2. Definition of the stress tensor associated with odd viscosity

In an incompressible liquid with broken time-reversal symmetry, the Cauchy stress tensor ${\boldsymbol {\sigma }}$ reads (Avron Reference Avron1998)

(2.1)\begin{equation} {\boldsymbol{\sigma}} ={-}p{\boldsymbol{I}}+ {\boldsymbol{\sigma}}^{e} + {\boldsymbol{\sigma}}^{o}. \end{equation}

In two dimensions

(2.2)\begin{equation} \sigma^{e}_{ik} = \eta_e \left( \frac{\partial u_i}{\partial x_k} + \frac{\partial u_k}{\partial x_i} \right) , \quad i=1,2, \ k=1,2, \end{equation}

is the standard (even) deviatoric part of the Cauchy stress tensor with standard (even) viscosity $\eta _e$, while ${\boldsymbol {\sigma }}^{o}$ is the odd part of the Cauchy stress tensor which arises in the presence of time-reversal symmetry breaking due, for instance, to a magnetic field. A general form for the Cauchy stress tensor (Lifshitz & Pitaevskii Reference Lifshitz and Pitaevskii1981, § 13) contains the odd (or anomalous) stress tensor as a special case and depends on the orientation of the magnetic field. We tacitly assume the presence of a magnetic field perpendicular to a flow in the $x$–$y$ plane. In this case, the odd part of the Cauchy stress tensor is characterized by a single odd viscosity coefficient and its components read (Avron Reference Avron1998; Lapa & Hughes Reference Lapa and Hughes2014)

(2.3)\begin{equation} \sigma^{o}_{ik} ={-} \eta_o (\delta_{i1}\delta_{k1} - \delta_{i2}\delta_{k2} ) \left( \frac{\partial u_1}{\partial x_2} +\frac{\partial u_2}{\partial x_1} \right) + \eta_o \left( \delta_{i1}\delta_{k2} + \delta_{i2} \delta_{k1} \right) \left( \frac{\partial u_1}{\partial x_1} - \frac{\partial u_2}{\partial x_2}\right), \end{equation}

for $i=1,2$ and $k=1,2$, where $\eta _o$ is the odd viscosity coefficient and $\delta _{ij}$ (with $i=1,2$ and $j=1,2$) is the Kronecker delta.

It is far more illuminating to collect (2.2) and (2.3) in a single compact formula in terms of the rate-of-strain tensor. Thus, the total stress (2.1) becomes

(2.4)\begin{align} {\sigma}_{ij} ={-}p\delta_{ij}+ {\eta}_{ik} \left(\frac{\partial u_k}{\partial x_j} + \frac{\partial u_j}{\partial x_k}\right), \quad i = 1,2,\ j = 1,2, \quad \text{where} \ {\eta}_{ik} = \left( \begin{array}{@{}cc@{}} \eta_e & -\eta_o \\ \eta_o & \eta_e \end{array} \right). \end{align}

We emphasize that the Cauchy stress tensor ${\boldsymbol {\sigma }}$ remains symmetric in the presence of odd viscosity. The components of the even and odd parts of the stress, in two coordinate systems, are collected in table 1.

Table 1.

Explicit formulas for even and odd stress defined in (2.2) and (2.3), respectively, in Cartesian coordinates with velocity $\boldsymbol {u}=(u,v)$ and in plane polar coordinates with velocity $\boldsymbol {u}=v_r\boldsymbol {\hat r}+v_\theta \boldsymbol {\hat {\theta }}$. The vorticity and continuity equations in the latter system are also included.

Because

(2.5)\begin{equation} \partial_j \sigma_{ij}^{o} = \eta_o \partial_i \omega, \end{equation}

odd viscosity appears in the Navier–Stokes equations as the coefficient of the gradient of vorticity (for an incompressible liquid) which can therefore be incorporated into a modified pressure $\tilde {p} = p + \eta _o \omega.$ With this notation, in the low-Reynolds-number regime we obtain the Stokes equations

(2.6)\begin{equation} -\boldsymbol{\nabla} \tilde{p} + \nabla^{2} \boldsymbol{u} = 0, \quad \boldsymbol{\nabla}{\,\boldsymbol{\cdot}\,}\boldsymbol{u}=0. \end{equation}

It can be shown (Anderson & Davis Reference Anderson and Davis1993) that the streamfunction satisfies the biharmonic equation and the pressure $\tilde {p}$ is determined from the first relation of equation (2.6).

