3.1 Geometric decomposition

In order to generate perturbative expansions of the conformation tensor $\unicode[STIX]{x1D63E}$ about the base state $\overline{\unicode[STIX]{x1D63E}}$ , we first define an appropriate fluctuating conformation. For this, we follow the approach by Hameduddin et al. (Reference Hameduddin, Meneveau, Zaki and Gayme2018), which we outline below. Later, we will also exploit this approach to generate the perturbative expansion we are seeking.

The conformation tensor, $\unicode[STIX]{x1D63E}$ , is the left Cauchy–Green tensor associated with the polymer deformation (Rajagopal & Srinivasa Reference Rajagopal and Srinivasa2000; Cioranescu, Girault & Rajagopal Reference Cioranescu, Girault and Rajagopal2016)

(3.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D63E}=\unicode[STIX]{x1D641}\boldsymbol{\cdot }\unicode[STIX]{x1D641}^{\mathsf{T}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D641}$ is the total deformation gradient that describes deformation with respect to the thermodynamic equilibrium. We decompose this deformation into two: a deformation about the thermodynamic equilibrium that yields the base state, and a deformation about the base state that yields the total deformation. Accordingly, we decompose the deformation gradient as

(3.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D641}=\overline{\unicode[STIX]{x1D641}}\boldsymbol{\cdot }\unicode[STIX]{x1D647}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D647}$ is the fluctuating deformation gradient, and $\overline{\unicode[STIX]{x1D641}}$ is the deformation gradient associated with $\overline{\unicode[STIX]{x1D63E}}$ ,

(3.3) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D641}}=\overline{\unicode[STIX]{x1D63E}}^{1/2}\boldsymbol{\cdot }\unicode[STIX]{x1D64D}. & & \displaystyle\end{eqnarray}$$

Here $\unicode[STIX]{x1D64D}\in \mathbf{SO}_{3}$ and $\mathbf{SO}_{3}$ is the special orthogonal group. It is readily verified that $\overline{\unicode[STIX]{x1D63E}}=\overline{\unicode[STIX]{x1D641}}\boldsymbol{\cdot }\overline{\unicode[STIX]{x1D641}}^{\mathsf{T}}$ . In practice, we set $\unicode[STIX]{x1D64D}=\unicode[STIX]{x1D644}$ (for example in § 5). The fluctuating deformation gradient has the associated tensor $\unicode[STIX]{x1D642}=\unicode[STIX]{x1D647}\boldsymbol{\cdot }\unicode[STIX]{x1D647}^{\unicode[STIX]{x1D64F}}$ , which satisfies the relationship

(3.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D63E}=\overline{\unicode[STIX]{x1D641}}\boldsymbol{\cdot }\unicode[STIX]{x1D642}\boldsymbol{\cdot }\overline{\unicode[STIX]{x1D641}}^{\mathsf{T}}. & & \displaystyle\end{eqnarray}$$

The tensor $\unicode[STIX]{x1D642}$ is positive definite and is equivalent to $\unicode[STIX]{x1D63E}$ , but is transformed so that $\unicode[STIX]{x1D63E}=\overline{\unicode[STIX]{x1D63E}}$ if and only if $\unicode[STIX]{x1D642}=\unicode[STIX]{x1D644}$ . Thus $\unicode[STIX]{x1D642}$ acts as a conformation tensor representing the fluctuation of $\unicode[STIX]{x1D63E}$ around $\overline{\unicode[STIX]{x1D63E}}$ .

The geometry of $\mathbf{Pos}_{3}$ can be used to quantify the fluctuating polymer deformation. Using (2.6), the geodesic distance between $\unicode[STIX]{x1D63E}$ and $\overline{\unicode[STIX]{x1D63E}}$ can be written as

(3.5) $$\begin{eqnarray}\displaystyle d(\overline{\unicode[STIX]{x1D63E}},\unicode[STIX]{x1D63E})=d(\unicode[STIX]{x1D644},\unicode[STIX]{x1D642})=\sqrt{\text{tr}\,\boldsymbol{{\mathcal{G}}}^{2}}, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{{\mathcal{G}}}\equiv \log \unicode[STIX]{x1D642}$ , and is a measure of the magnitude of the fluctuation. Similarly, we can evaluate whether a deformation is compressive or expansive with respect to the base state by examining the logarithmic volume ratio, $\text{tr}\,\boldsymbol{{\mathcal{G}}}$ .

3.2 Weakly nonlinear deformations

A weakly nonlinear expansion up to the $N$ th power of the velocity field is given by

(3.6) $$\begin{eqnarray}\displaystyle \boldsymbol{u}=\overline{\boldsymbol{u}}+\boldsymbol{u}^{\prime }=\overline{\boldsymbol{u}}+\mathop{\sum }_{k=1}^{N}\unicode[STIX]{x1D700}^{k}\boldsymbol{u}_{(k)}, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{u}_{(k)}(\boldsymbol{x},t)$ for $k\in [1,N]$ are velocities. A similar expansion for $\unicode[STIX]{x1D63E}$ is inappropriate because it is positive definite and there is no a priori guarantee on this property. In order to obtain an analogous expansion for the conformation tensor, we generalize the approach we used in the previous subsection by multiplicatively decomposing the fluctuating deformation gradient into $N$ separate components. The construction of this fluctuating deformation gradient, denoted $\unicode[STIX]{x1D647}_{wnl}$ , through a series of successively smaller deformations is illustrated in figure 2. Mathematically, we write

(3.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D647}_{wnl}=\unicode[STIX]{x1D647}_{(1)}^{\unicode[STIX]{x1D700}}\boldsymbol{\cdot }\unicode[STIX]{x1D647}_{(2)}^{\unicode[STIX]{x1D700}^{2}}\boldsymbol{\cdot }\cdots \boldsymbol{\cdot }\unicode[STIX]{x1D647}_{(N)}^{\unicode[STIX]{x1D700}^{N}}. & & \displaystyle\end{eqnarray}$$

We further assume that each $\unicode[STIX]{x1D647}_{(k)}^{\unicode[STIX]{x1D700}^{k}}$ in (3.7) is rotation-free. By the polar decomposition and the requirement that $\det \unicode[STIX]{x1D647}_{(k)}>0$ , this assumption implies that each $\unicode[STIX]{x1D647}_{(k)}$ is positive definite. Although each $\unicode[STIX]{x1D647}_{(k)}$ is rotation-free, the overall fluctuating deformation gradient $\unicode[STIX]{x1D647}_{wnl}$ given by (3.7) is not because the product of positive definite tensors is not necessarily positive definite. The rotation appears if the principal axes of $\unicode[STIX]{x1D647}_{(k)}$ and $\unicode[STIX]{x1D647}_{(k^{\prime })}$ , when $k\neq k^{\prime }$ , are misaligned. The deformation gradient $\unicode[STIX]{x1D647}_{wnl}$ is also not necessarily the same as $\unicode[STIX]{x1D647}$ defined previously. However, since $\unicode[STIX]{x1D647}_{wnl}\boldsymbol{\cdot }\unicode[STIX]{x1D647}_{wnl}^{\mathsf{T}}=\unicode[STIX]{x1D647}\boldsymbol{\cdot }\unicode[STIX]{x1D647}^{\mathsf{T}}=\unicode[STIX]{x1D642}$ , the polar decomposition can be used to show that $\unicode[STIX]{x1D647}_{wnl}=\unicode[STIX]{x1D651}\boldsymbol{\cdot }\unicode[STIX]{x1D647}$ for some rotation tensor $\unicode[STIX]{x1D651}$ .

Figure 1. Schematic of the geometric decomposition, adapted from Hameduddin et al. (Reference Hameduddin, Meneveau, Zaki and Gayme2018).

Figure 2. Schematic of a weakly nonlinear deformation, consisting of a sequence of successively smaller deformations.

