Computer drawings of typical equilibrium configurations show highly irregular interfaces, apparently caused by capillary waves which are pronounced in two dimensions.

Separating how near a static contact line: Slip at a wall and shape of a free surface

The effects of replacing the usual no-slip boundary condition by a slip-coefficient boundary condition on solid walls are determined by means of Galerkin finite element solutions of the Navier-Stokes equation system for steady two-dimensional discharge of liquid from a sharp-edged slot—the “die-swell” flow, in which menisci separate from contact lines on the die edges. A velocity-pressure formulation is used with a mixed interpolation basis of quadratic serendipity elements far velocity and bilinear functions for pressure. Alternate formulations of the free meniscus computation are examined; iteration on the normal stress condition proves superior except when surface tension is low. Even if slip is possible, the no-slip boundary condition proves accurate for Newtonian die-swell flow, provided the slip coefficient β is less than 10−3, bμ (μ viscosity; b, the channel half-width). Slip at the solid wall alleviates the apparent stress singularity at the exit. When β ⩾ 10−3, bμ, slip reduces die swell even of Newtonian liquid; local slip near the contact line is sufficient. Raising the Reynolds number decreases the upstream influence at the exit on the velocity profile and reduces die swell. Surface tension straightens the free meniscus profile and thus reduces die swell. Profile straightening is compounded by actions of surface tension and slip together.

Shapes of axisymmetric fluid interfaces of unbounded extent

Abstract The shape of an equilibrium fluid interface that extends far outward from a circular line of contact is calculated and tabulated as a function of radius of the contact circle, interface inclination or contact angle there, interfacial tension, density difference across the interface, and force of gravity. A novel and efficient method of calculation is described. The accuracy of the results is shown to exceed that generally needed in practice. Sample applications are made to problems of meniscus location, contact angle, flotation position, and dry patch radius.

Percolation theory of two phase flow in porous media

The simultaneous flow of two phases through porous media has long lacked a sound theoretical description, due, in large measure, to the complex geometry and topology of the pore space, and the chaotic, nearly random spatial distribution of the fluids. For some 20 years, a theory of flow through random media, percolation theory, has existed in the field of mathematical physics. In this paper, it is demonstrated that percolation theory describes the distribution of non-wetting fluid in common sandstones and limestones during capillary pressure and relative permeability measurements. In particular, the theory predicts the relative amounts of non-wetting fluid in each of three distinct fluid configurations: isolated, dead-end, and flow effective fluid. Porous media are herein idealized as random mixtures of solid and void to which concepts of percolation theory apply. For sinter-bodies a critical porosity, φc ⋍ 112, exists, below which the void space is not continuous across the sample. For many natural porous materials, e.g. sandstones and limestones, the dependence of residual non-wetting saturation upon porosity is roughly that which corresponds to the existence of a critical porosity, whose value is again 112. The distribution of the non-wetting phase during two phase flow, in particular, the amount of isolated and dead-end fluid, can be qualitatively predicted by simple graph models. The concepts of dead end and isolated fluid must be considered if relative permeability, capillary pressure, and other descriptors of two phase flow are to be understood.

Study of coating flow by the finite element method

Abstract Sillimans analysis of slot coating is extended to accommodate film flows with highly bent menisci, as in slide and curtain coating, by combining polar and Cartesian coordinate parametrizations of meniscus shape. The nonlinear algebraic equations from the subdomain/Galerkin weighted residual method and finite element basis functions are solved by Newtons method, which is perfected for the free boundary problem. Convergence behavior is examined. Slot-coating results reveal slow-flow zones and show that at high capillary number and low metering rate the downstream meniscus can no longer attach to the slot. Comparison of Coyne and Elrods approximate solutions with the finite element solutions shows that their assumption of a parabolic velocity profile leads to reasonable approximations of meniscus shapes.