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# Tunable viscous layers in Corbino geometry using density junctions

Ramal Afrose  Department of Physics, Faculty of Science, National University
of Singapore, Science Drive 3, Singapore 117542  Aydın Cem Keser  Australian
Research Council Centre of Excellence in Low-Energy Electronics
Technologies,The University of New South Wales, Sydney 2052, Australia  Oleg
P. Sushkov  Australian Research Council Centre of Excellence in Low-Energy
Electronics Technologies,The University of New South Wales, Sydney 2052,
Australia  School of Physics, University of New South Wales, Kensington, NSW
2052, Australia  Shaffique Adam  Department of Physics, Faculty of Science,
National University of Singapore, Science Drive 3, Singapore 117542  Centre
for Advanced 2D Materials, National University of Singapore, 6 Science Drive
2, Singapore 117546  Department of Materials Science and Engineering, National
University of Singapore, 9 Engineering Drive 1, Singapore 117575  Yale-NUS
College, 16 College Avenue West, Singapore 138527

###### Abstract

In sufficiently clean materials where electron-electron interactions are
strong compared to momentum-relaxing scattering processes, electron transport
resembles the flow of a viscous fluid. We study hydrodynamic electron
transport across density interfaces (n-n junctions) in a 2DEG in the Corbino
geometry. From numerical simulations in COMSOL using realistic parameters, we
show that we can produce tunable viscous layers at the density interface by
varying the density ratio of charge carriers. We quantitatively explain this
observation with simple analytic expressions together with boundary conditions
at the interface. We also show signatures of these viscous layers in the
magnetoresistance. Breaking down viscous and ohmic contributions, we find that
when outer radial region of the Corbino has higher charge density compared to
the inner region, the viscous layers at the interface serve to suppress the
magneto-resistance produced by momentum-relaxing scattering. Conversely, the
magneto-resistance is enhanced when the inner region has higher density than
the outer. Our results add to the repertoire of techniques for engineering
viscous electron flows, which hold a promise for applications in future
electronic devices.

##  I Introduction

In lieu of the Drude flow in conventional conductors, electrons flow like a
viscous fluid when collisions among them become the dominant scattering
mechanism [1, 2, 3, 4]. This hydrodynamic regime has long remained elusive in
experiments due to want of fabrication of sufficiently clean materials where
electron-electron interactions are strong compared to momentum-relaxing
scattering. However, the advent of ultra-high mobility 2D electron systems has
bridged this gap. Several effects of viscous electron flow like negative non-
local resistance [5, 6], electron-hole drag [7], vorticity [8, 9], Poiseulle
flow [10, 11], superballistic conductance through point contacts [12, 13, 14],
and violation of Widemann-Franz law [15, 16, 17, 18, 19] are predicted and
have also been observed. With a magnetic field, more unconventional effects
like negative magneto-resistance [20, 21, 22, 23, 24, 25], Hall viscosity [26,
27] and giant anomalous photoresistivity [28, 29, 30, 31, 32] have been seen.

Two main geometries that have been used to study electron transport at the
mesoscopic scale are the Hall bar and the Corbino ring. Unlike the Hall bar,
the Corbino does not have edges except for the source and drain terminals.
This distinctive feature makes it an attractive setup to study bulk states in
the quantum Hall regime, since quantum Hall transport measurements in the more
conventional Hall bar geometry are dominated by contribution from edge
currents. In addition, due to transverse Hall currents in a magnetic field,
the Corbino makes magneto-resistance a feasible probe to study hydrodynamics
[33, 34, 35, 36]. For example, Ref. [33] derives a quadratic-in-field magneto
resistance in a 2DEG Corbino ring (e.g. in GaAs heterojunctions), and shows
that an applied electric field is expelled from the bulk of the sample in
spite of viscous dissipation. However, at low carrier densities, due to non-
vanishing temperature gradients this is no longer true [37, 38]. Ref. [34]
shows how viscosity affects magneto-resistance in charge neutral graphene
Corbino ring assuming no-slip and no-stress leads. Ref. [35] extends the study
to low-density and the high-density limiting cases and show that although the
simple expression of Ref. [33] is valid in the high-doping Fermi liquid
regime, additional contributions appear near neutrality point. Thermoelectric
coefficients calculated in the ballistic limit [39] show signatures of
transition from quantum Hall transport to incoherent transport. Since we are
only concerned with the high charge density transport regime, we use the
formalism of Ref. [33].

[] [] [] []

Figure 1: (a) Schematic of the setup considered in this paper. (b), (c)
Colormap of $\tan\theta_{\text{Hall}}=v(r)/u(r)$ for uniform density
($n_{2}/n_{1}=1$), and a density gradient $n_{2}/n_{1}=2$, obtained from a
numerical simulation of the hydrodynamic equations (1) in COMSOL. The Hall
angle is expected to be approximately constant in the bulk where flow is
Drude-like, and change towards the contacts due to drag forces acting on the
electrons due to viscous layers. A density gradient induces formation of
additional viscous layers at the interface, sensitive to the density gradient.
(d) Radial profile (along a radial section of (c)) of
$\tan\theta_{\text{Hall}}$ for various outer densities $n_{2}$. The interface
induces a change in Hall angle in a region (shaded) approximately wide as the
Gurzhi length $D$ on either side. The jump in Hall angle at the interface also
increases for a higher density gradient.

The situation is very different in the Hall bar geometry, where magneto
resistance is either very weak (for small fields) or is complicated by several
factors such as change in viscosity [20, 26], size of cyclotron orbit compared
to channel width [40, 41], edge currents [42], etc. Moreover, the presence of
edges introduces an unknown boundary condition which effects the flow [43].
Recent efforts have been made to mitigate this problem by making samples with
perfect-slip boundaries. This was done by inducing an electron channel in a
GaAs heterostructure by applying a bias from a top gate instead of chemical
doping. There, viscous effects were artificially re-introduced by modifying
the geometry of the channel [22], or by means of magnetic modulation [44, 45,
46]. The Corbino geometry provides an alternative solution to the problem of
boundary conditions by completely eliminating boundaries except for contacts
at source and drain.

Experiments on the Corbino in graphene have directly probed the existence of
viscous layers localised at the input and output terminals [47]. These layers
are approximately as wide as the ‘Gurzhi length’ $D$, that is, the momentum
diffusion length arising from combination of momentum relaxing and momentum
conserving e-e scattering, and defined as $D=\sqrt{l_{\rm mr}l_{\rm ee}}/2$.
Due to the high mobility of 2DESs with respect to the metallic leads, and due
to rough surfaces of the leads over microscopic length scales, electron flow
into and out of the sample is almost always normal. In this Article, we
explore the possibility of a situation where the injected velocity has a
tunable tangential component, by considering electron flow through two regions
of different density (n-n junction) in the Corbino geometry. Away from the
boundaries, viscous forces cease to operate and the electron transport assumes
Drude character. The Drude flow depends on the charge density of carriers and
is different on either side of the junction. At a no-stress interface of two
densities, the tangential flow must be continuous, forcing the velocity to
reduce to a common value and leading to formation of viscous layers. The
strength of these layers is directly proportional to the mismatch between the
interface velocity and the Drude velocity in the bulk. Therefore, by tuning
the gate bias, we can easily realise viscous layers of varying lengths.

