1 Introduction↩︎

The study of fluid dynamics through the lens of quantum mechanics has a long history, dating back to the pioneering work of Madelung [1], [2], who showed that the linear Schrödinger equation for a free particle is equivalent to the equations of a compressible, inviscid, irrotational fluid at zero temperature and with zero pressure. This analogy has been extended and generalized in many directions over the decades [3]–[10], leading to the recent derivation [11] of a nonlinear Schrödinger-Navier-Stokes (SNS) equation that maps onto the Navier-Stokes equations for a viscous, irrotational fluid with a generic pressure at finite temperature. The key ingredient is a shift of the nonlinear potential of the nonlinear Schrödinger equation that removes the quantum pressure and introduces a viscous term, while preserving the conservation of mass. This reformulation naturally connects quantum pressure effects with Korteweg-type capillarity in classical fluid dynamics.

The SNS equation contains two dimensionless parameters: \(\kappa\), which controls the transition from the quantum (\(\kappa = 0\), Gross-Pitaevskii-like) to the classical (\(\kappa = 1\), Navier-Stokes) regime, and \(\gamma\), which controls the viscous dissipation. The standard Navier-Stokes equations are recovered for \(\kappa = 1\) and \(\gamma \neq 0\), while a Gross-Pitaevskii-like equation is recovered for \(\kappa = 0\) and \(\gamma = 0\). Intermediate values of \(\kappa\) and \(\gamma\) describe fluids that interpolate between these two limits and, as we show in this paper, this intermediate regime has a precise physical interpretation in terms of capillary fluids.

The paper is organized as follows. In Sec. II we introduce the SNS equation and its Madelung representation. In Sec. III we establish the connection with the Navier-Stokes-Korteweg equations for capillary fluids. In Sec. IV we derive the dispersion relation for sound modes with generic \(\kappa\) and \(\gamma\). In Sec. V we derive an effective one-dimensional SNS equation for a fluid confined in a narrow capillary tube.

2 The Schrödinger-Navier-Stokes equation↩︎

Inspired by previous works [3]–[7], [9], in a recent paper [11] we derived a Schrödinger-Navier-Stokes equation by shifting the nonlinear potential of the Gross-Pitaevskii equation. The central result is a mapping between a nonlinear Schrödinger equation for a complex field \(\psi({\boldsymbol{r}},t)\) and the incompressible Navier-Stokes equations for a viscous, irrotational fluid. The mapping is exact and is based on the generalized Madelung transformation \[\psi(\boldsymbol{r},t) = \sqrt{n(\boldsymbol{r},t)}\, e^{i\theta(\boldsymbol{r},t)} \label{mad1}\tag{1}\] with fluid number density \(n(\boldsymbol{r},t) = |\psi(\boldsymbol{r},t)|^2\) and velocity \[\boldsymbol{v} = D\boldsymbol{\nabla} \theta \, , \label{mad2}\tag{2}\] where \[D = {\hbar\over m}\] is the quantum diffusivity with \(\hbar\) the reduced Planck constant and \(m\) the mass of a particle of the fluid. The key structural limitation of that formulation is that, since the fluid velocity is defined as the gradient of the phase field, the flow is necessarily irrotational, wherever the phase \(\theta\) is regular. Vorticity can only appear in the form of quantized vortices at phase singularities (points where \(\psi = 0\)), which carry circulation in integer multiples of \(D\). However, as expected, it has been recently proved [12] that a coarse-graining procedure applied to a fully quantum fluid gives rise to an effectively rotational classical fluid. The SNS equation of Ref. [11] reads \[\begin{align} i \hbar \partial_t \psi = \left[-\frac{\hbar^2}{2m} \nabla^2 + \mu(|\psi|^2) + \kappa \frac{\hbar^2}{2m}\frac{\nabla^2 |\psi|}{|\psi|}\right.\nonumber\\\left. +i\gamma \, \frac{\hbar^2}{m} \nabla^2 \ln\left(\frac{\psi}{|\psi|}\right) \right] \, \psi \; , \label{eq:SNS} \end{align}\tag{3}\] where \(\mu(n)\) is the bulk chemical potential of the fluid, such that \[\mu(n) = {\partial \over \partial n} f(n)\] with \(f(n)\) the free energy density, the parameter \(\kappa\) controls the transition from quantum (\(\kappa = 0\)) to classical (\(\kappa = 1\)) regimes, and \(\gamma\) is the dimensionless dissipative coefficient. The standard Madelung picture is recovered in the limit \(\kappa = \gamma = 0\). Note that \(\nabla^2\ln(\psi/|\psi|) = i\nabla^2\theta\), so the last term reduces to \(-\gamma(\hbar^2/m)\nabla^2\theta\).

