Polymers in fluid flows
Abstract
The interaction of flexible polymers with fluid flows leads to a number of intriguing phenomena observed in laboratory experiments, namely drag reduction, elastic turbulence and heat transport modification in natural convection, and is one of the most challenging subjects in soft matter physics. In this paper we review our present knowledge on the subject. Our present knowledge is mostly based on direct numerical simulations performed in the last twenty years, which have successfully explained, at least qualitatively, most of the experimental results. Our goal is to disentangle as much as possible the basic mechanisms acting in the system in order to capture the basic features underlying different theoretical approaches and explanations.
Keywords: polymerflow interaction, drag reduction, elastic turbulence, heat transport modification
Posted with permission from the Annual Review of Condensed Matter Physics, Volume 9 ©2018 by Annual Reviews, http://www.annualreviews.org/.
I Introduction
The interaction of flexible polymers with fluid flows is one of the most challenging subjects in soft matter physics. Laboratory experiments show that even a minute concentration of polymers can dramatically change properties (drag reduction) of turbulent flows or, if the flow is laminar, can trigger a new form of turbulence named elastic turbulence. How to properly describe flowpolymer interactions and how to understand the basic physical phenomena observed in experiments is both a fundamental scientific issue and an important challenge to develop many industrial applications. In this paper we review our present knowledge on the subject. In the last twenty years, direct numerical simulations of polymerflow dynamics have successfully explained, at least qualitatively, most of the experimental results. Starting from this consideration, our goal is to disentangle as much as possible the basic mechanism acting in the system. Many different theories have been proposed and we feel confident that future work can provide a reasonably unified approach on the subject.
Ii Polymer Flow Interaction
ii.1 Polymer passive advection
We start our review by understanding the dynamics of polymers advected by a turbulent fluid flow. A flexible polymer can be considered as a chain of monomers. The size of a polymer in its stretched configuration is of the order of tens of micrometers. In most cases, this size is much smaller or at most equal to the smallest dissipative scale of turbulence, namely the Kolmogorov scale , where is the kinematic viscosity of the fluid and is the rate of turbulent energy dissipation. In this section we consider a polymer to be a passive object advected by the flow velocity and neglect its feedback to the flow. The important information comes from the endtoend vector of the polymer configuration. In its simplest form the Lagrangian dynamics of is described by Birds1987 (); lebedev1 (); lebedev2 ()
(1) 
where is the component of the velocity field, is the polymer relaxation time, is a parameter that takes into account the role of thermal fluctuations, and is independent white noise, deltacorrelated in time. The relaxation time depends on the chemical and physical properties of the polymer, and ranges from s up to tens of seconds. The function takes into account the finite extensibility of the polymer and that the maximum extension of polymer size is . Hereafter we choose , with . Using Equation 1, we can describe the statistical properties of polymers by considering the conformational tensor where the average is done on the noise. Upon using Ito calculus, the Eulerian equations for the conformational tensor are:
(2) 
First we consider the case , assuming that at any time. If we follow the polymer along its trajectory, the quantities are a function of only. Starting from Equation 1, we are interested in computing the statistical properties of . Following Referenceslebedev2 (); chertkov (); vincenzi1 (), we can compute the probability distribution from the knowledge of the long time behavior of the Lyapunov exponent of the Lagrangian trajectory. Let be the solution of the linear equation:
(3) 
with initial condition . For chaotic flows, the size grows exponentially in time as where is the finite time Lyapunov exponent, i,e, , being the Lyapunov exponent of the flow. It is also known benzi1 () that the probability of observing the value at time is given by , where is known as the Cramer entropy and is a convex function with a minimum at . The Cramer entropy can be used to estimate the large deviation properties of . More precisely, one can show that where is the Legendre transform of , i.e. . Asymptotically, we can estimate from Equation 1 that and that stretching properties of the polymer depend on the dimensionless number Wi. The quantity Wi is called Weissenberg number, although in many papers no reference to the Lyapunov exponent is made. Clearly there exists a critical value of for which the polymer tends to stretch indefinitely (for the case of ). This critical value signals the socalled coilstretch transition for the polymer. It is possible to show lebedev2 (); chertkov () that, for , the polymer extension shows a power law distribution where . The quantity is not known in general and depends on the intermittent features (if any) of the Lagrangian trajectory. In most cases one can assume as a first approximation and obtain
(4) 
From Equation 4 we can immediately see that : For the value of is positive and the probability distribution is normalizable; for the exponent becomes negative and is no longer normalizable. The ratio is a quantitative measure of intermittency for the Lagrangian trajectories: Strong intermittency implies and small values of ; small intermittency is equivalent to and large values of .