3. Moffatt vortices between a rigid boundary and a free surface

Consider a two-dimensional, slow, viscous flow in the corner region $0 \le \theta \le \alpha$, using plane polar coordinates $(r,\theta )$ with the origin at the corner and a velocity field $\boldsymbol {u}=v_r\boldsymbol {\hat r}+v_\theta \boldsymbol {\hat {\theta }}$. Here $\theta =0$ is a solid stationary boundary, while $\theta =\alpha$ is the liquid–gas interface (see figure 1). The flow is driven by a disturbance at a large distance. The boundary conditions for this flow are

(3.1)\begin{equation} v_r=v_\theta = 0, \quad \text{at} \ \theta=0, \quad \text{and} \quad v_\theta=0, \quad \sigma_{r\theta}=0,\quad \text{on} \ \theta = \alpha, \end{equation}

implied by the no-slip and no-penetration conditions at $\theta =0$, and by the zero normal velocity and zero shear stress at the liquid–gas interface $\theta = \alpha$. Solution of the Stokes equations leads to the following form for the streamfunction:

(3.2)\begin{equation} \psi = r^{n+1} \left[ A\cos (n+1)\theta + B\sin (n+1)\theta + C\cos (n-1)\theta + D \sin (n-1)\theta \right], \end{equation}

where $v_r=({1}/{r})({\partial \psi }/{\partial \theta })$ and $v_\theta =-({\partial \psi }/{\partial r})$. Here $A,B,C$ and $D$ are constants and $n$ is a real or complex number that has a real part $> 0$, so that the flow velocity vanishes at the corner. The exponent $n$ is determined by solving a transcendental equation that appears below (see (3.6)). While formally (3.2) is one term in an infinite series of terms whose index $n$ runs over all possible roots of the transcendental equation (3.6), we are only interested in how hydrodynamic fields behave close to the origin ($r\ll 1$). Thus, the dominant term out of the infinite sum of streamfunctions (3.2) over $n$ is the one with the smallest possible real part of $n$.

Figure 1.

Viscous flow of a liquid bounded by solid wall and a free surface. It is assumed that the flow is driven due to a disturbance at a large distance from the corner.

The pressure $p$ is given by $p = \tilde {p} - \eta _o \omega$, where (Anderson & Davis Reference Anderson and Davis1993)

(3.3)\begin{equation} \tilde{p} = 4\eta_e n r^{n-1}\left[ D\cos (n-1)\theta - C \sin (n-1)\theta \right] \end{equation}

and

(3.4)\begin{equation} \omega ={-} 4 n r^{n-1}\left[ C\cos (n-1)\theta + D \sin (n-1)\theta \right]. \end{equation}

The effect studied in this article is crucially dependent on the zero shear stress boundary condition at the free surface (last of (3.1)) which can be written in the form

(3.5)\begin{equation} \sigma_{r\theta}^{e} ={-} 2\eta_o \partial_r v_r \equiv \tau^{o} \quad \text{at} \ \theta = \alpha, \end{equation}

where $\sigma _{r\theta }^{e}$ is the standard shear stress of a Newtonian liquid (cf. table 1). This means that at the liquid–gas interface there is an extra stress $\tau ^{o}$ which will drive the flow. It can be understood as a normal stress-induced shear stress. This behaviour is not new. The odd-viscosity-induced shear stress has been observed in experiments and simulations to be out of phase with respect to interface undulations, acting at their inflection points and leading to unidirectional wave propagation (Soni et al. Reference Soni, Bililign, Magkiriadou, Sacanna, Bartolo, Shelley and Irvine2019, figure S22). A more comprehensive observation of the same effect was made during the expansion of a bubble in viscous flow (Ganeshan & Abanov Reference Ganeshan and Abanov2017). The odd-viscosity-mediated pressure-induced shear stress gave rise to new fluid flow behaviour, and, in particular, to an azimuthal velocity field transverse to the existing radial one.