It is important to recall the physical meaning of the \(\kappa\) term. The term \(\kappa(\hbar^2/2m)\nabla^2|\psi|/|\psi|\) is precisely the so-called Bohm quantum potential (more precisely a quantum pressure term) with sign chosen (for \(\kappa=1\)) to cancel the quantum pressure that arises from the kinetic term \(-(\hbar^2/2m)\nabla^2\). Indeed, for \(\kappa = 1\) the quantum pressure is completely cancelled and one recovers the purely classical NS equations without any density gradient effects. For \(\kappa = 0\) the quantum pressure is fully present and one recovers the Gross-Pitaevskii equation. For \(0 < \kappa < 1\) the quantum pressure is only partially cancelled, leaving a residual density-gradient term – which, as we show below, can be identified with the Korteweg capillary stress. We stress that the quantum pressure term is positive definite at the level of the energy functional and plays the role of a stabilizing interfacial energy. The terminology “negative surface tension” sometimes used in the literature refers instead to the effective dispersive character of the corresponding stress term in the dynamical equations, not to a negative sign of the underlying free energy contribution.

It is worth noting that microfluidics and the SNS equation share a common mathematical structure: both deal with incompressible flows at low Reynolds number, where viscous effects dominate and capillary forces at fluid interfaces play a crucial role. This structural affinity motivates the present investigation. The purpose of this paper is to discuss consequences of Eq. (3 ) that may be directly relevant to microfluidics. The first is a formal equivalence between the SNS equation with generic \(\kappa\) and the Navier-Stokes-Korteweg equations for capillary fluids, established at the level of an action functional. The second is the derivation of the dispersion relation for sound modes with generic \(\kappa\) and \(\gamma\).

3 Connection with Korteweg Fluids↩︎

3.1 The Navier-Stokes-Korteweg equations↩︎

By inserting Eqs. (1 ) and (2 ) into Eq. (3 ), and separating real and imaginary parts, one obtains two equations for the number density \(n({\boldsymbol{r}},t)\) and the velocity field \(\boldsymbol{v}(\boldsymbol{r},t)\). One is the continuity equation \[\partial_t n + {\boldsymbol{\nabla}} \cdot (n \boldsymbol{v}) = 0 \; . \label{eq:continuity}\tag{4}\] The other is the momentum equation \[(\partial_t + \boldsymbol{v}\cdot\boldsymbol{\nabla})\boldsymbol{v} = -\frac{\boldsymbol{\nabla} P}{mn} + \nu\nabla^2\boldsymbol{v} + \epsilon \boldsymbol{\nabla}\left(\frac{\nabla^2 n}{n} - \frac{|\boldsymbol{\nabla} n|^2}{2n^2}\right) \; , \label{eq:NSK}\tag{5}\] where \(P(n)=n \mu(n) - f(n)\) is the pressure, \[\nu = \gamma \, D = \gamma \, {\hbar\over m}\] is the kinematic viscosity and \[\epsilon = \frac{{1-\kappa}}{4}\,D^2 \label{eq:capillary}\tag{6}\] is the capillary coefficient. Eqs. (4 ) and (5 ) are called Navier-Stokes-Korteweg (NSK) equations [13], [14]. They describe classical fluids in which the free energy density depends not only on the local density but also on its gradients. This accounts for capillary effects and surface tension at fluid interfaces. This capillary coefficient \(\epsilon\) vanishes for \(\kappa = 1\) (purely classical NS fluid) and reaches its maximum \(\epsilon = D^2/4\) for \(\kappa = 0\) (Gross-Pitaevskii-like equation). Intermediate values \(0 < \kappa < 1\) describe classical capillary fluids in the NSK sense.

