Research

The stability analysis of two-fluid flows through channels and pipes attracted researchers from multi-disciplinary areas of science and technology owing to its wide range of applications like lubricated pipelines, mixing in static mixers, injectability of high concentration drugs through syringe, etc. These systems exhibit instabilities due to variation in density and viscosity of the underlying  fluids. 

Figure1: Schematic Representation of Two-Layer Flow

Figure 2: Schematic of Displacement flow

We explore two main configurations in our study: the Two-Layer configuration and the Displacement Flows scenario, depicted in Figures 1 and 2, respectively. In the Two-Layer setup, two fluids of varying viscosities are layered atop one another. Conversely, in Displacement Flows, one fluid completely occupies the channel, displacing another fluid. These flow phenomena are governed by fundamental equations: the continuity equation, Navier-Stokes equation, and convection-diffusion equation for solute concentration.

When fluid flows through a channel, boundary effects induce shear flow. This leads to differential velocities across the fluid layers, with slower movement near the walls compared to the center. Such shear effects can trigger Kelvin-Helmholtz instability, where different layers exhibit varying velocities. For single-fluid flows, a critical Reynolds number of 5772 marks the onset of instability. The question is how this critical value changes with the presence of two fluids. We investigate how differences in viscosity between the fluids influence the formation of Kelvin-Helmholtz instability. Our objective is to deepen the mathematical understanding of these instabilities.

Direct Numerical Simulations: In our study, we employed Direct Numerical Simulations (DNS) to capture the intricate dynamics at the interface. A significant challenge we encountered was effectively modeling the discontinuities in properties across the interface and the substantial jumps in viscosity stemming from concentration gradients. Furthermore, we faced the challenge of dealing with viscosity as an exponential function of solute concentration, necessitating the coupling of the convection-diffusion equation with the Navier-Stokes equation. To address these challenges, we adopted the diffuse interface model strategy, which builds upon the finite volume method initially introduced in previous studies.

Key Results: Contrary to previous studies that suggested a straightforward monotonic relationship between instability and viscosity ratio between fluids, our findings reveal a non-monotonic dependence. Additionally, we demonstrate that the onset time exhibits a non-monotonic behavior as a function of viscosity difference, R (see Figure). To substantiate our results, we conducted detailed parametric study, showing that maximum instability occurs at an intermediate viscosity ratio.

Linear Stability Analysis: In parallel with DNS, we conducted Linear Stability Analysis (LSA) to corroborate our DNS results. A prevalent simplification in previous LSA involved the Quasi-Steady State Approximation (QSSA), which assumes a fixed base state. However, we observed that the base state concentration undergoes diffusion over time, and early-time changes in base state concentration due to diffusion occur rapidly. This challenges the validity of QSSA, particularly during the initial stages of displacement when the base state changes swiftly. To address these concerns, we performed linear stability without invoking QSSA assumptions. Remarkably, our LSA results closely align with the findings from DNS, underscoring the efficacy of the QSSA method in capturing early time dynamics and instability onset. 

 

movie2.avi

Movie displays the temporal evolution of the instability at the interface for Two-layer Flows

DNS result showing onset time of instability is non-monotonic function of log-viscosity ratio , R

LSA result showing non-monotonic growth rate with viscosity ratio, R

Ongoing Work and Future Plan

In the realm of linear stability analysis for two miscible fluid layers, a commonly employed simplification is the Quasi-Steady State Approximation (QSSA). This approach entails investigating eigenvalues to compute the growth rate of instability. However, QSSA fails to anticipate early-time negative growth resulting from diffusion, thereby unable to determine the instability onset. In the case of two-layer flows, we conduct linear stability analysis by solving the linearized equations as an initial value problem (IVP). This allows us to capture the nuanced dynamics, especially during the critical early stages of instability onset. The application of linear stability analysis in the context of displacement flows remains relatively unexplored. Existing studies predominantly rely on experimental observations or non-linear simulations, overlooking discussions on instability onset. Hence, our objective is to fill this gap by conducting linear stability analysis for displacement flows.

Moreover, we are keen on investigating the effects of chemical reactions between the fluids and variations in fluid viscosity on instability and fluid mixing using linear stability analysis. By delving into these complexities, we aim to enhance our understanding of the underlying mechanisms driving fluid dynamics in these intricate systems.