2 Velocity field in the boundary layer

When considering the boundary layer properties, we treat the wall as flat. For the channel flow, it is correct if the thickness of the boundary layer $L$ is of the order of or smaller than the channel radius and its curvature. This condition also guarantees that correlation functions of the statistically stationary in time flow are homogeneous along the wall. Below, we consider the stationary case where the mean flow and the statistical properties of the flow fluctuations are time-independent. Note that our analysis is also applicable to the near-disk boundary layer in the von Kármán swirling flow, provided we are interested in a region far enough from the rotation axis (Burghelea et al. Reference Burghelea, Segre and Steinberg2006, Reference Burghelea, Segre and Steinberg2007).

The Reynolds decomposition for the fluid velocity $\boldsymbol{v}$ reads

(2.1) $$\begin{eqnarray}\boldsymbol{v}=\boldsymbol{U}+\boldsymbol{u},\end{eqnarray}$$

where $\boldsymbol{U}(\boldsymbol{r})$ and $\boldsymbol{u}(\boldsymbol{r},t)$ are the mean and fluctuating parts of the flow velocity, respectively. Due to the dominant role of viscosity inside the boundary layer, the velocity field can be regarded as a smooth and regular function of the coordinates. In particular, the mean velocity is approximated in the leading order by a shear flow

(2.2) $$\begin{eqnarray}U_{x}=sz,\end{eqnarray}$$

where $z\ll L$ and $s$ is the shear rate. Here and below we choose a Cartesian reference system in such a way that the $z$ -axis is perpendicular to the wall, and the $x$ -axis is directed along the velocity of the mean flow: see figure 1.

Figure 1.

Sketch of the polymer orientation geometry.

For the fluctuating velocity $\boldsymbol{u}$ the following proportionality laws are valid:

(2.3a,b ) $$\begin{eqnarray}u_{x,y}\propto z,\quad u_{z}\propto z^{2},\end{eqnarray}$$

provided $z\ll L$ . The laws are consequences of the fluid velocity smoothness, the no-slip boundary condition at the wall and the incompressibility condition $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0$ . Note also that $\boldsymbol{u}$ varies along the wall at distances of the order of $L$ .

3 Single polymer statistics

In the dilute limit, we neglect the mutual interactions of the polymer molecules. At $z\ll L$ a single polymer can be treated as a passive object whose dynamics is governed by the equations

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\boldsymbol{r}=\boldsymbol{v}(\boldsymbol{r},t)+\unicode[STIX]{x1D743}(t), & \displaystyle\end{eqnarray}$$

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\boldsymbol{R}=-\unicode[STIX]{x1D6FE}(R)\boldsymbol{R}+(\boldsymbol{R}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{v}(\boldsymbol{r},t)+\unicode[STIX]{x1D73B}(t), & \displaystyle\end{eqnarray}$$

where $\boldsymbol{r}$ is the coordinate of the mass centre of the polymer; $\boldsymbol{R}$ is the polymer end-to-end elongation vector; $\unicode[STIX]{x1D6FE}(R)$ is the extension-dependent spring constant of the polymer; $\unicode[STIX]{x1D743}$ and $\unicode[STIX]{x1D73B}$ are the Langevin forces having zero means and time-decorrelated second moments $\langle \unicode[STIX]{x1D709}_{i}(t_{1})\unicode[STIX]{x1D709}_{j}(t_{2})\rangle =2\unicode[STIX]{x1D705}\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D6FF}(t_{1}-t_{2})$ and $\langle \unicode[STIX]{x1D701}_{i}(t_{1})\unicode[STIX]{x1D701}_{j}(t_{2})\rangle =2\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FF}_{ij}\unicode[STIX]{x1D6FF}(t_{1}-t_{2})$ with $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D702}$ representing the diffusivity for translational and elongational degrees of freedom of polymer motion, respectively.

The physical picture behind (3.1) and (3.2) is very simple: the polymer is advected along the Lagrangian trajectory being stretched by the velocity gradient and relaxing to its equilibrium shape due to elasticity. Note that the specific form of the function $\unicode[STIX]{x1D6FE}(R)$ is not essential in the scope of this work. The only information on the elastic properties of the polymer required for our analysis is the spring constant in the Hookean (linear) limit $\unicode[STIX]{x1D6FE}_{0}=\unicode[STIX]{x1D6FE}(0)$ and the maximum polymer extensibility $R_{m}$ . We imply that $R_{m}$ is small compared with the boundary layer thickness $L$ .