Each deformation gradient $\unicode[STIX]{x1D647}_{(k)}^{\unicode[STIX]{x1D700}^{k}}$ in (3.7) has an associated left Cauchy–Green tensor, $\unicode[STIX]{x1D642}_{(k)}^{\unicode[STIX]{x1D700}^{k}}=\unicode[STIX]{x1D647}_{(k)}^{\unicode[STIX]{x1D700}^{k}}\boldsymbol{\cdot }(\unicode[STIX]{x1D647}_{(k)}^{\unicode[STIX]{x1D700}^{k}})^{\mathsf{T}}$ , which can be viewed as a geodesic of length $|\unicode[STIX]{x1D700}|^{k}\Vert \boldsymbol{{\mathcal{G}}}_{(k)}\Vert \sim |\unicode[STIX]{x1D700}|^{k}$ on $\mathbf{Pos}_{3}$ emanating from $\unicode[STIX]{x1D644}$ and can be expressed conveniently as

(3.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D642}_{(k)}^{\unicode[STIX]{x1D700}^{k}}=\text{e}^{\unicode[STIX]{x1D700}^{k}\boldsymbol{{\mathcal{G}}}_{(k)}}, & & \displaystyle\end{eqnarray}$$

where $\text{e}^{\boldsymbol{{\mathcal{A}}}}$ is the matrix exponential of $\boldsymbol{{\mathcal{A}}}$ , $\boldsymbol{{\mathcal{G}}}_{(k)}\in \mathbf{Sym}_{3}$ are tangents on $\mathbf{Pos}_{3}$ , and $\boldsymbol{{\mathcal{G}}}_{(0)}=\mathbf{0}$ . With (3.8), it is easy to show that $\det \unicode[STIX]{x1D647}_{wnl}>0$ , which means that $\unicode[STIX]{x1D647}_{wnl}$ is a physically admissible deformation gradient.

The tangents on $\mathbf{Pos}_{3}$ , $\boldsymbol{{\mathcal{G}}}_{(k)}$ , can be used to physically characterize the perturbation deformation. The associated deformation gradient is given by $\unicode[STIX]{x1D647}_{(k)}^{\unicode[STIX]{x1D700}^{k}}=\text{e}^{\unicode[STIX]{x1D700}^{k}\boldsymbol{{\mathcal{G}}}_{(k)}/2}$ . When $\boldsymbol{{\mathcal{G}}}_{(k)}$ is diagonal, $\unicode[STIX]{x1D647}_{(k)}^{\unicode[STIX]{x1D700}^{k}}$ is diagonal and thus the deformation is a shear-free distortion, or solely altering the volume ratio. On the other hand, when $\text{tr}\,\boldsymbol{{\mathcal{G}}}_{(k)}=0$ , then $\det \unicode[STIX]{x1D647}_{(k)}^{\unicode[STIX]{x1D700}^{k}}=1$ , and the deformation is purely shearing, or volume-ratio preserving.

Using the square root of (3.8) in (3.7), the left Cauchy–Green tensor $\unicode[STIX]{x1D642}$ is given by

(3.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D642} & = & \displaystyle \text{e}^{\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}/2}\boldsymbol{\cdot }\cdots \boldsymbol{\cdot }\text{e}^{\unicode[STIX]{x1D700}^{N-1}\boldsymbol{{\mathcal{G}}}_{(N-1)}/2}\boldsymbol{\cdot }\text{e}^{\unicode[STIX]{x1D700}^{N}\boldsymbol{{\mathcal{G}}}_{(N)}}\boldsymbol{\cdot }\text{e}^{\unicode[STIX]{x1D700}^{N-1}\boldsymbol{{\mathcal{G}}}_{(N-1)}/2}\boldsymbol{\cdot }\cdots \boldsymbol{\cdot }\text{e}^{\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}/2}\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle & = & \displaystyle \unicode[STIX]{x1D644}+\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}+\unicode[STIX]{x1D700}^{2}\left(\frac{\boldsymbol{{\mathcal{G}}}_{(1)}^{2}}{2}+\boldsymbol{{\mathcal{G}}}_{(2)}\right)+\unicode[STIX]{x1D700}^{3}\left(\frac{\boldsymbol{{\mathcal{G}}}_{(1)}^{3}}{6}+\text{sym}\,(\boldsymbol{{\mathcal{G}}}_{(1)}\boldsymbol{\cdot }\boldsymbol{{\mathcal{G}}}_{(2)})+\boldsymbol{{\mathcal{G}}}_{(3)}\right)+\cdots \,.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The second equality made use of the matrix exponential $\text{e}^{\unicode[STIX]{x1D700}^{k}\boldsymbol{{\mathcal{G}}}_{(k)}}=\sum _{p=0}^{\infty }\unicode[STIX]{x1D700}^{kp}\boldsymbol{{\mathcal{G}}}_{(k)}^{p}/p!$ . The form (3.10) is a series expansion of the conformation tensor that serves as an analogue to the weakly nonlinear expansion of the velocity in (3.6). In fact, the terms in (3.10) can be related to the standard weakly nonlinear expansion of the conformation tensor, $\unicode[STIX]{x1D63E}=\overline{\unicode[STIX]{x1D63E}}+\sum _{k=1}^{\infty }\unicode[STIX]{x1D700}^{k}\unicode[STIX]{x1D63E}_{(k)}$ . The difference between $\unicode[STIX]{x1D63E}_{(k)}$ and $\boldsymbol{{\mathcal{G}}}_{(k)}$ is that the latter can be related to a polymer perturbation deformation by means of the framework introduced above. Furthermore, equation (3.10) shows that the $\unicode[STIX]{x1D63E}_{(k)}$ are not independent of one another, even before the expansion is applied in the governing equations to examine the dynamics. This behaviour is consistent with the curved geometry of $\mathbf{Pos}_{3}$ because, unlike in Euclidean space, the characteristics of a particular perturbation depend on the location on the manifold where the perturbation is applied. In this view of the geometry, the deformation associated with $\unicode[STIX]{x1D647}_{(n)}^{\unicode[STIX]{x1D700}^{n}}$ is a perturbation to the deformation associated with $\unicode[STIX]{x1D647}_{(1)}^{\unicode[STIX]{x1D700}}\boldsymbol{\cdot }\unicode[STIX]{x1D647}_{(2)}^{\unicode[STIX]{x1D700}^{2}}\boldsymbol{\cdot }\cdots \boldsymbol{\cdot }\unicode[STIX]{x1D647}_{(n-1)}^{\unicode[STIX]{x1D700}^{n-1}}$ and thus the $n$ th-order term in the series expansion must depend on all $\boldsymbol{{\mathcal{G}}}_{(k)}$ with $k=1,\ldots ,n$ . The behaviour is also consistent with a physical understanding of successive deformations of the polymer; a deformation is only sensible with respect to an existing configuration and is thus dependent on it from the point of view of an independent observer.

Figure 3. Illustration of weakly nonlinear deformation when $N=2$ . In this case, the deformation corresponds to a piecewise geodesic on $\mathbf{Pos}_{3}$ . The thick black lines represent geodesics and dashed lines are Euclidean paths on $\mathbf{Pos}_{3}$ .

One aspect of the relationship between the present approach of decomposing the total deformation into a series of successive deformations (cf. figure 2) and the geometry of $\mathbf{Pos}_{3}$ is that the left Cauchy–Green tensor associated with each deformation is chosen to be a geodesic emanating from $\unicode[STIX]{x1D644}$ . Another direct connection between geodesics on $\mathbf{Pos}_{3}$ and the overall deformation represented by $\unicode[STIX]{x1D642}$ can be made when $N\leqslant 2$ . When $N=1$ , $\unicode[STIX]{x1D642}=\text{e}^{\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}}$ is simply a geodesic of length ${\sim}\unicode[STIX]{x1D700}$ emanating from $\unicode[STIX]{x1D644}$ in the direction $\boldsymbol{{\mathcal{G}}}_{(1)}$ . When $N=2$ , we have

(3.11) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D642}=\text{e}^{\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}/2}\boldsymbol{\cdot }\text{e}^{\unicode[STIX]{x1D700}^{2}\boldsymbol{{\mathcal{G}}}_{(2)}}\boldsymbol{\cdot }\text{e}^{\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}/2}, & & \displaystyle\end{eqnarray}$$

which implies that $\unicode[STIX]{x1D642}$ is ‘piecewise geodesic’: it consists of a geodesic of length ${\sim}\unicode[STIX]{x1D700}$ emanating from $\unicode[STIX]{x1D644}$ in the direction $\boldsymbol{{\mathcal{G}}}_{(1)}$ , followed by a geodesic of length ${\sim}\unicode[STIX]{x1D700}^{2}$ in the direction $\boldsymbol{{\mathcal{G}}}_{(2)}$ , as illustrated in figure 3. Such an interpretation is not generally possible for $N>2$ because then $\text{e}^{\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}/2}\boldsymbol{\cdot }\cdots \boldsymbol{\cdot }\text{e}^{\unicode[STIX]{x1D700}^{N-1}\boldsymbol{{\mathcal{G}}}_{(N-1)}/2}$ need not be in $\mathbf{Pos}_{3}$ . If we assume that $\boldsymbol{{\mathcal{G}}}_{(1)},\ldots ,\boldsymbol{{\mathcal{G}}}_{(N-1)}$ are commutative with respect to multiplication, then