For a large value of Wi, we must consider the case in Equation 1. In this case, the analysis, performed in vincenzi2 () and vincenzi3 () in some simplified case, shows that coilstretch transition does not change qualitatively, although the existence of a scaling range for depends on flows details. The overall message coming from the above discussion is rather clear: A single polymer passively advected in a chaotic flow undergoes a coilstretch transition when Wi is larger than a critical value . The precise value of depends on the flow properties [i.e. the function ], although it is reasonable to guess it is of order one. So far one has not been able to measure experimentally. For fully developed turbulence, one may use the simple estimate where is the Kolmogorov time. Alternatively, one can estimate as where is a characteristic velocity of the flow and its characteristic scale. Quite often the Weissenberg number defined by is also referred to as the Deborah number, . Not surprisingly, depending on the definition, the critical value of or changes considerably. However, the basic point of the above analysis is that the time criterion based on is the one to be used to capture the polymer coilstretch transition.
and coilstretch transition have been investigated numerically boffetta1 (); Gotoh2010 (); Schumaker2006 () and experimentally in Reference steinberg1 () (see Figure 1). In the experiment, polymers were advected in a swirling flow between the edge of a uniformly rotating glass of radius and angular velocity . The Weissenberg number is defined as where is the gap between plates. The authors considered polymers with m and m and they were able to measure the value by fluorescent techniques. A clear coilstretch transition is observed for . For , shows a powerlaw distribution, and for the smallest Wi studied. The experiment shows a rather remarkable qualitative agreement with the theoretical framework discussed in this section.
ii.2 When turbulence is affected by polymers
After discussing how polymers are affected by a fluid flow, we now discuss the change in the fluid flow due to the presence of polymers. We consider threedimensional homogeneous and isotropic turbulent flows far away from boundaries and review the ideas of Lumley Lumley73 () and de GennesTabordeGennes (); deGennes () on when turbulence would be affected by polymers.
For threedimensional homogeneous and isotropic turbulence, the Kolmogorov 1941 theory holds approximately with corrections due to intermittency. Energy is injected at the large scales, and cascaded down to smaller scales, and eventually dissipated at the Kolmogorov scale at which molecular viscosity directly acts. At each scale in the inertial range, there is a characteristic fluctuating velocity , related to by
(5) 
A crude estimate of the stretching at scale can be obtained by . Upon using Equation 5, we obtain . Our discussions in Section 2.1 lead one to think that polymers would affect turbulence at scales , where . Thus for scales , where is defined by , turbulence would be affected in a certain way. This is the idea originally proposed by Lumley. According to this idea, polymers would affect the turbulent flow as long as , regardless of the concentration of the polymers.
De Gennes followed a completely different approach. He thought that polymers could produce effect on turbulence only when the elastic energy stored by the polymers becomes comparable to the turbulent kinetic energy. He assumed that at scale , the elastic energy of polymers per unit volume would decrease with as
(6) 
where is the concentration of polymers per unit volume, is the Boltzmann constant, is the temperature of the polymer solution, and is some positive exponent depending on the specifics of the flow. Note that in Equation 1, the elastic energy is taken to be quadratic in , i.e. . The turbulent energy per unit volume at scale is given by , where is the density of the polymer solution and increases with . Thus polymers would affect turbulence at scales , where the scale is the scale at which the elastic energy and the turbulent kinetic energy become equal. Using Equations 5 and 6, and the expression for , we obtain
(7) 
where . For very small , is smaller than , and polymers cannot affect turbulence. Thus there exists a threshold concentration for the polymers to have any effect on turbulence. For dilute polymer concentrations above the threshold value, and as concentration increases approaches . De Gennes argued that polymers should be considered as passive for and the turbulent energy cascade would stop at , so can be thought of as an effective cutoff scale replacing the Kolmogorov scale . However, he provided no information on how the turbulent energy is eventually dissipated.
