Coriolis force-driven instabilities in stratified miscible layers on a rotationally actuated microfluidic platform

Stability analysis of stratified multiphase flow for a spanwise system of rotation plays a pivotal role in micromixing and micromachines. In several such systems, centrifugal actuation is the driving force, which creates a pressure gradient in a rotating channel and Coriolis force enhances mixing in a short span by destabilizing the flow. Here, we focus on the impact of the Coriolis force on a rotating two-fluid flow through a microchannel, which is miscible in nature, having small viscosity difference and thereby forming a thin diffusive interface between fluids due to viscosity stratification. Modal stability analysis is used to estimate the critical flow parameters which are, in turn, responsible for regulating the instability mechanism for different viscosity contrasts and mixed layer thicknesses. Usually, viscosity stratified flow with respect to streamwise disturbance becomes more unstable for a thinner mixed layer. On the contrary, our numerical computation confirms a completely discrepant scenario by considering Coriolis force-driven instability of a miscible flow system on account of spanwise disturbances. Possible physical mechanisms for the same are discussed in terms of base flow pattern and the energy fluctuation between the perturbed and base flow. Comparison of three-dimensional disturbances of the flow field, in both clockwise and anticlockwise directions (for two different viscosity ratios), is executed to provide an insight into the dynamics of the flow system. Distributions of the velocity perturbations display a critical bonding between the vortices near and away from the mixed layer. These vortices are, in turn, responsible for the variation in instability mechanism with respect to different viscosity ratios and rotational directions.

Figure 1

Schematic of the flow system considered. The entire system is rotating about the $z$ axis, which is taken to be perpendicular to the plane of the paper.

Figure 2

Neutral stability boundary for plane channel flow, applying single-fluid limit from the two-fluid miscible viscosity stratified Poiseuille type flow: (a) nonrotating single-fluid limit $(r=1.0,\mathrm{Ro}=0)$ for streamwise disturbances and (b) rotating single-fluid limit with nonzero rotation number $(r=1.0,\mathrm{Ro}=0.5)$ for spanwise disturbances. Solid line presents our computation result and circles are from Alfredsson and Persson [52] result.

Figure 3

Neutral stability boundaries for two different viscosity ratios $\left(r\right)$: in (a) and (b) panels $r=1.2$ and in (c) and (d) panels $r=2.0$. Right and left columns of the figure are for clockwise and anticlockwise rotation, respectively. Critical rotation number $\left(\mathrm{R}{\mathrm{o}}_{\mathrm{cr}}\right)$ for each configuration are $0.35,0.30,0.28$, and $0.20$ (red lines), respectively.

Figure 4

Neutral stability boundaries for both clockwise (dashed lines) and anticlockwise (solid lines) rotation in conjunction with more viscous fluid is near the upper wall: (a) for $r=1.2$ and (b) for $r=2.0$. Black lines are for miscible layer thickness $q=0.3$ and blue curves are for $q=0.1$. [In the inset $I$ results for general class of viscosity stratified flow with streamwise disturbance and without any rotation are given (dashed line for mixed layer thickness $q=0.1$ and solid line for $q=0.3$)].

Figure 5

Typical eigenfunctions for different rotational directions and for two different viscosity ratios $r=0.8$ [in (a)(ACW rotation), (b) (CW rotation)] and $r=1.2$ [in (c) (ACW rotation), (d) (CW rotation)]. Left pan and right pan of the sub-figures represent normal velocity and vorticity, respectively. Solid black line for real part and blue line for an absolute value of eigenfunction; dashed line is used for the imaginary part. Critical values of other parameters are used in this figure are given in Table 1.

Figure 6

Contours of maximum growth rate for (a) anticlockwise and (b) clockwise rotation with three different rotation numbers $Ro=0.1,0.2,0.3$. Other parameters are $\alpha =0.0,\mathrm{Re}=995,p=0,q=0.3$.

Figure 7

Eigenspectrum for $\mathrm{Re}=995.0,\alpha =0.15,\beta =1.0,\mathrm{Ro}=0.20$, with mixed layer thickness $q=0.3$ and $p=0.0$: in (a) $r=0.8$ and in (b) $r=1.2$. Different symbols are used to distinguish different modes based on their layer of sourcing. Modes, which are coming due to the presence of mixed layer are denoted by $*$ symbol and modes from upper/lower layers are denoted by the symbol $o/\mathrm{\Delta }.$ Black and red colours suggest anticlockwise and clockwise rotation, respectively.

Figure 8

Marginal stability boundaries for two different viscosity ratios (a) $r=1.2$ and (b) $r=2.0$ for two different wave numbers of streamwise disturbances $(\alpha =0.0,1.0)$. Solid lines are for ACW and dashed lines are for CW rotations. Here, $\alpha =0.0$ (black lines) corresponds to the two-dimensional disturbance. The other parameters are $\mathrm{Ro}=0.20$, $q=0.3$, and $p=0.0$.

Figure 9

Iso-contours of velocity field for three-dimensional disturbances with viscosity ratio (a) $r=1.2$ and (b) $r=0.8$ $(\mathrm{ACW}\mathrm{case}:\mathrm{Ro}=0.2,\mathrm{Re}=995,\alpha =0.15,\beta =1.0)$.The thickness of the miscible layer $q=0.3$ and miscible layer starts at $p=0.0$.

Figure 10

Isocontours of velocity field for three-dimensional disturbances with viscosity ratio (a) $r=1.2$ and (b) $r=0.8$ $(\mathrm{CW}\mathrm{case}:\mathrm{Ro}=0.2,\mathrm{Re}=995,\alpha =0.15,\beta =1.0)$. The thickness of the miscible layer $q=0.3$ and miscible layer starts at $p=0.0$.

Figure 11

Contours of maximum growth rate for (a) ACW and (b) CW rotation for three different rotation numbers $\mathrm{Ro}=0.1,0.2,0.3$. Other parameters are $\mathrm{Re}=995.0$, $p=0.0$, and $q=0.3$.

Figure 12

(a) Base velocity profile and (b) gradient of the base flow profile for viscosity ratio $r=1.2,2.0$, and mixed layer starting position $p=0.0$ with thickness $q=0.1$ and $0.3$.

Figure 13

Variation of the energy transfer term for viscosity ratio (a) $r=1.2$ and for (b) $2.0$. The other parameters are the critical values as mentioned in Table 1. The Reynolds number and rotation number considered for the flow is $\mathrm{Re}=995.0$ and $\mathrm{Ro}=0.2$, respectively.

Figure 14

Neutral stability boundaries for two different viscosity ratios $\left(r\right)$: in (a) and (b) panels $r=1.2$ and in (c) and (d) panels $r=2.0$. Right and left columns of the figure are for clockwise and anticlockwise rotation, respectively. Black line represents the results using average velocity and red line represents with maximum velocity.

Figure 15

The effect of layer thickness on neutral stability boundary for viscosity ratio (a) $1.2$ and (b) $2.0$. The other parameters are $\mathrm{Ro}=0.20$ (ACW), $q=0.3$, and $p=0.0$. Here, $\mathrm{Re}$ and $\mathrm{Ro}$ are defined using average velocity over the channel.

Figure 16

Comparison with unstratified flow for viscosity ratio $\left(r\right)$ (a) $1.2$ and (b) $2.0$. The blue and green line denotes unstratified flow for streamwise wave number $\alpha =0.0,1.0$. Dashed line represents CW rotation and solid line represents ACW rotation. Black and red line corresponds to viscosity stratified flow with streamwise wave number $\alpha =0.0,1.0$.