Energy transfer in simple and active binary fluid turbulence - a false friend of incompressible MHD turbulence

For active scalars, the kinetic energy is no longer conserved. The simplest active scalar flow is represented by a stably stratified fluid where the active scalar $\phi$ denotes the density perturbation and provides a feedback force density proportional to $\phi\hat{z}$, with $\hat{z}$ being the direction of stratification. For such flows, $\frac{1}{2}\langle\phi^{2}\rangle$ remains to be an inviscid invariant. However, instead of the kinetic energy, the total energy $\frac{1}{2}\langle u^{2}+\phi^{2}/N^{2}\rangle$ is found to be an inviscid invariant and the corresponding source-less differential exact relation is given by $\bm{\mathcal{F}}=\langle[(\delta\bm{u})^{2}+(\delta\phi/N)^{2}]\delta\bm{u}\rangle$ [22], where $N$ is the Brunt-Väisälä frequency.

A more complex feedback force arises if we consider a system of binary fluid mixtures. Binary fluid is a two-component system ranging from a mixture of two simple fluids $e.g.,$ oil and water, to complex systems such as ‘active’ binary fluids [23, 24, 25, 26] ¹¹1the word ‘active’ is used here with a different meaning from that of an ‘active’ scalar and will be explained later. For such systems, the active scalar $\phi$ represents the local molecular composition of the binary mixture [25]. Simple binary fluids contain microstructures like droplets or domains of one fluid into the sea of the other thus producing interfaces or domain walls. Sharp variation in $\phi$ across the interfaces generates diffusion currents which, in turn, drive the velocity field by exerting feedback stress proportional to $\kappa(\bm{\nabla}\phi\otimes\bm{\nabla}\phi)$ [28, 25], where $\kappa$ is a positive constant. However feedback stress does not necessarily take the same form if one of the binary-fluid components is ‘active’ i.e. the constituent particles are equipped with intrinsic mechanisms ($e.g.,$ self-propulsion, body deformation [29, 30] etc.,) of transmuting ambient free energy into directed motion thereby breaking the time reversal symmetry (TRS) at the scale of each constituent [31]. Nevertheless, if the active particles are either spherically symmetric or possess a weak orientational order ($e.g.,$ artificial active colloides, dilute bacterial suspensions etc.,) the feedback stress due to the activity of the particles can be written as $\eta(\bm{\nabla}\phi\otimes\bm{\nabla}\phi)$ where $\eta$ is the activity parameter [32]. The total feedback stress is finally obtained by adding the active stress to the stress due to the interfacial tension, and is given by $(\kappa+\eta)(\bm{\nabla}\phi\otimes\bm{\nabla}\phi)$ [26, 25].

Although individual components of a binary fluid tend to phase separate ( coarse-graning) below a critical temperature ($T_{c}$) [33], it is often useful to have them in the form of steady emulsions i.e. a homogeneous phase-mixed state [34]. This can be achieved through the generation of turbulence which, in effect, prevents the phase separation owing to its enhanced mixing property [35, 36, 37, 38]. In a physical system, turbulence can be attained in two ways, either by large non-linear perturbations ($e.g.,$ MHD, stably stratified flows etc.,) or due to the growth of linear instability ($e.g.,$ Rayleigh-Benárd convection). Simple binary fluid and active fluid with extensile stress ($\eta>0$) belong to the former category [24, 26] whereas active fluid with contractile stress ($\eta<0$) having large activity ($i.e.\,\,\xi=(\kappa+\eta)<0$) belongs to the latter one [39, 40]. Turbulence in an active binary fluid with $\xi<0$ finds its application in the study of bioconvective plums, accumulation of $Dinoflagellates$ in turbulent flows $etc.,$ [41]. Note that here, we are referring to high Reynolds number ($Re$) turbulence in active fluids which is different from the self-sustained pattern formation by dense active fluids at low $Re$ ($\sim 10^{-5}$) [42, 43, 44, 45, 46, 47, 48, 49]. The effect of turbulence on domain growth, energy spectra etc., have been explored both for simple and active binary fluids [24, 50, 51, 52, 26, 38]. However, till date, no exact relation has been derived for such systems.

In this paper, using two-point statistics, we derive exact relations for the inertial range energy transfer in binary fluid turbulence (BFT). Under the assumption of statistical homogeneity, first we derive a differential exact relation as in Eq. (1) in terms of the two-point fluctuations of ${\bf u}$ and ${\bm{\nabla}\phi}$. Then the same exact relation is also expressed in terms of (i) two-point correlators and (ii) newly defined upsilon variables. The derivation of the results are complemented by comparative statements on incompressible MHD turbulence which seems to have some interesting resemblances with our system [24].