This type of problem admits only specific solutions that relate $n$ with the admissible angle $\alpha$. The solution process proceeds as follows. The first two boundary conditions in (3.1) determine two of the arbitrary constants $A$ to $D$ in (3.2). The last two boundary conditions in (3.1) lead to a system of equations for the remaining two arbitrary constants in (3.2). This system has a non-trivial solution only if the determinant of the coefficient matrix vanishes. When odd viscosity is included, the vanishing of the determinant of the coefficient matrix leads to the condition

(3.6)\begin{equation} \left[\left({-}1+\cos \left(2 \alpha \right)\right) n^{2}-\cos \left(2 \alpha n \right)+1\right] \eta_o +\left[-\sin \left(2 \alpha \right) n +\sin \left(2 \alpha n \right)\right] {\eta_e} =0 \end{equation}

that determines the parameter $n$ for all admissible corner angles $\alpha$. In the absence of odd viscosity ($\eta _o\equiv 0$) the model is the same as that studied by Moffatt (Reference Moffatt1964) and Anderson & Davis (Reference Anderson and Davis1993), and relation (3.6) reduces to their conditions (3.20) and (2.27), respectively. In this case, as shown by Moffatt (Reference Moffatt1964), real solutions $n$ exist for angles $\alpha \gtrsim 78^{\circ }$.

For smaller angles, complex roots exist and the streamfunction represents eddies that are known as Moffatt vortices. We are interested in the effects of odd viscosity on these vortices. We therefore proceed by adapting the analysis of Moffatt (Reference Moffatt1964) to the case $\eta _o \ne 0$.

We seek complex solutions of (3.6) in the form $n=p + {\rm i}q$, where $p$ and $q$ are real numbers. Without loss of generality, we consider the roots with $q>0$ below. Setting $\xi = 2\alpha p$, $\zeta = 2\alpha q$, $k= {\sin 2\alpha }/{2\alpha }$ and the viscosity ratio

(3.7)\begin{equation} \eta = \frac{\eta_o}{\eta_e}, \end{equation}

and taking the real and imaginary parts of (3.6), we obtain two relations that may be written in the form

(3.8)\begin{align} \left( \begin{array}{@{}cc@{}} \cosh \zeta & -\eta\cosh \zeta \\ \eta\sinh \zeta & \sinh \zeta \end{array} \right) \left( \begin{array}{@{}c@{}} \sin \xi\\ \cos \xi \end{array} \right) = k \left( \begin{array}{@{}c@{}} \xi\\ \zeta \end{array} \right) + \eta \left( \begin{array}{@{}c@{}} \dfrac{\xi^{2}-\zeta^{2}}{2} \left(\dfrac{\sin\alpha}{\alpha}\right)^{2} -1\\ \xi\zeta\left(\dfrac{\sin\alpha}{\alpha}\right)^{2} \end{array} \right) . \end{align}

In the absence of odd viscosity (setting $\eta _o=0$), (3.8) reduces to the corresponding condition of Moffatt (Reference Moffatt1964, § 3.2). For a given angle $\alpha$ and viscosity ratio $\eta$ these equations are solved to give $\xi (\alpha,\eta )$ and $\zeta (\alpha,\eta )$.

Consider the ratio $\rho$ of distance from the corner between successive vortices and their relative intensity (or damping factor) $\varpi$:

(3.9a,b)\begin{equation} \rho = \frac{r_m}{r_{m+1}} = {\rm e}^{{2\alpha{\rm \pi}}/{\zeta}} \quad\text{and}\quad \varpi = \frac{v_{m+{1}/{2}}}{v_{m+{3}/{2}}} = {\rm e}^{{{\rm \pi}\xi}/{\zeta}}, \end{equation}

as defined and developed by Moffatt (Reference Moffatt1964), but where now $\xi$ and $\zeta$ may depend on odd viscosity. Here $r_m$ ($m=0,1,2,\ldots$) is the distance of the $m$th eddy from the corner and $v_{m+1/2}$ is its maximum velocity. In figures 2 and 3 we plot the logarithm of the strength and the distance of Moffatt vortices by increasing the value of odd viscosity through the viscosity ratio $\eta$.

