Beyond Ohm's law -- Bernoulli effect and streaming in electron hydrodynamics
BBeyond Ohm’s law - Bernoulli effect and streaming in electron hydrodynamics
Aaron Hui, Vadim Oganesyan,
2, 3 and Eun-Ah Kim School of Applied & Engineering Physics, Cornell University, Ithaca, New York 14853, USA Department of Physics and Astronomy, College of Staten Island, CUNY, Staten Island, NY 10314, USA Physics Program and Initiative for the Theoretical Sciences,The Graduate Center, CUNY, New York, NY 10016, USA Department of Physics, Cornell University, Ithaca, New York 14853, USA (Dated: October 2, 2020)Recent observations of non-local transport in ultraclean 2D materials raised the tantalizing possi-bility of accessing hydrodynamic correlated transport of many-electron state. However, it has beenpointed out that non-local transport can also arise from impurity scattering rather than interaction.At the crux of the ambiguity is the focus on linear effects, i.e. Ohm’s law, which cannot easilydifferentiate among different modes of transport. Here we propose experiments that can reveal richhydrodynamic features in the system by tapping into the non-linearity of the Navier-Stokes equation.Three experiments we propose will each manifest unique phenomenon well-known in classical fluids:the Bernoulli effect, Eckart streaming, and Rayleigh streaming. Analysis of known parameters con-firms that the proposed experiments are feasible and the hydrodynamic signatures are within reachof graphene-based devices. Experimental realization of any one of the three phenomena will pro-vide a stepping stone to formulating and exploring the notions of nonlinear electron fluid dynamicswith an eye to celebrated examples from classical non-laminar flows, e.g. pattern formation andturbulence.
I. INTRODUCTION
Electron hydrodynamics offers a powerful frameworkto understand transport in strongly correlated elec-tron systems.
The pursuit of electron hydrodynam-ics gained new impetus with the advent of recent ex-periments in a number of ultraclean 2D materials making a case for electron hydrodynamics through ob-servations of non-local transport, consistent with viscousflows familiar in classical fluids. The observations such asvortices, Poiseuille-like flow profiles, and unconventionalchannel width dependencies of resistance are indeed con-sistent with viscous effects in a linearized Navier-Stokesequation. However, these results are all in the linearresponse regime and can be ultimately described usinga non-local variant of Ohm’s law. Indeed, the linearizedNavier-Stokes equation can be simply recast using a non-local conductivity σ ( q ) . While non-local transportcan certainly be couched in the formalism of hydrody-namics, it is also clear that inherently finite length scalesof a realistic fermionic system can conspire to producenon-local transport indistinguishable from that impliedby the Navier-Stokes equation . Other ways of access-ing electron hydrodynamics are of great interest as weseek to understand and isolate competing effects.The overarching goal of this paper is to highlight theexistence of nonlinear electron phenomena that may beassociated with an effective hydrodynamic description.With that in mind, we directly adapt the Navier-Stokes(NS) equations of classical fluid dynamics to make thediscussion of the electron phenomenology explicit. Wedo not tackle the important and difficult question of aproper microscopic derivation of NS – indeed, there isevidence that many available electron devices are notquite in the asymptotic hydrodynamic regime . We do, however, find strong evidence in known material anddevice parameters to support feasibility of our propos-als. It is worth emphasizing that while the phenomenawe focus on in this work are leading deviations from lin-ear response, the NS results we obtain also suggest thepresence of instabilities at finite non-linearity. As in tra-ditional classical hydrodynamics, these different regimesare naturally demarcated using dimensionless Reynoldsnumbers.In Fig. I, we summarize the three proposals that wediscuss in this paper. The rest of the paper is organizedas follows. Section II sets up the notation and formal-ism of NS, paying particular attention to the spectrumof Reynolds numbers required to quantify nonlinear phe-nomena. Here, we also collect Reynolds number esti-mates from known parameters for graphene. Section IIIfocuses on the manifestation of the Bernoulli effect in thenonlinear current-voltage response of an electron funnel.Section IV derives the generation of downconverted DCcurrent from a localized finite-frequency excitation, anal-ogous to Eckart streaming or ”quartz wind”. Section Vdescribes the generation of static electron vortices (akinto Rayleigh streaming) from an extended AC excitation.Sections II-V are accompanied by Appendices A-D con-taining complete details of calculations. Finally, we closewith a summary of results and a discussion of open prob-lems, including the role of interactions. II. FORMALISM AND PARAMETERSA. Equations of fluid dynamics
The hydrodynamics of an electron fluid, as a long-wavelength effective theory, is described by a set of con- a r X i v : . [ c ond - m a t . s t r- e l ] S e p (a) (b) (c) FIG. 1. Proposed experimental setups and sketches of their observed effects. a) The Venturi geometry, comprised of a circularwedge of the hydrodynamic material in yellow. A nonlinear I-V characteristic with I ∼ √ V behavior is expected, markedin blue. The gray dashed line represents an unstable solution branch, while the gray region represents a possible instabilitytowards turbulent and/or intermittent flow. b) Eckart streaming. A voltage oscillation of zero mean is driven on one side of aback-gated device, leading to a rectified DC current I . For large l , the DC current scales as l − . For small l , oscillations due tointerference with the reflected wave become visible. c) Rayleigh streaming. In a similar back-gated geometry of (b), a standingwave of current oscillations of amplitude u and of period λ along x is imposed, leading to an oscillating magnetic field patternof period λ/ λ/ h/ servation laws for variables which decay slowly comparedto the coarse-graining scale of the system. The momen-tum (Navier-Stokes) and density continuity equations,which will be our primary interest in this paper, are ∂n∂t + ∇· ( n v ) = 0 (1) ∂ ( ρ v ) ∂t = F conv − ∇ p − ρ e ∇ φ + (cid:20) D ν + ˜ ζ (cid:21) ρ ∇∇ · v − ρν ∇ × ∇ × v − ργ v (2) F conv ≡ − ∇ · ( ρ v ⊗ v ) = ρ v · ∇ v + v ∇ · ( ρ v ) (3)where v is the velocity field, n is the number density fieldwith particles of mass m and charge e ( ρ and ρ e are themass and charge densities, respectively). The convectiveterm F conv is written to emphasize that it acts as aneffective force; this will be the primary source of nonlinearbehavior. The remaining terms may also be thought ofas (generalized) forces, and we can take their ratios fora particular flow pattern to characterize their relativeimportance. In addition to the conventional “viscous”Reynolds number Re ν corresponding to shear dissipation,a momentum-relaxation Reynolds number Re γ will be ofinterest. For simple non-singular flow profiles these may be expressedRe ν ≡ ∇ · ( ρ v ⊗ v ) ρν ∇ v = vLν = ILρ e hν (4)Re γ ≡ ∇ · ( ρ v ⊗ v ) ργ v = vLγ = Iρ e hLγ (5)with help of characteristic velocity v , gradient 1 /L , chan-nel width h and net current I = ρ e hv . In this paper,we primarily focus on the limit of low Reynolds num-bers Re γ , Re ν (cid:28)
1, i.e. leading corrections to linearresponse .Following standard practice, we make a further as-sumption of local equilibrium to write equations of statefor p and φ which closes the set of continuity equationsabove. We take a back-gated geometry as shown inFig. 1b, where the hydrodynamic metal and the back-gate separated by a distance d have a capacitance perunit area C = (cid:15)(cid:15) d . Therefore, we take the following localrelationships p = s ρ (6) φ = ρ e /C (7)where s FL is a constant corresponding to the speed ofsound in an uncharged, undamped fluid (i.e. a Fermiliquid). In Eq. 7, also called the “gradual channel ap-proximation,” the long-range Coulomb tail is screenedby the gate so that the longitudinal dispersion is gapless.This approximation is valid when the distance d betweenthe hydrodynamic metal and the gate is much smallerthan the typical wavelength of oscillations. There-fore, both p and φ obey the same functional form; if thedensity ρ = ρ (0) is constant, p can be absorbed into aneffective voltage φ eff ≡ φ + pρ (0) e in the momentum equa-tion. In particular, as a result of Eq. (7) there is alsoa electronic contribution s = n (0) e Cm to the undampedspeed of sound s ≡ (cid:113) s + s . B. Parameter Estimates
To estimate parameters, we consider a graphene-hBNstack with gate-channel separation d = 100 nm and av-erage carrier density n (0) ∼ cm − . In graphene,the relaxation rate γ ∼
650 GHz and ν ∼ . / s, so that the viscous length scale r d = (cid:113) νγ ∼ . µ m. The relative dielectric constant of hBN is (cid:15) ∼ . and we approximate m and e to be the bare electronmass and charge, respectively. Therefore, the electroniccontribution to sound is s cap ∼ . × m/s. Thespeed of sound of Fermi liquids is s FL ∼ v F , andFermi velocities for metals are generally v F ∼ m/s. Therefore, we will approximate the undamped speed ofsound s ∼ × m/s. Using the dispersion relationin Eq. A1, for ω = 1 THz we have the true speed ofsound s ∼ . × m/s and attenuation coefficient α ∼ / (6 µ m). As a rough estimate, for characteris-tic lengths h ∼ L ∼ µ m the Reynolds numbers areRe ν ∼ I/ (160 µ A) and Re γ ∼ I/ (26mA). The ratioRe ν / Re γ ∼ L /r d is controlled by the viscous lengthscale r d ∼ . µ m, so current micrometer-scale experimentswill be in a regime where Re γ tends to dominate the non-linear behavior. We remark that the apparent paradoxthat hydrodynamic effects could be dominated by mo-mentum relaxation is due to linear-response considera-tions; by tuning the sample width h such that r d (cid:28) h , ahydrodynamic description of the material remains validbut becomes indistinguishable from Ohm’s law in the ab-sence of convection. III. ELECTRONIC BERNOULLI EFFECT
We now apply the hydrodynamic formalism to derivea nonlinear contribution to the I-V characteristic V ∝ I in what we call the ‘Venturi’ geometry (see Fig. 2), firstanalytically in the limit ν →
0. For boundary conditions,we fix the voltage φ ( r ) = V and φ ( r ) = 0 and take no-slip (vanishing velocity) at the side walls θ = ± θ /
2. Wefind that stationary purely radial ”plug flow” ansatz v = v r ( r )Θ( θ − θ )ˆ r is a solution (with Θ the Heaviside step-function). Absence of viscosity is crucial as it allows fora zero-thickness boundary layer in this highly symmetric FIG. 2. A topview of the Venturi geometry, with inner radius r and outer radius r and total wedge angle θ . flow. The Navier-Stokes equation (Eq. (2)) reduces to asimple ordinary differential equation ∂∂r (cid:20) eφ + 12 mv r (cid:21) + mγv r = 0 , (8)where we have subsumed pressure into φ for simplicity. We further take the divergence-free (“incompressibleflow”) ansatz v r = Iρ (0) e θ r , where the yet-undeterminedconstant I is the total current and ρ (0) e is the averagecharge density. Substituting this ansatz into Eq. (8) andintegrating from r to r (see Fig. 2), we obtain the non-linear I-V characteristic V = 1 σ D (cid:20) l ln( h /h ) h − h I − (cid:18) h − h (cid:19) I ρ e γ (cid:21) (9)where σ D = n (0) e mγ is the Drude conductivity, l = r − r is the length, and h = θ r and h = θ r are thewidths at the contacts. The first term on the RHS cor-responds to the Ohmic contribution, while the secondterm is the nonlinear I contribution from convection.To further isolate the nonlinearity, we exploit the paritydifference between the two contributions. Because thenonlinearity is of even parity, a non-zero symmetrizedcurrent I sym ( V ) ≡ [ I ( V ) + I ( − V )] provides a directsignature of the nonlinearity. To estimate this effect, inFig. 3 we plot in blue the current fraction I sym /I and theI-V characteristic of Eq. (9) for wedge angle θ = π/ r = 5 µ m, r = 10 µ m, and graphene-hBN param-eters as discussed in Sec. II B. To incorporate a finiteshear viscosity, which is difficult to solve analytically (seeAppendix B), we solve the Navier-Stokes equations nu-merically and plot the results as points in Fig. 1a. Theexact ( ν = 0) result of Eq. (9) matches well with thenumerical result, as expected of the fact that the viscouslength scale r d ≡ (cid:113) νγ (cid:28) r θ is small for experimentallyrelevant parameters. As demonstrated by Fig. 1a, thisnonlinear effect ( I sym ∼
400 nA for I ∼ µ A) shouldbe experimentally measurable.This nonlinear I-V characteristic in electronic hydro-dynamics is the analogue of the Bernoulli effect in clas-