(3.12) $$\begin{eqnarray}\displaystyle \text{e}^{\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}/2}\boldsymbol{\cdot }\cdots \boldsymbol{\cdot }\text{e}^{\unicode[STIX]{x1D700}^{N-1}\boldsymbol{{\mathcal{G}}}_{(N-1)}/2}=\text{e}^{\frac{1}{2}\mathop{\sum }_{k=1}^{N-1}\unicode[STIX]{x1D700}^{k}\boldsymbol{{\mathcal{G}}}_{(k)}}\in \mathbf{Pos}_{3}. & & \displaystyle\end{eqnarray}$$

Thus, in this case, the interpretation of the successive deformations as a piecewise geodesic on $\mathbf{Pos}_{3}$ holds for arbitrary $N$ . The inability to extend the piecewise geodesic interpretation to arbitrary $N$ arises because successive deformations are, in general, physically misaligned and thus, by the polar decomposition, a rotation is imparted to the deformation gradient that depends on the order in which successive intermediate deformations were performed. This order of the intermediate deformations is only relevant when $N\geqslant 2$ . When the deformations are physically aligned, the overall deformation gradient is positive definite: it has no associated rotation and the order of the intermediate deformations can be arbitrarily changed.

An alternative to the present approach for generating a series expansion of $\unicode[STIX]{x1D642}$ is to expand the tangent vector on $\mathbf{Pos}_{3}$ :

(3.13) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D642}=\exp \left(\mathop{\sum }_{k=1}^{N}\unicode[STIX]{x1D700}^{k}\boldsymbol{{\mathcal{G}}}_{(k)}\right)=\unicode[STIX]{x1D644}+\mathop{\sum }_{k=1}^{N}\unicode[STIX]{x1D700}^{k}\boldsymbol{{\mathcal{G}}}_{(k)}+\frac{1}{2}\left(\mathop{\sum }_{k=1}^{N}\unicode[STIX]{x1D700}^{k}\boldsymbol{{\mathcal{G}}}_{(k)}\right)^{2}+\cdots \,. & & \displaystyle\end{eqnarray}$$

It is easy to show that our proposed approach (3.10) and the alternative (3.13) are equivalent if $\boldsymbol{{\mathcal{G}}}_{(1)},\ldots ,\boldsymbol{{\mathcal{G}}}_{(N)}$ are commutative with respect to multiplication. When commutativity is not satisfied, the two approaches are still equivalent up to $O(\unicode[STIX]{x1D700}^{4})$ . The drawback with using (3.13) is that the individual terms of the expansion cannot be associated with a polymer deformation because $\text{e}^{\unicode[STIX]{x1D63C}+\unicode[STIX]{x1D63D}}\neq \text{e}^{\unicode[STIX]{x1D63C}}\boldsymbol{\cdot }\text{e}^{\unicode[STIX]{x1D63D}}$ . The expansion (3.13) also cannot be related to the geometry of $\mathbf{Pos}_{3}$ in the same way as (3.10).

We developed an approach to generate a perturbation deformation of the polymers with arbitrarily many deformations. The magnitudes of the deformations are of successively higher order with respect to the distance metric on the manifold. We next consider the case of linear perturbations, where a single deformation is involved.

3.3 Linear perturbations

We can generate a small perturbation about $\overline{\unicode[STIX]{x1D63E}}$ by translating the conformation tensor along a geodesic emanating from $\overline{\unicode[STIX]{x1D63E}}$ . This can be accomplished by setting

(3.14) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D642}=\unicode[STIX]{x1D644}\#_{\unicode[STIX]{x1D700}}\text{e}^{\boldsymbol{{\mathcal{G}}}_{(1)}}=\text{e}^{\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}}, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{{\mathcal{G}}}_{(1)}$ is a prescribed tangent on $\mathbf{Pos}_{3}$ , $\unicode[STIX]{x1D700}\in \mathbb{R}$ and $d(\unicode[STIX]{x1D644},\text{e}^{\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}})=|\unicode[STIX]{x1D700}|\Vert \boldsymbol{{\mathcal{G}}}_{(1)}\Vert \sim \unicode[STIX]{x1D700}$ . The parameter $\unicode[STIX]{x1D700}$ here represents the amount of volumetric deformation encoded in the perturbation because the volume of $\unicode[STIX]{x1D642}$ is given by $\det \unicode[STIX]{x1D642}=\text{e}^{\unicode[STIX]{x1D700}\text{tr}\,\boldsymbol{{\mathcal{G}}}_{(1)}}$ , or equivalently $\det \unicode[STIX]{x1D63E}=c^{\unicode[STIX]{x1D700}}\det \overline{\unicode[STIX]{x1D63E}}$ for constant $c=\text{e}^{\text{tr}\,\boldsymbol{{\mathcal{G}}}_{(1)}}$ . The expression (3.14) is valid, in a kinematic sense, for all $\unicode[STIX]{x1D700}\in \mathbb{R}$ . In the case when $\unicode[STIX]{x1D700}\rightarrow 0$ , we may approximate the matrix exponential, $\text{e}^{\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}}=\sum _{k=0}^{\infty }\unicode[STIX]{x1D700}^{k}\boldsymbol{{\mathcal{G}}}_{(1)}^{k}/k!$ , as

(3.15) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D642}=\text{e}^{\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}}=\unicode[STIX]{x1D644}+\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}+O(\unicode[STIX]{x1D700}^{2}\text{e}^{\unicode[STIX]{x1D700}}), & & \displaystyle\end{eqnarray}$$

where the truncation error is based on the bounds derived by Suzuki (Reference Suzuki1976). The result (3.15) is the same as the weakly nonlinear expansion (3.10) with

(3.16) $$\begin{eqnarray}\displaystyle \boldsymbol{{\mathcal{G}}}_{(2)}=\boldsymbol{{\mathcal{G}}}_{(3)}=\cdots =\boldsymbol{{\mathcal{G}}}_{(N)}=\mathbf{0}. & & \displaystyle\end{eqnarray}$$

Multiplying (3.15) by $\overline{\unicode[STIX]{x1D641}}$ on both sides, we obtain

(3.17) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D63E}=\overline{\unicode[STIX]{x1D63E}}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D63E}_{(1)}+O(\unicode[STIX]{x1D700}^{2}\text{e}^{\unicode[STIX]{x1D700}}), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D63E}_{(1)}=\overline{\unicode[STIX]{x1D641}}\boldsymbol{\cdot }\boldsymbol{{\mathcal{G}}}_{(1)}\boldsymbol{\cdot }\overline{\unicode[STIX]{x1D641}}^{\mathsf{T}}$ , which is similar to the standard approach involving an additive perturbation to $\overline{\unicode[STIX]{x1D63E}}$ . However, now the fluctuation, $\unicode[STIX]{x1D63E}^{\prime }=\unicode[STIX]{x1D700}\unicode[STIX]{x1D63E}_{(1)}$ , has a clear interpretation as a tangent to the manifold at the base point $\overline{\unicode[STIX]{x1D63E}}$ . Furthermore, the normalization of $\unicode[STIX]{x1D63E}^{\prime }$ is proportional to the geodesic distance away from $\overline{\unicode[STIX]{x1D63E}}$ on $\mathbf{Pos}_{3}$ .

The geometric structure on $\mathbf{Pos}_{3}$ supplies us with the natural scalar product to be used in the analysis of linear perturbations. This scalar product, which depends on $\overline{\unicode[STIX]{x1D63E}}$ , is induced by the global distance metric on $\mathbf{Pos}_{3}$ and is given in (2.4). If we use the form of the additive perturbation given in (3.15), the natural scalar product reduces to the standard Frobenius norm. By taking the base point into account, equation (2.4) also allows us to compare the norm of tangent vectors at different base points.

4.1 Constraint on linear evolution

An initial condition consisting of a small-amplitude unstable mode will initially amplify exponentially as predicted by linear theory. Eventually, however, nonlinear effects will become significant because otherwise the conformation tensor will lose positive-definiteness. It is of interest to determine an estimate of the maximum time that the evolution of the perturbation can be well approximated by linear theory, i.e. along Euclidean lines. Such an estimate is a useful guide for selecting initial perturbation amplitude, e.g. by ruling out initial perturbations that have unacceptably small time for which the linear evolution holds.