Manipulating viscous electron flow at mesoscopic length scales is an actively
pursued endeavour. Recent works studying hydrodynamic flows through different
channel geometries [48] find that effective channel width can change for
different geometries while the microscopic scattering parameters are
unaffected. Although boundary conditions are never perfectly determined
microscopically, a perfect no-slip boundary can be realised over larger length
scales by considering current flowing through a series of constrictions [49].
Ref. [43] show that slip length at the boundaries can vary with temperature in
a non-trivial way, while other studies [50] demonstrate how non-linear
hydrodynamic effects like Bernoulli effect, Eckart streaming and Rayleigh
streaming can be realised in special scenarios. Our proposal of a gate-tunable
viscous layer could be used for easy electrical manipulation of thermal
dissipation at interfaces and adds to the growing repertoire of methods for
viscous flow engineering.

The plan of the paper is as follows. In Sec. II, we show the emergence of
distinct viscous layers at a density interface from numerical simulation of
compressible Stokes flow in COMSOL. We next present an analytical explanation
of this result in Sec. III using simple expressions derived from the more
complicated exact solution. We show that we can piecewise model the flow and
match them at the interface by using simple interface conditions, which we
derive in Sec IV. Then, in Section V, we show signatures of these viscous
layers in the magneto resistance. We also decompose the ohmic and viscous
contributions with analytic expressions. Finally, we end with a summary of our
findings in Section VI.

Table 1: Value of parameters of 2DEG used for simulation of hydrodynamic flow. Parameter | Description | Value | Parameter | Description | Value
---|---|---|---|---|---
$n_{1}$ | charge carrier density | $2\times 10^{11}\text{cm}^{-2}$ | $m^{*}$ | effective electron mass | $0.067m_{e}=6.1\times 10^{-32}\text{kg}$
$B$ | Applied magnetic field | $20\text{mT}$ | $l_{\text{ee}}$ | e-e scattering length for $n=n_{1}$ | $300\text{nm}$
$\mu$ | mobility | $20\times 10^{6}\text{cm}^{2}/\text{V}\cdot\text{s}$ | $l_{\text{mr}}$ | momentum-relaxing scattering length | $14.8\mu\text{m}$
$D_{1}$ | Gurzhi length for $n=n_{1}$ | $1.05\mu\text{m}$ | 2$d$ | width of interface | $0.2D_{1}=210\text{nm}$
$r_{1}$ | radius of inner contact | $4D_{1}=4.2\mu\text{m}$ | $r_{2}$ | radius of outer contact | $15D_{1}=15.8\mu\text{m}$

##  II Tunable viscous layers at density interface

In this section, we present results from numerical simulation of hydrodynamic
flow across a density gradient and show that viscous layers can be induced at
the interface of different density regions. In the parameter window favouring
hydrodynamic regime of transport, the local conservation laws of particle
number and momentum which govern the electron flow take the form of the
equation of continuity and the Stokes equation [22]

$\displaystyle\nabla\cdot\left(n\mathbf{u}\right)$ $\displaystyle=0$ (1a)
$\displaystyle ne\left(\mathbf{E}+\mathbf{u}\times\mathbf{B}\right)+$
$\displaystyle\nabla\cdot\bm{\sigma}-\frac{m^{*}n}{\tau}\mathbf{u}=0$ (1b)

where $\mathbf{u}$ is the macroscopic velocity of fluid elements,
$\bm{\sigma}=\eta\left[\nabla\otimes\mathbf{u}+\left(\nabla\otimes\mathbf{u}\right)^{T}-\left(\nabla\cdot\mathbf{u}\right)I\right]$
is the shear stress tensor. We have neglected pressure gradients because,
assuming the ‘gradual channel approximation’, they result in a negligible
correction to the capacitance between the 2DEG and the gates, and can be
absorbed into the electric potential. The bulk viscosity of electrons is
considered negligible [51, 22]. Equation (1) is supplemented by boundary
conditions, which are determined by the magnitude and direction of current
flowing in and out of the 2DEG at the inner and outer electrodes.

We also omitted the nonlinear convective derivative (Navier term)
$\mathbf{u}\cdot\nabla\mathbf{u}$ on account of the very low Reynolds number
($<0.1$) in the electron fluid. At temperatures when hydrodynamic effects
become prominent, thermal motion of electrons become sufficiently strong that
effects of Landau quantization are negligible. Furthermore, the Hall viscosity
compares to the shear viscosity by a factor of $2\omega_{c}\tau_{ee}$ [20].
From values in Table 1, this is $\approx 0.16$, which we assume is small and
therefore neglect Hall viscosity.

For the case of homogeneous charge density and assuming azimuthal symmetry,
with the convention $\mathbf{u}=u\hat{\mathbf{r}}+v\hat{\bm{\phi}}$, the exact
solution of (1) is given by [38, 52]

$\displaystyle u(r)=u_{1}r_{1}/r$ (2a) $\displaystyle
v\left(r\right)=c_{1}I_{1}\left(r/D\right)+c_{2}K_{1}\left(r/D\right)-\omega_{c}\tau
u_{1}r_{1}/r$ (2b)

where $\omega_{c}=eB/m^{*}$ is the cyclotron frequency, $D=\sqrt{\nu\tau}$ is
the Gurzhi length, $I_{1},K_{1}$ are first-order modified Bessel functions,
$c_{1},c_{2}$ are constants determined by fitting to boundary conditions, and
$u_{1}=u\left(r_{1}\right)$ is the input radial velocity.

Comparing this with the non-viscous, simple Drude solution

$v^{\mathsf{ohm}}\left(r\right)=-\omega_{c}\tau u_{1}\frac{r_{1}}{r}$ (3)

we can see that viscosity conspires with momentum-relaxing scattering in the
form of the Gurzhi length to affect the tangential velocity $v$. A simple way
to highlight this effect is to calculate the local angle between the flow
velocity and the radial vector, known as the Hall angle:
$\tan\theta_{\text{Hall}}=v(r)/u(r)$. For Drude flow (Eq. (3)), this is
constant, while viscous terms in Eq. (2) cause deviations from it. At the
source and drain, where flow is approximately normal to the surface of the
leads, the Hall angle is zero, therefore, viscous layers develop near these
terminals to accelerate the tangential flow. The appearance of these viscous
layers has already been shown in a Corbino sample with graphene by plotting
the Hall angle profile [47].

In our case, we consider a non-uniform density profile $n(r)$ varying over a
relatively small length scale $d\ll D$ at an interface at $r=r_{\star}$. In a
gated junction in GaAs quantum wells, the density interface is expected to
have a width $d\approx 100~{}\text{nm}-150~{}\text{nm}$. On solving the
electrostatic potential due to the gates, we find the density varies as

$n(r)=\frac{n_{1}+n_{2}}{2}+\frac{n_{2}-n_{1}}{2}\tanh([r-r_{\star}]/d)$ (4)

which interpolates between the two regions $r<r_{\star}$ and $r>r_{\star}$
with densities $n_{1},n_{2}$ respectively.