Thus the theory of de Gennes allows us to see clearly how and why the concentration of polymers must play a role in their effects on turbulence, whereas Lumley’s idea leads to the scale , which is a possible upperbound scale for polymers to affect turbulence. However, neither of the two theories gives any details on how turbulence is affected. Several experiments have been carried out to study the change in statistics of a turbulent counterrotating disk flow by polymers NJP2008 (); Ouellette2009 (); Xi2013 (); Quitry2016 (). Turbulence statistics such as Eulerian velocity structure functions are modified in the presence of polymers. It is found that the Eulerian velocity structure functions can be made to collapse into one master curve in Reference Xi2013 () but into two families, one for low polymer concentration and another for higher concentration in Reference Quitry2016 (), when the separation is rescaled by some length scale that depends on polymer concentration and relaxation time . The precise form of this length scale is different in these experiments. Suggestions to relate this length scale to or a new length scale at which the turbulent kinetic energy flux and the elastic energy flux are equal have been made Xi2013 (). However, we have yet to obtain a full understanding of all these experimental results.
Iii Polymer Effect on Flows: Experiments
iii.1 Drag Reduction
One wellknown effect is polymer reduces friction drag in turbulent wallbounded flows. This effect was discovered by Toms Toms (), who observed that an addition of about 10 parts per million by weight of a long chain polymer (polymethyl methacrylate) can lead to a significant reduction of friction drag in a turbulent flow of monochlorobenzene in a pipe while studying the degradation of polymers. Similar effect has been observed in turbulent pipe or channel flow of water with polyethylene oxide or polyacrylamide. A large number of experimental studies have been carried out to characterize this phenomenon (see e.g. Reference Virk75 () for a review). Below, we will summarize the key features of the phenomenon.
For a fluid of density flowing in a pipe of diameter , the friction drag is measured by the dimensionless Fanning friction factor , defined as the ratio between wall shear stress , which is the work done per unit volume due to the applied pressure gradient, and the kinetic energy density of the mean flow
(8) 
Here is the pressure drop across a distance in the pipe, and is the mean velocity, averaged over time and the crosssection of the pipe, and is given by , where is the streamwise velocity averaged over time and is the distance from the wall. When polymers are added, is not changed at low Reynolds number . At a certain value Re, the onset of drag reduction occurs and is reduced. Experimental measurements show that the onset value Re depends on polymer concentration Nadolink1987 (). The reduction of is tantamount to an increase in the mean streamwise velocity for a given pressure drop or a decrease in the pressure drop required for a fixed mean streamwise velocity. Such an enhancement of by polymers is shown in Figure 2 in terms of the commonly used wall units, defined by and , where is the friction velocity. After the onset, decreases when the polymer concentration is increased, and approaches an apparently universal limiting curve that cannot be exceeded by increasing the concentration further. This limit is referred to as the maximum drag reduction (MDR) asymptote (see Figure 2a). Theories of drag reduction will be discussed in section 5.2.
iii.2 Elastic Turbulence
Another interesting effect is that polymer with sufficiently long relaxation time or high Weissenberg number Wi can give rise to an irregular flow state with velocity fluctuations spanning a broad range of spatial and temporal scales even at low Reynolds number. This irregular flow state at high Wi and low Re is known as elastic turbulence Steinberg2000 () and is caused by an instability due to the polymer stresses. The first experiments of elastic turbulence were performed in dilute polymer solution in flows with curved streamlines such as swirling flow between two plates, TaylorCouette flow, and flow in a curvilinear channel Steinberg2004 (). More recently, elastic instability has also been observed experimentally in a long, straight microchannel Pan2013 ().
For example, in a swirling flow between two plates, the fluid in a stationary cylindrical cup of flat bottom plate is set into swirling motion by a coaxial rotating upper plate just touching the surface of the fluid. The average shear stress is measured as a function of the average shear rate . For small Weissenberg number, defined by , remains close to the value for a laminar shear flow at the same angular velocity , where is the viscosity of the polymer solution. It is found that when Wi is above some critical value, increases significantly and there is a transition from the laminar flow to elastic turbulence. For the same range of shear flow rates of the pure solvent without polymers, the ratio remains at unity within resolution of the measurements. The frequency power spectrum of the velocity fluctuations in elastic turbulence displays a powerlaw decay, which spans over about a decade in frequencies. The powerlaw dependence indicates a broad range of time scales of the motion and resembles that of developed turbulence of Newtonian fluid at high Re but the energy spectrum displays a steeper slope indicating that the flow is random but not as “rough” as in the case of fully developed turbulence of Newtonian fluid. The fluctuating velocity field can be visualized in the snapshots of the flow of the polymer solution above the transition as shown in Figure 2b.