FIG. 3. Main: A parametric plot of the voltage-symmetrizedcurrent I sym ( V ) ≡ [ I ( V ) + I ( − V )] against total current I ( V ). Inset: The I-V characteristic. The solid lines are ob-tained analytically from Eq. (9) in the ν → ν . Fixed-voltageboundary conditions are taken. The inner and outer radiusare 5 µ m and 10 µ m respectively, with wedge angle θ = π/ ν = . /s and γ = 650 GHz.Since r d ∼ . µ m and lengths are ∼ µ m, viscous correctionsto the analytic ν → ∼ sical hydrodynamics, the prototypical example of con-vective acceleration, which is traditionally demonstratedusing a Venturi tube. The Bernoulli effect is typicallydemonstrated in an inviscid fluid of divergence-free (in-compressible) flow, analogous to our assumptions. Infact, the classical Bernoulli (energy conservation) equa-tion is analogous to Eq. (8); the term in brackets corre-sponds to the classical Bernoulli contribution, while the γ term accounts for the additional dissipation from a fi-nite conductivity. As a result, the nonlinear term of theI-V characteristic Eq. (9) can be calculated exactly byclassical Bernoulli considerations.We turn to the subtle issue of solving for the total cur-rent I ( V ) given the input voltage V , i.e. verifying thatthe ansatz satisfies the boundary conditions. Becausethis requires solving a quadratic equation for I , the so-lution is generically multivalued and may not even havea solution. In the limit of small V , linear response mustprovide the correct answer on physical grounds; this se-lects the solution branch continuously connected to thesolution I = 0 at V = 0, where parity was broken by γ . The opposite branch is therefore expected to be un-stable to θ -dependent perturbations. The region wherethe purely radial solution does not exist corresponds toparticle flow in the divergent direction; for classical flu-ids, it is known that divergent flow eventually becomesunstable and develops turbulence. To estimate thescale of nonlinearity at which the radial ansatz fails, onecan define a Reynolds numberRe γ ≡ (cid:82) r r drF conv ,r − (cid:82) r r drργv r = − lh Iρ e γ (cid:34) h h − h h (cid:18) − h h (cid:19)(cid:35) (10) which is precisely the ratio of the two terms in Eq. (9).The instability point occurs at Re γ = − /
2. We summa-rize the resolution of these subtleties in Fig. 1a.Finally, we now highlight three aspects of the Bernoullinon-linearity that should help identify it unambiguouslyin experiments. To start, following Eq. (9) we notethat the quadratic term is independent of the momen-tum relaxation parameter γ , and hence may be identi-fied by comparing I-V traces taken at different temper-atures or even from different samples of the same ma-terial. Secondly, the simple charge density-dependencemay be probed by varying backgate voltage. After fac-toring out the density-dependent Drude resisitivity 1 /σ D (cf. Eq. 9), the nonlinear term only has an inverse de-pendence on charge density (and its sign depends on thecarrier charge). Lastly, Eq. (9) has a distinct geomet-ric dependence interpolating in a somewhat unusual waybetween conventional and ballistic transport. For a fixedaspect ratios h /h and l/h , we find that the Ohmic re-sistance contribution scales with the size of the device as1 /h while the nonlinear Bernoulli contribution scales as1 /h . In addition, the Ohmic resistance contribution hasthe conventional linear scaling with length l , while thenonlinear Bernoulli contribution has the l -independenthallmark of ballistic transport. IV. ECKART STREAMING: A“HYDRODYNAMIC SOLAR CELL”
A dramatic effect of nonlinearity occurs upon applyingan oscillatory drive: down-conversion. In a backgated de-vice of length l and width h (see Fig. 1b), we considersetting up a traveling longitudinal (sound) wave by ap-plication of a voltage oscillation φ ( x = 0) = V cos ωt atthe left contact with the right contact grounded ( φ ( x = l ) = 0). This will result in a DC current via the down-conversion sourced by the convective force (Eq. (3)).Such a device can be described as a “hydrodynamic so-lar cell” providing a DC photocurrent if the (localized)voltage oscillation is driven by EM radiation. For sim-plicity, we will focus on bulk dissipation (i.e. attenuationdue to α >
0) contributions to the convective force andneglect those of boundary dissipation, which only resultsin a quantitative underestimate of the DC current (seeAppendix C 5). This is the electronic analogue of Eckartstreaming in classical hydrodynamics, where the convec-tive force is primarily generated by bulk dissipation.
To see this, we need to solve the full Navier-Stokes equa-tion (Eq. (2)), whose nonlinearity precludes a single-mode ansatz. To handle this, we will seek a perturba-tive solution in the input voltage amplitude V (see Ap-pendix C for full mathematical detail). A. Perturbative Calculation
We begin by expanding the hydrodynamic variables ina power series expansion of V , e.g. ρ = ρ (0) + ρ (1) + ρ (2) + . . . ; ρ (0) corresponds to the equilibrium mass den-sity, while ρ (1) and ρ (2) are the first and second order so-lutions. At leading (linear) order, the single-mode ansatz φ (1) ∼ V e i ( ± k l x − ωt ) along x with wavenumber k l = k + iα is appropriate. Imposing the fixed-voltage boundary con-ditions, the solution of φ (1) is a traveling wave with re-flected component; the grounded edge acts as a mirror.Because of the backgate providing a capacitance per area C , the voltage oscillation of amplitude V sets up a chargedensity oscillation ρ (1) e = Cφ (1) of amplitude CV (seeEq. (7)). Via the density continuity equation (Eq. (1)),the density oscillations drive a longitudinal velocity os-cillation v (1) x , schematically written as v (1) x ∼ u (cid:60) (cid:104) e ( ik − α ) x − iωt + e ( ik − α )(2 l − x ) − iωt (cid:105) (11)where (cid:60) denotes real part and u = CV ρ (0) e ω | k l | is the velocityamplitude. We also take a no-slip boundary condition,which is not satisfied by v (1) x . However, as previouslystated we will neglect the boundary corrections to v (1) x forsimplicity (see Appendix C 5). As a result, the leadingorder solution v (1) x results in a DC convective force (seeEq. (3)) F (2)conv ,x = ρ (0) u α sinh[2 α ( l − x )] − k sin[2 k ( l − x )]cosh 2 αl − cos 2 kl (12)where the overbar denotes time-average. The first termin the numerator arises from the bulk dissipation α , whilethe second term arises from interference effects; in thelimit αl (cid:29)