Given this density profile, we solve the hydrodynamic equations (1) in the
numerical solver COMSOL, supplemented by no-slip conditions at the source and
drain. We assume a density gradient in Eq. (4) that smoothly connects the two
constant density regions over a characteristic length of $d=0.1D\approx
100\>\text{nm}$. Such a density setup can be created by using a dual gate
architecture with top and bottom gates [53]. We assume Fermi-liquid behaviour
of electrons with respect to density, i.e, the e-e scattering rate goes as
$\tau_{ee}^{-1}\propto 1/E_{F}\propto 1/n$. Also, given the high mobility of
2DEG, we assume the momentum-relaxing scattering is limited by phonons and not
by disorder, therefore, $\tau$ is independent of $n$ [22, 54, 55, 56]. The
Gurzhi length then varies as $D\propto n$. At hydrodynamic temperatures
$l_{ee}\sim n$ [22] or even stronger, therefore qualitatively $D\sim n$ holds
for a large range of parameters. We have assumed an electron-electron
scattering length $l_{ee}=300\text{nm}$ at $n_{1}=2\times
10^{11}\text{cm}^{-2}$, which corresponds to a temperature $T=20\text{K}$ in
2DEG in GaAs/AlGaAs [22]. A summary of the values of parameters is given in
Table 1.

In Fig. (1), we plot the Hall angle profile for uniform density and a finite
density gradient $n_{2}=2n_{1}$. We clearly see a change in the Hall angle
near the density interface, indicating the formation of viscous layers. We
also show the radial profile of $\tan\theta_{\text{Hall}}$ for different
density gradients. We find that a larger gradient causes a greater change in
Hall angle, thereby creating stronger viscous layers.

##  III Analytic derivation of tunable viscous layers

The hydrodynamic equations (1) for homogeneous charge density are solved by
Eqs. (2). This form, although known in literature, is very non-intuitive. We
show in appendix A that it can be approximated by the much simpler expression

$\displaystyle v\left(r\right)=$ $\displaystyle\left(\omega_{c}\tau
u_{1}+v_{1}\right)\sqrt{\frac{r_{1}}{r}}e^{-\left(r-r_{1}\right)/D}-\omega_{c}\tau
u_{1}\frac{r_{1}}{r}$ $\displaystyle+\left(\omega_{c}\tau
u_{2}+v_{2}\right)\sqrt{\frac{r_{2}}{r}}e^{-\left(r_{2}-r\right)/D},\qquad
r_{2}>r_{1}\gtrsim D$ (5)

where $r_{1},r_{2}$ are the radii of the inner and outer contacts. This shows
that there are two viscous layers exponentially localized over a length $D$ at
the inner and outer terminals at $r_{1},r_{2}$, and a Drude contribution which
dominates in the bulk $r_{1}+D\lesssim r\lesssim r_{2}-D$. Surprisingly, if
the injected velocity $v_{1}$ matches the non-viscous Drude value
$v^{\mathsf{ohm}}(r_{1})$ in (3), Eq. (5) predicts that the viscous layers
disappear completely. As noted earlier, metal-2DEG interfaces are almost
always no-slip ($v_{1},v_{2}\approx 0$), so this situation is never realized.

For flow through a density gradient, the continuity equation constrains the
radial velocity as $u(r)=n_{1}r_{1}u_{1}/n(r)r$. The bulk ohmic velocity
$v\sim-\omega_{c}\tau u(r)$ is therefore discontinuous at the interface, while
the boundary condition (10) derived in the next section states that $v$ must
be continuous. Therefore, viscous layers must develop at the interface at
$r_{\star}$ to force $v$ to a common value. Moreover, the viscous dissipation
in these layers is proportional to the mismatch between the Drude velocity in
the bulk and the interface velocity. By tuning the density ratio, we can tune
this mismatch and thereby produce viscous layers of varying strengths.

In Fig (2), we plot the velocity profile as a function of the radial distance
$r$ using the fit solution

$\displaystyle v\left(r\right)$ $\displaystyle=\left(\omega_{c}\tau
u_{1}+v_{1}\right)\sqrt{\frac{r_{1}}{r}}e^{-\left(r-r_{1}\right)/D_{1}}-\omega_{c}\tau
u\left(r\right)$
$\displaystyle+v^{\mathsf{int}}\left(r\right)+\left(\omega_{c}\tau
u_{2}+v_{2}\right)\sqrt{\frac{r_{2}}{r}}e^{-\left(r_{2}-r\right)/D_{2}}$ (6)

where the velocity near the interface ($u_{\star}\equiv
u(r_{\star}-d)=u_{1}r_{1}/r_{\star}$)

$\displaystyle
v^{\mathsf{int}}\left(r\right)=\begin{cases}\left(\omega_{c}\tau
u_{\star}+v_{\star}\right)\sqrt{\frac{r_{\star}}{r}}e^{-\left(r_{\star}-r\right)/D_{1}}&r<r_{\star}\\\
\left(\omega_{c}\tau
u_{\star}\frac{n_{1}}{n_{2}}+v_{\star}\right)\sqrt{\frac{r_{\star}}{r}}e^{-\left(r-r_{\star}\right)/D_{2}}&r>r_{\star}\end{cases}$

[] []

Figure 2: (a) Plot of tangential flow velocity as function of radial distance
from eq. (6). Numerical simulation of Eq. (1) using COMSOL with smooth density
gradient as in Eq. (4) with $d=0.1D_{1}~{}\sim 100\>\text{nm}$ shows that the
piecewise analytical solution with interface conditions approximates the flow
very well. Viscous layers approximately wide as the Gurzhi length $D$ act near
the input, output terminals and at the interface to match the bulk Ohmic
velocity to flow at the boundaries. Tuning the density ratio changes the
tangential velocity at the interface, $v_{\star}$, which, in turn modulates
the strength of viscous layers. (b) Velocity at interface as function of
density ratio, obtained using analytic solution (6) and using boundary
conditions (9), (10). $v_{\star}$ can be made to vary by a factor of 3 by
varying $n_{2}/n_{1}$ over experimentally feasible ranges. COMSOL simulation
of the Stokes flow with smooth interface gives identical values of $v_{\star}$
and verifies that our interface conditions are accurate.

We support this result with a numerical simulation of the Stokes flow in
COMSOL. As it can be seen, our approximate solution bears excellent agreement
with the exact result. Moreover, from boundary condition (9), we can equate
the off-diagonal stress tensor $\sigma_{r\phi}$ at $r_{\star}$ to get the
common velocity $v_{\star}$.

$-\frac{v_{\star}}{\omega_{c}\tau
u_{\star}}=\frac{1+2\rho_{\star}+2\rho_{\star}\overline{n}-\overline{n}^{2}}{-3+2\rho_{\star}+2\rho_{\star}\overline{n}^{2}+3\overline{n}^{3}}$
(7)

where $\rho_{\star}=r_{\star}/D_{1}$ and $\overline{n}=n_{2}/n_{1}$. A plot of
$v_{\star}$ vs $\overline{n}$ is also shown. Numerical simulation of the flow
across a smooth density gradient of width $d\ll D$ produces very similar
results to the piece wise analytical solution matched with interface
conditions.

