Numerical simulations of interfacial instabilities on a rotating miscible droplet in a time‐dependent gap Hele–Shaw cell with significant Coriolis effects

Abstract

SUMMARYInterfacial instability of a rotating miscible droplet with signiÿcant Coriolis force in a Hele-Shaw cell is simulated numerically. The in uences of the relevant control parameters are ÿrst discussed qualitatively by ÿngering patterns. More vigorous ÿngerings are found at higher rotational e ects, a lower viscosity contrast and a weaker e ective surface tension (Korteweg constant). For a time-dependent gap HeleShaw cell, a higher cell lifting rate makes the rotating droplet bear an inward straining ow, which leads to ÿngering enhancement. On the contrary, a higher pressing rate provides more stable e ects by additional squeezing outward ow. A quantitative analysis between the Coriolis e ects and tilting angles of ÿngers is addressed. For arbitrary combinations of all relevant control parameters, the values of tilting angles follow a nearly linear relationship with the Coriolis e ects. We estimate the correlation between the relevant control parameters (dimensionless Coriolis factor Re, viscosity parameter R, cell lifting rate a) and tilting angles (Â) of ÿngers that can be approximated as  = (0:0047 √ Pe=R+18:2a)Re for signiÿcant Korteweg stresses.

Fingering patterns on an expanding miscible drop in a rotating Hele‐Shaw cell

“…An exponentially increasing cell gap width [17,18] b(t) = b 0 eâ t is assumed in the present study, whereâ is a pressing control parameter with a negative value (â 0). These physical problems are governed by the set of following equations [6,8,[17][18][19]:…”

“…Fingering patterns of a rotating drop in a Hele-Shaw configuration have been investigated intensively [1][2][3][4][5][6][7][8][9] since the seminal work by Schwartz [10]. Based on the Hele-Shaw theory, Alvarez-Lacalle et al [2] investigated the interfacial instabilities of immiscible fluids in a rotating cell both experimentally and numerically, in which Coriolis forces are neglected.…”

“…They were confident that their experiments were properly described by the Hele-Shaw equations. On the other hand, in the studies of the miscible flow in a rotating cell, many simulations have been conducted [5][6][7][8][9]. However, without available experimental data for validation, some interesting comparisons between miscible and immiscible flows have been made on the basis of analogy in physical mechanisms of individual parameters and qualitative observations.…”

“…Another Hele-Shaw problem is a time-dependent gap cell, where the upper plate is lifted or pressed uniformly and the plates remain parallel to each other during the process. Chen and Chen [8] studied this miscible rotating cell with a time-dependent gap and reported that the higher pressing rate provides more stable effects by additional squeezing outward flow. However, these studies resulted from a drop without Korteweg stresses.…”

“…Though the coating flows are characterized by being free surface (3-D) flow and the contact angle effects are important, the approximation by lubrication theory yields a similar physical phenomenon to Hele-Shaw equations when the coating layer is extremely thin. During coating process, the layer of coating fluid keeps thinning and spreads outward, which bears similarities to a rotating drop in a pressing Hele-Shaw cell [8].…”