Iv Equation of Motion
Given Equation 2 for the conformational tensor , the effect of polymers on the flow can be computed using the momentum balance, which reads as
(9)  
(10) 
Here, is the kinematic viscosity of the solvent and (with components ) represents the momentum stress tensor due to polymer, where is the polymer contribution to the zeroshear viscosity of the polymer solution and increases with polymer concentration Birds1987 (). Equation 9 is often presented by writing and , which is convenient for numerical studies with a constant . The overall free energy in the system is defined as
(11) 
Using Equations 2 and 10, one can obtain the energy dissipation,
(12) 
Equations 2 and 9 with 10 are known as the finite extensible nonlinear elastic model with Peterlin’s approximation, or the FENEP model in short, for dilute polymer solutions. The FENEP model is based on the assumption of a simple nonlinear elastic force describing polymer stretching whereas more complex physical effects are not taken into account (see, for instance Reference Celani2006 ()) and it is always challenging to compare the relaxation time in the FENEP model against real experiments. Nevertheless the FENEP model can qualitatively capture the relevant experimental observations described in the previous section, namely elastic turbulence and drag reduction. If one ignores the finite extensibility of the polymer, and the resulting equations are usually referred to as OldroydB model. In the past few years, there have been a number of direct numerical simulations (DNSs) of the FENEP or OldroydB model for wall bounded turbulence Taegee2003 (); Sureshkumar1997 (); Dimitropoulos1998 (); DeAngelis2004 (); Ptasinski2003 () (see also White2008 () for a detailed review). In Figure 3a, we show the numerical results obtained in Taegee2003 () for the stream wise velocity profile for different values of Weissenberg number (here denoted by We). There is a clear increase in the velocity or a decrease in drag as We increases. The numerical results should be compared against the laboratory measurements shown in Figure 2a. This is clear evidence that the FENEP or OldroydB model captures the qualitative behavior of drag reduction found in laboratory, although a quantitative agreement may be available only after fine tuning of the model parameters. Twodimensional numerical simulations have been studied in References Boffetta2003 (); Boffetta2008new () for a Kolmogorov flow with periodic boundary conditions. The system is forced with a periodic forcing and attains a velocity field where is the size of the system. It is known that for smaller than the critical value , the flow is stable when there are no polymers. By increasing the Weissenberg number , the system develops an instability and the dynamics becomes chaotic in time and space, i.e. the system shows the characteristic features of elastic turbulence. Figure 3b, as well as a more detail analysis given in Reference Boffetta2008new (), clearly shows that elastic turbulence can be explained (at least qualitatively) by the constitutive Equations (2) and (9).
There is a simple but highly nontrivial argument that allows us to understand the role of the concentration parameter in the FENEP model Benzi2004 (); Procaccia2008 (). We consider the case where there is substantial stretching such that in Equation 10 can be neglected. Let us assume that the concentration changes as where is any positive real number. This is equivalent to saying that and, more importantly, the function changes as
(13) 
which shows that by changing the concentration we (formally) change the value of the maximum polymer extension in the FENEP equations. Next, we observe that it is possible to define an effective polymer relaxation time
(14) 
which gives
(15) 
Equation 15 tells us two pieces of information: (a) For we obtain , i.e. for high concentration the FENEP model becomes independent of the concentration itself; (b) for even a very small amount of stretching leads to . In Section 2.1, we see that polymer stretching occurs for , thus we can safely assume that the effect of polymers on the flow becomes relevant when is large enough. This is equivalent to say that there exists a critical value or a critical concentration, for which : For or concentration below the critical value, the effect of polymers is negligible, whereas for polymers can change the (turbulent) properties of the flow. In other words, the FENEP model seems to capture the basic argument due to de Gennes on the existence of a critical concentration for polymer effects on flows, without assuming any critical scale and/or any cascade argument for the energy flux. As a side result, for the OldroydB model with , the effect of the concentration disappears and any change in the concentration is equivalent to a redefinition of with no effect on the dynamics; the only relevant parameter being the polymer relaxation time . This implies that changing the polymer relaxation time in OldroydB model is qualitatively equivalent to changing concentration parameter for fixed in the FENEP model. The above discussion is independent of the way turbulence is generated and, in particular, of any wall effect.
