Flow induced by the rotation of two circular cylinders in a viscous fluid

The low-Reynolds-number Stokes flow driven by rotation of two parallel cylinders of equal unit radius is investigated by both analytical and numerical techniques. In Part I, the case of counterrotating cylinders is considered. A numerical (finite-element) solution is obtained by enclosing the system in an outer cylinder of radius $R_0\gg 1$, on which the no-slip condition is imposed. A model problem with the same symmetries is first solved exactly, and the limit of validity of the Stokes approximation is determined; this model has some relevance for ciliary propulsion. For the two-cylinder problem, attention is focused on the small-gap situation $\varepsilon \ll 1$. An exact analytic solution is obtained in the contact limit $\varepsilon =0$, and a net force $F_c$ acting on the pair of cylinders in this contact limit is identified; this contributes to the torque that each cylinder experiences about its axis. The far-field torque doublet (“torquelet”) is also identified. Part II treats the case of corotating cylinders, for which again a finite-element numerical solution is obtained for $R_0\gg 1$. The theory of Watson [Mathematika 42, 105 (1995)] is elucidated and shown to agree well with the numerical solution. In contrast to the counterrotating case, inertia effects are negligible throughout the fluid domain, however large, provided Re $\ll 1$. In the concluding section, the main results for both cases are summarized, and the situation when the fluid is unbounded ($R_0=\infty$) is discussed. If the cylinders are free to move (while rotating about their axes), in the counterrotating case they will then translate relative to the fluid at infinity with constant velocity, the drag force exactly compensating the self-induced force due to the counterrotation. In the corotating case, if the cylinders are free to move, then they will rotate as a pair relative to the fluid at infinity and the net torque on the cylinder pair is zero; the flow relative to the fluid at infinity is identified as a “radial quadrupole.” If, however, the cylinder axes are held fixed, then the Stokes flow in the counterrotating case extends only for a distance $r\sim {\mathrm{Re}}^{-1}log\left[{\mathrm{Re}}^{-1}\right]$ from the cylinders, and it is argued that the cylinders then experience a (dimensionless) force ${\widehat{F}}_y\sim 1/log[{\mathrm{Re}}^{-1}log\left[{\mathrm{Re}}^{-1}\right]]$; in the corotating case, the cylinder pair experiences a (dimensionless) torque $\widehat{\mathcal{T}}$, which tends to 17.2587 as $\varepsilon \downarrow 0$; this torque is associated with a vortex-type flow $\sim r^{-1}$ that is established in the far field. Situations that can be described by the condition $\varepsilon <0$ are treated for both counter- and corotating cases in the Supplemental Material.

Figure 1

Streamlines $\psi =$ const. (symmetric with respect to both $x$ axis and $y$ axis) from the numerical solution of the Stokes problem for counterrotating cylinders; the outer boundary condition is no-slip on $r=R_0$ (here $R_0=10$ and $\psi =0$ on $r=R_0$); the color code is the same for all four panels. (a) $\varepsilon =0.2$, full flow domain; (b) $\varepsilon =0.2$, zoom to neighborhood of inner cylinders; the sense of rotation is indicated by the arrows; (c) same zoom for $\varepsilon =0.1$; (d) same zoom for $\varepsilon =0.01$. Note that the two saddle points on the $y$-axis approach the origin as $\varepsilon$ decreases.

Figure 2

Streamlines $\psi (r,\theta )=$ const. for the flow (5); (a) with no slip on $r=R_0(here{R}_0=10)$; (b) with no outer boundary condition ($C=D=0$), and with $A=0$ thus showing the instantaneous (dipole) flow in a frame fixed to the fluid at infinity.

Figure 3

As for Figs. 1 and 1, further zoomed to near the origin; (a) $\varepsilon =0.1$; (b) $\varepsilon =0.01.$

Figure 4

(a) Normalized mass flux $Q$ through the small gap between the cylinders as a function of $\varepsilon$; the dashed asymptote is at $Q/\varepsilon =8/3=2.66\cdots$, as determined by lubrication theory; (b) showing that, as $\varepsilon$ increases to its limiting value (8 for $R_0=10$, when ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ make contact with ${\mathcal{C}}_0$), the normalized mass flux decreases to zero.