The paper is organized as follows: in Sec. II, we describe the model and the basic equations of dynamics followed by the derivation of total energy conservation in the inviscid limit. Sec. III contains the detailed derivation of the exact relation both in terms of the two-point increments and two-point correlators. We then introduce a set of variables ($\bm{\Upsilon^{\pm}}$) and express the inviscid invariant and the corresponding exact relation in terms of those variables in Sec. III.3. Finally in Sec. IV, we discuss our results and conclude.

A binary fluid, composed of two components $A$ and $B$, is usually defined in terms of their mean velocity field $\bm{u}(\bm{x},t)=(\rho_{A}\bm{u}_{A}+\rho_{B}\bm{u}_{B})/(\rho_{A}+\rho_{B})$ and the local molecular composition $\phi(\bm{x},t)=(\rho_{A}-\rho_{B})/(\rho_{A}+\rho_{B})$, where ($\rho_{A}$, $\rho_{B}$) and ($\bm{u}_{A}$, $\bm{u}_{B}$) are the densities and the velocities of the components $A$ and $B$ respectively [25]. Here we are considering systems where the velocity fluctuations are much less than the sound speed, thus justifying the assumption of incompressiblity for the resultant fluid [24]. The corresponding continuity and momentum evolution equations are given by

where $p$ is the fluid pressure, $\bm{\Sigma}$ the feedback stress tensor due to the active scalar $\phi$, $\nu$ the kinematic viscosity and finally $\bm{f}$ represents a large scale forcing. For a general phase-separating binary fluid mixture, $\phi$ satisfies Cahn-Hilliard equation given by

where $\mathcal{M}$ is the mobility coefficient, $\mu$ the chemical potential (or exchange potential) and $\mathcal{F}=\int\left[\frac{a}{2}\phi^{2}+\frac{{\color[rgb]{0,0,0}b}}{4}\phi^{4}+\kappa(\bm{\nabla}\phi)^{2}\right]d\tau$ represents a free energy functional, where $a<0$ while $b$ and $\kappa$ are positive constants and $\tau$ represents the volume [53, 23, 25]. For a phase-mixed binary fluid, $a>0$ and $\mathcal{F}=\int\left[\frac{a}{2}\phi^{2}+\kappa(\bm{\nabla}\phi)^{2}\right]d\tau$ and the Eq. (6) simply becomes

In addition to the force equation (5), one can also force Eq. (6) or Eq. (8) by an appropriate large scale forcing $g_{\phi}$ [38]. The corresponding evolution equation of $\phi(\bm{x},t)$ is therefore given by

The final step is to express the feedback stress tensor $\bm{\Sigma}$ in terms of the other field variables. Both for phase-separating and phase-mixed binary fluids, local inhomogeneity in $\phi$ leads to a feedback force density $\bm{\nabla}\cdot\bm{\Sigma}=-\phi\bm{\nabla}\mu$ [28, 25]. Using the expressions of $\mu$ and following a few steps of straightforward algebra, the feedback stress can be put in the form given below:

where $\sqrt{\kappa}$ is dimensionally homogeneous to the kinematic viscosity [24] and

with $\Gamma$ being the $\phi$ dependent part of the free energy density. Evidently, the term $\Lambda\mathbb{I}$ is proportional to the unit tensor and can therefore be absorbed under a gradient in the force equation (5) which can now be expressed as:

where $P^{*}=p+\Lambda$ can be understood as an effective pressure.

When one of the binary fluid components is active, the additional stress arises due to the intrinsic swimming mechanism of active particles [54]. Self-motility i.e., the ability of active particles to self-propel produces a dipolar force field which yields the additional stress for active fluids. If the particles are either spherically symmetric or having negligible orientational order, the active stress becomes proportional to $\eta\left(\bm{\nabla}\phi\otimes\bm{\nabla}\phi\right)$, where $\eta$ can be both positive or negative [38]. Eq. (12) now becomes:

where $\xi=\kappa+\eta$. Note that, unlike the feedback stress for simple binary fluids, the additional ‘active’ stress is not derived from any free-energy functional [25]. For the sake of algebraic manipulation, it is convenient to consider $\bm{\nabla}\phi$ as a field variable rather than $\phi$ itself. The respective evolution equation is given by