In this section we use (3.1) and (3.2) with the velocity field $\boldsymbol{v}$ given by (2.1)–(2.3) to describe the statistics of the polymer conformation inside the boundary layer in its dependence on the distance from the wall $z$ . Our analysis is based on several assumptions, to be justified: (i) the Lagrangian correlation time $\unicode[STIX]{x1D70F}_{c}$ of the flow in the near-wall region is much smaller than the characteristic time scale of the polymer stretching dynamics, (ii) the characteristic time scale of the stretching dynamics is much smaller than the characteristic time of polymer translational motion in the wall-normal direction, (iii) the thermal force $\unicode[STIX]{x1D73B}$ can be neglected in comparison with the effect of velocity gradient stretching and (iv) the polymer is strongly elongated along the mean velocity most of the time. A justification of these assumptions for the boundary layer region of high- $Wi$ elastic turbulence will be given below.

Due to time scale separation (i) and (ii), the statistics of the polymer conformation at a given spatial position is determined by the local intensity of the flow fluctuations which can be treated as shortly correlated in time. The conditions (iii) and (iv) in their turn mean that the main stochastic contribution to the polymer dynamics comes from the gradient of the $z$ -component of the fluctuating velocity. Then, one can apply the results of the theoretical works (Chertkov et al. Reference Chertkov, Kolokolov, Lebedev and Turitsyn2005; Turitsyn Reference Turitsyn2007), where exactly the same model assumptions have been exploited to examine the polymer dynamics in large mean shear with relatively weak flow fluctuations. For the sake of completeness we briefly reproduce the line of argument used in these works.

The orientation of the polymer end-to-end vector $\boldsymbol{R}$ can be parametrized by the spherical angles $\unicode[STIX]{x1D711}$ and $\unicode[STIX]{x1D703}$ , as shown in figure 1. As was discussed in Chertkov et al. (Reference Chertkov, Kolokolov, Lebedev and Turitsyn2005), in terms of these variables, equation (3.2) (with the Langevin force omitted due to assumption (iii)) transforms into the following set of equations:

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D711}=-s\sin ^{2}\unicode[STIX]{x1D711}+\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D711}}, & \displaystyle\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D703}=-s\sin \unicode[STIX]{x1D711}\cos \unicode[STIX]{x1D711}\sin \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D703}+\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D703}}, & \displaystyle\end{eqnarray}$$

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\ln R=-\unicode[STIX]{x1D6FE}(R)+s\cos ^{2}\unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D711}\sin \unicode[STIX]{x1D711}+\unicode[STIX]{x1D70D}_{\Vert }, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D711}}$ , $\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D703}}$ and $\unicode[STIX]{x1D70D}_{\Vert }$ are random terms related to the fluctuating component of the fluid velocity. Although these terms are spatially inhomogeneous, assumptions (i) and (ii) guarantee that statistics of the polymer elongation vector adjusts adiabatically to the local flow properties and, therefore, the internal degrees of freedom can be analysed separately from the translation motion.

The angular dynamics described by (3.3) and (3.4) is decoupled from the dynamics of the polymer length. Moreover, since the polymer is strongly elongated along the shear direction (assumption (iv)), the orientation angles are small most of the time, and the intensity of the random term $\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D711}}$ does not depend on $\unicode[STIX]{x1D703}$ in the leading-order approximation. This means that the angle $\unicode[STIX]{x1D711}$ becomes decoupled from both $R$ and $\unicode[STIX]{x1D703}$ . Treating $\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D711}}$ in (3.3) as a delta-correlated Gaussian process, we can write the Fokker–Plank equation

(3.6) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}P=s\unicode[STIX]{x2202}_{\unicode[STIX]{x1D711}}[\sin ^{2}\unicode[STIX]{x1D711}P]+2D\unicode[STIX]{x2202}_{\unicode[STIX]{x1D711}}^{2}P,\end{eqnarray}$$

in which $P(t,\unicode[STIX]{x1D711})$ is the probability density function of $\unicode[STIX]{x1D711}$ , and $D$ denotes the angular diffusion coefficient. Equation (3.6) should be supplemented by the periodic boundary condition $P(t,-\unicode[STIX]{x03C0}/2)=P(t,\unicode[STIX]{x03C0}/2)$ and the normalization condition $\int _{0}^{\unicode[STIX]{x03C0}}P(t,\unicode[STIX]{x1D711})\,\text{d}\unicode[STIX]{x1D711}=1$ . In the strong shear limit, $s\gg D$ , the stationary solution of (3.6) is localized at small angles and the steady-state average value of $\unicode[STIX]{x1D711}$ is given by (see Turitsyn (Reference Turitsyn2007) for details of the derivation)

