Boundary-integral method for drop deformation between parallel plates

Abstract

A new boundary-integral method is proposed to study the deformation of drops between two parallel walls. The free-space Green's functions are extended to obey the no-slip condition at the walls. The current formulation is limited to drops with viscosity equal to the matrix fluid, but can be extended to study the effect of nonunit viscosity ratio systems. With this method, the influence of the capillary number and the degree of confinement on drop deformation is investigated. Results for small capillary are compared with small-deformation theory and large capillary results with recent experiments. In both cases, an excellent match is observed. Drops undergoing shear flow deform stronger and align themselves more in the flow direction as the distance between the walls becomes smaller relative to the drop size. Furthermore, the shapes of the drops start to divert significantly from the normal ellipsoidal shapes found, as they show more pointed tips closer to the walls. The transient deformation behavior for more confined systems shows that the drops stretch out to a maximum value, and they slowly retract again to a steady situation. For larger capillary numbers even damped, oscillatory behavior is observed. Investigating the critical capillary number reveals that a minimum is found at a mediocre degree of confinement, after which the critical capillary number increases again to values even larger than the unconfined system. The breakup mode also makes a significant change as it goes from binary to ternary breakup, where the breakup occurs as the drop is retracting.

Two-dimensional vesicle dynamics under shear flow: Effect of confinement

“…Although confining walls were considered in the above-mentioned studies their effect on the dynamics was not studied. We believe that it is of interest to study the impact of the walls on the dynamics of vesicles, a question that, to the best of our knowledge, has not been treated in the literature so far for vesicles, capsules, or red blood cells but only for a droplet [17] and a hard sphere [18].…”

Microconfined equiviscous droplet deformation: Comparison of experimental and numerical results

“…Only a few studies were conducted including the effects of two parallel walls on the behavior of single droplets. [21][22][23][24] For instance, using an integral solution, Shapira and Haber calculated a first-order correction for wall effects that took into account the deviation from sphericity and the drag force acting upon a droplet in confined shear flow. 21 It was shown for equal viscosities that the obtained deformation parameter matched experimental results quite well.…”

“…Recently, for a viscosity ratio of 1, numerical simulations were developed to study droplet behavior in confined flows. [22][23][24] For instance, a boundary integral method ͑BIM͒ was used to study the behavior of a single droplet confined between two parallel walls. 23 The inclusion of wall effects in a boundary integral method was done by modification of the Green's functions.…”

“…[22][23][24] For instance, a boundary integral method ͑BIM͒ was used to study the behavior of a single droplet confined between two parallel walls. 23 The inclusion of wall effects in a boundary integral method was done by modification of the Green's functions. 23,25,26 This way, it is possible to predict the shape, dimensions, and orientation of confined droplets in shear flow for all viscosity ratios.…”

“…23 The inclusion of wall effects in a boundary integral method was done by modification of the Green's functions. 23,25,26 This way, it is possible to predict the shape, dimensions, and orientation of confined droplets in shear flow for all viscosity ratios.…”