In the current study, we are interested in the turbulent energy transfer which necessarily involves the fluctuations with respect to the mean fields. By choosing an appropriate Galilean transformation, one can eliminate the mean velocity field. However, the mean composition gradient field can not be eliminated by such transformations. It is then useful to decompose $\bm{\nabla}\phi$ as

where $S\hat{z}$ denotes the mean composition gradient field and $\bm{\nabla}\psi$ or $\bm{q}$ denotes the corresponding fluctuating field. The corresponding evolution equations of $\bm{u}$ and $\bm{q}$ will then be given by

where $\bm{\nabla}\cdot\bm{u}=0$, $\bm{\nabla}\times\bm{q}=0$, $\bm{\nabla}g_{\phi}=\bm{g}$, $P=P^{*}+\xi(q^{2}/2+Sq_{z})$ and $\bm{\nabla}\cdot[(S\hat{z}+\bm{q})\otimes(S\hat{z}+\bm{q})]=(S\hat{z}+\bm{q})\bm{\nabla}\cdot\bm{q}+\bm{\nabla}(q^{2}/2+Sq_{z})$. For such a system, the total turbulent energy is composed of the kinetic and the ‘active’ energy and can be written as

In the following, we show E is an inviscid invariant of the flow. For that we simply neglect the large scale forcing and small scale dissipation terms in Eqs. (16) and (17), which gives

Now combining Eqs. (18)–(20), we obtain

Finally using Gauss-divergence theorem with periodic or vanishing boundary conditions, one can show the total energy to be an inviscid invariant of the flow.

Here we derive the two-point exact relation corresponding to the inertial range energy transfer in statistically homogeneous BFT. Following [15, 10], one can first define the two point correlator associated with the total energy (Eq. (18)) as

where the unprimed and primed quantities represent the corresponding field properties at point $\bm{x}$ and $\bm{x}^{\prime}=\bm{x+r}$ respectively.

Now, we calculate the time evolution of the energy correlators. Similar to Eq. (16), we can also write the evolution equation for $\bm{u}^{\prime}$. Now combining $\bm{u}\cdot\partial_{t}\bm{u}^{\prime}$ and $\bm{u}^{\prime}\cdot\partial_{t}\bm{u}$, we obtain

where, $D_{u}=\langle\bm{u}^{\prime}\cdot\nu\nabla^{2}\bm{u}+\bm{u}\cdot\nu\nabla^{2}\bm{u}^{\prime}\rangle$ and $F_{u}=\langle\bm{u}^{\prime}\cdot\bm{f}+\bm{u}\cdot\bm{f}^{\prime}\rangle$ represent the effective dissipation and forcing contributions in $\partial_{t}\left\langle\bm{u}\cdot\bm{u}^{\prime}\right\rangle$. To obtain Eq. (25), we also use the property of statistical homogeneity

and the following relations:

Again, similar to Eq. (17), one can also obtain an evolution equation for $\bm{q}^{\prime}$. Combining $\bm{q}\cdot\partial_{t}\bm{q}^{\prime}$ and $\bm{q}^{\prime}\cdot\partial_{t}\bm{q}$, we get

where $D_{q}=\langle\bm{q}^{\prime}\cdot\mathcal{M}\nabla^{2}\left(\bm{\nabla}\mu\right)+\bm{q}\cdot\mathcal{M}{\nabla^{\prime}}^{2}\left(\bm{\nabla}^{\prime}\mu^{\prime}\right)\rangle$ and $F_{q}=\langle\bm{q}^{\prime}\cdot\bm{g}+\bm{q}\cdot\bm{g}^{\prime}\rangle$. In the following, we derive the exact relation in two ways [55]:

In this paper we derived several exact relations for fully developed, three dimensional, homogeneous binary fluid turbulence. Using this relation, we can calculate the scale-to-scale transfer rate of total energy (kinetic plus active energy) within the so-called inertial range. Previously, it has been argued [24] that simple binary fluids and incompressible MHD are structurally similar on account of the equations of dynamics and linear wave modes. Drawing analogy with IK phenomenology, they also predicted a $-3/2$ power law for turbulent energy spectra. However, for fully developed turbulence, our paper shows that the generic form of the exact relation derived in Eq. (32) differs from that of incompressible MHD turbulence [8]. In particular, the flux $\bm{\mathcal{F}}$ in Eq. (32) has two sign reversals in comparison with that of Eq. (3). Interestingly, active binary fluids with contractile stress ($\xi<0$) is found to be algebraically identical to the exact relation of incompressible MHD turbulence if we replace the magnetic fields in Eq. (3) with $\sqrt{|\xi|}\bm{q}$. Nevertheless, this two systems are categorically different with respect to the linear stability analysis. Under weak perturbations, an active binary fluid with contractile stress leads to linear instability whereas an incompressible MHD fluid responds to weak perturbations in terms of Alfvén waves. From Eq. (32), we retrieved the exact relation for passive scalar turbulence in the limit of vanishing activity parameter. In addition, we have predicted a possible inverse cascade of energy in 3D active binary fluid turbulence with extensile stress when the correlation between the $\bm{u}$ and the $\bm{q}$ fields are sufficiently weak. Inspired by the Elsässer variables, here we introduced the ‘upsilon’ variables and wrote the dynamical equations for binary fluids in a more symmetric form by the introduction of upsilon variables. Finally, from Eq. (32), we also wrote the exact relation in terms of the upsilon variables. Interestingly, this exact relation looks exactly similar to that of incompressible MHD turbulence when expressed in terms of the Elsässer variables. However unlike incompressible MHD, here the cross-helicity $\int(\bm{u}\cdot\bm{q})d\tau$ is not an inviscid invariant. A helicity cascade is therefore not guaranteed [57] and hence the derivation of the corresponding exact relation is not useful in BFT.