(3.7) $$\begin{eqnarray}\langle \unicode[STIX]{x1D711}\rangle \approx \frac{3^{1/3}\sqrt{\unicode[STIX]{x03C0}}}{\unicode[STIX]{x1D6E4}(1/6)}\left(\frac{D}{s}\right)^{1/3}.\end{eqnarray}$$

Here and in what follows, the angle brackets denote averaging over time or over realizations of the chaotic flow.

The near-wall shear rate can be estimated as $s\sim U_{L}/L$ where $U_{L}$ is the mean velocity of the flow at the border of the boundary layer. Next, the angular diffusion coefficient is determined by the gradient of the wall-normal component of velocity fluctuations: $D=(1/2)\int _{-\infty }^{0}\langle \unicode[STIX]{x2202}_{x}u_{z}(\boldsymbol{r}(t),t)\unicode[STIX]{x2202}_{x}u_{z}(\boldsymbol{r}(0),0)\rangle \,\text{d}t$ . Using (2.3) one obtains $\unicode[STIX]{x2202}_{x}u_{z}\sim u_{L}z^{2}/L^{3}$ where $u_{L}$ measures the characteristic (wall-normal) fluctuating velocity at $z=L$ , and, therefore, $D\sim (\unicode[STIX]{x2202}_{x}u_{z})^{2}\unicode[STIX]{x1D70F}_{c}\sim u_{L}^{2}\unicode[STIX]{x1D70F}_{c}z^{4}/L^{6}$ . Inserting the estimates of $s$ and $D$ into (3.7), we find the conditional average of $\unicode[STIX]{x1D711}$ at a given distance $z$ from the wall:

(3.8) $$\begin{eqnarray}\langle \unicode[STIX]{x1D711}\rangle \sim \frac{u_{L}^{2/3}\unicode[STIX]{x1D70F}_{c}^{1/3}z^{4/3}}{U_{L}^{1/3}L^{5/3}}.\end{eqnarray}$$

Next let us consider the dynamics of the polymer elongation $R$ . Since typical orientation angles are small, $\unicode[STIX]{x1D711},\unicode[STIX]{x1D703}\ll 1$ , we can replace $\sin \unicode[STIX]{x1D711}$ in (3.5) by $\unicode[STIX]{x1D711}$ , while $\cos \unicode[STIX]{x1D711}$ and $\cos \unicode[STIX]{x1D703}$ can be replaced by unity. Further, we neglect the term $\unicode[STIX]{x1D70D}_{\Vert }$ in comparison with $s\unicode[STIX]{x1D711}$ because the diffusion coefficient $D_{\Vert }\sim (\unicode[STIX]{x2202}_{x}u_{x})^{2}\unicode[STIX]{x1D70F}_{c}\propto z^{2}$ associated with $\unicode[STIX]{x1D70D}_{\Vert }$ is much smaller than $s\langle \unicode[STIX]{x1D711}\rangle \propto z^{4/3}$ at $z\ll L$ . Then in the main approximation the dynamics of $R$ is described by the following equation:

(3.9) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\ln R=-\unicode[STIX]{x1D6FE}(R)+s\unicode[STIX]{x1D711}.\end{eqnarray}$$

For the statistically stationary state we immediately find

(3.10) $$\begin{eqnarray}\langle \unicode[STIX]{x1D6FE}\rangle =s\langle \unicode[STIX]{x1D711}\rangle \sim \frac{u_{L}^{2/3}U_{L}^{2/3}\unicode[STIX]{x1D70F}_{c}^{1/3}z^{4/3}}{L^{8/3}}.\end{eqnarray}$$