Figure 2.

The logarithm of the relative strength $\varpi$ between two adjacent Moffatt vortices as a function of the corner angle $\alpha$ in radians (cf. (3.9a,b)). The leftmost (solid) curve corresponds to the results of Moffatt (Reference Moffatt1964, figure 5) where odd viscosity is absent. The dashed and dotted curves depict the logarithm of the vortex strength when the viscosity ratio $\eta = {\eta _o}/{\eta _e}$ assumes the values $0.2$ and $2$, respectively. Two effects of odd viscosity can be seen. First, for a given angle $\alpha$, odd viscosity has decreased the intensity of vortices and thus they can now be observed with greater facility. For example, at $\alpha = {\rm \pi}/4$, $\varpi (\eta =0) \sim 30\,288$ while $\varpi (\eta =2) \sim 442$. Second, it is apparent that odd viscosity has extended the range of corner angles $\alpha$ over which these vortices can be observed.

Figure 3.

The logarithm of the ratio of distances $\rho$ in (3.9a,b) between two adjacent Moffatt vortices as a function of corner angle $\alpha$ in radians. The leftmost (solid) curve corresponds to the results of Moffatt (Reference Moffatt1964, figure 5), where odd viscosity is absent. The dashed and dotted curves depict the logarithm of the distance ratio when the viscosity ratio $\eta = {\eta _o}/{\eta _e}$ assumes the values $0.2$ and $2$, respectively. Odd viscosity has mediated to reduce the size of Moffatt vortices and has extended the range of corner angles $\alpha$ over which these vortices can be observed.

In figure 2 two effects of odd viscosity can be seen. First, for a given corner angle $\alpha$, odd viscosity has decreased the intensity of vortices. For example, for $\alpha = {\rm \pi}/4$ the reduction is by almost two orders of magnitude (from $\varpi (\eta =0) \sim 30\,288$ to $\varpi (\eta =2) \sim 442$) and thus they can now be observed with greater facility. Second, it is apparent that odd viscosity has extended the range over which Moffatt vortices are present. When $\eta =0$ Moffatt vortices can be observed in corners with angle $\alpha$ up to $78^{\circ }$. When $\eta =2$ the angles can be up to $136^{\circ }$.

In figure 3 two effects of odd viscosity can be seen: odd viscosity has mediated to reduce the size of Moffatt vortices and has extended the range of corner angles $\alpha$ over which these vortices can be observed. Figures 4(a) and 4(b) display Moffatt vortices in the absence and presence of odd viscosity, respectively. Odd viscosity has decreased the size of the Moffatt vortices.

Figure 4.

Moffatt vortices in a corner of angle $\alpha = {\rm \pi}/6$. (a) In the absence of odd viscosity ($\eta =0$). (b) Moffatt vortices for viscosity ratio $\eta \equiv \eta _o/\eta _e= 2$. Odd viscosity has squeezed and decreased the size of the vortices.

Why does the intensity of vortices and their size decrease in the presence of odd viscosity? To answer this question we calculate the radial velocity $v_r$ at the liquid–gas interface $\theta = \alpha$:

(3.10)\begin{equation} v_r = \mathrm{Re} \left\{ r^{n} f'(\theta)\right\}, \end{equation}

where $f(\theta )$ is the angle-dependent factor of the streamfunction $\psi$ in (3.2). Letting $n = p + {\rm i} q$ and $f = f_R + {\rm i}f_I$ we calculate that

(3.11)\begin{equation} v_r = {\rm e}^{p\ln r} \left[ f_R' \cos(q\ln r) - f_I' \sin(q\ln r) \right] \sim r^{p}\sin (q \ln r + \epsilon_1(\theta)) \end{equation}

for an angle-dependent phase $\epsilon _1(\theta )$. Evaluating $v_r$ at the liquid–gas interface, $\theta = \alpha$

(3.12)\begin{equation} v_r(r,\alpha) \sim r^{p} \sin ( q \ln r + \epsilon_1 (\alpha)), \end{equation}

we see that when $r$ satisfies

(3.13)\begin{equation} q \ln r + \epsilon_1 ={-}m{\rm \pi}, \quad m = 0,1,2,\ldots \end{equation}

the velocity changes sign an infinite number of times as one approaches the point $r=0$.

