S. S. Sadhal

Stokes flow past bubbles and drops partially coated with thin films. Part 1. Stagnant cap of surfactant film – exact solution

In this investigation the creeping flow due to the motion of a liquid drop or a bubble in another immiscible fluid is examined when the interface is partially covered by a stagnant layer of surfactant. The associated boundary-value problem involves mixed boundary conditions at the interface, which lead to a set of dual series equations. An inversion of these equations yields the exact solution to the stagnant cap problem. Several useful results are obtained in closed form. Among these are the expressions for the drag force, the difference between the maximum and the minimum interfacial tensions, and the amount of adsorbed surfactant. A shifting of the centre of the internal vortex is observed.

Forthcoming Lab on a Chip tutorial series on acoustofluidics: Acoustofluidics—exploiting ultrasonic standing wave forces and acoustic streaming in microfluidic systems for cell and particle manipulation

Forthcoming lab on a chip tutorial series on acoustofluidics : Acoustofluidics - Exploiting ultrasonic standing wave forces and acoustic streaming in microfluidic systems for cell and particle manipulation

Flow past a liquid drop with a large non-uniform radial velocity

In this analysis, the translation of a liquid drop experiencing a strong non-uniform radial velocity has been investigated. The situation arises when a moving liquid drop experiences condensation, evaporation or material decomposition at the surface. By simultaneously treating the flow fields inside and outside the drop, we have obtained physical results relevant to the problem. The magnitude of the radial velocity is allowed to be very large, but the drop motion is restricted to slow translation. The solution to the problem has been developed by considering a uniform radial flow with the translatory motion introduced as a perturbation. The role played by the inertial terms due to the strong radial field has been clearly delineated. The study has revealed several interesting features. An inward normal velocity on a slowly moving drop increases the drag. An increasing outward normal velocity decreases the drag up to a minimum beyond which it increases. The total drag force not only consists of contributions from the viscous and the form drags but also from the momentum transport at the interface. Since the liquid drop admits a non-zero tangential velocity, the tangential momentum convected by the radial velocity forms a part of this drag force. The circulation inside the drop decreases (increases) with an outward (inward) normal velocity. A sufficiently large non-uniform outward velocity causes the circulation to reverse. In the limit of the internal viscosity becoming infinite, our analysis collapses to the simple case of a translating rigid sphere experiencing a large non-uniform radial velocity. By letting the radial velocity become vanishingly small the Stokes-flow solution is recovered. An important contribution of the present study is the identification of a new singularity in the flow description. It accounts for both the inertial and the viscous forces and displays Stokeslet-like characteristics at infinity. Disciplines Engineering | Mechanical Engineering Comments Suggested Citation: Sadhal, Satwindar S. and Portonovo S. Ayyaswamy. (1983). Flow past a liquid drop with a large non-uniform radial velocity.. Journal of Fluid Mechanics. Vol. 133, p. 65-81. Copyright 1983 Cambridge University Press. This journal article is available at ScholarlyCommons: http://repository.upenn.edu/meam_papers/187 J . Fluid Meek (1983), vol. 133, p p . 65-81 Printed in Great Britain 65 Flow past a liquid drop with a large non-uniform radial velocity By S. S. SADHAL Department of Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1453 AND P. s. AYYASWAMY Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104 (Received 1 October 1982 and in revised form 1 February 1983) In this analysis, the translation of a liquid drop experiencing a strong non-uniform radial velocity has been investigated. The situation arises when a moving liquid drop experiences condensation, evaporation or material decomposition at the surface. By simultaneously treating the flow fields inside and outside the drop, we have obtained physical results relevant to the problem. The magnitude of the radial velocity is allowed to be very large, but the drop motion is restricted to slow translation. The solution to the problem has been developed by considering a uniform radial flow with the translatory motion introduced as a perturbation. The role played by the inertial terms due to the strong radial field has been clearly delineated. The study has revealed several interesting features. An inward normal velocity on a slowly moving drop increases the drag. An increasing outward normal velocity decreases the drag up to a minimum beyond which it increases. The total drag force not only consists of contributions from the viscous and the form drags but also from the momentum transport at the interface. Since the liquid drop admits a non-zero tangential velocity, the tangential momentum convected by the radial velocity forms a part of this drag force. The circulation inside the drop decreases (increases) with an outward (inward) normal velocity. A sufficiently large non-uniform outward velocity causes the circulation to reverse. In the limit of the internal viscosity becoming infinite, our analysis collapses to the simple case of a translating rigid sphere experiencing a large non-uniform radial velocity. By letting the radial velocity become vanishingly small the Stokes-flow solution is recovered. An important contribution of the present study is the identification of a new singularity in the flow description. It accounts for both the inertial and the viscous forces and displays Stokeslet-like characteristics at infinity.

Stokes flow past compound multiphase drops: the case of completely engulfed drops/bubbles

In this paper the translatory motion of a compound drop is examined in detail for low-Reynolds-number flow. The compound drop, consisting of a liquid drop or a gas bubble completely coated by another liquid, moves in a third immiscible fluid. An exact solution for the flow field is found in the limit of small capillary numbers by approximating the two interfaces to be spherical. The solution is found for the general case of eccentric configuration with motion of the inner sphere relative to the outer together with the motion of the system in the continuous phase. The results show that the viscous forces tend to move the inner-fluid sphere towards the front stagnation point of the compound drop. For equilibrium of the inner sphere with respect to the outer there must, therefore, be a body force towards the front. This can only be achieved with the necessary condition that there be a buoyant force on the inner sphere, opposite to that of the compound drop in the continuous phase. For a given set of fluids, two or four equilibrium configurations may be found. Of these only one or two, respectively, are stable. The others are unstable. For the special case of concentric configuration, the equilibrium is always metastable.