3.2 Action functional and dissipation↩︎

The last term in Eq. (5 ) is the Korteweg stress and it is responsible for surface tension at fluid interfaces. It derives from the Korteweg free energy functional [13], [14] \[\mathcal{F}_K = \int d^3r\left[f(n) + \frac{\epsilon}{2n}|\boldsymbol{\nabla} n|^2\right] . \label{eq:FK}\tag{7}\] The SNS equation, as we show below, is associated precisely with the Korteweg functional Eq. (7 ), that is closely related to the Cahn-Hilliard functional [15].

Within a generalized Lagrange-d’Alembert-Rayleigh framework for dissipative systems, the SNS equation can be written as the variational condition: \[\delta\mathcal{S} + \delta\mathcal{S}_{\gamma} = 0 ,\] where \[\begin{align} \mathcal{S} = \int dt\,d^3r\left[-\hbar \, n\, \partial_t\theta - \frac{\hbar^2 n}{2m}|\boldsymbol{\nabla}\theta|^2 - n \, U\right.\nonumber\\\left. - (1-\kappa)\frac{\hbar^2}{8mn}|\boldsymbol{\nabla} n|^2 - f(n) \right] \label{eq:action} \end{align}\tag{8}\] is the conservative action functional in terms of number density \(n\) and phase \(\theta\) of the Schrödinger field \(\psi\) with \(\delta S\) its first variation, while \[\delta \mathcal{S}_{\gamma} = \int dt \, d^3r\; \left( - \gamma \, \frac{\hbar^2}{m} \, |\boldsymbol{\nabla}\theta|^2 \right) \, \delta n \label{eq:rayleigh}\tag{9}\] is the first variation of dissipation. In (8 ) the first term is the Berry phase (kinetic term for the phase field), the second is the kinetic energy of the flow, the third is the energy density associated with the external potential \(U\), the forth is the residual quantum pressure (proportional to \(1-\kappa\)), and the fifth is the bulk interaction energy. The first variation \(\delta \mathcal{S}_{\gamma}\) is the non-exact 1-form for dissipation, that is strictly related to the dissipative Reyleigh energy functional \[{\cal R} = \int d^3r \, \gamma \, {\hbar^2\over m} |\boldsymbol{\nabla}\theta|^2 \, n \: .\]

For a classical fluid, \(D\) should be interpreted not as the quantum diffusivity but as an effective parameter with dimensions of a diffusivity, related to the capillary coefficient by \(D = \sqrt{4\epsilon/(1-\kappa)}\). In this interpretation, the SNS equation provides a quantum wave representation of a purely classical capillary fluid, in which \(\hbar\) and \(m\) are a convenient parametrization of \(\epsilon\). This is not fully surprising: all classical friction forces have quantum origins.

3.3 Spherical bubble↩︎

A key advantage of the SNS formulation over the NSK equations emerges in problems where the density vanishes at the interface, such as the nucleation of a spherical bubble. Consider a bubble of radius \(R\) embedded in a fluid of bulk density \(n_0\). We look for a stationary, spherically symmetric solution \(\psi = \Phi(r) \, e^{-i{\bar \mu} t/\hbar}\), with \(\Phi(r) = \sqrt{n(r)}\) and \(\bar{\mu}\) the chemical potential of the non uniform system. With \(U = 0\) and \(\gamma = 0\), the SNS equation after Taylor-expanding \(\mu(n)\) around \(n_0\), reduces to: \[\frac{d^2\Phi}{dr^2} + \frac{2}{r}\frac{d\Phi}{dr} = \frac{1}{\xi_\kappa^2}(\Phi^3/n_0 - \Phi), \label{eq:bubble}\tag{10}\] where \(\xi_\kappa = \xi \sqrt{1-\kappa}\) is the \(\kappa\)-dependent healing length, with \(\xi = \sqrt{D^2 m/(2 n_0 \mu'(n_0))}\) the standard healing length, and \({\bar \mu}=\mu(n_0)\). This equation is regular even at \(\Phi = 0\), i.e., in the bubble core where \(n \to 0\).