Thus, we see that by changing the density ratio, we can tune the velocity
$v_{\star}$ at the interface, by which we can control the appearance of
viscous layers. This also has implications in the electric resistance, as we
explore in Section V.

##  IV Boundary conditions of flow at interface

We derive the boundary conditions on which our previous results are based. We
consider a sharp interface between two regions of densities $n_{1}$ and
$n_{2}$. By sharp, it is implied that the variation of density is over length
$d$ much smaller than the dimensions of the sample, but larger than the Fermi
wavelength. To derive boundary conditions, we integrate the equations of flow
(1a), (1b) over a patch with faces parallel to the interface. Because of the
$\phi$ symmetry of the system, this is the same as integrating the equations
from $r_{\star}-d$ to $r_{\star}+d$, where $2d$ is the thickness of the
interface.

For the continuity equation,

$\int_{r_{\star}-d}^{r_{\star}+d}dr\;\partial_{r}\left(rnu\right)=0$

The radial velocity is therefore discontinuous, as

$\left(nu\right)\big{|}_{r_{\star}-d}^{r_{\star}+d}=0$ (8)

Integrating the tangential component of (1b),

$\displaystyle-
eB\int_{r_{\star}-d}^{r_{\star}+d}rdr\;nu+\int_{r_{\star}-d}^{r_{\star}+d}dr\frac{\partial_{r}\left(r^{2}\sigma_{r\phi}\right)}{r}$
$\displaystyle-\frac{m^{\star}}{\tau}\int_{r_{\star}-d}^{r_{\star}+d}rdr\;nv$
$\displaystyle=0$

From the continuity equation, $r\;nu=\mathrm{const}$, so, in the limit of
small $d$, the first term gives a vanishing contribution. Similarly, assuming
the tangential flow $v$ does not diverge at the interface, the third term
contributes negligibly in limit of small $d$. Therefore,

$\displaystyle
0\approx\frac{1}{r_{\star}}\int_{r_{\star}-d}^{r_{\star}+d}dr\;\partial_{r}\left(r^{2}\sigma_{r\phi}\right)=\frac{1}{r_{\star}}\left(r^{2}\sigma_{r\phi}\right)\bigg{|}_{r_{\star}-d}^{r_{\star}+d}$

In other words, the off-diagonal stress tensor, $\sigma_{r\phi}$, is
continuous at the interface:

$\sigma_{r\phi}\big{|}_{r_{\star}-d}^{r_{\star}+d}=0$ (9)

Given $\sigma_{r\phi}=r\partial_{r}\left(v/r\right)$, this implies that the
tangential flow velocity $v$ is also continuous.

$v\big{|}_{r_{\star}-d}^{r_{\star}+d}=0$ (10)

Multiplying the radial component of (1b) by $u$ and integrating,

$\displaystyle-e\int_{r_{\star}-d}^{r_{\star}+d}rdr\;nu\partial_{r}\Phi+eB\int_{r_{\star}-d}^{r_{\star}+d}rdr\;nuv$
$\displaystyle+\int_{r_{\star}-d}^{r_{\star}+d}dr\frac{u}{r}\partial_{r}\left(r^{2}\sigma_{rr}\right)-\frac{m^{*}}{\tau}\int_{r_{\star}-d}^{r_{\star}+d}rdr\;nu^{2}$
$\displaystyle=0$

Using the fact that $rnu=I/2\pi e=\text{const}$ across the interface, as
before, we find that the terms proportional to $B,\tau^{-1}$ have a vanishing
contribution in the limit of small $d$. The net condition reduces to

$\displaystyle\Phi\big{|}_{r_{\star}+d}^{r_{\star}-d}\times\frac{I}{2\pi}=\left(ru\sigma_{rr}\right)\bigg{|}_{r_{\star}+d}^{r_{\star}-d}+\int_{r_{\star}-d}^{r_{\star}+d}r\;dr\;\frac{\sigma_{rr}^{2}}{\eta}$
(11)

where integration by parts has been used for the right hand side. Given
$\sigma_{rr}=\eta r\partial_{r}\left(u/r\right)$, we find

$\displaystyle\left(ru\sigma_{rr}\right)\bigg{|}_{r_{\star}+d}^{r_{\star}-d}=\left(\frac{I}{2\pi
e}\right)^{2}\frac{2}{r_{\star}^{2}}\left(\frac{\eta_{2}}{n_{2}^{2}}-\frac{\eta_{1}}{n_{1}^{2}}\right)$
(12)

Together with potential jump at the inner and outer contacts, this covers
viscous dissipation in the homogeneous regions. The remaining term in Eq. (11)
is just the viscous dissipation due to compressive flow at the interface.
Neglecting derivative of $r$ compared to $n$ at the interface,

$\displaystyle\int_{r_{\star}-d}^{r_{\star}+d}r\;dr\;\frac{\sigma_{rr}^{2}}{\eta}$
$\displaystyle=\left(\frac{I}{2\pi e}\right)^{2}\int
dr\;{r^{3}}\eta\left(\frac{d}{dr}\frac{1}{nr^{2}}\right)^{2}$
$\displaystyle\approx\left(\frac{I}{2\pi
e}\right)^{2}\int\frac{\eta}{n^{4}r}\left(\frac{dn}{dr}\right)^{2}dr$

Finally, in the limit $d\to 0$, we can approximate

$\displaystyle\left(\frac{dn}{dr}\right)^{2}=\left(\frac{\Delta
n}{2d}\right)^{2}\text{sech}^{4}\left(\frac{r-r_{\star}}{d}\right)\approx\frac{\left(\Delta
n\right)^{2}}{3d}\;\delta\left(r-r_{\star}\right)$ (13)

Using this in the integral for viscous dissipation, the total potential drop
at the interface is

$\displaystyle-\frac{\Delta\Phi^{\mathsf{int}}}{I}=$
$\displaystyle\frac{\eta_{2}}{\pi\left(n_{2}e\right)^{2}r_{\star}^{2}}-\frac{\eta_{1}}{\pi\left(n_{1}e\right)^{2}r_{\star}^{2}}$
$\displaystyle+\frac{\eta_{\star}}{\pi\left(n_{\star}e\right)^{2}}\frac{1}{6r_{\star}d}\left(\frac{\Delta
n}{n_{\star}}\right)^{2},$ (14)

where $\Delta n=n_{2}-n_{1}$ is the difference in densities,
$n_{\star}=(n_{1}+n_{2})/2$ is the density in the middle of the junction, and
$\eta_{\star}$ is the corresponding shear viscosity.

Eqs. (8), (9), (10) and (IV) are the required boundary conditions of the
problem.