V Drag Reduction
v.1 Drag reduction without drag
The basic feature of fully developed threedimensional turbulence is the Reindependence of the energy dissipation rate . This feature, sometimes referred to as the zeroth law of turbulence, is often written as
(16) 
where represents the typical turbulent velocity and its corresponding scale. Notwithstanding the arbitrary definition of and , the constant has the physical meaning of the drag coefficient. To see that this is the case, we can compute as where is the forcing per unit volume acting on the system. From Equation 16 we obtain:
(17) 
which is precisely the definition of the friction factor in wallbounded turbulence (see Equation 8). Using Equation 16 and/or Equation 17, one can study the phenomenon of drag reduction for systems with no boundaries. Here we show some of the many cases studied in recent years. We start by reviewing the results obtained for three dimensional turbulent flow for a Kolmogorov forcing with periodic boundary conditions and mean flow . We considered the limit of large Re=, and relatively small . In Figure 4a, we show the main result obtained in Boffetta2005a () for and . The comparison between the Newtonian case () and the effect of polymers () clearly shows that there is an increase of the mean flow due to polymers. A direct computation of given by Equation 17 shows a significant amount of drag reduction by almost a factor of two. Similar results have been obtained for turbulent flows forced by constant shear Brasseur2010 (). Following Reference Boffetta2005b (), the case of Kolmogorov forcing can be studied analytically for small Re and De. The stability diagram for the OldroydB model is displayed in Figure 4b. Two clear regions are present: For small Re and De, the critical value of the Reynolds number increases with increasing De, i.e., the flow is more stable due to the effect of polymers; for small Re and large De a second region of unstable flows appears and corresponds to elastic turbulence discussed in Section 3.2. Using multiplescale analysis, it is possible to compute analytically the drag coefficient for relatively small Re and De (see Reference Boffetta2005c ()). One obtains with a positive constant, i.e. drag tends to decrease as De is increased. Although the above analytical results do not refer to turbulent flows, they clearly show that the phenomenon of drag reduction should be considered in more general terms and not only restricted to wallbounded turbulent flows. The computation of the drag coefficient can also be done for homogeneous and isotropic turbulence. In Reference Deangelis2005 (), DNSs of the FENEP model have been performed for different values of De. In Table 1, we show the value of (rightmost column) as a function of De (first column): Clearly for large enough De, we observe the effect of drag reduction (in the way defined in this section). To perform a scale by scale analysis for homogeneous and isotropic turbulence with polymers Deangelis2005 (), we consider , where is a vector field lebedev2 (). The equation for and are given by
(18)  
(19) 
where with the repeated index summation notation used. The relative advantage in using Equations 18 and 19 is that we can easily compute the analog of the equation in this case, namely:
(20) 
where , and . We identify the de Gennes length scale as , the value of at which the two terms on the lefthand side of Equation 20 are equal. Similarly we can define the scale as the value of at which the two terms on the righthand side are equal. If , we obtain
(21) 
where we have used Equation 5. In other words the de Gennes scale satisfies the generalised Lumley criteria given by Equation 21. Using DNS Deangelis2005 () it is possible to show that , as reported in Table 1. The overall picture is that, within the FENEP model, there exists a unique scale satisfying both the de Gennes criterion and the generalized Lumley criterion Equation 21. Hereafter, we shall refer to the scale as the Lumleyde Gennes scale. As discussed in the previous section, depends on the concentration. We remark that something similar to the scale has been extracted from experimental data in Reference Xi2013 () as discussed in Section 2.2.
