Steady flow of one uniformly rotating fluid layer above another immiscible uniformly rotating fluid layer

The steady laminar flow of two immiscible, uniformly rotating fluid layers is studied and exact similarity solutions of the axisymmetric Navier-Stokes equations in cylindrical polar coordinates are found. The similarity solutions occur with a flat interface at $z=0$ under the parameter restriction that ${\sigma }^2\rho =1$, where $\sigma$ is the ratio of the fluid angular velocities at $z=\pm \infty$ and $\rho$ is the density ratio of the two fluids. Under this restriction the problem reduces to one with two independent parameters $\sigma$ and $\mu$, the viscosity ratio of the fluids. Numerical results of the resulting system of ordinary differential equations are found for selected values of $\mu$ and $\sigma$, and it is shown that similarity solutions exist for ${\sigma }_c\left(\mu \right)\le \sigma \le 1$, where ${\sigma }_c\left(\mu \right)<0$ (i.e., counterrotating flows). For $\sigma <0$ the lower fluid can become divided into two distinct recirculation regions between which fluid cannot transfer.

Figure 1

Schematic diagram of the two-fluid rotating flow system considered. This figure depicts a counterrotating case.

Figure 2

Variation of upper interfacial radial stress parameter $f_1^{\prime \prime }\left(0\right)$ with positive values of $\sigma$ at $\mu =\{0.1,0.2,0.5,1.0,2.0,5.0,10.0\}$ with the arrow in the direction of increasing values of $\mu$.

Figure 3

Variation of upper interfacial azimuthal stress parameter $g_1^{\prime }\left(0\right)$ with positive values of $\sigma$ at $\mu =\{0.1,0.2,0.5,1.0,2.0,5.0,10.0\}$ with the arrow in the direction of increasing values of $\mu$.

Figure 4

Upper layer interfacial radial velocity $f_1^{\prime }\left(0\right)$ for positive values of $\sigma$ at $\mu =\{0.1,0.2,0.5,1.0,2.0,5.0,10.0\}$ with the arrow in the direction of increasing values of $\mu$.

Figure 5

Upper layer interfacial azimuthal velocity $g_1\left(0\right)$ for positive values of $\sigma$ at $\mu =\{0.1,0.2,0.5,1.0,2.0,5.0,10.0\}$ with the arrow in the direction of increasing values of $\mu$.

Figure 6

Variation of far-field behavior of the axial velocity parameter in the upper layer $f_1\left(\infty \right)$ with positive values of $\sigma$ at $\mu =\{0.1,0.2,0.5,1.0,2.0,5.0,10.0\}$ with the arrow in the direction of increasing values of $\mu$.

Figure 7

Variation of far-field behavior of axial velocity parameter in the lower layer $f_2(-\infty )$ with positive values of $\sigma$ at $\mu =\{0.1,0.2,0.5,1.0,2.0,5.0,10.0\}$ with the arrow in the direction of increasing values of $\mu$.

Figure 8

Variation of upper interfacial radial stress parameter $f_1^{\prime \prime }\left(0\right)$ with negative values of $\sigma$ at $\mu =\{0.1,0.2,0.5,1.0,2.0,5.0,10.0\}$ with the arrow in the direction of increasing values of $\mu$.

Figure 9

Variation of upper interfacial azimuthal stress parameter $g_1^{\prime }\left(0\right)$ with negative values of $\sigma$ at $\mu =\{0.1,0.2,0.5,1.0,2.0,5.0,10.0\}$ with the arrow in the direction of increasing values of $\mu$.

Figure 10

Upper layer interfacial radial velocity $f_1^{\prime }\left(0\right)$ for negative values of $\sigma$ at $\mu =\{0.1,0.2,0.5,1.0,2.0,5.0,10.0\}$ with the arrow in the direction of increasing values of $\mu$.

Figure 11

Upper layer interfacial azimuthal velocity $g_1\left(0\right)$ for negative values of $\sigma$ at $\mu =\{0.1,0.2,0.5,1.0,2.0,5.0,10.0\}$ with the arrow in the direction of increasing values of $\mu$.

Figure 12

Variation of far-field behavior of the axial velocity parameter in the upper layer $f_1\left(\infty \right)$ with negative values of $\sigma$ at $\mu =\{0.1,0.2,0.5,1.0,2.0,5.0,10.0\}$ with the arrow in the direction of increasing values of $\mu$.

Figure 13

Variation of far-field behavior of axial velocity parameter in the lower layer $f_2(-\infty )$ with negative values of $\sigma$ at $\mu =\{0.1,0.2,0.5,1.0,2.0,5.0,10.0\}$ with the arrow in the direction of increasing values of $\mu$.

Figure 14

Variation of the limiting values ${\sigma }_c$ for negative values of $\sigma$ as a function of $\mu$.

Figure 15

Plot of both upper and lower layer radial velocity profiles $f^{\prime }\left(\eta \right)$ for $\mu =1$ at $\sigma =\{-0.1,0.0,0.2,0.5,0.8\}$ with the $\sigma =-0.1$ profile plotted as the dashed line. The arrow denotes the direction of increasing $\sigma$.

Figure 16

Plot of both upper and lower layer azimuthal velocity profiles $g\left(\eta \right)$ for $\mu =1$ at $\sigma =\{-0.1,0.0,0.2,0.5,0.8\}$ with the $\sigma =-0.1$ profile plotted as the dashed line. The arrow denotes the direction of increasing $\sigma$.

Figure 17

Plot of both upper and lower layer axial velocity profiles $-2f\left(\eta \right)$ for $\mu =1$ at $\sigma =\{-0.1,0.0,0.2,0.5,0.8\}$ with the $\sigma =-0.1$ profile plotted as the dashed line. The arrow denotes the direction of increasing $\sigma$.