[] [] []

Figure 3: (a) Scaled magnetoresistance
$\left[R(B)-R(0)\right]/(R_{1}^{\textsf{Dru}}\omega_{c}^{2}\tau^{2})$ with
density ratio. The circles are numerical simulation values from COMSOL,
obtained by calculating the potential difference between inner and outer
electrodes. The red solid line is an analytical plot of sum of viscous and
ohmic dissipation (see eq. (16)). For non-viscous flow, scaled magneto
resistance for different mobilities collapse onto a single curve (the Ohmic
limit, eq. (15)). For high-mobility sample, viscous layers at the terminals
and at interface suppress magnetoresistance (red curve). $x$ axis is set to
log scale to give equal emphasis to the cases $n_{2}>n_{1}$ and $n_{2}<n_{1}$.
(b) Resistance due to viscous dissipation, calculated from COMSOL and from
analytic solution (6) (expression in table 2). Because viscosity is
proportional to density, viscous dissipation goes up with $n_{2}/n_{1}$. At
very high $n_{2}$, velocity gradient in the bulk is low, resulting in a
decrease in viscous resistance. The opposite happens for very low $n_{2}$. (c)
Resistance due to Ohmic dissipation. In the bulk, the magnitude of current
density, $|J|\sim J^{\mathsf{in}}(1+\omega_{c}^{2}\tau^{2})$ is approximately
constant, so the ohmic resistance per unit area $R^{\mathsf{ohm}}=J^{2}/\mu
ne$ goes down monotonically with $n_{2}$.

##  V Signature of viscous layers in magnetoresistance

In Fig (3), we plot the magneto-resistance $\Delta
R=R\left(B\right)-R\left(0\right)$ vs. $n_{2}/n_{1}$, calculated numerically
for a high mobility $\mu=2\times 10^{6}\text{cm}^{2}/\text{V}\cdot\text{s}$
and a low mobility $\mu=2000\;\text{cm}^{2}/\text{V}\cdot\text{s}$ Corbino
ring. Assuming a ultra-high mobility sample where momentum-relaxing scattering
is due to phonons only, the mobility should be close to our assumed high
mobility value at $T\approx 20\text{K}$ [22]. It must be noted that in the
Corbino, voltage applied divided by current yields the inverse magneto-
conductance $1/G\left(B\right)$, which is not equal to the magneto-resistance
$R\left(B\right)$ as the conductivity tensor in a magnetic field is non-
diagonal. However, in the following, we use inverse magneto-conductance and
magneto-resistance interchangeably for convenience.

For non-viscous flow, described by the Drude equation, the resistance for the
density junction is simply

$\Delta
R^{\mathsf{ohm}}=\left(\omega_{c}\tau\right)^{2}\frac{R_{1}^{\mathsf{Dru}}}{2\pi}\left[\ln\left(\frac{r_{\star}}{r_{1}}\right)+\frac{n_{1}}{n_{2}}\ln\left(\frac{r_{2}}{r_{\star}}\right)\right]$
(15)

where the Drude resistivity
$R_{1}^{\mathsf{Dru}}=m^{*}/n_{1}e^{2}\tau=15.3\Omega$ for values of
parameters in Table 1. When scaled by
$\left(\omega_{c}\tau\right)^{2}R_{1}^{\mathsf{Dru}}(=244\Omega)$, the magneto
resistance vs $n_{2}/n_{1}$ for samples with different mobility should
collapse onto a single curve. This is clearly reflected in the result for the
low mobility simulation in Fig (3). However, we find that the presence of
viscous layers serve to suppress this magneto-resistance. This is counter
intuitive, given that viscous dissipation increases, but by decelerating the
velocity at the interface, the viscous layers reduce the ohmic dissipation as
well.

To describe this analytically, we start with the power dissipation for
hydrodynamic flow:

$I\Delta\Phi^{\textsf{bulk}}+\oint\mathbf{u}\cdot\bm{\sigma}\cdot
d\mathbf{S}=\frac{1}{\tau}\int
m^{*}n\mathbf{u}^{2}dV+\int\frac{\bm{\sigma}^{2}}{2\eta}dV$ (16)

The first term on the left is the rate of work done by the electric potential
to drive the current, while the second term is the rate of work done against
the boundary stress. The energy provided by these terms are dissipated by
Ohmic and viscous forces, described by the terms on the right hand side. The
importance of boundary stress is apparent if we consider the particular case
of pure radial flow with no ohmic dissipation [33]. It can be shown that in
this case, the potential drop $\Delta\Phi^{\textsf{bulk}}=0$, whereas power
dissipation by viscous forces is finite. The only way the dissipated power can
be compensated is by the boundary term on the left hand side. Here, we expand
this to a broader framework involving the effect of tangential velocity
(caused by magnetic field) and effect of disorder.

When $B=0$, the flow is radial and the viscous and ohmic dissipation are
decoupled in the electric resistance. By solving the Stokes equation, we find

$\displaystyle R\left(0\right)$
$\displaystyle=R^{\mathsf{vis}}\left(0\right)+R^{\mathsf{ohm}}\left(0\right)$
(17) $\displaystyle R^{\mathsf{vis}}\left(0\right)$
$\displaystyle=\frac{\eta_{1}}{\pi\left(n_{1}e\right)^{2}}\frac{1}{r_{1}^{2}}-\frac{\eta_{2}}{\pi\left(n_{2}e\right)^{2}}\frac{1}{r_{2}^{2}}+R^{\textsf{vis,int}}$
$\displaystyle R^{\mathsf{ohm}}\left(0\right)$
$\displaystyle=\frac{R_{1}^{\mathsf{Dru}}}{2\pi}\ln\frac{r_{\star}}{r_{1}}+\frac{R_{2}^{\mathsf{Dru}}}{2\pi}\ln\frac{r_{2}}{r_{\star}}$

The first term in $R^{\textsf{vis}}(0)$ is due to the potential drop at the
inner lead, the second term is from the outer lead and $R^{\textsf{vis,int}}$
is due to the potential difference at the interface, determined by Eq. (IV).
This energy is dissipated by viscous forces in the regions of homogeneous
charge density and due to compressive flow at the interface. For parameters in
Table 1 and $n_{2}=3n_{1}$, we find the boundary resistance at the inner and
outer leads is $0.24\Omega$ while the resistance arising due to potential jump
at the interface is $1.76\Omega$, i.e, the total viscous resistance is
$R^{\textsf{vis}}(0)=2.0\Omega$. On the other hand, the zero-field ohmic
dissipation $R^{\textsf{ohm}}(0)$ is $2.6\Omega$. This conforms well with
numerical values of viscous and ohmic dissipation (in Eq. (16)) from our
COMSOL simulation ($2.3\Omega$ and $2.6\Omega$ respectively).