To proceed we need to relate the correlation time $\unicode[STIX]{x1D70F}_{c}$ to the spatial scale $L$ and the characteristic velocities $U_{L}$ and $u_{L}$ . It is natural to expect that $\unicode[STIX]{x1D70F}_{c}$ is of the same order as the inverse Lyapunov exponent $\unicode[STIX]{x1D706}_{L}^{-1}$ at the edge of the boundary layer. Whereas the distance between two fluid parcels evolves accordingly to (3.9) with $\unicode[STIX]{x1D6FE}=0$ , the Lyapunov exponent defined as the mean logarithmic rate of divergence of neighbouring Lagrangian trajectories is given by $\unicode[STIX]{x1D706}_{L}=s\langle \unicode[STIX]{x1D711}\rangle$ , where $\langle \unicode[STIX]{x1D711}\rangle$ is taken at $z=L$ . Imposing the condition $\unicode[STIX]{x1D70F}_{c}\sim \unicode[STIX]{x1D706}_{L}^{-1}$ , we find from (3.8) that $\unicode[STIX]{x1D70F}_{c}\sim L/\sqrt{U_{L}u_{L}}$ . That yields the following estimates:

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \unicode[STIX]{x1D711}\rangle \sim \left(\frac{u_{L}}{U_{L}}\right)^{1/2}\left(\frac{z}{L}\right)^{4/3}, & \displaystyle\end{eqnarray}$$

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \unicode[STIX]{x1D6FE}\rangle \sim \frac{(u_{L}U_{L})^{1/2}}{L}\left(\frac{z}{L}\right)^{4/3}. & \displaystyle\end{eqnarray}$$

According to (3.11) and (3.12), the resulting effect of the hydrodynamic stretching cannot be decomposed into the additive contributions of the stationary shear part and the fluctuating part of the flow. Instead, the polymer dynamics in the boundary layer is determined by the combined effect of regular and chaotic components: fluctuations of the velocity gradient provide polymer stretching in the wall-normal direction required for the polymer to ‘feel’ the presence of the mean shear.

4 Numerical simulations

To check the scaling laws (3.11) and (3.12) we performed the simulations with synthetic random flow which mimics the boundary layer of elastic turbulence. In the case of strong shear the stretching dynamics becomes essentially two-dimensional, so we restrict ourselves to two-dimensional simulations. Namely, the components of the planar velocity field $(v_{x},v_{z})$ in the periodic domain $x\in [0,L]$ , $z\in [-L,L]$ are chosen to be $v_{x}=U_{x}+u_{x}$ , $v_{z}=u_{z}$ , where

(4.1) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle U_{x}=\frac{U_{L}}{\unicode[STIX]{x03C0}}\sin \frac{\unicode[STIX]{x03C0}z}{L},\\ \displaystyle u_{x}=\frac{L^{2}}{\unicode[STIX]{x03C0}^{2}}\left[a_{1}(t)\cos \frac{2\unicode[STIX]{x03C0}x}{L}+a_{2}(t)\sin \frac{2\unicode[STIX]{x03C0}x}{L}\right]\sin \frac{\unicode[STIX]{x03C0}z}{L},\\ \displaystyle u_{z}=\frac{2L^{2}}{\unicode[STIX]{x03C0}^{2}}\left[a_{1}(t)\sin \frac{2\unicode[STIX]{x03C0}x}{L}-a_{2}(t)\cos \frac{2\unicode[STIX]{x03C0}x}{L}\right]\left(1-\cos \frac{\unicode[STIX]{x03C0}z}{L}\right).\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The independent random variables $a_{1}$ and $a_{2}$ are telegraph processes, that is $a_{1}$ and $a_{2}$ remain constant during time slot $\unicode[STIX]{x1D70F}_{c}$ and their values are chosen from identical normal distributions with zero mean and variances $\langle a_{1}^{2}\rangle =\langle a_{2}^{2}\rangle =u_{L}^{2}/L^{4}$ . Let us stress that the velocity field $(v_{x},v_{z})$ is incompressible ( $\unicode[STIX]{x2202}_{x}v_{x}+\unicode[STIX]{x2202}_{z}v_{z}=0$ ) and that it reproduces the proportionality laws $U_{x},u_{x}\propto z$ , $u_{z}\propto z^{2}$ when $|z|\ll L$ . In our numerics we chose $L=1$ , $U_{L}=1$ , $u_{L}=0.1$ and $\unicode[STIX]{x1D70F}_{c}=L/\sqrt{U_{L}u_{L}}$ . The polymer dynamics was modelled via a two-dimensional version of (3.1) and (3.2) supplemented by the finitely extendible nonlinear elastic (FENE) model: $\unicode[STIX]{x1D6FE}(R)=\unicode[STIX]{x1D6FE}_{0}(1-R^{2}/R_{m}^{2})^{-1}$ . The parameters $\unicode[STIX]{x1D6FE}_{0}$ , $R_{m}$ , $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D702}$ were adjusted in such a way that $R_{m}/R_{0}=10^{2}$ , $Wi=10^{4}$ and $Pe=10^{6}$ , where $R_{0}=\sqrt{3\unicode[STIX]{x1D702}/\unicode[STIX]{x1D6FE}_{0}}$ is the radius of gyration of a polymer and $Pe=U_{L}L/\unicode[STIX]{x1D705}$ is the Péclet number.