Figure 5

(a) Plot of ${\varepsilon }^{3/2}\widehat{p}\left(Y\right) v\mathrm{s} Y=y/\sqrt{\varepsilon }$ as given by Eq. (25); (b) velocity profiles [Eq. (26)] for $\varepsilon ={10}^{-2}$ at two sections, $y=0.7{\varepsilon }^{1/2}$ (blue) and $y=0.9{\varepsilon }^{1/2}$ (red); the boundaries are at $x=\pm h\left(y\right)$ in each case, and diverge as $y$ increases; the viscous wall stress is positive for $y<0.816{\varepsilon }^{1/2}$, negative for $y>0.816{\varepsilon }^{1/2}$.

Figure 6

(a) Streamlines ${\psi }_L(x,y)=$ const. as given by Eq. (27) for the case $\varepsilon =0.01$ [compare with Fig. 3]; (b) vertical velocity on the midplane $v(0,Y)$, where $Y=y/\sqrt{\varepsilon }$, for $\varepsilon =0.2,0.1,0.05,0.025,$ with correspondingly decreasing line thickness; the solution determined by lubrication theory [Eq. (28)] is shown in black, dash-dotted; (c) same as panel (a), but closer to the origin and with $x$ coordinate stretched by a factor 10; (d) the same from the full numerical solution, with the same color code; the difference is almost imperceptible. The critical levels $y=\pm 0.816{\varepsilon }^{1/2}$ and $y=\pm 2.45{\varepsilon }^{1/2}$ are shown by the dashed lines; (e) and (f) the same for the case $\varepsilon =0.001$; (e) correct aspect ratio for this value of $\varepsilon$; (f) zoomed near the origin and stretched (by a factor 15) in the $x$ direction, to show more details of the structure; the critical levels $y=\pm 0.816{\varepsilon }^{1/2}$ and $y=\pm 2.45{\varepsilon }^{1/2}$ are again shown by the dashed lines.

Figure 7

Distribution of the vertical forces on ${\mathcal{C}}_2$ as functions of the angle $\varphi$; the narrowest point of the gap is at $\varphi =\pi$; $R_0=10$; viscous term (blue), pressure term (green), total force (red); (a) $\varepsilon ={10}^{-1}$; (b) $\varepsilon ={10}^{-2}$.

Figure 8

(a) Integrated contributions to the vertical force on ${\mathcal{C}}_2$, viscous (blue), pressure (green) and the total (red), as functions of $\varepsilon$; (b) the same, but with the viscous and pressure contributions rescaled by $\sqrt{\varepsilon }$ (the total force is not rescaled); (c) dependence of total force on $R_0, \varepsilon =0.1$ (solid), $\varepsilon =0.01$ (dashed); no-slip condition on ${\mathcal{C}}_0$ (red), stress-free condition on ${\mathcal{C}}_0$ (black).

Figure 9

(a) Plot of the function $(4\varepsilon -2y^2)/{(y^2+2\varepsilon )}^2$, for $\varepsilon ={10}^{-5}$ (blue), $5\times {10}^{-6}$ (green), and $\varepsilon =0$ (red). As $\varepsilon \downarrow 0$, the spike at $y=0$ becomes longer and narrower, and ultimately disappears in the limit $\varepsilon =0$; (b) contour (shown in blue) used for calculation of the contact force when $\varepsilon =0$; the short segment is at the level $y=y_1$, and the limit $y_1\downarrow 0$ is considered.

Figure 10

(a) Contours $\xi =$ const. (red) and $\eta =$ const. (blue), given by Eq. (35); the contours $\xi =0,\pm 1/2$ are shown in black; (b) corresponding contours given by the conformal mapping $\zeta =log\left[\right(z+c)/(z-c\left)\right]$, as used by Jeffery [2] here $c=1$ and the cylinders of unit radius are shown by the contours $\xi =\pm {sinh}^{-1}c\approx \pm 0.881$.