Table 2: Magneto resistance due to bulk ohmic flow and due to viscous layers, using approximate solution (6). Notation: If $\tilde{v}_{i}=-\frac{v\left(r_{i}\right)}{u\left(r_{i}\right)\omega_{c}\tau}$, $i=1,2$ for the inner and outer contacts, respectively, and $\tilde{v}_{\star\pm}=-\frac{v_{\star}}{u\left(r_{\star}\pm d\right)\omega_{c}\tau}$, then $\alpha_{1}=\left(\tilde{v}_{1}-1\right)\left(1+\frac{3D_{1}}{2r_{1}}\right)$, $\alpha_{2}=\left(\tilde{v}_{2}-1\right)\left(1-\frac{3D_{2}}{2r_{2}}\right)$, $\alpha_{\star\mp}=\left(\tilde{v}_{\star\mp}-1\right)\left(1\pm\frac{3D_{1}}{2r_{\star}}\right)$, $\beta_{i}=\frac{1}{2}\left(\tilde{v}_{i}-1\right)+2\left(\tilde{v}_{i}-1\right)^{2}$, $i=1,2,\star\pm$. $\tilde{v}_{1},\tilde{v}_{2}=0$ in our simulations, $\tilde{v}_{\star}$ is given by Eq. (7). The expression for $I_{1}$ is given in Eq. (B1) in appendix. $\Delta R\left(B\right)$ | Viscous | Ohmic
---|---|---
bulk | $\left(\omega_{c}\tau\right)^{2}\times R^{\mathsf{vis}}\left(0\right)$ | $\left(\omega_{c}\tau\right)^{2}\times R^{\mathsf{ohm}}\left(0\right)$
bdy layer | $\begin{array}[]{c}\frac{\eta_{1}\left(\omega_{c}\tau\right)^{2}}{\pi\left(n_{1}e\right)^{2}}\left[I_{1}+\frac{\alpha_{\star-}^{2}}{4r_{\star}D_{1}}+\frac{2\alpha_{\star-}}{r_{\star}^{2}}\right]\\\ +\frac{\eta_{2}\left(\omega_{c}\tau\right)^{2}}{\pi\left(n_{2}e\right)^{2}}\left[\left(\frac{\alpha_{\star+}^{2}}{4r_{\star}}+\frac{\alpha_{2}^{2}}{4r_{2}}\right)\frac{1}{D_{2}}+\frac{2\alpha_{\star+}}{r_{\star}^{2}}+\frac{2\alpha_{2}}{r_{2}^{2}}\right]\end{array}$ | $\begin{array}[]{c}\frac{\left(\omega_{c}\tau\right)^{2}}{2\pi}\times\bigg{[}R_{1}^{\mathsf{Dru}}D_{1}\left(\frac{\beta_{1}}{r_{1}}+\frac{\beta_{*-}}{r_{*}}\right)\\\ +R_{2}^{\mathsf{Dru}}D_{2}\left(\frac{\beta_{*+}}{r_{*}}+\frac{\beta_{2}}{r_{2}}\right)\bigg{]}\end{array}$

A magnetic field couples the viscous and ohmic dissipation in the electric
potential and writing an analytical expression for resistance becomes
difficult. However, based on our simplified solution (6), we can still derive
approximate analytic expressions, as summarised in Table 2. We find that the
magnetic field contribution is proportional to $(\omega_{c}\tau)^{2}=16$ and
is therefore larger than the zero-field resistance ($7.15\Omega$ for viscous
and $19.2\Omega$ for ohmic dissipation, from COMSOL). From our simplified
expressions, we can quantitatively break up the power dissipation into
spatially localized channels. From inner to outer, they are: (i) boundary
resistance at the inner lead due to viscous stresses acting on radial flow,
(ii) resistance from viscous layer located $\sim D_{1}$ from the inner lead,
arising due to tangential velocity, (iii) potential drop in bulk, primarily
due to ohmic scattering, (iv) resistance from viscous layers near the density
interface, due to tangential velocity, (v) viscous dissipation due to radial
compressive flow at the interface, (vi) bulk resistance due to ohmic
scattering in the outer region, followed by the outer viscous layer and the
boundary resistance at the outer lead.

Our expressions in Table 2 allows us to separately calculate the contributions
to the total resistance from the different viscous layers, something that
would not be possible from either a global measurement or a numerical
simulation which probes the total voltage drop and current across the entire
device. In Fig (4), we plot the dissipation from the expressions in Table 2.
We find that for $n_{2}>n_{1}$, the net contribution from the interface
viscous layers is negative, while for $n_{2}<n_{1}$, it is positive. A
heuristic explanation can be given as follows. Because viscosity is
proportional to density, near $n_{2}/n_{1}=1$, viscous dissipation at
interface layers increases with $n_{2}$. For very high densities, however, the
velocity gradients in these layers is small, resulting in decrease in
dissipation. The converse happens when $n_{2}<n_{1}$. On the other hand, the
Ohmic dissipation, which goes as $n|\mathbf{u}|^{2}$, drops with increasing
density because the velocity of charge carriers decreases. For
$n_{2}/n_{1}<1$, the increase in $|\mathbf{u}|$ due to decrease in $n$
saturates, and the Ohmic dissipation starts to decrease due to decrease in
$n_{2}$. The Ohmic contribution dominates over the viscous one, therefore, the
net contribution to magneto resistance from the viscous layers is negative
when $n_{2}/n_{1}>1$ and positive when $n_{2}/n_{1}<1$.

Figure 4: Viscous, ohmic and total contribution to magnetoresistance due to
viscous layers at interface, obtained from analytic expressions in table 2.
Negative values indicate that the viscous layers decrease magneto resistance
arising from momentum relaxing scattering. Because viscosity is proportional
to density, viscous dissipation increases with $n_{2}$ near $n_{2}/n_{1}=1$.
For high densities, velocity gradients in the viscous layers become small,
therefore, viscous dissipation decreases. The opposite is true for very small
$n_{2}/n_{1}$. Also because the viscous layers decelerate the flow more at the
interface when $n_{2}/n_{1}$ is large, the ohmic dissipation, which is
proportional to $n|\mathbf{u}|^{2}$, decreases. For $n_{2}/n_{1}<1$, the
opposite should be true, however, at very low $n_{2}/n_{1}$, the decrease in
$n$ becomes the overriding factor in ohmic dissipation, and the ohmic
contribution starts to fall.

##  VI Summary

We have studied hydrodynamic electron flow across density junctions in the
Corbino geometry. Starting from the Stokes and continuity equations, we have
derived boundary conditions of flow across the interface. Using these
conditions, we have shown that we can make tunable viscous layers at the
interface by varying the density ratio of the junction. We have also
calculated the viscous and Ohmic dissipations by these viscous layers and we
have found opposing behaviour when electrons flow from regions of lower or
higher density to the other.

Density modulation in Corbino using top gates have been done in experiments
[53]. Also, by using temperature as a tuning parameter, we can smoothly switch
from the ballistic to hydrodynamic regimes. It would be an interesting problem
to address the transport characteristics at this crossover as Ref. [12] did
for point contacts. Study of thermal transport due to viscous layers is
another possible avenue of investigation, especially for sample near charge
neutrality [34, 35]. Additionally, one could thread a magnetic flux through
the Corbino like in the quantum Hall setup and study its effect on transport,
as in Ref. [36]. Therefore, our results presented in this paper open up
possibilities for exploring novel transport phenomena in the Corbino geometry,
and in general in the field of viscous electronic engineering.

###### Acknowledgements.

We are grateful to Giovanni Vignale for valuable discussions and for
collaboration on a closely related project. This work was supported by the
Singapore National Research Foundation Investigator Award (NRF-
NRFI06-2020-0003) and the Australian Research Council Centre of Excellence in
Future Low-Energy Electronics Technologies (CE170100039).