1, where interference effects are small, theRHS of Eq. (12) simplifies to αe − αx . This rectified DCforce will result in a DC current.We now solve for the second-order DC current I (2) .The DC current density J (2) ≡ ρ (0) e v (2) + ρ (1) e v (1) must bedivergence-free to satisfy current conservation (i.e. den-sity continuity Eq. (1)). With the ansatz v (2) y = 0, thisimplies that the current density J (2) = J (2) x ( y ) ˆx onlyvaries along y . However, the convective force given byEq. (12) varies along x . This paradox is resolved bystatic screening, where the x -dependence of convectionwill be canceled by contributions from the effective volt-age φ (2)eff ≡ φ (2) + ρ (0) e p (2) . Utilizing separation of vari-ables in the NS equation (Eq. (2)), we can solve for φ (2)eff by applying the voltage-fixed boundary conditions φ (2) ( x = 0) = φ (2) ( x = l ) = 0. Therefore, the “screened”convective force (which is no longer spatially dependent)becomes F (2)conv ,x − ρ (0) e ∂φ (2)eff ∂x = 1 l (cid:90) l dxF (2)conv . (13) FIG. 4. Main: A plot of I (2) at fixed input currentamplitude I for device length l = 30 µ m and graphene-hBN parameters stated in Sec. II B, in units of A = I ρ (0) e h lγ (cid:16) − r d h tanh h r d (cid:17) . We remark that this is also ascaled plot of the Reynolds number Re γ . Inset: A blowup ofthe yellow highlighted portion. At high frequencies, Re γ sat-urates to a constant A = I ρ (0) e h lγ , while at sufficiently lowfrequencies the interference oscillations become more visible.The gray box demarcates the low frequency region ω (cid:28) γ ,where perturbation theory in V breaks down for a fixed I . Solving NS for the current density J (2) x and integratingacross the channel to get the total current I (2) , we get I (2) = I ρ (0) e h lγ (cid:20) − − kl cosh 2 αl − cos 2 kl (cid:21) × (cid:18) − r d h tanh h r d (cid:19) (14)where I ≡ ρ (0) e hu is the input current amplitude andhave assumed that convection provides the dominant DCforce (see Appendix C 3). The term in parentheses isa viscous correction, reflecting the y -dependence of thecurrent flow due to no-slip. The bracketed terms corre-spond to dissipation and interference contributions fromthe convective force (Eq. (3)), respectively. The effect ofthese contributions is demonstrated in Fig. 1b, where wehave schematically plotted the dependence of DC currenton the channel length l . In the limit αl (cid:28)
1, the inter-ference term dominates, leading to oscillatory behaviorcontrolled by kl . In the opposite limit αl (cid:29)
1, the in-terference term becomes negligible, and the DC currentscales as I (2) ∼ l − . Other than the device length l , onecould also study the frequency dependence of Eq. (14)(via k l ( ω ) = k + iα ), which is plotted in Fig. 4 for afixed I . Similarly, interference effects appear at lowfrequencies and become negligible at high frequencies.
B. Discussion and Estimates
An effect similar to Eckart streaming was previouslydiscussed by Dyakonov and Shur and extended inRef. 34. They envisaged operating with zero DC currentbias I = 0 instead of zero DC voltage drop, so that onegenerates a DC voltage instead of a DC current. Thesetheoretical treatments similarly neglected boundarydissipation, which only leads to quantitative correctionsto DC voltage. However, for their case boundary dis-sipation leads to qualitative flow corrections (see Ap-pendix C 5); further discussion is deferred to Sec. V.We point out that, in either case, if the voltage oscil-lation is driven by an impingent EM wave, the device isa “hydrodynamic solar cell” generating a DC photocur-rent (photovoltage). In contrast to typical solar cells (e.g.a p-n junction), the hydrodynamic solar cell does notbreak parity by construction; parity is intrinsically bro-ken by dissipation, setting the direction of the photocur-rent. Therefore, Eckart streaming provides a novel mech-anism for photocurrent (photovoltage) generation. Sig-natures of downconverted DC voltage generation by THzradiation have been measured in ultraclean 2DEGs. One can define Reynolds numbers to estimate thestrength I (2) /I of the nonlinearity. The Reynolds num-ber Re γ for this system can be defined asRe γ ≡ l (cid:82) l F (2)conv ,x ρ (0) e γu = I ρ (0) e h lγ (cid:20) − − kl cosh 2 αl − cos 2 kl (cid:21) (15)which explicitly appears in Eq. (14). The viscousReynolds number can be similarly defined such thatRe ν = h r d Re γ , where we approximate the viscous gra-dients to have length scale L = h (see Eq. (4)). Thecontribution from Re ν is hidden within r d ; in the limit r d (cid:29) h where viscous contributions dominate, Re ν canbe made manifest by perturbatively expanding Eq. (14)in h/r d . Since r d (cid:29) h for the experimental systems of in-terest, the Reynolds number Re γ ∼ I (2) /I correspondsto the scale of DC current (up to a small viscous correc-tion).We now estimate the size the DC current in exper-iment (see Appendix A for dispersion relations). Wetake device size l = 50 µ m and h = 5 µ m and operateat ω = 1 THz, with graphene-hBN parameters fromSec. II B; for these choices, the interference effects aresmall since αl ∼
5. Therefore, we find Re γ ∼ I / (312mA)and therefore I (2) / nA ∼ ( I / µ A) . Observing the os-cillatory effects is more difficult, requiring smaller l andmore measurement precision. Despite this, in an opti-mistically sized device of length l = 20 µ m, we plot thefrequency dependence of Re γ in Fig. 4. The oscillationsare suppressed by a factor of 0 .
01; if one asks for a stream-ing current I (2) ∼ V. RAYLEIGH STREAMING
We now turn to the limit where boundary dissipationdominates, i.e. the bulk dissipation α is negligible. Here,the no-slip condition is critical. In a rectangular back-gated device of width h (see Fig. 1c), we consider settingup a longitudinal standing wave of wavelength λ (cid:29) α − along x . In this case, the system cannot support a finiteDC current due to reflection symmetry in y . Therefore,down-converted DC current flows sourced by the convec-tive force (see Eq. (3)) must circulate. The circulatingcurrent leads to a measurable orbital magnetization ofwavelength λ/ x with reflection-symmetric mod-ulation along y (see Fig. 1c). This is the analogue ofRayleigh streaming in classical hydrodynamics, wherethe convective force is primarily generated by bound-ary dissipation. Remarkably, localized boundaryeffects lead to nontrivial flows throughout the bulk (seeAppendix D for full mathematical detail).