Figure 2.

(a,b) The average orientation angle $\langle \unicode[STIX]{x1D711}\rangle$ and the spring constant $\langle \unicode[STIX]{x1D6FE}\rangle$ dependences on the distance from the wall $z$ . The solid lines represent the results of numerical simulations and the dashed lines are the theoretical power-law profiles: see (3.11) and (3.12). (c) The mean square of the polymer elongation $\langle R_{x}^{2}\rangle$ along the shear direction as a function of distance from the wall $z$ . The dashed line represents the asymptotic value corresponding to the limit of infinitely large shear. (d) The statistical moment $C=\langle \unicode[STIX]{x1D6FE}R_{x}R_{z}\rangle$ (solid line) versus the distance from the wall $z$ . This plot demonstrates that $\langle \unicode[STIX]{x1D6FE}R_{x}R_{z}\rangle$ depends on $z$ in the same manner as $\langle \unicode[STIX]{x1D6FE}\rangle \langle \unicode[STIX]{x1D711}\rangle R_{m}^{2}$ (dashed line).

Figure 2 shows the steady-state expected values of the orientation angle $\langle \unicode[STIX]{x1D711}\rangle$ and spring constant $\langle \unicode[STIX]{x1D6FE}\rangle$ in their dependence on $z$ . In agreement with theoretical prediction, $\langle \unicode[STIX]{x1D711}\rangle$ and $\langle \unicode[STIX]{x1D6FE}\rangle$ exhibit power-law behaviour with exponent $4/3$ starting from some small distance from the wall and continuing for a scale of the order of $L$ . The mean square of the polymer elongation along the wall is approximately constant in this region, $\langle R_{x}^{2}\rangle \approx 0.62R_{m}^{2}$ : see figure 2(c). This is because $\langle R_{x}^{2}\rangle$ is close to its limiting value, corresponding to the limit of infinitely large shear, which is numerically found to be approximately $0.65R_{m}^{2}$ for the FENE model. Note that the limiting value differs from $R_{m}^{2}$ due to the ongoing tumbling of a polymer molecule (Turitsyn Reference Turitsyn2007).

5 Relation between mean flow and fluctuations

Having described the single polymer statistics inside the boundary layer, we now pass to the macroscopic level of description. The hydrodynamics of the low- $Re$ polymer solution is governed by the interplay of viscous forces and elastic forces, while fluid inertia can be neglected. Namely, if the flow is incompressible, i.e. the mass density of the neat fluid $\unicode[STIX]{x1D70C}$ is a constant and $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}=0$ , the momentum conservation law (with the nonlinear inertial term omitted) reads

(5.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}v_{i}=-\frac{1}{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D735}_{i}p+\unicode[STIX]{x1D735}_{j}\unicode[STIX]{x1D6F1}_{ij}^{v}+\unicode[STIX]{x1D735}_{j}\unicode[STIX]{x1D6F1}_{ij}^{e}. & & \displaystyle\end{eqnarray}$$

Here $p$ is the pressure, $\unicode[STIX]{x1D6F1}_{ij}^{v}$ is the viscous stress tensors and $\unicode[STIX]{x1D6F1}_{ij}^{e}$ is the tensor of elastic stresses created by the polymer stretching in the flow. Equation (5.1) is applicable at scales much larger than the inter-polymer distance where the polymer solution can be regarded as a continuous medium.

The elastic stress is the source of instability leading to the turbulent-like state of a polymer solution. In the bulk region of elastic turbulence, where polymers provide strong back reaction on the velocity field, the elastic stresses are of the order of or larger than the viscous stresses existing in the flow. However, in the peripheral region, the polymers are relatively weakly stretched since fluctuations of the wall-normal velocity gradient diminish towards the no-slip wall. This is why a boundary layer is formed, where viscosity overcomes elasticity.

