Stokes Flow Conditions Research Articles

Dynamic Surface Tension Effects in Impact of a Drop with a Solid Surface

Recent research on free surface flows in the presence of surface-active species in which a fluid interface undergoes very large deformations, e.g., as in the deformation and breakup of drops in extensional flows under conditions of Stokes flow (Stone, H. A., and Leal, L. G.,J. Fluid Mech.220,161 (1990)) (19) and the formation of drops from capillaries (Zhang, X., and Basaran, O.A.,Phys. Fluids7,1184 (1995)) (20) has shown that dynamic surface tension (DST) effects can radically alter the dynamics compared to situations in which the fluid interface is clean. In this paper, we present results of an experimental study that examines the impact with a solid substrate of drops of Newtonian liquids containing two commonly used surfactants. In the experiments, an ultra high-speed video and associated image analysis system is used to monitor the dynamics of the impact process. On account of the extremely large deformations that a drop exhibits and the large-amplitude oscillations that it undergoes upon impacting and spreading on the substrate, DST plays a complex and dominant role in determining the dynamics and the asymptotic state that is approached at large times. A major consequence of the presence of surfactant is that on the one hand its accumulation on the fluid interface reduces the surface tension and thereby enhances the spreading of the drop across the substrate. On the other hand, the non-uniform distribution of surfactant along the fluid interface gives rise to Marangoni stresses that inhibit drop spreading. Given the fact that many liquids used in atomization coating applications ranging from the spraying of agricultural chemicals to painting of substrates contain surfactants and/or other surface-active species, the fundamental results to be reported in this paper have important practical ramifications.

Asymptotic solutions of miscible displacements in geometries of large aspect ratio

Asymptotic solutions are developed for miscible displacements at Stokes flow conditions between parallel plates or in a cylindrical capillary, at large values of the geometric aspect ratio. The single integro-differential equation obtained is solved numerically for different values of the Péclet number and the viscosity ratio. At large values of the latter, the solution consists of a symmetric finger propagating in the middle of the gap or the capillary. Constraints on conventional convection-dispersion-equation approach for studying miscible instabilities in planar Hele–Shaw cells are obtained. The asymptotic formalism is next used to derive—in the limit of zero diffusion— a hyperbolic equation for the cross-sectionally averaged concentration, the solution of which is obtained by analytical means. This solution is valid as long as sharp shock fronts do not form. The results are compared with recent numerical simulations of the full problem and experiments of miscible displacement in a narrow capillary.

Shear flow over a liquid drop adhering to a solid surface

The hydrostatic shape, transient deformation, and asymptotic shape of a small liquid drop with uniform surface tension adhering to a planar wall subject to an overpassing simple shear flow are studied under conditions of Stokes flow. The effects of gravity are considered to be negligible, and the contact line is assumed to have a stationary circular or elliptical shape. In the absence of shear flow, the drop assumes a hydrostatic shape with constant mean curvature. Families of hydrostatic shapes, parameterized by the drop volume and aspect ratio of the contact line, are computed using an iterative finite-difference method. The results illustrate the effect of the shape of the contact line on the distribution of the contact angle around the base, and are discussed with reference to contact-angle hysteresis and stability of stationary shapes. The transient deformation of a drop whose viscosity is equal to that of the ambient fluid, subject to a suddenly applied simple shear flow, is computed for a range of capillary numbers using a boundary-integral method that incorporates global parameterization of the interface and interfacial regriding at large deformations. Critical capillary numbers above which the drop exhibits continued deformation, or the contact angle increases beyond or decreases below the limits tolerated by contact angle hysteresis are established. It is shown that the geometry of the contact line plays an important role in the transient and asymptotic behaviour at long times, quantified in terms of the critical capillary numbers for continued elongation. Drops with elliptical contact lines are likely to dislodge or break off before drops with circular contact lines. The numerical results validate the assumptions of lubrication theory for flat drops, even in cases where the height of the drop is equal to one fifth the radius of the contact line.

Theory of airflow through filters modelled as arrays of parallel fibres

Existing theories of fluid flow through fibrous filters modelled as arrays of parallel fibres are reviewed and a new theory is described, which has the unique property of giving, in a simple analytical form, a continuous stream function through arrays of fibres. The method, which depends on Stokes flow conditions, is based on the variational principle and minimization procedures and is first applied to the simplest array, which is rectangular in structure. The theory is then generalized to arrays of lower symmetry predicting, in accordance with experiment, that the flow pattern is highly dependent on the structure but the pressure drop is only weakly dependent.

A model for fluid flow in dies

A new method is presented for modeling fluid flow inside dies under conditions of Stokes flow. The dies of interest consist of a distribution channel and a narrow slit. Flow in these regions is treated simultaneously and the two-dimensional flow field in the slit is modeled as that in a Hele—Shaw cell. The dependence of flow uniformity on the die geometry is demonstrated explicitly, and the die dimensions and the shape of the distribution channel are correlated with the exit velocity profiles. It is shown that optimum die design can be achieved by allowing for sufficient length of the slit region, and by selecting the proper size and orientation of the distribution channel. Comparisons of the model predictions with the simplified analysis of unidirectional flow in the slit indicate the limited applicability of the latter approach. Experimental data are also presented which validate the accuracy of the proposed model.

The lubrication force between two viscous drops

The hydrodynamic force resisting the relative motion of two unequal drops moving along their line of centers is determined for Stokes flow conditions. The drops are assumed to be in near-contact and to have sufficiently high interfacial tension that they remain spherical. The squeeze flow in the narrow gap between the drops is analyzed using lubrication theory, and the flow within the drops near the axis of symmetry is analyzed using a boundary integral technique. The two flows are coupled through the nonzero tangential stress and velocity at the interface. Depending on the ratio of drop viscosity to that of the continuous phase, and also on the ratio of the distance between the drops to their reduced radius, three possible flow situations arise, corresponding to nearly rigid drops, drops with partially mobile interfaces, and drops with fully mobile interfaces. The results for the resistance functions are in good agreement with an earlier series solution using bispherical coordinates. These results have important implications for droplet collisions and coalescence.

Stokes drag on hollow cylinders and conglomerates

An experimental study of the drag on hollow cylinders and conglomerates falling in a viscous fluid under Stokes flow conditions is described. The experiments were carried out in a tank of square cross section using silicone oil as the Newtonian fluid. Settling velocities of the free falling objects were measured and corrected to conditions of zero Reynolds number flow in an unbounded fluid. The results reveal that all objects tested have Stokes settling velocities smaller than that of a sphere of equal mass and volume. Measurements are reported in terms of the settling speed ratio defined as the ratio of the Stokes settling speed to that of a sphere of equal mass and volume. For the hollow cylinders two parameters are varied: the aspect ratio (length to outside diameter) and the radius ratio (inner to outer radius). Measurements show that the settling speed ratio decreases markedly as the hollowness of the cylinder increases. Each fixed radius ratio data set exhibits a maximum settling speed ratio near an aspect ratio of 1.65. For conglomerates composed of n spheres two trends appear: one for planar configurations and the other for globular clusters. Experimental data for two spheres and three spheres in point contact are in good agreement with recent theoretical results.