Figure 11

(a) Streamlines $\psi =$ const., as given by Eq. (38); (b) zoom near the point of contact; (c) contours $\widehat{p}=$ const. in the half-plane $x>0$ as given by Eq. (43).

Figure 12

(a) Streamlines associated with the torquelet ${\psi }_{\mathcal{T}}=-(\partial /\partial x)(log{r}^2+{sin}^2\theta )$; (b) streamlines associated with the vortex dipole ${\psi }_{\mathcal{V}}=-(\partial /\partial x)(log{r}^2)$. The torquelet streamlines appear to be more compressed towards the plane $y=0$.

Figure 13

(a) Contour of integration (shown in red) for calculating the total resultant vertical force on both cylinders when they make contact ($\varepsilon =0$); the integral round the right-hand cylinder ${\mathcal{C}}_2$ runs from $\varphi =-{\varphi }_1$ to $\varphi ={\varphi }_1$, where $0<\pi -{\varphi }_1\ll 1$ (and this is equal, by symmetry, to the contribution from ${\mathcal{C}}_1$); and the integral on the small horizontal segments at $y=\pm y_1$ run from $x=-y_1^2/2$ to $y_1^2/2$, where $y_1=sin{\varphi }_1=sin(\pi -{\varphi }_1)\sim \pi -{\varphi }_1$; we then take the limit ${\varphi }_1\to \pi$ (i.e., $y_1\to 0$); (b) zoom near the origin, and expanded by a factor of 2 in the $x$ direction to make the short sections at $y=\pm y_1$ more clearly visible.

Figure 14

(a) The torque ${\widehat{\mathcal{T}}}_2\left(\alpha \right)$ where $\alpha ={cosh}^{-1}(1+\varepsilon )\sim {\left(2\varepsilon \right)}^{1/2}$, computed from Jeffery's solution (see Appendix pp1, in black), and from the full numerics with $R_0=10$ (in blue), and $R_0=15$ (in red); (b) The function $\alpha {\widehat{\mathcal{T}}}_2\left(\alpha \right)$ with the same color code; the level $2\pi$ is shown by the dashed line, showing that ${\widehat{\mathcal{T}}}_2\left(\alpha \right)\sim 2\pi /\alpha =\pi {(2/\varepsilon )}^{1/2}$ as $\alpha \to 0$.

Figure 15

(a) Streamlines ${\psi }_{J2}(x,y,\alpha )=$ const. as given by Eq. (69), for the choice $\epsilon =1(\alpha =1.317)$; the flow asymptotes rapidly to rigid body rotation $\mathrm{\Omega }\left(\alpha \right)=-0.275$; (b) streamlines in rotating frame, exhibiting the radial flow in the far field with $\mathrm{\Lambda }\left(\alpha \right)=-1.449$; (c) contours ${\psi }_L(x,y)=$ const. in the narrow gap, for corotating cylinders with $\varepsilon =0.001$.

Figure 16

(a) Streamlines ${\displaystyle \psi (r,\theta )=\frac{1}{4}{r}^2-{cos}^2\theta =}$ const. (in the contact limit $\varepsilon =0$); (b) streamlines ${\psi }_Q(r,\theta )=-{cos}^2\theta =$ const; (c) velocity profile for the radial quadrupole, defined in Part II C 2.

Figure 17

(a) Contours ${\psi }_W(x,y)=$ const. as given by Eq. (B5) in Appendix pp2, for the situation when $r_1=1,\varepsilon =0.1$ (so $\alpha =0.4436$); (b) zoom of the same near the origin, showing in red the expected saddle point where ${\psi }_W(0,0)=0.0491$.