Consider an initial condition which is a perturbed base flow, $\boldsymbol{u}|_{t=0}=\overline{\boldsymbol{u}}+\unicode[STIX]{x1D700}\boldsymbol{q}$ , $\unicode[STIX]{x1D63E}|_{t=0}=\overline{\unicode[STIX]{x1D63E}}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D64C}$ with $\unicode[STIX]{x1D700}\ll 1$ , and where $(\boldsymbol{q},\unicode[STIX]{x1D64C})$ is an unstable eigenmode of the linear stability equations, with growth rate $\unicode[STIX]{x1D714}_{i}>0$ . For the purpose of the current derivation, it is helpful to rewrite the conformation tensor by pre-multiplying $\unicode[STIX]{x1D63E}|_{t=0}$ by $\overline{\unicode[STIX]{x1D641}}^{-1}$ and post-multiplying by $\overline{\unicode[STIX]{x1D641}}^{-\mathsf{T}}$ (see (3.4)). This operation yields $\unicode[STIX]{x1D642}|_{t=0}=\unicode[STIX]{x1D644}+\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{Q}}}$ , where $\boldsymbol{{\mathcal{Q}}}=\overline{\unicode[STIX]{x1D641}}^{-1}\boldsymbol{\cdot }\unicode[STIX]{x1D64C}\boldsymbol{\cdot }\overline{\unicode[STIX]{x1D641}}^{-\mathsf{T}}$ . If we assume that the mode grows according to linear theory for some time and $\unicode[STIX]{x1D642}$ evolves along Euclidean lines, then $\unicode[STIX]{x1D642}(t)=\unicode[STIX]{x1D644}+\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{Q}}}\text{e}^{\unicode[STIX]{x1D714}_{i}t}$ . If any eigenvalue of $\boldsymbol{{\mathcal{Q}}}$ is negative, $\unicode[STIX]{x1D644}+\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{Q}}}\text{e}^{\unicode[STIX]{x1D714}_{i}t}$ will eventually lose positive-definiteness.

Suppose $\boldsymbol{{\mathcal{Q}}}$ is not zero and is harmonic in a spatial direction, then $\boldsymbol{{\mathcal{Q}}}$ has a strictly negative eigenvalue somewhere in the domain. For positive-definiteness of $\unicode[STIX]{x1D642}$ , we require $1+\unicode[STIX]{x1D700}\unicode[STIX]{x1D70E}_{i}(\boldsymbol{{\mathcal{Q}}})\text{e}^{\unicode[STIX]{x1D714}_{i}t}>0$ for each $i=1,2,3$ , where $\unicode[STIX]{x1D70E}_{i}(\boldsymbol{{\mathcal{Q}}})$ denotes the $i$ th largest eigenvalue of the tensor $\boldsymbol{{\mathcal{Q}}}$ . Wherever $\unicode[STIX]{x1D70E}_{i}(\boldsymbol{{\mathcal{Q}}})<0$ , the dynamics must induce a curvature on the evolution along $\mathbf{Pos}_{3}$ before a time $t_{max}$ when the eigenvalue crosses zero. This $t_{max}$ is given by

(4.28) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{i}t_{max}=-(\log \unicode[STIX]{x1D700}+\log \max _{i}|\unicode[STIX]{x1D70E}_{i}(\boldsymbol{{\mathcal{Q}}})|), & & \displaystyle\end{eqnarray}$$

and determines an upper bound on the time for which evolution of $\unicode[STIX]{x1D642}$ along Euclidean lines does not violate the positive-definiteness constraint on the conformation tensor. The condition (4.28) is a guide for selecting the initial perturbation amplitude $\unicode[STIX]{x1D700}$ , based on $t_{max}$ ; reducing $\unicode[STIX]{x1D700}$ , one can arbitrarily increase $t_{max}$ to the desired value.

Instead of evolving along Euclidean lines, if one assumes that the perturbation evolves along a geodesic, then $\unicode[STIX]{x1D642}=\text{e}^{\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}}$ as in equation (3.14). Such an evolution remains on the manifold of $\mathbf{Pos}_{3}$ for any perturbation amplitude. We would then formally have the superexponential evolution $\unicode[STIX]{x1D642}=\text{e}^{\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{Q}}}\text{e}^{\unicode[STIX]{x1D714}_{i}t}}$ . Expanding the exponentials it can be easily shown that such an evolution is equivalent to evolution along Euclidean lines for sufficiently small $\unicode[STIX]{x1D700}$ .

Physically, a perturbation that is harmonic in space leads to regions of the flow where the polymers are compressed much more, in the sense of a volumetric change, than the maximum expansion. This is because positive and negative additive perturbations to $\overline{\unicode[STIX]{x1D63E}}$ with equal magnitudes are not of equal magnitude with respect to the natural distance on $\mathbf{Pos}_{3}$ .

5.1 Background

The growth and saturation of Tollmien–Schlichting waves in the channel flow considered here were simulated previously by Lee & Zaki (Reference Lee and Zaki2017). That study motivates the present focus on TS waves and, for this reason, its main relevant findings are outlined.

Lee & Zaki (Reference Lee and Zaki2017) performed linear stability analyses of channel flow of FENE-P fluid, with $\unicode[STIX]{x1D6FD}=0.90$ and $L_{max}=100$ . They reported the growth rates of the instability waves in $(Re,Wi)$ -space, and identified the neutral curve which exhibited non-monotonic behaviour: The critical $Re$ initially decreased and subsequently increased with increasing $Wi$ . At $Re=4667$ , they examined the two-dimensional mode which is most unstable in the Newtonian configuration, and showed that its growth rate increased with Weissenberg number up to a maximum at $Wi\approx 1.83$ . Further increasing $Wi$ to 4.50 decreased the maximum growth rate to the Newtonian value, and even lower for $Wi>4.50$ . The authors simulated the nonlinear evolution of Tollmien–Schlichting waves at $Re=4667$ for Newtonian fluid and viscoelastic fluids at $Wi\in \{1.83,4.50,6.67\}$ , i.e. cases with modal growth rates below, at and above the Newtonian value.

Lee & Zaki (Reference Lee and Zaki2017) examined the evolution of the modes using the (spatially averaged) perturbation kinetic energy and found, for all cases, that it initially grew at a rate consistent with predictions from linear theory. In all but the cases with $Wi=4.50$ , the kinetic energy eventually saturated; our focus is on conditions where an equilibrium solution is possible, and hence $Wi=4.50$ is not discussed further. The $Wi=1.83$ and $Wi=6.67$ will hereafter be referred to as $W1.83$ and $W6.67$ . The saturation energy for the Newtonian case and $W1.83$ coincided, while $W6.67$ saturated at a much lower energy. Superficially, the saturation appeared to be similar, but Lee & Zaki (Reference Lee and Zaki2017) found that the behaviour of the flow before and after saturation was distinctly different in $W1.83$ compared to $W6.67$ .

Just prior to saturation, Lee & Zaki (Reference Lee and Zaki2017) reported that the growth rate of the average kinetic energy in $W1.83$ increases substantially beyond the linear rate – qualitatively similar to behaviour in the Newtonian case. Qualitative similarity was also reported for the mean-velocity distortion as well as the term-by-term breakdown of the perturbation kinetic-energy budget in the saturated state. The flow behaviour for $W6.67$ was markedly different: there was no increase in the growth rate of average kinetic energy prior to saturation; the mean-velocity distortion in the saturated state was astonishingly small; the term-by-term breakdown of the perturbation kinetic-energy budget in the saturated state bore more resemblance to the linear growth stage, such as predominance of the critical-layer peak in the production term, rather than to the saturated states for $W1.83$ and the Newtonian case. We refer the reader to the original paper (Lee & Zaki Reference Lee and Zaki2017) for detailed figures documenting these distinctions. In the present work, we will leverage our new framework to shed more light on the differences between the polymer deformation in the two cases $W1.83$ and $W6.67$ as they saturate, as opposed to focusing on the velocity field and its modification by the polymer, which was the focus of Lee & Zaki (Reference Lee and Zaki2017).