##  Appendix A Derivation of approximate solution of Stokes flow

The exact solution of Stokes equation (1b) is given by (2). Here, we derive
the simplified form (5). Assuming $r_{1},r_{2}\gtrsim D$, we approximate the
Bessel functions appearing in the solution by their asymptotic forms. $K_{1}$
is a decreasing function, hence its contribution towards the outer boundary is
small, i.e,

$v\big{|}_{r\rightarrow r_{2}}\approx
c_{1}I_{1}(r/D)+v^{\mathsf{ohm}}(r)\approx c_{1}\frac{e^{r/D}}{\sqrt{2\pi
r/D}}+v^{\mathsf{ohm}}(r)$

Fitting this to the output flow at $r_{2}$,

$v\big{|}_{r\rightarrow r_{2}}\approx\left(\omega_{c}\tau
u_{2}+v_{2}\right)\sqrt{\frac{r_{2}}{r}}e^{-(r_{2}-r)/D}-\omega_{c}\tau\frac{u_{2}r_{2}}{r}$

We find that viscous correction to the Ohmic flow predominates in a region of
width $\approx D$ from the outer boundary.

Similarly, because $I_{1}$ is an increasing function of $r$, and given that
inner and outer boundaries are largely separated, we expect the relative
contribution of $I_{1}$ towards inner boundary is small.

$v\big{|}_{r\rightarrow r_{1}}\approx
c_{2}K_{1}(r/D)+v^{\mathsf{ohm}}(r)\approx c_{2}\sqrt{\frac{\pi
D}{2r}}e^{-r/D}+v^{\mathsf{ohm}}(r)$

Fitting to input flow at $r_{1}$:

$v\big{|}_{r\rightarrow r_{1}}\approx\left(\omega_{c}\tau
u_{1}+v_{1}\right)\sqrt{\frac{r_{1}}{r}}e^{-(r-r_{1})/D}-\omega_{c}\tau\frac{u_{1}r_{1}}{r}$

Stitching these together in (2), we get the required expression.

Although the above approximations hold for $D\lesssim r_{1}$, a similar
approach can be made for $r_{1}\lesssim D$ ($r_{2}$ large). This situation,
seemingly impractical, is now a possibility with the fabrication of ultra-
clean semiconductor heterojunctions. In recent experiments, mobilities as high
as $50\times 10^{6}\text{cm}^{2}/\text{V}\cdot\text{s}$ have been reached
[57], for which the Gurzhi length is of the order of $10\mu m$. To simplify
the Stokes solution, we assume a hypothetical interface at $r_{\star}=r_{1}+D$
and divide the solution into two regions. The outer region, similar to before,
has the form

$\begin{split}v^{\mathsf{out}}\approx&\left(\omega_{c}\tau
u_{\star}+v_{\star}\right)\sqrt{\frac{r_{\star}}{r}}e^{-(r-r_{\star})/D}-\omega_{c}\tau\frac{u_{1}r_{1}}{r}\\\
&+\left(\omega_{c}\tau
u_{2}+v_{2}\right)\sqrt{\frac{r_{2}}{r}}e^{-(r_{2}-r)/D}\end{split}$ (18)

For the inner region $r_{1}<r<r_{\star}$, we make small-argument expansion of
the Bessel functions in the exact solution (2):

$v^{\mathsf{in}}=A\frac{r}{r_{1}}+A^{\prime}\frac{r_{1}}{r}+A^{\prime}\frac{r_{1}}{2}r\ln\frac{r}{r_{1}}+O(r^{2}/D^{2})$

where coefficients $A,A^{\prime}$ have been suitably defined to impose
boundary conditions. Matching the flow at the input terminal,

$\begin{split}v^{\mathsf{in}}&=\left(v_{1}-A\right)\frac{r_{1}}{r}+A\frac{r}{r_{1}}+\frac{r_{1}}{2D^{2}}\left(v_{1}-A+\omega_{c}\tau
u_{1}\right)r\ln\frac{r}{r_{1}}\end{split}$ (19)

The remaining unknown $A$ can be determined numerically by imposing continuity
of vorticity at $r_{\star}$. The complete solution (18), (19) is plotted in
Fig (5) for the hypothetical case of different injected tangential velocities.
The approximate analytic solution is also compared with the exact solution
(2).

Figure 5: Approximate solution of Stokes flow (circles) for different injected
velocities when $r_{1}\lesssim D$. Solid lines represent the exact solution

##  Appendix B Derivation of magnetoresistance expressions in Table 2

###  B.1 Ohmic dissipation

From the power budget equation (16), the Ohmic power dissipation is

$P^{\mathsf{ohm}}=2\pi m^{*}\tau^{-1}\int rn\left(u^{2}+v^{2}\right)dr$

First, let us consider homogeneous charge density. The radial component is
simply $u=u_{1}r_{1}/r$, hence, the corresponding contribution to
$P^{\mathsf{ohm}}$ is

$P^{\mathsf{ohm},r}=I^{2}\frac{R^{\mathsf{Dru}}}{2\pi}\ln\frac{r_{2}}{r_{1}}$

where the total current $I=2\pi r_{1}neu_{1}$ and
$R^{\mathsf{Dru}}=m^{*}/ne^{2}\tau$.

From approximate solution (5), we can split the tangential velocity into a
bulk ohmic contribution and a boundary contribution as

$v=v^{\mathsf{ohm}}+v^{\mathsf{bdy}}$

where $v^{\mathsf{bdy}}\propto e^{-\left(r-r_{1}\right)/D}$ near $r_{1}$ and
$v^{\mathsf{bdy}}\propto e^{-\left(r_{2}-r\right)/D}$ near $r_{2}$. From this,
the tangential flow contribution to $P^{\mathsf{ohm}}$ is

$\displaystyle P^{\mathsf{ohm},t}=$ $\displaystyle\frac{2\pi
m^{*}n}{\tau}\bigg{\\{}\int r\left(v^{\mathsf{ohm}}\right)^{2}dr$
$\displaystyle+\int
r\left[2v^{\mathsf{ohm}}v^{\mathsf{bdy}}+\left(v^{\mathsf{bdy}}\right)^{2}\right]dr\bigg{\\}}$

The first term, which is an ohmic contribution and comes from the bulk, is:

$\displaystyle P^{\mathsf{ohm},t}\bigg{|}_{\mathsf{bulk}}$
$\displaystyle=I^{2}\left(\omega_{c}\tau\right)^{2}\frac{R^{\mathsf{Dru}}}{2\pi}\ln\frac{r_{2}}{r_{1}}$
$\displaystyle=\left(\omega_{c}\tau\right)^{2}P^{\mathsf{ohm},r}=\left(\omega_{c}\tau\right)^{2}P^{\mathsf{ohm}}\bigg{|}_{B=0}$

The remaining contribution, coming from the viscous boundary layers, is

$\displaystyle P^{\mathsf{ohm},t}$
$\displaystyle\bigg{|}_{\mathsf{inner\;bdy}}\times\frac{\tau}{2\pi m^{*}n}$
$\displaystyle=-2\omega_{c}\tau u_{1}r_{1}\left(v_{1}+\omega_{c}\tau
u_{1}\right)\int_{r_{1}}^{\infty}e^{-\left(r-r_{1}\right)/D}dr$
$\displaystyle+r_{1}\left(v_{1}+\omega_{c}\tau
u_{1}\right)^{2}\int_{r_{1}}^{\infty}e^{-2\left(r-r_{1}\right)/D}dr$
$\displaystyle=\left(\omega_{c}\tau
u_{1}\right)^{2}r_{1}D\left\\{2\left(\tilde{v}_{1}-1\right)+\frac{1}{2}\left(\tilde{v}_{1}-1\right)^{2}\right\\}$

with the notation $\tilde{v}_{1}=-v_{1}/\omega_{c}\tau u_{1}$. An exactly
similar calculation gives

$\displaystyle P^{\mathsf{ohm},t}$
$\displaystyle\bigg{|}_{\mathsf{outer\;bdy}}\times\frac{\tau}{2\pi m^{*}n}$
$\displaystyle=\left(\omega_{c}\tau
u_{2}\right)^{2}r_{2}D\left\\{2\left(\tilde{v}_{2}-1\right)+\frac{1}{2}\left(\tilde{v}_{2}-1\right)^{2}\right\\}$

with $u_{2}=u\left(r_{2}\right)=u_{1}r_{1}/r_{2}$, and
$\tilde{v}_{2}=-v_{2}/\omega_{c}\tau u_{2}$.