Figure 18

(a) Plots of $u_W(0,y)$ for $r_1=1$, and $\varepsilon =0.1$ (solid black), $\varepsilon =0.01$ (solid blue), $\varepsilon =0.001$ (dashed blue); the last two curves are indistinguishable on the scale shown. These graphs show the expected behavior $u_W(0,|y\left|\right)\sim {\left|y\right|}^{-1}$ for large $\left|y\right|$. The red line has slope $1/2$, and correctly represents the flow in the lubrication region $y=O\left({\varepsilon }^{1/2}\right)$; (b) Corresponding plots of $y{u}_W(0,y)$ for positive $y$, showing that in fact $y{u}_W(0,y)\to k\left(\varepsilon \right)$ as $y\to \infty$. (c) The function $k\left(\varepsilon \right)={lim}_{\left|y\right|\to \infty }\left|y{u}_W(0,y)\right|$, which asymptotes to 1.3734 as $\varepsilon \downarrow 0$; this asymptote is shown by the dashed line, which coincides with the level of $\mathcal{T}\left(0\right)/4\pi$ as determined by Eq. (75).

Figure 19

Streamlines $\psi =$ constant from the numerical solution of the Stokes problem for corotating cylinders with $\varepsilon =0.01$; the outer boundary condition on $r=R_0$ is either no slip (a), (b); stress-free (c), (d); or (e), (f) with $u_{\theta }=k\left(0.01\right)/R_0$, from Watson [10] and Fig. 18.

Figure 20

Axial velocity distributions corresponding to the streamline plots of Fig. 19; $\varepsilon =0.01$: (a) $v(x,0)$ ($2+\varepsilon <x<10$); (b) $u(0,y)(-10<y<10)$. The outer boundary condition on $r=R_0$ is either no slip in blue; stress-free in green; or with $u_{\theta }=k\left(0.01\right)/R_0$ in red. The influence of the outer boundary condition is evident: in the no-slip case, the far-field flow is quite close to that of a point vortex ($\sim r^{-1}$ for $r\gg 1$), except near to $r=R_0$ where the no-slip condition is imposed; in the stress-free case, the far field is rigid body rotation ($\sim r$).

Figure 21

Torque on the bounding cylinder ${\mathcal{C}}_0$ (with no-slip boundary condition) as a function of $R_0$ for $\varepsilon =0.04$ (red), 0.02 (green), 0.01 (blue). The double limit $R_0\to \infty$ and $\varepsilon \downarrow 0$ (taken numerically in either order) is needed to approach the asymptotic value 17.2587, marked by the dashed line, as derived in Eq. (75).

Figure 22

(a) The function ${\widehat{\mathcal{T}}}_2\left(\alpha \right)$ [cf. Fig. 14] showing the large-$\alpha$ asymptote (dashed) at the level $4\pi \approx 12.566$; (b) streamlines ${\psi }_J(x,y,\alpha )=$ const. as given by Eq. (A2), for the choice $\varepsilon =1(\alpha =1.317)$; for this choice, the saddle points are at $y=\pm 3$ and the flow asymptotes to the uniform stream ${\displaystyle (-\frac{1}{4},0)}$ as $r\to \infty$; (c) the velocity $v_J(0,y)$ for $\varepsilon =$ 1 (black), 0.1 (blue), 0.2 (red), and 0.001 (green); in the limit $\varepsilon =0$, the asymptotic uniform stream is $(-\frac{1}{2},0)$, as shown by the dashed line.

Figure 23

(a) Log-log plots of the functions $S[N,\alpha ]$ for $N=10,{10}^2,{10}^3,{10}^4$, indicating that $S[N,\alpha ]\sim k{\alpha }^2$ over a range of $\alpha$ that extends towards $\alpha =0$ with increasing $N$; (b) plots of ${\alpha }^{-2}S[N,\alpha ]$ for $N={10}^3,{10}^4$, indicating that $k\approx 0.7281$.

Figure 24

Verification of the impermeability and no-slip conditions (B12) on $\xi =\alpha$; (a), (b) $\varphi (\alpha ,\eta )/\varphi (0,\pi )$ and ${sinh\alpha \partial \varphi /\partial \xi |}_{\xi =\alpha }$; $\alpha =0.4583,N=10,\varphi (0,\pi )=-0.4681$; (c), (d) the same for $\alpha =0.1413,N=20,\varphi (0,\pi )=-0.4972$.