This is a crucial advantage over the NSK formulation. In the NSK equations, the Korteweg stress contains terms proportional to \(|\boldsymbol{\nabla}n|^2/n^2\) and \(\nabla^2 n/n\), which are singular when \(n \to 0\). The substitution \(\Phi = \sqrt{n}\) in the SNS formulation removes these singularities, yielding the regular Eq. (10 ). For \(R \gg \xi_\kappa\), the curvature term \(2\Phi'/r\) is negligible at the interface \(r \simeq R\), and Eq. (10 ) has the approximate solution \[n(r) \simeq n_0\tanh^2\left(\frac{r-R}{\sqrt{2}\xi_\kappa}\right). \label{eq:bubble95profile}\tag{11}\] This approximation predicts a vanishing density on the spherical shell \(r=R\). However, treating the curvature term \(2\Phi'/r\) perturbatively we find a correction of the density such that we obtain at the interface \(r=R\) \[n(r)\simeq \left( \sqrt{n_0}\tanh\left(\frac{r-R}{\sqrt{2}\xi_\kappa}\right)-\sqrt{n_0}\frac{\xi_\kappa}{R} \right)^2, \label{eq:bubble95profile95corrected}\tag{12}\] which regularized the density and prevents it from reaching zero. Therefore, the approximate solution Eq. (11 ) is not valid directly at the interface \(r=R\) as corrections become relevant.

The surface tension \(\sigma\) of the bubble interface follows from the profile of Eq. (11 ): \[\sigma = 2 \, \epsilon \int \left(\frac{d\Phi}{dr}\right)^2 dr = \frac{\sqrt{2}}{6} \sqrt{1-\kappa} \frac{D^2 n_0}{\xi}, \label{eq:sigma}\tag{13}\] which shows that \(\sigma\) vanishes for \(\kappa \to 1\) (classical NS limit, no capillary effects) and reaches its maximum for \(\kappa = 0\) (Gross-Pitaevskii limit). The Young-Laplace pressure difference across the bubble interface is as follows: \[\Delta P = \frac{2\sigma}{R},\] which together with Eq. (13 ) gives a closed expression for \(\Delta P\) in terms of \(\kappa\), \(D\), \(n_0\), \(R\) – all measurable quantities in a microfluidic experiment.

Finally, it is instructive to compute the Rayleigh dissipation function for the bubble profile. With \(\gamma = \gamma_0\) constant and \(\boldsymbol{v} = D\boldsymbol{\nabla}\theta\), for a stationary solution with \(\theta = \mathrm{const}\) one has \(\boldsymbol{v} = \boldsymbol{0}\) and therefore: \[\mathcal{R} = \int d^3r\, m\gamma_0 n|\boldsymbol{v}|^2 = 0.\] This result is physically transparent: a stationary bubble with no flow has zero viscous dissipation. The Rayleigh function \(\mathcal{R}\) vanishes identically because the velocity field is zero, not because \(\gamma_0 = 0\). This confirms that \(\mathcal{R}\) correctly captures dissipative physics: dissipation requires flow, and a static bubble does not flow.

The situation changes if one considers a bubble that grows or shrinks radially with velocity \(\dot{R}(t)\). For an incompressible fluid with spherical symmetry, the continuity equation \(\boldsymbol{\nabla}\cdot\boldsymbol{v}=0\) gives a velocity field in the exterior region \(r > R\): \[\boldsymbol{v} = \dot{R}\frac{R^2}{r^2}\hat{r} \; ,\] which satisfies the boundary condition \(v_r(R) = \dot{R}\) at the bubble interface and vanishes at large distances. The Rayleigh dissipation function is then: \[\mathcal{R} \, \simeq \, 4\pi m\gamma_0 n_0 \dot{R}^2 R^4 \int_R^{\infty} \frac{dr}{r^2} = 4\pi m\gamma_0 n_0 \dot{R}^2 R^3 \; ,\] which is finite and proportional to \(R^3\dot{R}^2\). The dissipation therefore increases with the bubble volume and the radial velocity, consistently with the expected phenomenology of dissipative interface dynamics.