A. Perturbative Calculation
We begin by working perturbatively in the input cur-rent amplitude u , where at linear order we take the lon-gitudinal wave ansatz v (1) l,x = u sin kx cos ωt (16)This is consistent with a current-fixed boundary condi-tion J x ( x = 0) = 0 (i.e. DC current I = 0). Forsimplicity, we work in a semi-infinite strip of width h (i.e. | y | ≤ h/ x ≥
0) with the above current-fixed boundary condition. To satisfy no-slip, a trans-verse mode v (1) t is necessary to correct the total flow v (1) = v (1) l + v (1) t . This transverse correction dispersesalong y with wavenumber k t = k (cid:48) t + ik (cid:48)(cid:48) t , and hence formsa “boundary layer” of size 1 /k (cid:48)(cid:48) t exponentially localizedto the wall. We will work in the thin boundary layer andlong wavelength limit k (cid:48)(cid:48)− t (cid:28) h (cid:28) λ . In this limit, theresulting convective force (see Eq. (3)) can be schemati-cally written as F (2)conv ,x ∼ ρ (0) u k e − k (cid:48)(cid:48) t y + sin 2 kx + ( y ↔ − y ) (17)where y + = y + h is the distance from the lowerboundary. As a result of the quadratic non-linearity,the wavelength of the convective force is halved to λ/ r d ≡ νγ (cid:28) h . The convective forcelocalized to the boundary layer of size 1 /k (cid:48)(cid:48) t leads to alocalized flow along x . Because of the shear viscosity ν ,the boundary layer momentum propagates into the bulkwith the viscous length scale r d . Therefore, the boundarylayer “screens” the no-slip condition, providing instead aslip velocity for the bulk flow. This slip velocity canbe written as v (2)slip sin 2 kx , where schematically v (2)slip ∼ u k γ e − /k (cid:48)(cid:48) t r d . Equipped with the slip boundary, we nowsolve the NS equation (Eq. (2)) for the bulk flow wherethe convective force vanishes and obtain J (2)bulk ,x = J (2)slip sin 2 kx (cid:34) h r d cosh yr d − sinh h r d h r d cosh h r d − sinh h r d (cid:35) (18) J (2)bulk ,y = J (2)slip kr d cos 2 kx (cid:34) − h r d sinh y r d + yr d sinh h r d h r d cosh h r d − sinh h r d (cid:35) (19)The slip current J (2)slip ≡ ρ (0) e v (2)slip results from boundaryconvection, while the term in brackets is a geometric fac-tor resulting from satisfying the slip velocity boundarycondition. The DC current flow is plotted in Fig. 1c,where it is clear that the current circulates in cells oflength λ/ h/ B. Discussion and Estimates
A previous related proposal by Dyakonov and Shur and its recent extension discussed downconversion ef-fects with a current-fixed boundary J ( x = 0) = 0, sim-ilar to this case. However, they instead took a stress-free boundary condition which has no boundary dissipa-tion. In their case, there is no circulating current; with-out boundary-layer contributions, the convective forceonly leads to an excess of DC voltage (see Appendix D).Therefore, Rayleigh streaming is qualitatively distinctfrom previous nonlinear proposals in electron hydrody-namics.Since the effect of the convective force is to generate aslip velocity v (2)slip , we can estimate the scale v (2)slip /u byan appropriate Reynolds number. The Reynolds numberRe γ is defined in this case to beRe γ ≡ max F (2)conv ρ (0) γu = I ρ (0) e h k γ f ( ω/γ ) (20)where f is a dimensionless function of ω/γ described inAppendix D. We remark that f develops an interestingresonance at ω = √ γ where perturbation theory breaksdown, but we operate away from this point and will notdiscuss it further. It turns out Re γ e − /k (cid:48)(cid:48) t r d = v (2)slip /u ,i.e. slip velocity is given by the Reynolds number up to anexponential factor controlled by the viscous length scale r d . However, the viscous Reynolds number Re ν does notcontribute to the effect; in the limit γ →
0, the scale
FIG. 5. A plot of the bulk vorticity distribution Ω (2)bulk ≡∇ × J (2)bulk induced by Rayleigh streaming for h = 5 µ m and ω = 2 THz with graphene-hBN parameters as in Sec. II B.The local bulk vorticity corresponds to a Coulomb-like pointsource of magnetic field due to Ampere’s law. v (2)slip /u is instead set by the Mach number u k/ω . De-spite the necessity of a finite shear viscosity ν to generatea convective force, Re ν does not set the scale v (2)slip of theresult; this curious fact was first remarked by Rayleigh (see Appendix D for additional discussion).We propose that the circulating flow profile could bedetected via magnetometry. To estimate the effect in re-alistic systems, we set ω = 2 THz and channel width h = 5 µ m with graphene-hBN parameters as in Sec.II B(see Appendix A for dispersion relations). We first verifythe assumptions we made: k (cid:48)(cid:48)− t (cid:28) h (cid:28) λ , r d (cid:28) h , and α (cid:28) k . These are k (cid:48)(cid:48) t h ∼ h/λ ∼ . r d /h ∼ . α/k ∼ .
2, so we expect our solution to be roughlycorrect. For the scale of the DC effect, we find Re γ ∼ I / (23mA) and k (cid:48)(cid:48) t r d ∼ .
1, so that v slip ∼ ( I / u .Since Ampere’s law implies −∇ B z = µ ∇ × J δ ( z ), thevorticity Ω ≡ ∇ × J acts as a Coulomb-like point sourceof magnetic field. The vorticity is plotted for these pa-rameters in Fig. 5, where it is concetrated near the edgessince the viscous length scale r d (cid:28) h is small. To makea rough estimate of the magnetic field strength, we take B z ∼ µ z (cid:82) cell ∇ × Ω (2)bulk at a height z from the sample;we approximate the magnetic field to be sourced by thenet circulation in the nearest vortical cell. This gives B z ∼ ( I / . µ A) z/µ m × − T. Therefore, the magnetic fieldsshould be detectable for I ∼ . µ A by scanning SQUIDmagnetometers.
VI. SUMMARY AND OUTLOOK
This paper argues for using non-linear DC transportand other manifestations of convective nonlinearity toidentify and study electron hydrodynamics. We havelaid out three electronic analogues of nonlinear classi-cal phenomena - the Bernoulli effect, Eckart streaming,and Rayleigh streaming - which lead to an experimentallymeasurable nonlinear I-V characteristic, down-convertedDC current, and DC current vortices, respectively (seeFig. I). We have opted to derive and discuss all threeeffects using the familiar Navier-Stokes formalism, leav-ing a more complete microscopic treatment for futurework. All three effects result from the interplay of thenon-dissipative and nonlinear convection force with otherdissipative contributions in Navier-Stokes from viscosityand momentum relaxation.It is interesting to note that interactions do not playan explicit role in our results – both convection and mo-mentum relaxation (the dominant form of relaxation)are well understood in the non-interacting limit of themany-electron problem. Instead, strong electron-electroninteractions justify the coarse-grained effective descrip-tion, removing the need to consider the complications ofquasi-particle physics. In particular, local equilibration(assumed throughout) is likely to be violated in the limitof weak interactions, requiring a more systematic micro-scopic treatment. This will be required, for example,before extrapolating our results to low temperatures.To obtain stronger nonlinear signatures, one would liketo make the Reynolds numbers Re ν and Re γ as large aspossible. Since the viscous length scale r d = ν/γ is typ-ically smaller than the characteristic lengths in experi-ment, Re γ is the limiting factor. In addition to reducingthe momentum relaxation rate γ , one could also reducethe density n at fixed current to improve the Reynoldsnumbers; particles must move more rapidly to maintainthe current. Therefore, nonlinear effects should be mostprominent in clean, low-density hydrodynamic materials.Our focus has been away from linear response, which is a bedrock foundation of experimental condensed matterphysics. Nonlinear phenomena are comparatively moredifficult to interpret and tend to be less explored, espe-cially with the purpose of extracting basic information,e.g. where in the phase diagram a given material hap-pens to be. However, since our primary focus has been on leading deviations from linear response, we are nonethe-less optimistic that identifying electron hydrodynamicsfrom nonlinear behavior is feasible.In particular, the detection of the AC-generated staticcurrent described above would provide strong evidencefor the presence of hydrodynamic behavior. Addition-ally, hydrodynamic nonlinearities should also generateupconverted 2 f signals, which we leave to future work.This also tantalizingly suggests the possible utility of hy-drodynamic materials as a novel platform for creatingnonlinear electronic devices. The nonlinear I-V char-acteristic of the Venturi wedge device clearly displays theonset of instability phenomena far separated from linearresponse. Such convective instabilities are a known routeto classical turbulence , i.e. in the absence of momen-tum relaxation. In the electronic system, where momen-tum relaxation dominates and viscous length scale r d isshort, we suspect that the behavior may be qualitativelydistinct from turbulence. These and other phenomenapose a fertile frontier for near-term exploration of elec-tron hydrodynamics.