Based on our previous analysis, here we propose a plausible phenomenology and predict a power law for the turbulent energy spectrum in simple and active binary fluids with extensile stress. Writing the Eqs. (49) and (50) in a compact form, we obtain

In the above equations, possible nonlinear interactions can be obtained from the terms $(\bm{u}\cdot\bm{\nabla})\Upsilon^{\pm}$ and $(\bm{Q}\times\bm{\nabla})\times\bm{\Upsilon^{\pm}}$. Whereas $(\bm{u}\cdot\bm{\nabla})\Upsilon^{\pm}$ represent the advection of $\bm{\Upsilon}^{\pm}$ by the velocity field, the terms $(\bm{Q}\times\bm{\nabla})\times\bm{\Upsilon^{\pm}}$ represent the variation of $\bm{\Upsilon}^{\pm}$ in a plane perpendicular to $\bm{Q}$. Expressing $\bm{Q}$ as a sum of the mean field $\bm{Q}_{0}$ and the fluctuation $\widetilde{\bm{Q}}$ in Eq. (54), we get three types of interactions. While we associate two kinds of non-linear time scales $\tau^{u}_{\ell}$ and $\tau^{Q}_{\ell}$ corresponding to the terms $(\bm{u}\cdot\bm{\nabla})\widetilde{\bm{\Upsilon}}^{\pm}$ and $(\widetilde{\bm{Q}}\times\bm{\nabla})\times\widetilde{\bm{\Upsilon}}^{\pm}$ respectively, one linear time-scale $\tau_{0\ell}$ corresponds to the term $(\bm{Q}_{0}\times\bm{\nabla})\times\widetilde{\bm{\Upsilon}}^{\pm}$, with $\widetilde{\bm{a}}$ representing the fluctuating part of the vector $\bm{a}$. This linear time can be associated with the concentration waves as defined in [24] and the corresponding dispersion relation is given by $\omega(k)=\pm\sqrt{\xi}|\bm{k}\times\bm{Q}_{0}|$. If the two non-linear interactions are assumed to be independent, the effective distortion time-scale $\tau_{\ell}$ becomes

where $\ell(\equiv|\bm{r}|)$ is the characteristic size of the eddies, $u_{\ell}\sim|\delta\bm{u}|$, $Q_{\ell}\sim|\delta\bm{Q}|$, $\tau^{u}_{\ell}\sim\ell/u_{\ell}$ and $\tau^{Q}_{\ell}\sim\ell/\widetilde{Q}_{\ell}$. Analogous to the Alfvén time scale in incompressible MHD, here we can also define the linear time scale $\tau_{0\ell}\sim{\ell}/Q_{0}$. In the presence of strong $\bm{Q}_{0}$, we have $\tau_{0\ell}<<\tau_{\ell}$. In that case, the energy transfer time-scale $\tau^{tr}_{\ell}\sim\tau_{\ell}^{2}/{\color[rgb]{0,0,0}\tau_{0\ell}}$ and the energy flux rate $\varepsilon$ would scale as [58, 2, 3]

Under the assumption of weak correlations between $u_{\ell}$ and $Q_{\ell}$, we have $\widetilde{\Upsilon}^{+}_{\ell}\sim\widetilde{\Upsilon}^{-}_{\ell}\sim\widetilde{\Upsilon}_{\ell}\sim u_{\ell}$ and hence $\tau_{\ell}\sim\tau^{u}_{\ell}\sim{\ell}/\widetilde{\Upsilon}_{\ell}$. Combining all, finally we can write

Again, by definition of energy spectrum $E(k)$

where $k$ is the wavenumber corresponding to $\ell$. This is in agreement with the predictions of [24], which was done for simple binary fluids only.

Similar types of studies can be generalized to the compressible binary fluids as well as to the mixtures of more than two fluids.

The authors acknowledge Jayanta K. Bhattacharjee for useful discussions. NP acknowledges the grant under PMRF scheme (Government of India). SB acknowledges the DST-Inspire faculty research grant (DST/PHY/2017514).