4 Sound Modes with Generic \(\kappa\) and \(\gamma\)↩︎

Working with \(U=0\) we consider small perturbations around a uniform stationary state \(\psi_0 = \sqrt{n_0}\,e^{-i\mu(n_0)t/\hbar}\): \[n = n_0 + \delta n, \qquad \theta = \delta\theta,\] with \(\delta n\) and \(\delta\theta\) small. The linearized continuity and momentum equations are as follows: \[\partial_t \delta n + n_0 D\nabla^2\delta\theta = 0,\] \[\partial_t\delta\theta + \frac{\mu'(n_0)}{m D}\delta n - (1-\kappa)\frac{D}{4n_0}\nabla^2\delta n - \gamma D\nabla^2\delta\theta = 0,\] where \(\mu'(n_0) = d\mu/dn|_{n_0}\). Note the sign of the quantum pressure term: it contributes positively to the restoring force, stabilizing the system at short wavelengths.

Substituting plane wave solutions \(\delta n,\,\delta\theta \sim e^{i(\boldsymbol{k}\cdot\boldsymbol{r}-\omega t)}\) and requiring non-trivial solutions, one obtains: \[\omega^2 + i\gamma Dk^2\omega - c_s^2 k^2 - (1-\kappa)\frac{D^2k^4}{4} = 0, \label{eq:dispersion}\tag{14}\] where \[c_s = \sqrt{\frac{n_0 \mu'(n_0) D}{\hbar}}\] is the speed of sound. Three limiting cases confirm the correctness of this result. When \(\kappa = 1\) and \(\gamma = 0\), Eq. (14 ) gives \(\omega = \pm c_s k\) — linear phonons, as expected for a purely classical fluid without capillary effects. When \(\kappa = 0\) and \(\gamma = 0\), one obtains: \[\omega^2 = c_s^2 k^2 + \frac{D^2k^4}{4},\] which is precisely the Bogoliubov dispersion relation of a weakly interacting Bose gas. The crossover from the phonon regime (\(\omega \simeq c_s k\)) to the free-particle regime (\(\omega \simeq Dk^2/2\)) occurs at \(k_* = 2c_s/D = \sqrt{2}/\xi\). When \(\kappa = 1\) and \(\gamma \neq 0\), one obtains attenuated phonons. In the long-wavelength limit \(\gamma Dk \ll c_s\): \[\omega \simeq \pm c_s k - \frac{i\gamma Dk^2}{2},\] with spatial attenuation length \(\ell = 2c_s^3/(\gamma D\omega^2)\), growing as \(k^{-2}\) — consistent with viscous damping in classical fluids.

For generic \(\gamma\) and \(0<\kappa<1\), the solution of Eq. (14 ) is: \[\omega = -\frac{i\gamma Dk^2}{2} \pm \sqrt{c_s^2k^2 + \frac{(1-\kappa-\gamma^2)D^2k^4}{4}}. \label{eq:solution}\tag{15}\] The quantum pressure \((1-\kappa)\) contributes positively to \(\omega^2\) – it is a stabilizing dispersive effect that stiffens the fluid at short wavelengths, consistent with capillary stabilization of interfaces. The dissipation \(\gamma^2\) contributes negatively – it tends to suppress short-wavelength oscillations. The crossover from propagating to purely diffusive behavior occurs when \((1-\kappa-\gamma^2)D^2k^4/4 > c_s^2k^2\), i.e., for: \[k > k_c = \frac{2c_s}{D\sqrt{\gamma^2-(1-\kappa)}},\] which exists only if \(\gamma^2 > 1-\kappa\), i.e., when dissipation dominates over capillary stiffness. Using Eq. (6 ), this condition reads \(\gamma^2 D^2/4 > \epsilon\), i.e. when viscous dissipation dominates over capillary effects set by \(\epsilon\).

We finally consider the case \(\kappa>1\) (and generic \(\gamma\)), which leads to a negative capillary coefficient \(\epsilon\). Under this condition, \((1-\kappa)\) contributes negatively to \(\omega^2\), resulting in a destabilizing effect that weakens the propagation at short wavelengths. Since \(\gamma^2\) contributes negatively, both dissipation and dispersion oppose propagation. Thus, the crossover from propagating to diffusive behavior occurs, once again, for values larger than \(\kappa_c\) which, unlike the previous case, always exists. This regime may be relevant for building an analog of quantum mechanics to active matter, where negative capillary coefficient (or surface tension) has been discussed in continuum theories of dry [16], [17] and wet systems [18], [19].