5.2 Simulation set-up

The set-up of the direct numerical simulations performed herein is identical to the one described by Lee & Zaki (Reference Lee and Zaki2017). The flow in the channel is maintained at a constant mass rate with bulk Reynolds number $Re\equiv hU_{b}/\unicode[STIX]{x1D708}=4667$ , where $U_{b}$ is the bulk velocity, $h$ is the channel half height and $\unicode[STIX]{x1D708}$ is the total kinematic viscosity of the fluid. The two conditions simulated correspond to $Wi\equiv \unicode[STIX]{x1D70F}_{relaxation}U_{b}/h\in \{1.83,6.67\}$ , both with $\unicode[STIX]{x1D6FD}=0.9$ and $L_{max}=100$ . The flow is composed of a laminar base state at these parameters, plus a small-amplitude two-dimensional Tollmien–Schlichting wave.

The domain and grid sizes are listed in table 1. The simulations assume all fields are periodic in the streamwise $x$ and spanwise $z$ directions, and imposes no-slip condition on the velocity field at the two walls ( $y=\pm 1$ ). There are no boundary conditions on the conformation tensor. The numerical scheme used in the present simulations is similar to that used by Lee & Zaki (Reference Lee and Zaki2017), except that here we use second-order Runge–Kutta time stepping for the conformation tensor and a variant of the slope-limiting approach developed by Vaithianathan et al. (Reference Vaithianathan, Robert, Brasseur and Collins2006) for the conformation tensor advection. The latter is designed to ensure positive-definiteness of the conformation tensor (see § 2.2 of Lee & Zaki (Reference Lee and Zaki2017) and appendix B of Hameduddin et al. (Reference Hameduddin, Meneveau, Zaki and Gayme2018)). The algorithm has been extensively validated by comparing exponential growth rates of instability waves to predictions from linear theory (Lee & Zaki Reference Lee and Zaki2017, and present study) and non-modal amplification of disturbances (Agarwal, Brandt & Zaki Reference Agarwal, Brandt and Zaki2014).

Table 1. Parameters of the simulation set-up and the viscoelastic Tollmien–Schlichting wave. The characteristic length and velocity are the channel half-height and bulk flow speed.

The laminar base state can be derived from the governing equations by assuming that the flow is fully developed, so that the state variables are only functions of $y$ . The $x$ , $y$ and $z$ components of the laminar velocity field, respectively, are given by

(5.1) $$\begin{eqnarray}\displaystyle \overline{u}(y)=\frac{1}{2}\frac{Re}{\unicode[STIX]{x1D6FD}}\frac{\text{d}\overline{p}}{\text{d}x}(y^{2}-1)-\frac{1-\unicode[STIX]{x1D6FD}}{\unicode[STIX]{x1D6FD}}\int _{-1}^{y}\overline{\unicode[STIX]{x1D61B}}_{xy}(s)\,\text{d}s,\quad \overline{v}=0,~\overline{w}=0 & & \displaystyle\end{eqnarray}$$

and the laminar pressure gradient $\text{d}\overline{p}/\text{d}x$ is a fixed constant chosen so that the bulk velocity is unity, $(1/2)\int _{-1}^{1}\overline{u}(y)\,\text{d}y=1$ . The laminar velocity profiles for $Re=4667$ , $\unicode[STIX]{x1D6FD}=0.90$ , $L_{max}=100$ and $Wi\in \{1.83,6.67\}$ are shown in figure 5. The two curves are indistinguishable, and are as similar to the Newtonian profile as they are to one another.

The polymer stress component $\overline{\unicode[STIX]{x1D61B}}_{xy}$ is a solution to the following cubic equation:

(5.2) $$\begin{eqnarray}\displaystyle \frac{1}{L_{max}^{2}}\overline{\unicode[STIX]{x1D61B}}_{xy}^{3}+\frac{f(3)}{2Wi^{2}}\left(1+\frac{2f(3)+1}{L_{max}^{2}}+\frac{1-\unicode[STIX]{x1D6FD}}{\unicode[STIX]{x1D6FD}}f(3)\right)\overline{\unicode[STIX]{x1D61B}}_{xy}-\frac{Re\,f^{2}(3)}{2\unicode[STIX]{x1D6FD}Wi^{2}}\frac{\text{d}\overline{p}}{\text{d}x}y=0. & & \displaystyle\end{eqnarray}$$

The discriminant of (5.2) is negative and thus there is only one real root, which can be obtained using standard methods such as the analytical approach by Cardano & Witmer (Reference Cardano and Witmer1993). The remaining components of the stress are

(5.3a,b ) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D61B}}_{xx}=\frac{2Wi}{f(3)}\overline{\unicode[STIX]{x1D61B}}_{xy}^{2},\quad \overline{\unicode[STIX]{x1D61B}}_{yy}=\overline{\unicode[STIX]{x1D61B}}_{zz}=\overline{\unicode[STIX]{x1D61B}}_{xz}=\overline{\unicode[STIX]{x1D61B}}_{yz}=0, & & \displaystyle\end{eqnarray}$$

and the associated conformation tensor can be calculated using the stress relation (4.4). Figure 6 shows the non-zero components of the laminar base-state conformation tensor, normalized so their magnitudes are comparable. The normalized profiles are very similar but their absolute magnitudes are dramatically different, highlighting the importance of correctly ascertaining the relative size of a perturbation to these profiles.

Figure 5. Laminar streamwise velocity profile for $W1.83$ (black line with symbols,

) and $W6.67$ (grey line with symbols,

).

Figure 6. Components of the laminar base-flow conformation tensor $\overline{\unicode[STIX]{x1D63E}}$ , each normalized by its value at the bottom wall ( $y=-1$ ). Black lines are for $W1.83$ and grey lines are for $W6.67$ . (a) $\overline{\unicode[STIX]{x1D60A}}_{xx}$ (

,

) and $\overline{\unicode[STIX]{x1D60A}}_{yy}$ (

,

); (b) $\overline{\unicode[STIX]{x1D60A}}_{xy}$ (

,

). For $W1.83$ , $\overline{\unicode[STIX]{x1D60A}}_{xx}|_{y=-1}=60.15$ , $\overline{\unicode[STIX]{x1D60A}}_{yy}|_{y=-1}=0.99$ and $\overline{\unicode[STIX]{x1D60A}}_{xy}|_{y=-1}=5.42$ . For $W6.67$ , $\overline{\unicode[STIX]{x1D60A}}_{xx}|_{y=-1}=656.48$ , $\overline{\unicode[STIX]{x1D60A}}_{yy}|_{y=-1}=0.93$ and $\overline{\unicode[STIX]{x1D60A}}_{xy}|_{y=-1}=17.50$ .

The two-dimensional initial perturbation is evaluated separately by solving the eigenvalue problem associated with the stability of the laminar base flow. The streamwise wavenumber is prescribed, $k_{x}=1$ , and the eigenmode with the highest growth rate is selected at each flow condition. The modal complex frequencies, $\unicode[STIX]{x1D714}_{r}+\text{i}\unicode[STIX]{x1D714}_{r}$ , are listed in table 1. The temporal growth rate is given by the complex part of the eigenvalue, $\unicode[STIX]{x1D714}_{i}$ , and the phase velocity is given by $\unicode[STIX]{x1D714}_{r}/k_{x}$ .

5.3 Initial perturbation

The initial condition is constructed from the superposition of the base state and the instability mode,

(5.4a,b ) $$\begin{eqnarray}\displaystyle \boldsymbol{u}|_{t=0}=\overline{\boldsymbol{u}}+\text{Re}\{\hat{\boldsymbol{u}}^{\prime }|_{t=0}\text{e}^{\text{i}k_{x}x}\},\quad \unicode[STIX]{x1D63E}|_{t=0}=\overline{\unicode[STIX]{x1D63E}}+\text{Re}\{\hat{\unicode[STIX]{x1D63E}}^{\prime }|_{t=0}\text{e}^{\text{i}k_{x}x}\}, & & \displaystyle\end{eqnarray}$$

where $(\overline{\boldsymbol{u}},\overline{\unicode[STIX]{x1D63E}})$ is the laminar flow, $(\boldsymbol{u}^{\prime },\unicode[STIX]{x1D63E}^{\prime })$ is the (additive) perturbation and hats denote quantities that are Fourier transformed in the $x$ direction – only the $k_{x}=1$ Fourier component is non-zero for the initial perturbation. The initial perturbation magnitude is fixed at 0.01 % of the bulk velocity. The non-zero components of $\hat{\unicode[STIX]{x1D63E}}^{\prime }|_{t=0}$ are shown in figures 7 and 8. The figures also show the non-zero components of $\hat{\boldsymbol{{\mathcal{G}}}}|_{t=0}$ which is evaluated by pre- and post-multiplying the second equation in (5.4) with $\overline{\unicode[STIX]{x1D641}}^{-1}=\overline{\unicode[STIX]{x1D641}}^{-\mathsf{T}}$ , and is the perturbation tangent along $\mathbf{Pos}_{3}$ ,

(5.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D642}|_{t=0}=\unicode[STIX]{x1D644}+\text{Re}\{\hat{\boldsymbol{{\mathcal{G}}}}|_{t=0}\text{e}^{\text{i}k_{x}x}\}. & & \displaystyle\end{eqnarray}$$

The correct form of the tangent on $\mathbf{Pos}_{3}$ , $\hat{\boldsymbol{{\mathcal{G}}}}|_{t=0}$ , reveals details about the perturbation that are not apparent from $\hat{\unicode[STIX]{x1D63E}}^{\prime }|_{t=0}$ . We describe the most salient of these points below.