As discussed above, the zero shear stress condition $\sigma _{r\theta } = 0$ at $\theta = \alpha$ implies that an extra shear stress $\tau ^{o}$, due to odd viscosity, is active at the liquid–gas interface. It reads

(3.14)\begin{equation} \tau^{o} ={-} 2\eta_o \frac{\partial v_r}{\partial r}. \end{equation}

Thus, differentiating (3.11) with respect to $r$ and evaluating at $\theta =\alpha$ we obtain the following expression for the extra shear stress:

(3.15)\begin{equation} \tau^{o}\sim{-}\eta_o r^{p-1}\sin(q \ln r + \epsilon_1(\alpha) + \epsilon_2(\alpha)). \end{equation}

This means that the extra shear stress $\tau ^{o}$ acts on the liquid–gas interface with the same periodicity as the radial velocity but with an opposite sign and is out of phase by an angle

(3.16)\begin{equation} \epsilon_2 = \tan^{{-}1}\frac{q}{p}. \end{equation}

For a corner of angle $\alpha = {\rm \pi}/6$ and viscosity ratio $\eta = 2$, we calculate the ratio $q/p \sim 0.535$ and thus $\epsilon _2 \sim 0.491$ rad or approximately $28^{\circ }$. In figure 5 we display the main features of the foregoing analysis.

Figure 5.

Odd viscosity gives rise to the ‘extra’ shear stress $\tau ^{o} =- 2\eta _o({\partial v_r}/{\partial r})$ at the liquid–gas interface (cf. (3.14)). Filled circles denote the locations at the liquid–gas interface where the radial velocity $v_r$ is zero, while the arrows on the liquid–gas interface denote the direction of the radial velocity $v_r$. The odd shear stress $\tau ^{o}$ in (3.14) is out of phase with respect to the radial velocity by approximately $28^{\circ }$ (3.16) and has an opposite sign. Thus, the shear stress is predominantly directed opposite to the radial interfacial velocity. This squeezes the vortices and damps their rotation.

Figure 6 displays the normalized interfacial radial velocity and extra stress calculated from the full solution of the boundary-value problem. Both agree well with the trend depicted in figure 5 and discussed in the previous paragraph. Both curves in figure 6 have the same periodicity determined by $q \ln r$ as in expression (3.12). Using the locations of two adjacent zeros of these curves (located at $r \sim 1.38$ and $r\sim 1.22$ for $v_r$ and $\tau ^{o}$, respectively) we can calculate the phase difference $180 q \ln (1.38/1.22)/ {\rm \pi}$, which reproduces the value of $28^{\circ }$ found from (3.16).

Figure 6.

Normalized radial velocity $v_r(r, \alpha )$ (equation (3.12)) and extra shear stress $\tau ^{o}(r, \alpha )$ (equation (3.14)) evaluated at the liquid–gas interface $\theta = \alpha ={\rm \pi} /6$ with $\eta = {\eta _o}/{\eta _e} = 2$ versus radial distance $r$ from the origin. The horizontal axis is displayed in logarithmic scale. The radial velocity and stress follow the trend displayed in figure 5: the odd shear stress $\tau ^{o}$ in (3.14) is out of phase with respect to the radial velocity by approximately $28^{\circ }$ (equation (3.16)) and has an opposite sign.

4. Moffatt vortices in a flat liquid layer

A liquid lying on a horizontal substrate at $y=0$ with a flat free surface at $y=h$ can be considered as a special case of the previous discussion when the corner angle $\alpha \rightarrow 0$. Moffatt vortices form in this case and their wavelength is given by the expression

(4.1)\begin{equation} \lim_{\alpha\rightarrow 0} \frac{2{\rm \pi} h}{\zeta (\eta)} , \end{equation}

where $\zeta$ is the solution of (3.8) when $\alpha \rightarrow 0$. In figure 7 we display the dependence of the dimensionless size of eddies versus the viscosity ratio $\eta = {\eta _o}/{\eta _e}$. At $\eta \sim 1$, odd viscosity has significantly reduced the size of the eddies.