5 Effective 1D SNS equation for capillary tubes↩︎

A natural application of the SNS equation to microfluidics is the derivation of an effective one-dimensional equation for a fluid confined in a narrow capillary tube of radius \(a\). This is the fluid analog of the dimensional reduction from 3D to 1D of the nonlinear Schrödinger equation for Bose-Einstein condensates [20], [21] or electromagnetic waves confined in quasi-one-dimensional waveguides [22]

Consider a cylindrical capillary of radius \(a\) and axis along \(z\). We write the wavefunction as: \[\psi(\boldsymbol{r},t) = \phi(r)\,\chi(z,t),\] where \(\phi(r)\) is the transverse profile, normalized as \(2\pi\int_0^a \phi^2(r)\,r\,dr = 1\), and \(\chi(z,t)\) is the effective 1D wavefunction. The transverse profile \(\phi(r)\) is determined by the boundary condition at the capillary wall \(r = a\) and by the equilibrium condition in the transverse direction.

Substituting into the SNS equation (3 ) with \(U = 0\) and integrating over the transverse degrees of freedom, one obtains the effective 1D SNS equation: \[\begin{align} i\hbar\,\partial_t\chi = \left[-\frac{\hbar^2}{2m}\partial_z^2 + \mu_{1D}(|\chi|^2) + \kappa\frac{\hbar^2}{2m} \frac{\partial_z^2|\chi|}{|\chi|}\right.\nonumber\\\left. + i\gamma_{1D}(|\chi|^2)\frac{\hbar^2}{m}\partial_z^2 \ln\left(\frac{\chi}{|\chi|}\right)\right]\chi, \label{eq:SNS1D} \end{align}\tag{16}\] where the effective 1D chemical potential is: \[\mu_{1D}(n_{1D}) = \mu(n_{1D}/A_{\rm eff}) + \frac{\hbar^2}{2m} \langle\phi|-\nabla_\perp^2|\phi\rangle,\] with \(n_{1D} = |\chi|^2\) the 1D number density, \(A_{\rm eff} = \pi a^2\) the effective cross-sectional area, and \(\langle\phi|-\nabla_\perp^2|\phi\rangle = 2\pi\int_0^a |\boldsymbol{\nabla}_\perp\phi|^2 r\,dr\) the transverse kinetic energy. The effective 1D dissipative coefficient is: \[\gamma_{1D}(n_{1D}) = \gamma(n_{1D}/A_{\rm eff}).\] Eq. (16 ) has exactly the same structure as the 3D SNS equation (3 ), with a renormalized chemical potential and a dissipative coefficient that depend on the transverse confinement via \(A_{\rm eff}\) and the transverse kinetic energy.

The effective 1D capillary coefficient is the following: \[\epsilon_{1D} = (1-\kappa)\frac{D^2}{4},\] which is independent of \(a\) — the capillary coefficient along the tube axis is not renormalized by the transverse confinement. However, transverse confinement enters through \(\mu_{1D}\), which depends on \(a\) through the transverse kinetic energy \(\hbar^2/(2m)\langle\phi|-\nabla_\perp^2|\phi\rangle \sim \hbar^2/(2ma^2)\).

The effective 1D speed of sound is: \[c_{s,1D} = \sqrt{\frac{n_{1D}\,\mu_{1D}'(n_{1D})\,D}{\hbar}} .\] The 1D dispersion relation for sound modes takes the same form as Eq. (14 ): \[\omega^2 + i\gamma_{1D} Dk^2\omega - c_{s,1D}^2 k^2 - (1-\kappa)\frac{D^2k^4}{4} = 0,\] with the crossover wavevector: \[k_c = \frac{2c_{s,1D}}{D\sqrt{\gamma_{1D}^2-(1-\kappa)}}.\] This shows that the competition between capillary stiffness and viscous damping in the capillary tube is governed by the same dimensionless condition \(\gamma_{1D}^2 > 1-\kappa\) as in the 3D case (except for \(\kappa>1\)), but with the 1D speed of sound \(c_{s,1D}\) that depends on the radius of the tube \(a\) through \(\mu_{1D}\).