Figure 7. $Wi=1.83$ . Components of the initial perturbation tensor Fourier mode: (a–d) in the native form, $\hat{\unicode[STIX]{x1D63E}}^{\prime }|_{t=0}$ , and (e–h) in the form of a tangent on $\mathbf{Pos}_{3}$ , $\hat{\boldsymbol{{\mathcal{G}}}}|_{t=0}$ . In all panels, solid black lines are the absolute magnitudes of the modes, solid grey lines are the phase angles $\unicode[STIX]{x1D703}$ . The horizontal red dashed line is the location of the critical layer, and the vertical thin black dotted line marks zero phase angle. Note: as indicated in the axes labels, the perturbation values are normalized to optimize the clarity of the plots.

Figure 8. $Wi=6.67$ . Components of the initial perturbation tensor Fourier mode: (a–d) in the native form, $\hat{\unicode[STIX]{x1D63E}}^{\prime }|_{t=0}$ , and (e–h) in the form of a tangent on $\mathbf{Pos}_{3}$ , $\hat{\boldsymbol{{\mathcal{G}}}}|_{t=0}$ . In all panels, solid black lines are the absolute magnitudes of the modes, solid grey lines are the phase angles $\unicode[STIX]{x1D703}$ . The horizontal red dashed line is the location of the critical layer, and the vertical thin black dotted line marks zero phase angle. Note: as indicated in the axes labels, the perturbation values are normalized to optimize the clarity of the plots.

We first consider $W1.83$ . The perturbation streamwise normal stretch $\hat{\unicode[STIX]{x1D60A}}_{xx}^{\prime }|_{t=0}$ is shown in figure 7(a), and suggests that the polymer perturbation rapidly tapers off above the critical layer (the location where the instability phase speed equals the local mean velocity). Figure 7(e), however, shows that $|(\hat{{\mathcal{G}}}_{xx}|_{t=0})|$ remains relatively constant from the critical layer up to 0.84 channel half-heights away from the wall. Since $\text{tr}\,\hat{\boldsymbol{{\mathcal{G}}}}|_{t=0}\approx \hat{{\mathcal{G}}}_{xx}|_{t=0}$ , the large values of $\hat{{\mathcal{G}}}_{xx}|_{t=0}$ imply that the change in the volume ratio induced by the perturbation, relative to the mean, is similar deep in the channel and at the critical layer. This is a reflection of the fact that the mean volume is smaller closer to the centreline and therefore deformations with respect to it appear, in general, larger than those with respect to the near-wall configuration. This accurate account of the perturbation magnitude is only feasible by examining it in the correct form along the tangent to $\mathbf{Pos}_{3}$ , i.e.  $\boldsymbol{{\mathcal{G}}}$ , and also highlights the drawback of an additive, or vector space, view.

While the maximum $|(\hat{\unicode[STIX]{x1D60A}}_{yy}^{\prime }|_{t=0})|$ is located at the critical layer, the largest $y$ -direction normal stretch actually occurs at the wall as shown by $|(\hat{{\mathcal{G}}}_{yy}|_{t=0})|$ . Thus, examining the perturbation in its correct form, $\boldsymbol{{\mathcal{G}}}$ , dispels possible confusion regarding the dominance of the perturbation at the critical layer. The component $\hat{{\mathcal{G}}}_{xy}|_{t=0}$ captures the shearing deformation induced by perturbation on the mean configuration (see the discussion in § 3.2). A peak in $|(\hat{{\mathcal{G}}}_{xy}|_{t=0})|$ appears below the critical layer, which demonstrates that the perturbation induces the most shearing of the mean configuration at that location. This effect is missing from $|(\hat{\unicode[STIX]{x1D60A}}_{xy}^{\prime }|_{t=0})|$ .

Unlike $W1.83$ , at the higher Weissenberg number ( $W6.67$ ), figure 8(a–e) shows that both $\hat{\unicode[STIX]{x1D60A}}_{xx}^{\prime }|_{t=0}$ and $|(\hat{{\mathcal{G}}}_{xx}|_{t=0})|$ taper off above the critical layer. In this case, the peak $|(\hat{{\mathcal{G}}}_{xx}|_{t=0})|$ is no longer at the wall, but at the critical layer. On the other hand, while the maximum of $|(\hat{\unicode[STIX]{x1D60A}}_{yy}^{\prime }|_{t=0})|$ is located at the critical layer, the maximum of $|(\hat{{\mathcal{G}}}_{yy}|_{t=0})|$ is slightly below.

In the classical approach, $\unicode[STIX]{x1D63E}^{\prime }$ does not explain the changes in the volume ratio of the perturbed to the base conformation or the shearing deformation, because the principal axes of $\unicode[STIX]{x1D63E}^{\prime }$ and $\overline{\unicode[STIX]{x1D63E}}$ are not necessarily aligned. Thus, the geometry and relevant interpretations laid out in this paper must be kept in mind when studying perturbations to the conformation tensor.

The longest time, $t_{max}$ , for which the conformation tensor can evolve along a Euclidean path on $\mathbf{Pos}_{3}$ was derived (4.28). This time was evaluated from the initial condition at each wall-normal plane in the channel, and is shown in figure 9. The minimum value of $t_{max}$ represents the upper bound on the duration of linear evolution for the entire domain. As per (4.28), choosing a larger initial perturbation magnitude $\unicode[STIX]{x1D700}$ decreases $t_{max}$ . For $W1.83$ the minimum $t_{max}$ is located at the wall, while for $W6.67$ it is located at the critical layer, which suggests that the critical layer plays an important role in the latter case. This result is consistent with figures 7 and 8: The dominant perturbation for $W1.83$ is at the wall, while that for $W6.67$ is closer to the critical layer.

Figure 9. Upper bound, for each wall-normal plane, on the time for which evolution of $\unicode[STIX]{x1D642}$ along Euclidean lines remains positive definite, as defined in (4.28). The time is normalized by the growth rate, ${t_{max}}_{\ast }=\unicode[STIX]{x1D714}_{i}t_{max}$ . The black line is $W1.83$ and the grey line is $W6.67$ . The solid red circle markers (●) indicate the lowest ${t_{max}}_{\ast }$ for each curve: ${t_{max}}_{\ast }=6.48$ for $Wi=1.83$ , and ${t_{max}}_{\ast }=5.63$ for $Wi=6.67$ . The location of the critical layer for $W6.67$ , which is approximately the same for $W1.83$ , is shown as a red dashed line.

