David S. Rumschitzki

On the surfactant mass balance at a deforming fluid interface

The amount of surfactants ~surface active agents! adsorbed onto a fluid interface affects its surface tension. Thus the distribution of surfactants must be determined to find the jump in the normal and tangential stresses across the interface. Scriven ~see also Aris, Slattery, and Edwards et al.! uses differential geometry to derive the correct surface balance equation for an arbitrary surface coordinate system. Also invoking differential geometry, Waxman develops a correct form in ~‘‘fixed’’! surface coordinates that advance only normal to the surface. To arrive at this balance without appealing to differential geometry, Stone presents a simple physical derivation which leads to a form of the mass balance which is easy to solve numerically. Unfortunately, Stone’s derivation leaves the nature of the unsteady time derivative ambiguous. Here we follow the spirit set in Stone to derive geometrically the surface balance in a way that keeps the nature of the time derivative explicit. We verify that in Stone’s form the time derivative must hold the fixed coordinates constant, as the numerical implementation of this form of the mass balance actually do. We also derive a new form valid in an arbitrary surface coordinate system. Consider a fixed point A on a fluid surface with local normal n as in Fig. 1. We locate any two perpendicular planes which intersect along n. The intersection of each of these planes with the surface near the point A define curves whose unit tangents are t1 and t2 . By construction ]t1/]s152~1/R1!n and ]t2/]s252~1/R2!n, where ds1 and ds2 are differential arcs and R1~.0! and R2~.0! are the radii of curvature of the curves. Geometrically, these differential arcs are ds15R1df1 and ds25R2df2 , where df1 and df2 are the differential angles in the figure, and ]t1/]f152n and ]t2/]f252n. Thus in this locally orthogonal system, the components of the surface metric tensor aab are: Aa115R1 , a1250, and Aa225R2 and the diagonal elements simply act as scale factors. These arcs define a patch of area dA5Aa11Aa22df1df25Aadf1df2 where a is the determinant of the metric tensor. The diagonal components of the curvature tensor bab are defined by @]ta /]fa#–n 5 baa /Aaaa ~no sum on a!; so b1152R1 and b2252R2 . The curvatures are negative because as drawn in Fig. 1 both arcs are concave down with respect to the normal. If U is the instantaneous material velocity vector at the fixed point, its components along

Nonlinear interfacial stability of core‐annular film flows

n,t1 ,t2% are U5Us(1)t11Us(2)t21Wn, where W is the normal component and Us(1) and Us(2) are the physical components tangent to the surface. The fixed point advances along the normal ~n! as shown in the Fig. 1 a distance WDt so that the patch perimeters have lengths (R11WDt)df1 and (R21WDt)df2 at the time t1Dt; thus the change in area of the patch is WDt(R11R2)df1df2 and the per unit area per unit time rate of change is

Temporal instability of compound threads and jets

In this paper the weakly nonlinear stability of two‐phase core‐annular film flows in the limit of small film thickness and in the presence of both viscosity stratification and interfacial tension is examined. Rational asymptotic expansions are used to derive some novel nonlinear evolution equations for the interface between the phases. The novel feature of the equations is that they include a coupling between core and film dynamics thus enabling a study of its effect on the nonlinear evolution of the interface. The nonlinear interfacial evolution is governed by modified Kuramoto–Sivashinsky equations in the cases of slow and moderate flow [the former also developed by Frenkel, Sixth Symposium on Energy Engineering Sciences (Argonne Lab. Pub. CONF‐8805106, 1988), p.100, using different asymptotic methods], which include new nonlocal terms that reflect core dynamics. These equations are solved numerically for given initial conditions and a range of parameters. Some interesting behavior results, such as tran...

An asymptotic theory for the linear stability of a core-annular flow in the thin annular limit

Compound threads and jets consist of a core liquid surrounded by an annulus of a second immiscible liquid. Capillary forces derived from axisymmetric disturbances in the circumferential curvatures of the two interfaces destabilize cylindrical base states of compound threads and jets (with inner and outer radii R 1 and aR 1 respectively). The capillary instability causes breakup into drops; the presence of the annular phase allows both the annular- and core-phase properties to influence the drop size. Of technological interest is breakup where the core snaps first, and then the annulus. This results in compound drops. With jets, this pattern can form composite particles, or if the annular fluid is evaporatively removed, single drops whose size is modulated by both fluids. This paper is a study of the linear temporal instability of compound threads and jets to understand how annular fluid properties control drop size in jet breakup, and to determine conditions which favour compound drop formation. The temporal dispersion equation is solved numerically for non-dimensional annular thicknesses a of order one, and analytically for thin annuli ( a – 1 = e [Lt ] 1) by asymptotic expansion in e. There are two temporally growing modes: a stretching mode, unstable for wavelengths greater than the undisturbed inner circumference 2π R 1 , in which the two interfaces grow in phase; and a squeezing mode, unstable for wavelengths greater than 2π aR 1 , which grows exactly out of phase. Growth rates are always real, indicating that in jetting configurations disturbances convect downstream with the base velocity. For order-one thicknesses, the growth rate of the stretching mode is higher for the entire range of system parameters examined. The drop size scales with the wavenumber of the maximally growing wave ( k max ). We find that for the dominant stretching mode and a = 2, variations from 0.1 to 10 in the ratios of the annulus to core viscosity, or the tension of the outer surface to that of the inner interface, can result in changes in k max by a factor of approximately 2. However, for these changes in the system ratios, the growth rate ( s max ) and the ratio of the amplitude of the outer to the inner interface ( A max ) for the fastest growing wave only change marginally, with A max near one. The system appears most sensitive to the ratio of the density of the annulus to the core fluid. For a variation between 0.1 and 10, k max again changes by a factor of 2, but A max and s max vary more significantly with large amplitude ratios for low density ratios. The amplitude ratio of the stretching mode at the maximally growing wave ( A max ) indicates whether the film or core will break first. When this ratio is near one, linear theory predicts that the core breaks with the annulus intact, forming compound drops. Except for low values of the density ratio, our results indicate that most system conditions promote compound drop formation. For thin annuli, the growth rate disparity between modes becomes even greater. In the limit e → 0, the squeezing growth rate is roughly proportional to e 2 while the stretching mode growth rate is roughly proportional to e 0 and asymptotes to a single jet with radius R 1 and tension equal to the sum of the two tensions. Thus, in this limit the growth rate and k max are independent of the film density and viscosity. The amplitude ratio of the stretching mode becomes equal to one for all wavenumbers; so thin films break as compound drops. Our results compare favourably with previously published measurements on unstable waves in compound jets.