6 Conclusions and perspectives↩︎

We have shown that the Schrödinger-Navier-Stokes equation with generic parameters \(\kappa\) and \(\gamma\) provides a genuine variational formulation of capillary fluid dynamics. The action functional Eq. (8 ) naturally decomposes into a conservative part – containing the Korteweg capillary stress with capillary coefficient \(\epsilon = (1-\kappa)D^2/4\) – and a dissipative part given by Eq. (9 ). The parameter \(\kappa\) thus acquires a precise physical meaning: it measures the degree of capillarity of the fluid, interpolating between the Gross-Pitaevskii equation (\(\kappa = 0\), maximum capillary effects) and the classical Navier-Stokes equations (\(\kappa = 1\), no capillary effects). The dispersion relation Eq. (15 ) shows that \(\kappa\) and \(\gamma\) play complementary roles: \((1-\kappa)\) controls capillary stiffness, while \(\gamma\) controls viscous damping, with the Bogoliubov dispersion relation recovered in the quantum limit and viscously damped phonons in the classical limit. Finally, the dimensional reduction to an effective one-dimensional equation for capillary tubes opens a natural avenue for applications to confined microfluidic geometries.

From a computational perspective, it would be interesting to assess whether the SNS formulation offers advantages over standard Navier-Stokes solvers, particularly in the presence of capillary effects, an issue that deserves further investigation. Moreover, possible connections with analog systems, including active or quantum-inspired fluids, may offer further perspectives [23]. For example, the role of nearcontact (\(10\) nm) interactions between macroscopic objects (\(100\) micron droplets) discussed in [24] is potentially amenable to investigations of quantum macroscopic effects in soft matter. Another example is that effective negative surface tension terms, obtained with \(\epsilon<0\), arise in active matter systems, where additional non-variational contributions to the stress tensor may lead to contractile behavior [18], [19]. In this context, Eq.(3 ) could provide a promising quantum-mechanical analog for describing quorum-sensing bacteria, where rotational contributions are not relevant in the overdamped limit [25]. Note also that our theory correctly incorporates both Korteweg tensor and dissipative effects, thus it differs from other formulations, such as that discussed in Ref.[26] where the dissipation is included via the drag term of the Langevin equation. This suggests that the SNS framework may offer a useful perspective to interpret such non-equilibrium fluid systems.

It is finally important to emphasize that the search for quantum-like wave formulations of the Navier-Stokes equations has attracted considerable attention in recent years as a potentially promising route towards the formulation of quantum algorithms for classical fluid dynamics [27], [28]. It is estimated that current leading edge computational fluid dynamics simulations on near exa-scale computers, featuring \(Re \sim 10^{8}\), could be accommodated within \(q \sim 80\) qubits, well within the nominal capabilities of current quantum hardware. On the same line, numerical weather forecast on a global scale, featuring \(Re \sim 10^{13}\) (hence totally unfeasible on a classical computer) could be accommodated within a quantum computer with \(q \sim 130\), again within the current capabilities of current quantum technology (once perfect error correction is in place). However, in order for such a blue-sky scenario to become real, two major obstacles need to be overcome, i.e. non-linearity and dissipation. Several strategies have been devised to bypass these two barriers, typically based on Carleman embedding of the nonlinear fluid problem in an infinite-dimensional linear space [29], [30]. However, the resulting quantum algorithms are still far from achieving quantum advantage. In this context, it has been argued that casting fluid dynamics into a quantum-like wave formalism may help bridge this gap, whence a keen interest in SNS formulations [10]. By showing that the SNS formulation seamlessly extends to non-ideal fluids characterized by complex interface dynamics, we lay down a potential bridge to the quantum simulation of a broad class of complex states of flowing matter.

Acknowledgements↩︎

LS thanks Cesare Vianello and Flavio Toigo for useful discussions. LS acknowledges the project “Frontiere Quantistiche” (Dipartimenti di Eccellenza) of the Italian Ministry of University and Research (MUR), the “Iniziativa Specifica Quantum” of INFN, the European Union-NextGenerationEU within the National Center for HPC, Big Data and Quantum Computing (Project No. CN00000013, CN1 Spoke 10: “Quantum Computing”), the EU Project PASQuanS 2. AT acknowledges support from GNFM-INdAM.

References↩︎