Effects of soluble and insoluble surfactants on the motion of drops

The fluid dynamics of moving drops in the presence of a soluble surfactant and an insoluble impurity is examined in detail. The main purpose of this analysis is to establish a fairly general theory that agrees with experimental measurements. Particular attention is paid to situations involving a stagnant cap which arise when low-solubility surfactants are present. Earlier theories on stagnant caps have not satisfactorily explained the experimental results and a two-impurity model is therefore proposed. The analysis is carried out semi-analytically using a matched asymptotic analysis of the Proudman-Pearson type for weakly inertial flows. The results seem to be in good agreement with the available data at a Peclet number of about 700 for the soluble surfactant. In particular the predicted flow field within the drop is found to be consistent with the experimental measurements of Horton. The concentration profiles graphically exhibit the physical phenomena involved in the mass transport. Another new result is the analytical expression for the drag force corrected up to O(Re) for the case involving only the insoluble surfactant.

Internal circulation in a drop in an acoustic field

An investigation of the internal flow field for a drop at the antinode of a standing wave has been carried out. The main difference from the solid sphere case is the inclusion of the shear stress and velocity continuity conditions at the liquid-gas interface. To the leading order of calculation, the internal flow field was found to be quite weak. Also, this order being fully time dependent has a zero mean flow. At the next higher order, steady internal flows are predicted and, as in the case of a solid sphere, there is a recirculating layer consisting of closed streamlines near the surface. In the case of a liquid drop, however, the behavior of this recirculating Stokes layer is quite interesting. It is predicted that the layer ceases to have recirculation when [formula: see text], where [symbol: see text] is the liquid viscosity, mu is the exterior gas-phase viscosity, and M is the dimensionless frequency parameter for the gas phase, defined by M = i omega a2 rho/mu, with a being the drop radius. Thorough experimental confirmation of this interesting new development needs to be conducted. Although it seems to agree with many experiments with levitated drops where no recirculating layer has been clearly observed, a new set of experiments for specifically testing this interesting development need to be carried out.

Boundary conditions for stokes flows near a porous membrane

A theoretical development is carried out to model the boundary conditions for Stokes flows near a porous membrane, which, in general, allows non-zero slip as well as normal flow at the surface. Two types of models are treated: an infinitesimally thin plate with a periodic array of circular apertures and a series of parallel slits. For Stokes flows, the mean normal flux and slip velocity are proportional to the pressure difference across the membrane and the average shear stress at the membrane, respectively. The appropriate proportionality constants which depend on the membrane geometry are calculated as functions of the porosity. An interesting feature of the results is that the slip at the membrane has, in general, a direction different from that of the applied shear for these models.

Thin-Flame Theory for the Combustion of a Moving Liquid Drop: Effects Due to Variable Density

The combustion of a moving liquid fuel drop has been investigated. The drop experiences a strong evaporation-induced radial velocity while undergoing slow translation. In view of the high evaporation velocity, the flow field is not in the Stokes regime. The combustion process is modelled by an indefinitely fast chemical reaction rate. While the flow and the transport in the continuous phase and the drop internal circulation are treated as quasisteady, the drop heat-up is regarded as a transient process. The transport equations of the continuous phase require analysis by a singular perturbation technique. The transient heat-up of the drop interior is solved by a series-truncation numerical method. The solution for the total problem is obtained by coupling the results for the continuous and dispersed phases. The enhancement in the mass burning rate and the deformation of the flame shape due to drop translation have been predicted. The initial temperature of the drop and the subsequent heating influence the temporal variations of the flamefront standoff ratio and the flame distance. The friction drag, the pressure drag and the drag due to interfacial momentum flux are individually predicted, and the total drag behaviour is discussed. The circulation inside the drop decreases with evaporation rate. A sufficiently large non-uniform evaporation velocity causes the circulation to reverse.

Laminar Condensation on a Moving Drop. Part 1. Singular Perturbation Technique

In this paper, laminar condensation on a spherical drop in a forced flow is investigated. The drop experiences a strong, radial, condensation-induced velocity while undergoing slow translation. In view of the high condensation velocity, the flow field, although the drop experiences slow translation, is not in the Stokes-flow regime. The drop environment is assumed to consist of a mixture of saturated steam (condensable) and air (non-condensable). The study has been carried out in two different ways. In Part 1 the continuous phase is treated as quasi-steady and the governing equations for this phase are solved through a singular perturbation technique. The transient heat-up of the drop interior is solved by the series-truncation numerical method. The solution for the total problem is obtained by matching the results for the continuous and dispersed phases. In Part 2 both the phases are treated as fully transient and the entire set of coupled equations are solved by numerical means. Validity of the quasi-steady assumption of Part 1 is discussed. Effects due to the presence of the non-condensable component and of the drop surface temperature on transport processes are discussed in both parts. A significant contribution of the present study is the inclusion of the roles played by both the viscous and the inertial effects in the problem treatment.

Thermal constriction resistance: effects of boundary conditions and contact geometries

Abstract The problem of steady-state thermal constriction resistance is modeled by means of various spatially periodic arrangements of circular disks (contact regions) on the surface of a semi-infinite solid. Three cases of disk boundary condition are considered : uniform flux, the ‘equivalent isothermal flux’, and the condition of isothermal disks. Analytical expressions for the resistance are derived as power series of K 1 4 , where k is the fraction of the solid surface occupied by the disks. The behavior of the resistance is then studied as a function of disk boundary condition, spatial arrangement and concentration.