The generalization to a system with two different densities is
straightforward: the power dissipated is the sum of the dissipations from each
of the uniform density regions. The result is given in Table 2.

###  B.2 Viscous dissipation

$P^{\mathsf{vis}}=2\pi\int\frac{\bm{\sigma}^{2}}{2\eta}r\;dr$

Under azimuthal symmetry, $\sigma_{rr}=-\sigma_{\phi\phi}=\eta
r\partial_{r}\left(u/r\right)$, while $\sigma_{r\phi}=\sigma_{\phi r}=\eta
r\partial_{r}\left(v/r\right)$. The viscous dissipation due to radial flow

$P^{\mathsf{vis},r}=\frac{2\pi}{\eta}\int\sigma_{rr}^{2}r\;dr=I^{2}\frac{\eta}{\pi\left(ne\right)^{2}}\left(\frac{1}{r_{1}^{2}}-\frac{1}{r_{2}^{2}}\right)$

This is the same as power supplied due to boundary stresses in the clean
limit, when the electric field driving the current disappears [33]. The
magnetic field correction, and also the contribution from disorder, comes from
the azimuthal flow, as

$\displaystyle P^{\mathsf{vis},t}$
$\displaystyle=\frac{2\pi}{\eta}\int\sigma_{r\phi}^{2}r\;dr$
$\displaystyle=2\pi\eta\bigg{\\{}\int_{r_{1}}^{r_{2}}r^{3}\left[\partial_{r}\left(\frac{v^{\textsf{ohm}}}{r}\right)\right]^{2}dr$
$\displaystyle+\int_{r_{1}}^{r_{2}}r^{3}\left[2\partial_{r}\left(\frac{v^{\textsf{ohm}}}{r}\right)\partial_{r}\left(\frac{v^{\mathsf{bdy}}}{r}\right)+\partial_{r}\left(\frac{v^{\textsf{bdy}}}{r}\right)^{2}\right]dr\bigg{\\}}$

Like for ohmic dissipation, the first term is proportional to the zero $B$
field (bulk) resistance

$P^{\mathsf{vis},t}\bigg{|}_{\mathsf{bulk}}=\left(\omega_{c}\tau\right)^{2}P^{\mathsf{vis},r}$

The contribution from the boundary layers comes from the second term. At the
inner boundary layer,

$\displaystyle P^{\mathsf{vis},t}\bigg{|}_{\mathsf{inner\;bdy}}$
$\displaystyle=I^{2}\frac{\eta(\omega_{c}\tau)^{2}}{\pi\left(ne\right)^{2}}\bigg{\\{}\int_{r_{1}}^{\infty}F_{1}\left(r\right)e^{-\left(r-r_{1}\right)/D}dr$
$\displaystyle+\int_{r_{1}}^{\infty}F_{2}\left(r\right)e^{-2\left(r-r_{1}\right)/D}dr\bigg{\\}}$
$\displaystyle F_{1}\left(r\right)$
$\displaystyle=\frac{4}{r_{1}^{1/2}r^{3/2}}\left(\tilde{v}_{1}-1\right)\left(1+\frac{3D}{2r}\right)$
$\displaystyle F_{2}\left(r\right)$
$\displaystyle=\frac{1}{r_{1}}\left(\tilde{v}_{1}-1\right)^{2}\left(1+\frac{3D}{2r}\right)^{2}$

with $\tilde{v}_{1}=-v_{1}/\omega_{c}\tau u_{1}$ as before. If $r_{1}\gg D$,
$F_{1},F_{2}$ are slowly varying with $r$ compared to the exponential, hence,
we may set their values as fixed at $r_{1}$.

$\displaystyle P^{\mathsf{vis},t}\bigg{|}_{\mathsf{inner\;bdy}}$
$\displaystyle=I^{2}\frac{\eta}{\pi\left(ne\right)^{2}}\left\\{\frac{2\alpha_{1}}{r_{1}^{2}}+\frac{\alpha_{1}^{2}}{4r_{1}D}\right\\}$
$\displaystyle\alpha_{1}$
$\displaystyle=\left(\tilde{v}_{1}-1\right)\left(1+\frac{3D}{2r_{1}}\right)$

A very similar expression holds at $r_{2}$:

$\displaystyle P^{\mathsf{vis},t}\bigg{|}_{\mathsf{outer\;bdy}}$
$\displaystyle=I^{2}\frac{\eta(\omega_{c}\tau)^{2}}{\pi\left(ne\right)^{2}}\left\\{\frac{2\alpha_{2}}{r_{2}^{2}}+\frac{\alpha_{2}^{2}}{4r_{2}D}\right\\}$
$\displaystyle\alpha_{2}$
$\displaystyle=\left(\tilde{v}_{2}-1\right)\left(-1+\frac{3D}{2r_{1}}\right)$

The dissipation due to two density regions is the sum of dissipation from each
region; however, for our choice of parameters $r_{1}=4D$, the assumption
$r_{1}/D\gg 1$ is not valid and naively using the above expression gives
inaccurate estimates near $r_{1}$. In this case, we directly integrate the
full expression keeping the generic form of
$F_{1}\left(r\right),F_{2}\left(r\right)$. The result can be expressed in
terms of the exponential integral function:

$\displaystyle
P^{\mathsf{vis},t}\bigg{|}_{\mathsf{inner\;bdy}}=I^{2}\frac{\eta\left(\omega_{c}\tau\right)^{2}}{\pi\left(ne\right)^{2}}I_{1}$
$\displaystyle
I_{1}=\bigg{\\{}\frac{2\left(\tilde{v}_{1}-1\right)}{r_{1}^{2}}+\frac{\left(\tilde{v}_{1}-1\right)^{2}}{4r_{1}D}\left(1+\frac{3D}{2r_{1}}\right)+\lambda\bigg{\\}}$
(20)
$\displaystyle\lambda=\frac{3}{4r_{1}^{2}}\left(\tilde{v}_{1}-1\right)^{2}\left(1+\frac{r_{1}}{D}e^{2r_{1}/D}\mathrm{Ei}\left[-\frac{2r_{1}}{D}\right]\right)$
$\displaystyle\mathrm{Ei}\left[z\right]=-\int_{-z}^{\infty}dt\;e^{-t}/t$

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