Temporal and spatial instability of an inviscid compound jet

We study the linear stability of a vertical, perfectly concentric, core–annular flow in the limit in which the gap is much thinner than the core radius. The analysis includes the effects of viscosity and density stratification, interfacial tension, gravity and pressure-driven forcing. In the limit of small annular thicknesses, several terms of the expression for the growth rate are found in order to identify and characterize the competing effects of the various physical mechanisms present. For the sets of parameters describing physical situations they allow immediate determination of which mechanisms dominate the stability. Comparisons between the asymptotic formula and available full numerical computations show excellent agreement for non-dimensional ratio of undisturbed annular thickness to core radius as large as 0.2. The expansion leads to new linear stability results (an expression for the growth rate in powers of the capillary number to the

An experimental investigation of the convective instability of a jet

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The effects of insoluble surfactants on the linear stability of a core-annular flow

power) for wetting layers in low-capillary-number liquid–liquid displacements. The expression includes both capillarity and viscosity stratification and agrees well with the experimental results of Aul & Olbrich (1990). Finally, we derive Kuramoto–Sivashinsky-type integro-differential equation for the later nonlinear stages of the interfacial dynamics, and discuss their solutions.

The linear stability of a core–annular flow in an asymptotically corrugated tube

This paper examines the linear hydrodynamic stability of an inviscid compound jet. We perform the temporal and the spatial analyses in a unified framework in terms of transforms. The two analyses agree in the limit of large jet velocity. The dispersion equation is explicit in the growth rate, affording an analytical solution. In the temporal analysis, there are two growing modes, stretching and squeezing. Thin film asymptotic expressions provide insight into the instability mechanism. The spatial analysis shows that the compound jet is absolutely unstable for small jet velocities and admits a convectively growing instability for larger velocities. We study the effect of the system parameters on the temporal growth rate and that of the jet velocity on the spatial growth rate. Predictions of both the temporal and the spatial theories compare well with experiment.

The weakly nonlinear interfacial stability of a core–annular flow in a corrugated tube

This paper is an experimental study of the convective instability of a jet. It is well known that a jet issuing forth from a nozzle is unstable due to surface tension forces that cause it to break downstream into drops. We apply a disturbance of a given frequency at the nozzle tip. This applied frequency determines the wavelength and the growth rate of the growing disturbances and, thereby, the drop size. We measure the wavelength and the growth rate by fitting the entire digitized image of a jet to the functional form suggested by the linear theory. Thus, it makes use of the entire profile instead of the small number of points used in previous studies. Also, in contrast to previous work, we independently measure the jet velocity and the wave speed. At high non-dimensional jet velocity, the experimental results for the growth rates and the wave numbers agree with the linear stability theory of an infinite jet in the absence of gravity. At very low velocity (low Froude number) gravity is important and the agreement is not good.

Marangoni effects on the motion of an expanding or contracting bubble pinned at a submerged tube tip

The effects of an insoluble surfactant on the linear stability of a two-fluid core-annular flow in the thin annulus limit, for axisymmetric disturbances with wavelengths large compared to the annulus thickness, h 0 , are the focus of this investigation. A base shear flow affects the interfacial surfactant distribution, thereby inducing Marangoni forces that, along with capillary forces, affect the fluid-fluid interface stability. The resulting systems stability differs markedly from that of the same system with zero base flow. In the thin-annulus limit (the ratio e of the undisturbed annulus thickness to core radius tends to zero), common in applications, a scaling and asymptotic analysis yields a coupled set of equations for the perturbed fluid-fluid interface shape and surfactant concentration. The linear dynamics of the annular film fully determine these equations, i.e. the core dynamics are slaved to the film dynamics. The theory provides a unified view of the mechanism of stability in three different regimes of capillary number Ca (defined as the product of the core viscosity, μ 1 , and the centreline velocity, W 0 , divided by the interface tension, σ * 0 , that corresponds to an undisturbed (signified by the subscript 0) uniform surfactant concentration, Γ* 0 ). In the absence of a base flow or in the limit of small Ca(«e 2 ), Marangoni forces deriving from non-uniformities in the interface concentration of insoluble surfactants oppose the net capillary forces. These latter forces normally stabilize the longitudinal curvature and destabilize the circumferential curvature of perturbations to the interface. In the limit of large Ca(> e 2 ), Marangoni forces destabilize disturbances with wavelengths that are large compared to the annulus thickness