Acknowledgements
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Here we study the hydrodynamic modes at linear order (without boundary conditions), where the convective term F conv is neglected. Because of linearity, the harmonic modes will not mix; the linear-order ansatz v (1) ∝ e i ( kx − ωt ) isappropriate. We eliminate the variables p and φ in Navier-Stokes (Eq. (2)) by using density continuity (Eq. (1)) aswell as the equations of state (Eq. (6) and Eq. (7)). The resulting dispersion relation can be separated in longitudinal( ∇ × v (1) = 0) and transverse ( ∇ · v (1) = 0) contributions, and are given by ω l = (cid:16) s − iω l (cid:104) ν + ˜ ζ (cid:105)(cid:17) k l − iω l γ (A1) ω t = iνk t − iγ (A2)where s = s + s . The longitudinal dispersion describes a damped sound wave with undamped speed s ; bothpressure and electric forces contribute additively to s as a result of the equations of state. In particular, the electronic0contribution relies on backgate screening of the Coulomb interaction to achieve this form. The transverse dispersiondescribes the propagation of incompressible shear oscillations, whose spatial extent is controlled by the viscous lengthscale r d ; a finite shear viscosity is necessary for the transfer of momentum into adjacent layers. In contrast to thelongitudinal case, the transverse modes do not drive density oscillations and therefore do not generate pressure orelectric forces. Therefore, the transverse result is independent of the equations of state, and in particular it does notdepend on the presence of a backgate.We remark that measuring the attenuation of longitudinal and transverse oscillations would provide direct,boundary-independent measures of both shear and bulk viscosity, as opposed to DC flow profiles which require theboundary or inhomogenous current injection profiles to enforce velocity gradients. A careful experimentalstudy of finite-frequency behavior of hydrodynamic materials has yet to be done even at linear order, as far as theauthors are aware; in particular, this could provide new cross-checks of previous viscosity measurements. A proposalfor for a shear viscometer utilizing oscillatory motion was made in Ref. 56. Appendix B: Electronic Venturi Effect - Treating Viscosity
The full problem, with both finite (kinematic) shear viscosity ν and momentum relaxation γ is challenging. Becauseviscous effects are controlled by a length-scale r d = (cid:113) νγ , one expects a crossover from viscous-dominated to relaxation-dominated flow as a function of local channel width h = rθ . In particular, the resistance of the thin h (cid:28) r d regionshould scale as 1 /h (Gurzhi/Poiseuille regime), while the resistance of the h (cid:29) r d region should scale as 1 /h (Ohmicregime). Even in the viscous-dominated regime γ →
0, a radial flow assumption is inconsistent with the fixed-voltageboundary conditions as described in the main text; angular components of velocity must contribute. Therefore, forfinite ν we expect the exact solution of Eq. (9) to also break down for strong particle flows in the convergent direction,possibly towards turbulence.
1. Purely viscous limit - Jeffrey-Hamel flow
In the purely viscous limit γ →
0, the leading order flow is a generalization of Poiseuille flow to non-parallelwalls. This case also admits an exact solution of the Navier-Stokes equation, known as Jeffrey-Hamel flow.
However, as we are only interested in low-velocity flows, a perturbative treatment will suffice. In contrast to fixed-voltage boundary conditions, where one cannot assume purely radial flow and therefore is more difficult to solve, wewill assume fixed-current boundary conditions where the θ -dependent radial flow v = v r ( θ )ˆ r is a good ansatz. Inaddition, we take the divergence-free (incompressible) ansatz v (1) r = F ( θ ) /r for an yet-undetermined function F . Onsubstitution and integration of the ˆ θ NS equation (Eq. (2)), we find that the NS equations give em ∂φ (1) ∂r = νr d Fdθ (B1) em φ (1) = 2 νr F ( θ ) + S ( r ) (B2)where S ( r ) is determined from the boundary conditions. Substituting for φ (1) , we find that S ( r ) = K ν r + const forsome constant K by separation of variables. The leading order solution is v (1) r = Ine r θ − θ (cid:18) cos 2 θ cos θ − (cid:19) (B3) em φ (1) = Ine νr θ − θ cos 2 θ cos θ (B4)Since v (2) r = 0, the pressure gradient must balance the convective force. Therefore, the total potential is given by em φ = νIne r θ − θ (cid:32) cos 2 θ cos θ + I neν θ − θ (cid:18) cos 2 θ cos θ − (cid:19) (cid:33) (B5)We see that φ (2) is suppressed by a viscous Reynolds number Re ν ∼ Ineν , as expected. Analogous to the purely Ohmiccase discussed in the main text, it is known that divergent Jeffrey-Hamel flow is unstable towards turbulence. Appendix C: Eckart Streaming
In this section, we lay out the mathematical calculation of Sec. IV in full detail.
1. Leading order solution
As mentioned in the main text, we take the ansatz that the leading order solution is described by a longintudinalsound mode with wavevector k l = k + iα (see Eq. (A1)). Applying the voltage-fixed boundary conditions and usingthe density continuity equation (see Eq. (1)), we find φ (1) = V (cid:60) (cid:20) e ( ik − α ) x − e ( ik − α )(2 l − x ) − e ( ik − α )2 l e − iωt (cid:21) (C1) v (1) x = u (cid:60) (cid:20) e ( ik − α ) x + e ( ik − α )(2 l − x ) − e ( ik − α )2 l e − i Arg k l e − iωt (cid:21) (C2)where u = CV ρ (0) e ω | k l | and (cid:60) denotes real part. To satisfy the no-slip boundary, we must also include a divergence-free(incompressible) contribution to the flow corresponding to a boundary layer correction, as is done in Sec. V. We deferthe discussion of this correction to the end of this section, assuming that its contribution is small.
