Instability of a liquid sheet with viscosity contrast in inertial microfluidics

Eur. Phys. J. E (2021)44:144 https://doi.org/10.1140/epje/s10189-021-00147-1 THE EUROPEAN PHYSICAL JOURNAL E Regular Article - Flowing Matter Instability of a liquid sheet with viscosity contrast in inertial microfluidics Kuntal Patela and Holger Stark Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany Received 31 July 2021 / Accepted 11 November 2021 © The Author(s) 2021 Abstract Flows at moderate Reynolds numbers in inertial microfluidics enable high throughput and inertial focusing of particles and cells with relevance in biomedical applications. In the present work, we consider a viscosity-stratified three-layer flow in the inertial regime. We investigate the interfacial instability of a liquid sheet surrounded by a density-matched but more viscous fluid in a channel flow. We use linear stability analysis based on the Orr–Sommerfeld equation and direct numerical simulations with the lattice Boltzmann method (LBM) to perform an extensive parameter study. Our aim is to contribute to a controlled droplet production in inertial microfluidics. In the first part, on the linear stability analysis we show that the growth rate of the fastest growing mode ξ ∗ increases with the Reynolds number Re and that its wavelength λ∗ is always smaller than the channel width w for sufficiently small interfacial tension Γ. For thin sheets we find the scaling relation ξ ∗ ∝ mt2.5 s , where m is viscosity ratio and ts the sheet thickness. In contrast, for thicker sheets ξ ∗ decreases with increasing ts or m due to the nearby channel walls. Examining the eigenvalue spectra, we identify Yih modes at the interface. In the second part on the LBM simulations, the thin liquid sheet develops two distinct dynamic states: waves traveling along the interface and breakup into droplets with bullet shape. For smaller flow rates and larger sheet thicknesses, we also observe ligament formation and the sheet eventually evolves irregularly. Our work gives some indication how droplet formation can be controlled with a suitable parameter set {λ, ts , m, Γ, Re}. 1 Introduction Miniaturized flow devices in the form of a lab-on-achip [1] are often employed for processing fluid flows on the micron scale [2]. Lab-on-a-chip microfluidic applications are used in cell biology [3], chemical synthesis [4], and for manipulating multi-component flows [5], to name but a few. Standard microfluidic devices operate in the Stokes flow regime, while only recently inertial microfluidic platforms have emerged [6]. Their flows at moderate Reynolds numbers enable high throughput and inertial focusing [7,8] in order to develop manipulation techniques for biomedical applications. Motivated by this, a plethora of research has been carried out on inertial microfluidics in the last decade [9–13] including our own studies on the manipulation of soft capsules and solid particles using the inertial lift force [14–16]. Recently, instabilities of single-phase flow in different geometries have also been investigated in the inertial regime with the aim to enhance fluid mixing [17,18]. In this article, we use linear stability analysis and lattice Boltzmann simulations to investigate the viscositydriven instability of a multi-component microfluidic flow at finite Reynolds numbers. We let a liquid sheet a e-mail: author) (corresponding 0123456789().: V,-vol stream at the center of a microchannel surrounded by a flowing liquid of larger viscosity and same density and monitor its instability towards modulated interfaces and droplet breakup. Figure 1a shows how the instability develops along the flow direction in a sufficiently long channel. In contrast, in our theoretical investigation we will assume periodic boundary conditions. Such three-layer configurations with two interfaces are commonly encountered in two-phase microfluidic flows [19]. Yih [20] first showed that the fluid–fluid interface in two-layer Couette and Poiseuille flows with viscosity contrast is unstable irrespective of the value of the Reynolds number. Later studies concentrated on interface perturbations with small wavelengths [21,22]. In general, instabilities in viscosity-stratified flows can occur either due to the direct presence of the fluid interface but also due to bounding walls. Boomkamp and Miesen presented an energy budget analysis for the unstable Yih or interface mode, which is triggered by the discontinuity of the shear rate at the interface [23]. Already single-phase flows become unstable at sufficiently large Reynolds numbers due to the presence of bounding walls which cause destabilizing Reynolds stresses. The resulting shear or Tollmien–Schlichting modes also exist in viscosity-stratified flows. Different energy contributions in the energy budget analysis of 123 144 Page 2 of 19 Fig. 1 a Schematic of how an interfacial instability develops along a three-layer flow with viscosity contrast resulting in a steady interfacial wave or the formation of droplets. b Typical design of a channel inlet to generate a three-layer flow. The dashed green line indicates, where the channel walls separating fluid 1 and 2 ends and where the fluid–fluid interface begins Boomkamp and Miesen [23] were quantified for twolayer channel flows by Valluri et al. [24] using linear stability analysis. Various nonlinear mechanisms governing the instability of viscosity-stratified flows were reported by Ó Náraigh et al. [25] using three-dimensional direct numerical simulations. Recently, Kalogirou et al. presented the interface dynamics of a thin viscous film adjacent to a wall in a two-layer channel flow with small viscosity contrast [26]. In addition to planar configurations, also core-annular flows in cylindrical channels have been investigated [27– 31]. A recent linear stability analysis of core annular flows by Sahu [32] showed the existence of an unstable mode different from Yih and Tollmien–Schlichting modes, which Mohammadi and Smits [33] had also reported earlier in their linear stability analysis of twolayer Couette flows. Redapangu et al. [34] considered a two-phase flow in an inclined channel with the fluid– fluid interface of two immiscible fluids normal to the channel walls. In their numerical simulations they then studied how one fluid intrudes the other so that a very irregular three-layer flow arose. For more details on the instability of viscosity-stratified flows, we refer the reader to the comprehensive review article by Govindarajan and Sahu [35]. Viscosity-stratified flows naturally occur in microfluidics when droplets are generated. We review some relevant work. Kurdzinski et al. [36] working in the inertial regime reported different behavior of the central stream in a three-layer configuration of miscible fluids. With increasing Reynolds number they observed a disturbed, a broken, an oscillating, and a stable central stream. In their experiments at low to moderate Reynolds numbers, Hu and Cubaud [37] studied two-layer flows of 123 Eur. Phys. J. E (2021)44:144 misc (...truncated)