De  

0  0.156      1.87 
0.18  0.174  1.23  0.603  1.90 
0.54  0.238  2.20  0.973  1.55 
0.54  0.232  2.20  0.923  1.60 
Equations 18 and 19 can be used to define a shell model Benzi2004 (); Benzi2003 (); Pandit2005 (); Benzi2004b (). It is worth noting that shell models have been extensively used to successfully understand some features of the statistical properties of intermittency in three dimensional turbulence. In this case, shell models are able to reproduce the relevant statistical features of DNS of homogenous and isotropic turbulence for FENEP model, including drag reduction as previously defined. Also shell models can be used to demonstrate the existence of elastic turbulence at extremely low Re, as recently shown in Ray2016 () (see Figure 5a. In short, shells model can be used as a simple numerical tool to understand the basic features of energy fluxes end energy exchange in the FENEP model and provide a clear cut definition for the Lumleyde Gennes scale .
v.2 Drag reduction in wall bounded flows
We now turn our attention on the phenomenon of drag reduction in wallbounded turbulence. Our major interest is to understand how the average velocity profile changes due to polymer effects in turbulent flows, as described in section 2.1. Our starting point is to briefly review the von Karman theory for wallbounded turbulence. Using wall units defined in section 3.1, we can define , the dimensionless shear of the average flow and the (absolute) turbulent momentum flux. Then momentum conservation reads:
(22) 
Hereafter we shall neglect the last term on the righthand side of Equation 22. To close the problem, we need a relation between and in agreement to the von Karman hypothesis. It turns out Procaccia2008 () that such a phenomenological equation can be derived using the energy balance in the form of
(23) 
Upon multiplying Equation 23 by the turbulent kinetic energy , we can interpret as follows: the first term on lefthand side is the turbulent dissipation near the viscous layer ; the second term on the lefthand side is the turbulent energy dissipation obtained by von Karman original argument while the term on the righthand side is the turbulent energy production. Solution of Equations 22 and 23 gives the Newtonian profile (black line) shown in Figure 2 panel (b) with and . Asymptotically, we have where . The region between the asymptotic behavior of and is usually referred to as the buffer layer, i.e., a region near the wall where the maximum of turbulent energy production is located.
Lumley Lumley73 () and de Gennes deGennes () provided different theoretical interpretations for drag reduction in wallbounded turbulent flows ( see also White2008 () for a recent review). Lumley attributed drag reduction to an increase of the buffer layer: “The conclusion is that, granted the law of the wall and the defect law, a drag reduction must appear as a thickened sublayer…The sensitivity is such that doubling the sublayer thickness about halves the skin friction” (Lumley73 (), p. 376). According to Lumley, the increase of the buffer layer is due to an overall increase in the fluid viscosity because of polymer stretching and there should exist a critical polymer relaxation time for drag reduction to occur. He also argued, without any quantitative derivations, that the increase should be proportional to polymer concentration for small concentration. At variance with Lumley’s interpretation, de Gennes suggested that polymers changes the Kolmogorov cascade of turbulence at a critical scale where most of the turbulent energy goes to polymer elastic stretching. The de Gennes scale depends on polymer concentration and there should exist a critical concentration for drag reduction to occur. Nevertheless de Gennes argued: “But the net result is still an enhancement of the intermediate ‘buffer layer’. We expect drag reduction from this, although we have not carried out detailed analog Lumley’ matching” (deGennes (), p. 35). It is worthwhile to mention that de Gennes argued that the critical scale for polymer stretching should act as effective cutoff scale for turbulent fluctuations, i.e., an effective Kolmogorov scale, although de Gennes stated “Thus we do not know the ultimate fate of the turbulent energy” (deGennes (), p. 44). Finally, de Gennes never mentioned drag reduction as induced by an energy flux that went from small scale polymer fluctuations to turbulent fluctuations. At any rate, neither de Gennes nor Lumley was able to provide the explanation of why the buffer layer should increase due to polymer stretching. Also, at the time when Lumley and de Gennes theories were proposed, no DNSs based on the FENEP or OldroydB models were available. Our task is now to provide a quantitative explanation for drag reduction and MDR within the framework of the FENEP or OldroydB models, in which the ultimate fate of turbulent kinetic energy, i.e., the energy dissipation, is given by Equation 12. One possible answer to our question is provided in References Procaccia2008 () and Benzi2006 (), and it is based on the idea that the polymer stretching produces an effective spacedependent viscosity. The phenomenological theory is based on the following observations:

In wallbounded turbulent flows, where the local shear is large, the polymer conformational tensor satisfies the inequalities .

To first order approximation one can estimate and where is given in Equation 14.

Turbulent kinetic energy is fixed by the Lumleyde Gennes criteria .

Turbulent energy dissipation is the sum of the viscous effect and .