2. Second-order density continuity equation
We now turn to the time-averaged second-order hydrodynamic equations, where we have assumed v (2) y = 0. Thedensity continuity (i.e. current conservation) equation (see Eq. (1)) gives ∂J (2) x ∂x ≡ ∂∂x (cid:104) ρ (0) e v (2) x + ρ (1) e v (1) x (cid:105) = 0 (C3)which tells us that J (2) x ( y ) only depends on y . We remark that it is crucial that v (2) is not divergence-free (incompress-ible); because the “drift” contribution ρ (1) e v (1) x is non-zero and x -dependent, divergence-ful (compressive) contributionsof v (2) x are necessary to satisfy current conservation.
3. Second-order Navier-Stokes equation - DC forces and screening
Replacing v (2) x in favor of J (2) x in the Navier-Stokes equation (see Eq. (2)), we get me (cid:20) − ν ∂ ∂y + γ (cid:21) J (2) x = F (2)eff (C4) − ρ (0) e ∂φ (2)eff ∂x + F (2)conv ,x + F (2)elec ,x + F (2)comp ,x ≡ F (2)eff (C5)where we used separation of variables with constant F eff to split the momentum equation, and ρ (0) e φ (2)eff ≡ ρ (0) e φ (2) + p (2) .We remark that Eq. (C4) is an Ohmic-Poiseuille equation describing steady, divergence-free (incompressible) flowin rectangular channel, where F eff can be interpreted as the effective force driving the flow. The convective force isdefined in Eq. (3), while the terms F (2)elec ,x and F (2)comp ,x are given by F (2)elec ,x = ρ (1) e ∂φ (1) ∂x (C6) F (2)comp ,x =(2 ν + ˜ ζ ) ρ (1) ∂ v (1) x ∂x − ∂ (cid:16) ρ (1) v (1) x (cid:17) ∂x (C7)2where in the second line we have used ∂∂x ( ρ (0) v (2) x ) = − ∂∂x ( ρ (1) v (1) x ). These provide nonlinear contributions to F (2)eff in addition to the convective force. The first term comes from the backreaction of the electric force; we remark thatthe presence of this nonlinearity was also noted by Ref. 34. The second term comes from compressive dissipation. Bysolving for φ (2)eff with the zero-voltage boundary conditions, we find the simple result F eff = 1 l (cid:90) l dxF (2)conv ,x + F (2)elec ,x + F (2)comp ,x (C8)The action of the effective voltage is to “screen” all the forces via a spatial average, rendering the resulting effectiveforce x -independent. We comment that l (cid:82) l dxF (2)elec ,x = CV l has no α or k dependence, and therefore no interferencebehavior; the value of F (2)elec ,x is fixed at the ends by the voltage boundary conditions. By dimensional analysis,these contributions are small relative to the convective force when s ω | k l | (cid:28) (2 ν +˜ ζ ) | k l | ω (cid:28)
1, respectively. Forparameters as discussed in the main text, we find s | k l | ω ∼ .
24 and (2 ν +˜ ζ ) | k l | ω ∼ .
06 are small, so that ignoring F (2)elec ,x and F (2)comp ,x is valid.
4. Rectified DC solution
The solution of the Ohmic-Poiseuille equation (Eq. C4) is J (2) x = ρ (0) e u F (2)eff ρ (0) γu (cid:32) − cosh yr d cosh h r d (cid:33) (C9) I (2) = I F (2)eff ρ (0) γu (cid:18) − r d h tanh h r d (cid:19) (C10)The term in square brackets is suggestively written to resemble momentum-relaxation Reynolds number Re γ , whichis indeed true when the convective force dominates (see Eq. (15)). We remark that the convective contribution to I (2) /I is largely α -independent (see Eq. (14)); in the limit αl (cid:29)
1, where the interference term can be neglected, theresult is surprisingly α -independent even though α was necessary to generate convective gradients. Instead, the scaleof the convective gradient is screened, being controlled by the device length l − . This α -independence has an analoguein Rayleigh streaming, where the shear viscosity ν does not set the scale of the rectified bulk flow even though it wasnecessary to set up convective forces.
5. Revisiting Boundary Dissipation (Rayleigh Streaming)
We return to the issue of the no-slip condition and boundary layer corrections (i.e Rayleigh streaming), whichwe ignored for the leading order solution. For simplicity, we will neglect contributions from the reflected wave (i.e. αl (cid:29) k t = k (cid:48) t + ik (cid:48)(cid:48) t ,decaying exponentially from the wall with length 1 /k (cid:48)(cid:48) t . For parameters as discussed in the main text, we find k (cid:48)(cid:48) t h ∼ . > x -direction. Uponsolving the Ohmic-Poiseuille equation (Eq. (C4)) with a voltage-fixed boundary condition φ ( x = l ) = 0 (as in themain text), we get an additional contribution J (2)Rayleigh ,x = v (2)slip cosh yr d cosh h r d (C11) I (2)Rayleigh = v (2)slip tanh h r d (C12)Therefore, the no-slip boundary (i.e. Rayleigh streaming) only provides a quantitative correction to the DC current.By estimating v (2)slip ∼ u e − /k (cid:48)(cid:48) t r d I | k l | ρ (0) e hγ from the Rayleigh Reynolds number in Eq. (20) with exponential decay arising3from the viscous length scale r d , we find that boundary dissipation contributes additively to the bulk dissipationcontribution.If instead one takes the current-fixed boundary condition J ( x = l ) = 0, a rectified DC voltage will develop asdiscussed in previous works. However, these previous works did not consider the effect of a no-slip boundary. Asa result of no-slip, we expect only a quantitative change to the DC voltage analogous to the previous case. However,a qualitative change occurs in the current flow - a circulating current must develop in the channel as in Sec. V. Thelength and width of the circulation will be set by the device dimensions, as opposed that of Sec. V where the length isset by the wavelength. Surprisingly, the bulk current density flows in an opposite direction to that of the convectiveforce; because convective forces are stronger near the boundary than the bulk, the forward DC flow along x must benear the boundary while the counterflow is in the bulk. This reversed bulk counterflow would be also be interestingevidence for hydrodynamic behavior, though measuring the local current density may prove challenging.
Appendix D: Rayleigh Streaming
In this section, we fill out the mathematical details of Sec. V.