For the statistically stationary flow, the only component of the viscous stress tensor remaining non-zero after averaging is $\langle \unicode[STIX]{x1D6F1}_{xz}^{v}\rangle =\unicode[STIX]{x1D708}\unicode[STIX]{x2202}_{z}U_{x}$ , where $\unicode[STIX]{x1D708}$ is Newtonian viscosity of the neat fluid. The corresponding component of the polymer-induced elastic stress is given by (Bird et al. Reference Bird, Hassager, Armstrong and Curtiss1987)

(5.2) $$\begin{eqnarray}\langle \unicode[STIX]{x1D6F1}_{xz}^{e}\rangle =\frac{n}{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6E4}}\langle \unicode[STIX]{x1D6FE}(R)R_{x}R_{z}\rangle ,\end{eqnarray}$$

where $n$ is the concentration of the polymers in solution, and $\unicode[STIX]{x1D6E4}$ is the kinetic coefficient, which determines the polymer relaxation. In the two-bead-one-spring model of the polymer molecule one has $\unicode[STIX]{x1D6E4}\sim (\unicode[STIX]{x1D70C}\unicode[STIX]{x1D708}R_{0})^{-1}$ .

It is difficult to derive a closed-form expression for the statistical moment $\langle \unicode[STIX]{x1D6FE}R_{x}R_{z}\rangle$ entering (5.2). However, for the boundary layer region one can safely write the estimate $\langle \unicode[STIX]{x1D6FE}R_{x}R_{z}\rangle \sim \langle \unicode[STIX]{x1D6FE}\rangle \langle \unicode[STIX]{x1D711}\rangle R_{m}^{2}$ , which is confirmed by numerical simulations: see figure 2(d). Then, using (3.11), (3.12) and (5.2), we obtain $\langle \unicode[STIX]{x1D6F1}_{xz}^{e}\rangle \sim n\unicode[STIX]{x1D708}R_{0}R_{m}^{2}u_{L}z^{8/3}/L^{11/3}$ at $z\ll L$ . By the definition of the boundary layer, the elastic stress must become of the order of the viscous stress $\langle \unicode[STIX]{x1D6F1}_{xz}^{v}\rangle \sim \unicode[STIX]{x1D708}U_{L}/L$ at $z=L$ . This condition produces the following relation:

(5.3) $$\begin{eqnarray}\frac{U_{L}}{u_{L}}\sim nR_{0}R_{m}^{2},\end{eqnarray}$$

which is the central result of this work. Equation (5.3) shows that in contrast to high- $Re$ turbulence, where the fluctuating velocity is always of the order of the mean flow in the viscous sublayer, wall-bounded elastic turbulence possesses a non-trivial relation between the mean and the fluctuating velocity components.

It should be stressed that the line of argument presented here cannot give us the value of the numerical factor in (5.3) which may significantly differ from unity in hydrodynamic problems of this sort. This remark must be taken into account when making numerical predictions based on the above result. At the same time, equation (5.3) uncovers the parametric relation between the quantities $U_{L}$ , $u_{L}$ , $n$ , $R_{m}$ and $R_{0}$ which can be tested experimentally.

Another important concern that may restrict the applicability of our results is the difference between the properties of the real polymer molecule and that of the idealized dumbbell-like model adopted here. In particular, one may expect that in high- $Wi$ flow, the kinetic coefficient $\unicode[STIX]{x1D6E4}$ in (5.2) should be estimated as $(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D708}R_{m})^{-1}$ since for the strongly stretched polymer molecule the force of hydrodynamic friction affecting the dynamics of the elongation vector $\boldsymbol{R}$ is determined by the scale $R_{m}$ . This leads to the relation $U_{L}/u_{L}\sim nR_{m}^{3}$ which, in contrast to (5.3), does not contain the equilibrium size $R_{0}$ of the polymers.