5.4 Time evolution of instability waves

In order to track the nonlinear temporal evolution of the unstable modes, we consider the following scalar quantities,

(5.6a,b ) $$\begin{eqnarray}\displaystyle E\equiv \frac{1}{L_{x}L_{y}L_{z}}\int |\boldsymbol{u}^{\prime }|^{2}\,\text{d}x\,\text{d}y\,\text{d}z,\quad J\equiv \frac{1}{L_{x}L_{y}L_{z}}\int d^{2}(\overline{\unicode[STIX]{x1D63E}},\unicode[STIX]{x1D63E})\,\text{d}x\,\text{d}y\,\text{d}z, & & \displaystyle\end{eqnarray}$$

where $d^{2}(\overline{\unicode[STIX]{x1D63E}},\unicode[STIX]{x1D63E})=d^{2}(\unicode[STIX]{x1D644},\unicode[STIX]{x1D642})=\text{tr}\,\boldsymbol{{\mathcal{G}}}^{2}$ and $\boldsymbol{{\mathcal{G}}}=\log \unicode[STIX]{x1D642}$ . The quantity $E$ is the volume-averaged perturbation kinetic energy, or the Euclidean norm associated with $\boldsymbol{u}^{\prime }$ , and was used by Lee & Zaki (Reference Lee and Zaki2017) to characterize viscoelastic Tollmien–Schlichting waves. While their fluctuation was defined with respect to the instantaneous mean flow, here the reference laminar state is used. We propose the quantity $J$ to evaluate the evolution of the polymer deformation, which is the volume-averaged squared geodesic distance of $\unicode[STIX]{x1D63E}$ away from $\overline{\unicode[STIX]{x1D63E}}$ (Hameduddin et al. Reference Hameduddin, Meneveau, Zaki and Gayme2018). The geodesic distance is the natural way to measure the size of the fluctuating deformation in $\unicode[STIX]{x1D63E}$ because we cannot define a norm on $\mathbf{Pos}_{3}$ due to the lack of a vector space structure. In the classical approach, one would perhaps use the Frobenius norm of $\unicode[STIX]{x1D63E}^{\prime }$ to quantify the fluctuating polymer deformation, which would lead to a wide variety of difficulties, e.g. the Frobenius norm would predict that regions of negative $\unicode[STIX]{x1D63E}^{\prime }$ are equivalent to regions of positive $\unicode[STIX]{x1D63E}^{\prime }$ . However, this is manifestly not the case (cf. discussion in § 2): regions of positive (negative) $\unicode[STIX]{x1D63E}^{\prime }$ represent polymer expansion (compression) and while expansions may be arbitrarily large, compressions cannot be. The difficulties cited above arise when Euclidean concepts are foisted upon a non-Euclidean manifold and can be completely eliminated by adopting the mathematically consistent viewpoint that treats $\mathbf{Pos}_{3}$ as a Riemannian manifold. The quantity $J$ is proposed in such a spirit.

The time evolutions of $E$ and $J$ for both cases are shown in figure 10(a) as a function of the normalized time $t_{\ast }=\unicode[STIX]{x1D714}_{i}t$ . For $W1.83$ , the evolution of $E$ matches the prediction by linear theory for $t_{\ast }\lesssim 5$ , then shows super-exponential growth and finally saturates at $t_{\ast }\approx 10$ . For $W6.67$ , the evolution agrees with linear theory for $t_{\ast }\lesssim 4$ , shows no super-exponential growth and saturates at $t_{\ast }\approx 8$ . The different behaviours when the curves deviate from linear theory suggest that different physical mechanisms are at play at the two Weissenberg numbers. The evolutions of normalized $J$ closely match those of normalized $E$ . They are also consistent with an assumption that $\unicode[STIX]{x1D642}$ initially evolves along a linear approximation of the geodesic on $\mathbf{Pos}_{3}$ emanating from $\unicode[STIX]{x1D644}$ because for $t_{\ast }\lesssim 5$ ,

(5.7) $$\begin{eqnarray}\displaystyle d^{2}(\overline{\unicode[STIX]{x1D63E}},\unicode[STIX]{x1D63E})=\text{tr}\,\log ^{2}\unicode[STIX]{x1D642} & {\approx}\text{tr}\,\log ^{2}(\unicode[STIX]{x1D644}+\boldsymbol{{\mathcal{G}}})\approx \text{tr}\,\boldsymbol{{\mathcal{G}}}^{2}\sim \unicode[STIX]{x1D700}^{2}\text{e}^{2\unicode[STIX]{x1D714}_{i}t}, & \displaystyle\end{eqnarray}$$

where we used the matrix Mercator series (Higham Reference Higham2008), and assumed $\Vert \boldsymbol{{\mathcal{G}}}\Vert \sim \unicode[STIX]{x1D700}\ll 1$ .

Figure 10. (a) Evolution of $E$ and $J$ , as defined in (5.6), normalized by the initial values. For $W1.83$ , $E(0)=2.05\times 10^{-9}$ and $J(0)=1.53\times 10^{-7}$ ; for $W6.67$ , $E(0)=1.97\times 10^{-9}$ and $J(0)=2.03\times 10^{-6}$ . Solid lines (——,

) are $J(t)/J(0)$ while dotted lines (- - - - - -,

) are $E(t)/E(0)$ . The red dash-dot line (– - – -) is an extrapolation based on the growth rate predicted by linear theory. (b) Deviation from linear theory: solid lines (——,

) are the deviations defined by (5.8) and dotted lines (- - - - - -,

) are the deviations defined by (5.9). In both (a) and (b), black lines are $W1.83$ , grey lines are $W6.67$ , thin dashed red lines (– – –) mark $t_{\ast }=4$ and $t_{\ast }=5$ , and the solid red circle markers (●) indicate the minimum ${t_{max}}_{\ast }=t_{max}\unicode[STIX]{x1D714}_{i}$ as defined in (4.28) and also marked in figure 9 ( ${t_{max}}_{\ast }=6.48$ for $W1.83$ and ${t_{max}}_{\ast }=5.63$ for $W6.67$ ).

As the Tollmien–Schlichting waves evolve in time, nonlinear effects become appreciable. For the velocity field, the role of nonlinearity can be quantified using the Euclidean norm of the deviation of the velocity field from the prediction by linear theory. The question then becomes: how do we quantify the deviation of the conformation tensor obtained using direct numerical simulations from the prediction by linear theory? In the spirit of geometric consistency, we evaluate the squared geodesic distance between the two,

(5.8) $$\begin{eqnarray}\displaystyle \frac{1}{L_{x}L_{y}L_{z}}\int d^{2}(\unicode[STIX]{x1D642},\unicode[STIX]{x1D644}+\text{Re}\{\hat{\boldsymbol{{\mathcal{G}}}}|_{t=0}\text{e}^{\text{i}(k_{x}x+\unicode[STIX]{x1D714}t)}\})\,\text{d}x\,\text{d}y\,\text{d}z, & & \displaystyle\end{eqnarray}$$

which is plotted in figure 10(b). The deviation in (5.8) is a measure of the importance of nonlinear effects. Predictably, the deviation from linear theory is only valid as long as the linear approximation $\unicode[STIX]{x1D644}+\text{Re}\{\hat{\boldsymbol{{\mathcal{G}}}}|_{t=0}\text{e}^{\text{i}(k_{x}x+\unicode[STIX]{x1D714}t)}\}$ remains positive definite, and hence $t<t_{max}$ . The upper bounds on that time, calculated using (4.28), were shown in figure 9 and are also indicated as solid red circle markers in figure 10. For longer times, we can instead evaluate

(5.9) $$\begin{eqnarray}\displaystyle \frac{1}{L_{x}L_{y}L_{z}}\int d^{2}(\unicode[STIX]{x1D642},\text{e}^{\text{Re}\{\hat{\boldsymbol{{\mathcal{G}}}}|_{t=0}\text{e}^{\text{i}(k_{x}x+\unicode[STIX]{x1D714}t)}\}})\,\text{d}x\,\text{d}y\,\text{d}z. & & \displaystyle\end{eqnarray}$$

The quantity (5.9) is also shown in figure 10. The evolution of (5.8) is virtually indistinguishable from (5.9) because the difference between them is $O(\unicode[STIX]{x1D700}^{2}\text{e}^{\unicode[STIX]{x1D700}})$ . As illustrated in the figure, however, the latter can be extended indefinitely since we are evaluating the deviation away from a perturbation along a geodesic, which is guaranteed to remain on the manifold.

For $W1.83$ , the deviation between DNS and linear theory increases at a relatively slow rate up to $t_{\ast }\approx 5$ , when it begins to grow faster until $t_{\ast }\approx 6.5$ . The initial growth in the deviation is associated with part of the region matching linear theory in the evolution of $J$ , $0\lesssim t_{\ast }\lesssim 5$ , while the later growth may be associated with super-exponential growth in $E$ and $J$ that appears before saturation. As discussed earlier, at even longer times than $t_{\ast }\approx 6.5$ , the linear approximation does not remain positive definite everywhere in the domain and thus the deviation cannot be calculated further. For $W6.67$ , we see a similar slow initial growth in the deviation from linear theory until $t_{\ast }\approx 4$ , where the growth abruptly becomes faster.

The above discussion emphasizes the importance and value of the geometry of $\mathbf{Pos}_{3}$ . It allowed us to formulate measures of the deviation away from linear theory in (5.8) and (5.9). It also enabled the interpretation of a perturbation as an excursion along a geodesic, and furnished us with a measure of the deviation (5.9) that remains valid at large times.