Figure 7.

Vortex size in a flat liquid layer of thickness $h$ on a solid substrate and surrounded by an ambient gas phase (4.1) versus the viscosity ratio $\eta = {\eta _o}/{\eta _e}$. Odd viscosity reduces the size of vortices.

In figure 8 we display the damping factor $\varpi$ as described in (3.9a,b) versus the viscosity ratio $\eta \equiv {\eta _o}/{\eta _e}$. In the absence of odd viscosity one obtains the result of Moffatt (Reference Moffatt1964), where the damping factor is ${\rm e}^{8.53}\sim 5000.$ On the other hand, at a viscosity ratio equal to one, the damping factor is ${\rm e}^{5.8}\sim 330.$ This means that odd viscosity equips us with a greater facility to observe Moffatt vortices in an experiment.

Figure 8.

The logarithm of the relative strength $\varpi$ between two adjacent Moffatt vortices versus the viscosity ratio $\eta = {\eta _o}/{\eta _e}$ (cf. (3.9a,b)) for a flat liquid layer of thickness $h$ on top of a solid substrate, surrounded by an ambient gas phase. The vortex relative strength in the absence of odd viscosity is $\varpi \sim 5000$. It reduces to $\varpi \sim 330$ even for a small viscosity ratio, $\eta \equiv {\eta _o}/{\eta _e}=1$. Thus odd viscosity increases the possibility of observation of Moffatt vortices in active matter.

The plateau reached in figures 7 and 8 can be understood from the bifurcation condition (3.8). As odd viscosity increases, the factors multiplying $\eta$ ($= \eta _o/\eta _e$) in (3.8) become dominant. Explicitly, they read $\cos \xi \cosh \zeta + ({\xi ^{2} -\zeta ^{2}})/{2} =1$ and $\sin \xi \sinh \zeta = \xi \zeta$. These relations determine $\xi$ and $\zeta$ for a large viscosity ratio $\eta = \eta _o/\eta _e$ but are essentially independent of viscosity.

5. The tendency of odd viscosity to moderate singular behaviour

A certain regularization is observed in flow endowed with odd viscosity. For instance, Soni et al. (Reference Soni, Bililign, Magkiriadou, Sacanna, Bartolo, Shelley and Irvine2019) observed experimentally the levelling off of liquid surface undulations due to odd viscosity. Here we show analytically that odd viscosity mediates to moderate unphysical behaviour (cf. figure 9).

Figure 9.

(a) Unphysical liquid flow behaviour when the corner is of angle $\alpha >128^{\circ }$ as depicted in figure 10 by considering the streamfunction (5.3) and radial velocity (5.4) at the liquid–gas interface $\theta =\alpha$ (cf. Moffatt Reference Moffatt1964, figure 2b). (b) Odd viscosity regularizes the flow and restores its physical behaviour at the liquid–gas interface. (a) Unphysical behaviour. (b) Physically realistic.

Figure 10.

Variation of interface velocity $v_r$ at $\theta =\alpha$ (see figure 9) versus angle $\alpha$ of the liquid wedge (at radius $r=1$ and shear strength $\tau =1$). In the absence of odd viscosity (solid curve, $\eta \equiv \eta _o/\eta _e =0$) the flow is singular at $\alpha \sim 128^{\circ }$. Odd viscosity mediates to moderate the singular behaviour by moving it towards the $\alpha = {\rm \pi}$ axis (dashed and dash-dotted curves for $\eta =1$ and $\eta =10$, respectively).

Consider the flow between a wall and a free surface under shear $\tau$, like a shear ‘wind’ (Davis et al. Reference Davis, Gratton and Davis2010). The boundary conditions for this flow are (Moffatt Reference Moffatt1964, figure 2b)

(5.1)\begin{equation} v_r=v_\theta = 0, \quad \text{at} \ \theta=0, \quad \text{and} \quad v_\theta=0, \quad \sigma_{r\theta}=\tau\quad \text{on} \ \theta = \alpha, \end{equation}

implying no slip and no penetration at $\theta =0$ and zero normal velocity and a surface shear wind $\tau$ at the liquid–gas interface $\theta = \alpha$. These conditions are applied on the streamfunction $\psi = \tau r^{2} [ A\cos 2\theta +B \sin 2\theta + C\theta +D ]$. This is identical to the problem considered by Moffatt (Reference Moffatt1964), but here the shear stress condition also accounts for odd viscosity:

(5.2)\begin{equation} \sigma_{r\theta} \equiv \sigma_{r\theta}^{e}+2\eta_o \partial_r v_r = \tau \quad \text{on}\ \theta=\alpha, \end{equation}

where $\sigma _{r\theta }^{e}$ is the standard (even) shear stress (see table 1). As is the case in § 3, here we also have an extra stress $\tau ^{o} =-2\eta _o \partial _r v_r$ which will drive the flow. The same interpretation, that of a normal stress-induced shear stress, can also be applied here.

We can thus determine the streamfunction, which now depends on $\eta _o$:

(5.3)\begin{equation} \psi ={-}\frac{r^{2} \tau \left({-}2 \alpha +2 \theta -\sin \left(2 \alpha -2 \theta \right)-2 \theta \cos \left(2 \alpha \right)+\sin \! \left(2 \alpha \right)+2 \cos \left(2 \theta \right) \alpha -\sin \left(2 \theta \right)\right)}{4 \left(\left(2 \sin \left(2 \alpha \right) \alpha +2 \cos \left(2 \alpha \right)-2\right) \eta_o +\left(2 \cos \left(2 \alpha \right) \alpha -\sin \left(2 \alpha \right)\right) \mathit{\eta_e} \right)}. \end{equation}

To check how realistic this result is, one can consider the velocity of the free interface at $\theta = \alpha$:

(5.4)\begin{equation} v_r(\alpha) = \frac{r \tau \left(\sin \left(2 \alpha \right) \alpha +\cos \left(2 \alpha \right)-1\right)}{\left(2 \sin \left(2 \alpha \right) \alpha +2 \cos \left(2 \alpha \right)-2\right) \eta_o +\left(2 \cos \left(2 \alpha \right) \alpha -\sin \left(2 \alpha \right)\right) {\eta_e}}. \end{equation}

In the absence of odd viscosity the radial flow at the interface has a singularity when $\tan (2\alpha )=2\alpha$, i.e. for the critical angle $\alpha \approx 2.24$ rad (approximately $128^{\circ }$). This is shown by the solid curve in figure 10. Clearly the infinite velocity at this singularity is unrealistic and the assumption of low-Reynolds-number flow also breaks down here. For larger angles the velocity at the free surface acquires a sign that is opposite to the applied shear, which is also unphysical. However, incorporating the odd viscosity contribution moves the singularity towards $\alpha = {\rm \pi}$. As shown in figure 10, when $\eta =1$ the singularity appears at $\alpha \approx 2.73$ rad (approximately $156^{\circ }$) and when $\eta = 10$ the singularity has been pushed to $\alpha \approx 3.08$ rad (approximately $176^{\circ }$).

The reason for the removal of the unphysical behaviour again lies in the odd viscosity. Assume a corner angle $\alpha >128^{\circ }$ (which, in the absence of odd viscosity, leads to the unphysical behaviour depicted in figure 9a). Condition (5.2) can be written as

(5.5)\begin{equation} \sigma_{r\theta}^{e} = \tau - \frac{2\eta_o}{r}v_r(\alpha; \eta_o) \sim \tau - \frac{2\eta_o}{r}v_r(\alpha; 0) + \,\cdots\, , \end{equation}

where the function $v_r(\alpha ;\eta _o)$ was expanded in a Taylor series about $\eta _o=0$. Since by assumption we lie in the unphysical region $\alpha >128^{\circ }$ (when odd viscosity is absent; cf. figure 9a), this implies that $v_r(\alpha ;0)<0$ thus making the last term in (5.5) positive. Thus, at least with respect to the accuracy provided by the leading-order term of the expansion, odd viscosity superposes an extra positive shear stress to the existing shear wind $\tau$. The effect of this process is to reverse the sign of the interfacial velocity, which is depicted in figure 9(b) and by the dashed and dash-dotted curves in figure 10.