We are ready to justify the set of assumptions (i)–(iv) underlying the analytical procedure presented above. First of all, we note that the characteristic time of the extensional dynamics is $\langle \unicode[STIX]{x1D6FE}\rangle ^{-1}$ . Then, as follows from (3.12) and from the estimate $\unicode[STIX]{x1D70F}_{c}\sim L/\sqrt{U_{L}u_{L}}$ , assumption (i), which can be written as $\unicode[STIX]{x1D70F}_{c}\ll \langle \unicode[STIX]{x1D6FE}\rangle ^{-1}$ , is self-consistent for any $z\ll L$ . There is a simple explanation as to why the time scales $\unicode[STIX]{x1D70F}_{c}$ and $\langle \unicode[STIX]{x1D6FE}\rangle ^{-1}$ are well separated inside the boundary layer. In the bulk region of elastic turbulence, where fluid motion is strongly influenced by polymer back reaction, the velocity correlation time $\unicode[STIX]{x1D70F}_{c}$ must be of the same order as the relaxation time of the elastic stresses $\langle \unicode[STIX]{x1D6FE}\rangle ^{-1}$ . Since the flow fluctuations in the boundary layer are induced by bulk turbulence, then the correlation time $\unicode[STIX]{x1D70F}_{c}$ near the wall is also determined by the value of $\langle \unicode[STIX]{x1D6FE}\rangle ^{-1}$ in the bulk. At the same time, hydrodynamic stretching of polymers in close vicinity to the wall is much weaker than their stretching in the bulk. We thus conclude that $\unicode[STIX]{x1D70F}_{c}\ll \langle \unicode[STIX]{x1D6FE}\rangle ^{-1}$ in the boundary layer.

Next, let us examine the range of validity for the ‘adiabatic’ assumption (ii). Since the centre of mass of the polymer is passively advected by the flow, the probability density function $n(z,t)$ of its $z$ -coordinate obeys the standard advection–diffusion equation $\unicode[STIX]{x2202}_{t}n=\unicode[STIX]{x2202}_{z}[D_{zz}(z)\unicode[STIX]{x2202}_{z}n]$ , where $D_{zz}(z)=\unicode[STIX]{x1D707}z^{4}+\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D707}\sim u_{L}^{2}\unicode[STIX]{x1D70F}_{c}/L^{4}$ (Lebedev & Turitsyn Reference Lebedev and Turitsyn2004; Chernykh & Lebedev Reference Chernykh and Lebedev2011). The time required for a polymer at distance $z$ from the wall to ‘feel’ the inhomogeneity of the fluctuations is estimated as $\tilde{\unicode[STIX]{x1D70F}}(z)\sim z^{2}/D_{zz}(z)$ . This time must be much larger than the characteristic time of the stretching dynamics, i.e. $\tilde{\unicode[STIX]{x1D70F}}(z)\gg \langle \unicode[STIX]{x1D6FE}\rangle ^{-1}$ . Given equation (3.12), this leads to the following condition:

(5.4) $$\begin{eqnarray}z\gg r_{\unicode[STIX]{x1D705}}=\left(\frac{U_{L}}{u_{L}}\right)^{3/20}\frac{L}{Pe^{3/10}}.\end{eqnarray}$$

As for the assumption (iii), the Langevin force $\unicode[STIX]{x1D73B}$ produces the correction of the order of $\unicode[STIX]{x1D702}/R_{m}^{2}$ to the angular diffusion coefficient $D$ . The correction is relatively small if

(5.5) $$\begin{eqnarray}z\gg r_{\unicode[STIX]{x1D702}}=\left(\frac{R_{0}}{R_{m}}\right)^{1/2}\left(\frac{U_{L}}{u_{L}}\right)^{3/8}\frac{L}{Wi^{1/4}},\end{eqnarray}$$

where $Wi=U_{L}\unicode[STIX]{x1D6FE}_{0}^{-1}/L$ is the Weissenberg number.

Finally, the assumption (iv) is equivalent to the inequalities $\langle \unicode[STIX]{x1D711}\rangle \ll 1$ and $\langle \unicode[STIX]{x1D6FE}\rangle \gg \unicode[STIX]{x1D6FE}_{0}$ . The former is satisfied for any $z\ll L$ provided $u_{L}\lesssim U_{L}$ , while the latter can be rewritten as

(5.6) $$\begin{eqnarray}z\gg r_{\unicode[STIX]{x1D6FE}_{0}}=\left(\frac{U_{L}}{u_{L}}\right)^{3/8}\frac{L}{Wi^{3/4}}.\end{eqnarray}$$

Taking into account (5.3), we conclude from (5.4)–(5.6) that the spatial scales $r_{\unicode[STIX]{x1D705}}$ , $r_{\unicode[STIX]{x1D702}}$ and $r_{\unicode[STIX]{x1D6FE}_{0}}$ are small in comparison with the width of the boundary layer $L$ in the limit when both the Weissenberg number $Wi$ and the Péclet number $Pe$ are large. That ensures the self-consistency of our asymptotic theory.