Point 2 implies that that terms can be neglected, where is the turbulent velocity fluctuation in direction. Based upon the previous points, one can show that Equations 22 and 23 are modified as
(24)  
(25) 
where , and is a constant depending on the ratio and , and is a constant that approaches in the limit concentration . We remark that the very existence of a spacedependent viscosity can be numerically checked by computing the quantity , where is the total energy dissipation in the system and is the energy dissipation due to velocity fluctuations. It turns out that is a linear function of with very good accuracy (see, for instance Figure 5b). Also, the existence of a spacedependent viscosity can be derived in the generalization of the classical PrandtlBlasius boundary layer theory for the OldroydB or FENEP models, as recently shown in References Blasiusjfm (); FENEPjfm () (see Section 6). The approach described in Reference Procaccia2008 () justifies two independent questions: (a) Does spacedependent viscosity produce drag reduction? (b) What is the functional form of that maximizes drag reduction and, eventually, provide a quantitative prediction of the MDR asymptote? In principle, both questions can be answered using numerical simulations. In Reference Deangelis2004b (), linear viscosity profiles are shown to produce drag reduction in wallbounded turbulent flows and, for increasing slope of the spacedependent viscosity, one observes a decrease in the drag. Concerning the second question, using Equations 24 and 25, Reference Benzi2005b () showed that the MDR can be achieved when is linear and that the resulting velocity profile is given by
(26) 
assuming . Equation 26 is plotted in Figure 2b and it is extremely close to the MDR behavior discussed in Reference Virk75 (). It is interesting to remark that the asymptotic behavior given by Equation 26 is achieved in the limit , i.e., in the limit turbulence tends to vanish. In this limit, therefore, the Lyapunov exponent of the system becomes extremely small and, consequently, the correlation time of any fluctuating quantity becomes very long. This phenomenon is observed in DNS of the FENEP model as discussed in Reference Graham2012 (), where periods of hibernation and activation of turbulence activity are observed near MDR. We emphasize that the formulation of the MDR as a variational problem for the spacedependent viscosity implies the universality of the MDR behavior, independent of polymer characteristics. For very large Wi there may be nonnegligible contributions from correlations between the fluctuating strain rates and the conformational tensor . These contributions may support turbulent fluctuations near the MDR asymptote in a way similar to the effect of elastic turbulence, i.e., there may be on average a flux of energy from polymer to turbulence. This possibility was pointed out in References Dubief2012 () and Dubief2013 () and referred to as elastoinertial turbulence. However, this effect may be consistent with the existence of a spacedependent viscosity described in Equations 24 and 25 . Also, for homogeneous shear flow simulations reported in Brasseur2010 () as well as in the numerical simulations of wallbounded turbulence discussed in Reference Ptasinski2003 (), no energy flux from polymer to turbulence has been found.
It is worth noting that asymptotic behavior similar to MDR is observed in numerical simulations of the FENEP model in homogeneous shear turbulence Brasseur2010 (), where the concept of a spacedependent viscosity cannot work. However, the concept of spacedependent viscosity can be generalized to scaledependent viscosity as discussed in Reference Benzi2004c () and a scaledependent viscosity may explain drag reduction for the cases reviewed in Section 5.1.
Vi Polymer in Heat Transfer Flows
The effect of polymers on heat transfer has been studied in turbulent RayleighBénard (RB) convection. In the RB system, a fluid is constrained between two horizontal plates that are heated from below and cooled from above, and the system is controlled by two parameters: the Rayleigh number, Ra , which measures the thermal forcing due to the temperature difference between the two plates, and the Prandtl number, Pr , which is the ratio between the kinematic viscosity and the thermal diffusivity of the fluid. In addition, is the isobaric volume expansion coefficient of the fluid, the acceleration due to gravity, and is the vertical distance between the top and bottom plates. In turbulent RB convection, there are distinct flow regions, namely viscous boundary layers near all rigid walls and two thermal boundary layers, one above the bottom plate and one below the top plate, and an approximately homogeneous bulk flow in the central region of the convection cell. In the Boussinesq approximation, the equations of motion for RB convection with polymers are
(27)  
(28) 
where is the temperature field, is the mean temperature averaged over time and the whole system, is along the vertical direction, and the polymer stress tensor is given by Equation 10 with the conformation tensor governed by Equation 2. Heat transport is measured by the Nusselt number (Nu), which is the normalized heat flux defined by
(29) 
where denotes an average over a horizontal plane of the convection cell and time.