1. Leading order solution - Boundary corrections
Recall that we work in the limit k (cid:48)(cid:48)− t (cid:28) h (cid:28) λ of a thin boundary layer and long wavelength. In this limit, we canseparate the flow into bulk and boundary regions, stitching the flow together at the interface. We first focus on theboundary layer region, concentrating on the lower boundary layer near y = − h/
2; flow at the upper boundary layer isgiven by reflection symmetry about y = 0. In the lower boundary layer, the leading-order longitudinal (irrotational)and transverse (incompressible) velocity components of v (1)wall are v (1)wall ,l,x = v (1) l,x = u sin kx (cid:60) e iωt (D1) v (1)wall ,t,x = − u sin kx (cid:60) (cid:2) e ik t y + e − iωt (cid:3) (D2) v (1)wall ,t,y = − u k cos kx (cid:60) (cid:20)(cid:0) − e ik t y + (cid:1) e − iωt ik t (cid:21) (D3)where y + = y + h is the distance from the lower wall, we take k (cid:48)(cid:48) t >
0, and (cid:60) denotes real part. Although v (1)wall ,y is smallcompared to v (1)wall ,x , the y -gradients of v (1)wall ,y are large and must be included when computing the convective force.The longitudinal contribution v (1)wall ,l,x is inherited from the longitudinal ansatz of Eq. (16). We remark that we havenot assumed that v (1)wall is divergence-free (incompressible) unlike classic discussions ; that the divergence-free(incompressible) ansatz is not correct has been previously pointed out, , though it has no consequence in the limit γ →
0. In the limit k (cid:48)(cid:48) t y + (cid:29)
1, we find that v wall ,x returns to our longitudinal ansatz v (1) l,x as the boundary-layercorrections exponentially vanish. However, v (1)wall ,t,y is non-zero in this limit and requires correction in the bulk. Wewill not concern ourselves with the bulk corrections to v (1) y , as they are small and do not contribute substantially tothe convective force. Therefore, the convective force in the bulk and boundary layers are F (2)conv, bulk ,x = ρ (0) u k
14 sin 2 kx ( −
2) (D4) F (2)conv, wall ,x = ρ (0) u k
14 sin 2 kx (cid:104) − e iθ t ) e ik t y + − e − k (cid:48)(cid:48) t y + cos θ t (cid:105) (D5)where θ t ≡ Arg k t .
2. Second-order Navier-Stokes
We now study the DC second-order flow. We begin by noting that the assumption k (cid:48)(cid:48)− t (cid:28) h (cid:28) λ implies that v y (cid:28) v x , i.e. flow is primarily along x because the channel is thin. By using the NS equations (Eq. (2)), this implies4that the effective voltage φ eff = φ + ρ (0) e p satisfies ∂φ eff ∂y (cid:28) ∂φ eff ∂x , i.e. voltage gradients (and density gradients) are alsoprimarily along x .Next, we simplify the NS equation (Eq. (2)). First, we note that the backreactive electric force F (2)elec ≡ ρ (1) e ∇ φ (1) = 0.We will also assume that compressional dissipation F comp ≡ (2 ν + ˜ ζ ) ρ ∇∇ · v is negligible, which is consistent withour assumption that the longitudinal attenuation α is small. Finally, for simplicity we neglect the additional term νρ (1) e ∇ × ∇ × v (1) as is done in classical treatments of Rayleigh streaming; this term depends on thedensity dependence of ν , where classical works assumed that the dynamic viscosity µ ≡ ρν is constant. Therefore,the NS equation becomes me (cid:20) − ν ∂ ∂y + γ (cid:21) J (2) x = F (2)conv ,x − ρ (0) e ∂φ (2)eff ∂x (D6)where we have used k (cid:48)(cid:48)− t (cid:28) h (cid:28) λ to drop the x -derivatives (cf. Eq. (C4) and Eq. (C5)). Note that this form isequivalent to assuming that v (2) is divergence-free (incompressible).Since the convective force is only x -dependent in the bulk, we must have ρ (0) e ∂φ (2)eff ∂x = F conv,bulk ,x (D7)upon imposing I = 0 (i.e. J x ( x = 0) = 0). More concretely, the boundary conditions for v (2) x ( y = ± h/
2) will fix the y -dependent homogeneous solutions of Eq. (D6), leaving φ (2)eff to enforce I (2) = 0. Since ∂φ (2)eff ∂y is small, this expressionfor φ (2)eff is also valid in the boundary layer. Therefore, after “screening” from the effective voltage, the resultant forceis only non-zero in the boundary layer.
3. Second-order boundary layer solution
We first solve Eq. (D6) in the boundary layer, where the “screened” convective force is not negligible. Assuming r d (cid:28) h , the solution for the lower boundary layer is J (2)wall ,x = ρ (0) e u sin 2 kx (cid:60) (cid:34) v slip u − u k γ (cid:32) − (3 + e iθ t ) e ik t y + k t r d + 1 − (2 cos θ t ) e − k (cid:48)(cid:48) t y + k (cid:48)(cid:48) t r d − (cid:33)(cid:35) (D8) v (2)slip = u k γ e − y + rd (cid:60) (cid:34) − (3 + e iθ t )( i ˜ ω + 2)4 + ˜ ω − θ t − √ ω (cid:35) (D9)where v (2)slip enforces the no-slip boundary conditions and have rewritten k t r d in terms of ˜ ω = ω/γ using Eq. A2. Awayfrom the wall where the convective force vanishes, the velocity v (2)wall ,x → v (2)slip sin 2 kx achieves a non-zero limitingvalue if k (cid:48)(cid:48) t r d is sufficiently large; the boundary layer sets up a slip boundary for the bulk flow. In the main text, we(optimistically) approximate the size of the boundary to be 1 /k (cid:48)(cid:48) t so that we evaluate v (2)slip at y + /r d = 1 / ( k (cid:48)(cid:48) t r d ). Theresulting bulk flow is solved from Eq. (D6) with a vanishing RHS and with the slip boundary generated from theboundary layer; the solution is given in the main text (Eq. (18) and Eq. (19)).We make three remarks on v slip . First, in the limit ν →
0, the flow becomes increasingly singular at the wallsso the boundary layer will no longer by described by hydrodynamics. Second is the surprising fact that ν is largely ν -independent. In the limit γ →
0, we recover the classical result v slip = − u u kω which is ν -independent, despitethe necessity of ν to set up convective gradients. Instead of the viscous Reynolds number Re ν , the slip velocity iscontrolled by the Mach number u ω/k . This was first noted by Rayleigh in the classical situation. Finally, v slip hasa resonance at ω = √ γ corresponding to − k (cid:48)(cid:48) t r d + 1 = 0. We leave further study of this interesting convectiveinstability to future work; for this paper we only work in the limit v slip (cid:28) u0