The bulk flow of turbulent RB convection without the boundary layers is believed LohseToschiPRL () to be a good approximation of the ultimate regime at large Ra. The effect of polymer in this regime has been studied by DNS of the equations of motion with periodic boundary conditions BCEPRL (). In this case, polymer enhances the length scale of thermal plumes and heat transfer increases as Nu , where Wi and . A similar enhancement in heat transfer by polymers has been found in RayleighTaylor turbulence Boffetta2010 (); Boffetta2017 () (see Figure 6). In contrast, an experimental investigation Ahlers () of turbulent RB convection at moderate Ra reported a small but clear reduction of Nu in the presence of polymers, and the amount of heat reduction increases with polymer concentration. At low Ra, most of the energy dissipation is concentrated in the boundary layers, and thus the numerical results for the ultimate regime do not apply. In later experimental studies WeiPRE (); XiaJFM (), both a convection cell with usual smooth top and bottom plates and a convection cell with rough top and bottom plates have been used. The reduction of heat transport by polymers at moderate Ra has been confirmed in the smooth cell, whereas an enhancement in heat transfer is observed in the rough cell when the polymer concentration is not too small (see Figure 7). It is believed that the pyramidal structures of the rough plates perturb the boundary layers and make the flow resembling that of the bulk flow even at moderate Ra.
In an attempt to understand the experimental results at moderate Ra in the usual smooth convection cell, the effect of polymers in the framework of the classical PrandtlBlasius boundary layer theory was analyzed Blasiusjfm (); FENEPjfm (), which is known GL2000 (); Revmodlohse () to be a good approximation for moderate Ra and stable boundary layers. The outcomes of the approach discussed in References Blasiusjfm (); FENEPjfm () are the following: First, the effect of polymers induces an effective spacedependent viscosity in the equation of motion, which peaked in the boundary layer and vanishes away from the boundary layer. Second, upon increasing the polymer concentration, heat reduction is observed, i.e., Nu decreases with respect to corresponding to the Newtonian flow with no polymer at the same zeroshear viscosity. Finally, it can be shown that if the effective viscosity extends into the bulk of the system, i.e., the center of the cell, the value of Nu increases against . Physically this means that heat enhancement should be observed if a substantial stretching of the polymer in the bulk of the system exists. The last statement has been confirmed by a very recent DNS of turbulent RB convection with polymers BCEPRE () at moderate Ra and .
Vii Conclusions
Polymerflow interactions lead to a number of nontrivial and counterintuitive phenomena observed in laboratory experiments, namely drag reduction, elastic turbulence and modification of heat transport in natural convection. Our present knowledge is mostly based on direct numerical simulations, which provide a qualitative (and in some cases quantitative) explanation of experimental results. Still many different questions should be answered in order to define a possible unified scenario (if it exists). In this review, we have highlighted the most relevant results obtained in the last twenty years trying to capture the basic features underlying different theoretical approaches and explanations. It is fair to say that much progress have been made by suitable analyses of both laboratory and numerical investigations. Such progress has been able to foster new theoretical interpretations and to open a new view on the subject. The same approach can be applied to study the interaction of turbulent and/or laminar flows for other complex fluids as in the case of rigid polymers and surfactants. The outcomes of such investigations will be relevant for both our basic scientific understanding of complex fluid dynamics and technological applications.
Summary Points

FENEP and OldroydB models can explain qualitatively most of the experimental results on drag reduction and elastic turbulence.

A clear cut definition of coilstretch transition has been successfully provided in terms of Lagrangian properties of polymer dynamics.

The notion of drag reduction can be generalized to flows without boundaries.

Polymerflow interaction can change the heat transport in natural and forced convection.
Future Issues

More realistic models of flexible polymers are needed to make quantitative predictions of polymerflow interactions.

Inertial effects on polymer dynamics, due to density mismatch between polymer and flow, need to be investigated.
Disclosure Statement
The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.
Acknowledgments
R.B. acknowledges funding from the European Research Council under the European Union’s Seventh Framework Programme, AdG ERC Grant Agreement No. 339032 and E.S.C.C. acknowledges support from the Hong Kong Research Grants Council (Grants no 400304, 400311).
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