Nonlinear effects eventually lead to the saturated state shown in figure 10. It is instructive to compare the differences in polymer deformation between the initial linear stage and the saturated condition. We quantify this difference using the logarithmic volume ratio (2.9), here abbreviated as LVR, and the geodesic distance from the laminar base flow (2.6), which we will refer to here simply as the geodesic distance. The LVR is the ratio of the volume of $\unicode[STIX]{x1D63E}$ to the volume of $\overline{\unicode[STIX]{x1D63E}}$ and describes whether a deformation is volumetrically expansive or compressive, and by how much relative to the reference state. The geodesic distance is analogous to the norm of the velocity, but for the conformation tensor. These two measures together allow for a succinct, yet rich picture of the deformation. For example, a volume-preserving purely shearing deformation can be detected since it will lead to zero LVR but non-zero geodesic distance.

The isocontours of the LVR for $W1.83$ differ significantly between the initial condition and the saturated state (figure 11 a). For the initial condition, the most significant variations occur near the wall below the critical layer, with only very weak volume-ratio changes elsewhere. On the other hand, the main activity in the saturated state is focused away from the wall: a thin, elongated region of large volume-ratio expansion is centred at $y\approx -0.5$ , and a large region of volume-ratio expansion is near $y\approx -0.1$ and is connected to the wall via a filament of expansive stretch. The geodesic distance shown in figure 11(b) is consistent with these volumetric changes. However, the geodesic distance peaks at $x\approx 0.3\unicode[STIX]{x03C0}$ in the critical layer, even though this region only shows relatively small LVR in figure 11(a). These results imply that the polymer is undergoing a shearing deformation at the this location. The significant differences between the flow structures in the linear and saturated stages demonstrate that the mechanism for the initial growth of the Tollmien–Schlichting wave is disrupted in the latter stage. Only at saturation do we observe large volume-ratio changes near the centreline, and the localized largely shearing deformation near the critical layer.

Figure 11. Case $W1.83$ at $t_{\ast }\in \{0,17\}$ . Isocontours of (a) logarithmic volume ratio, $\text{tr}\,\boldsymbol{{\mathcal{G}}}$ , (b) geodesic distance from the laminar, $\sqrt{\text{tr}\,\boldsymbol{{\mathcal{G}}}^{2}}$ . In both panels, the top half ( $y\in (0,1]$ ) of the channel is at $t_{\ast }=0$ and the bottom half ( $y\in [-1,0)$ ) of the channel is at $t_{\ast }=17$ . The solid black line is the channel centreline and the dashed red lines are the locations of the critical layers.

Figure 12. Case $W6.67$ at $t_{\ast }\in \{0,8\}$ . Isocontours of (a) normalized logarithmic volume ratio, $\text{tr}\,\boldsymbol{{\mathcal{G}}}$ , (b) normalized geodesic distance from the laminar, $\sqrt{\text{tr}\,\boldsymbol{{\mathcal{G}}}^{2}}$ . In both panels, the top half ( $y\in (0,1]$ ) of the channel is at $t_{\ast }=0$ and the bottom half ( $y\in [-1,0)$ ) of the channel is at $t_{\ast }=8$ . The solid black line is the channel centreline and the dashed red lines are the locations of the critical layers.

The isocontours of the LVR for $W6.67$ , shown in figure 12(a), are strikingly similar at the initial and saturated states. The main volume-ratio change is near the critical layer, and remains so in the saturated state. The geodesic distance in figure 12(b) confirms this finding, with little shearing deformation evident. Both measures also show that there is a spreading of the stretched/compressed region from the critical layer towards the centreline. Both the initial condition and the saturated state are dominated by the $k_{x}=1$ mode, suggesting that the nonlinear terms in the governing equations do not significantly alter the perturbation spectra.

Figure 13. Components of $\unicode[STIX]{x1D700}^{k}\boldsymbol{{\mathcal{G}}}_{k}$ with $k\in \{1,2\}$ for $W1.83$ at $t_{\ast }=4$ , i.e. the tangents on $\mathbf{Pos}_{3}$ that appear in the weakly nonlinear deformation expansion (3.10). (a) $\unicode[STIX]{x1D700}^{k}{\mathcal{G}}_{k_{xx}}$ , (b) $\unicode[STIX]{x1D700}^{k}{\mathcal{G}}_{k_{yy}}$ , (c) $\unicode[STIX]{x1D700}^{k}{\mathcal{G}}_{k_{xy}}$ . In all panels, the top half ( $y\in (0,1]$ ) of the channel is $k=1$ and the bottom half ( $y\in [-1,0)$ ) of the channel is $k=2$ . The solid black line is the channel centreline and the dashed red lines are the locations of the critical layers.

Figure 14. Components of $\unicode[STIX]{x1D700}^{k}\boldsymbol{{\mathcal{G}}}_{k}$ with $k\in \{1,2\}$ for $W6.67$ at $t_{\ast }=4$ , i.e. the tangents on $\mathbf{Pos}_{3}$ that appear in the weakly nonlinear deformation expansion (3.10). (a) $\unicode[STIX]{x1D700}^{k}{\mathcal{G}}_{k_{xx}}$ , (b) $\unicode[STIX]{x1D700}^{k}{\mathcal{G}}_{k_{yy}}$ , (c) $\unicode[STIX]{x1D700}^{k}{\mathcal{G}}_{k_{xy}}$ . In all panels, the top half ( $y\in (0,1]$ ) of the channel is $k=1$ and the bottom half ( $y\in [-1,0)$ ) of the channel is $k=2$ . The solid black line is the channel centreline and the dashed red lines are the locations of the critical layers.

5.5 Weakly nonlinear deformation

In order to examine the initial deviation from linear theory, we adapt the approach that was used by Benney & Lin (Reference Benney and Lin1960) to study secondary instabilities in Newtonian parallel shear flows. For sufficiently small deviations from linear evolution, we assume that $\unicode[STIX]{x1D642}$ can be expressed as a polynomial expansion, here as the weakly nonlinear deformation in (3.10), because then we can endow the terms in the expansion with physical meaning. The growth rate and phase speed predicted by linear theory can be used to directly calculate the first variable $\boldsymbol{{\mathcal{G}}}_{(1)}$ in the series (3.10),

(5.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}=\text{Re}\{\hat{\boldsymbol{{\mathcal{G}}}}|_{t=0}\text{e}^{\text{i}(k_{x}x-\unicode[STIX]{x1D714}t)}\}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D714}$ is the eigenvalue listed in table 1. Once $\boldsymbol{{\mathcal{G}}}_{(1)}$ is available, it is straightforward to rearrange (3.10) and obtain an approximation to the second variable $\boldsymbol{{\mathcal{G}}}_{(2)}$ :

(5.11) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}^{2}\boldsymbol{{\mathcal{G}}}_{(2)}=\unicode[STIX]{x1D642}-(\unicode[STIX]{x1D644}+\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D700}^{2}\boldsymbol{{\mathcal{G}}}_{(1)}^{2})+O(\unicode[STIX]{x1D700}^{3}). & & \displaystyle\end{eqnarray}$$

For both Weissenberg numbers, we will perform the analysis at $t_{\ast }=4$ , since the deviation from linear theory at that time is still relatively small but not negligible (see figure 10). The $xx$ , $yy$ and $xy$ components of $\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}$ and $\unicode[STIX]{x1D700}^{2}\boldsymbol{{\mathcal{G}}}_{(2)}$ are reported in figure 13, for $W1.83$ . The $xx$ components of both $\boldsymbol{{\mathcal{G}}}_{(1)}$ and $\boldsymbol{{\mathcal{G}}}_{(2)}$ in figure 13(a) show similarly shaped isocontours, but are out of phase with each other. Recall from the discussion in § 3.2 and (4.23) that the LVR is the sum of the trace of the terms of the weakly nonlinear deformation. For both $\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}$ and $\unicode[STIX]{x1D700}^{2}\boldsymbol{{\mathcal{G}}}_{(2)}$ , the $xx$ component contributes more to LVR than any other component. The ability to tease out the volume-ratio change by simply examining the terms in the expansion highlights a major benefit of using a weakly nonlinear deformation expansion, rooted in physical notions.

The isocontours of the $xy$ component of $\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}$ and $\unicode[STIX]{x1D700}^{2}\boldsymbol{{\mathcal{G}}}_{(2)}$ are shown in figure 13(c). This component represents the shearing deformation (see the discussion in § 3.2). For the prediction by linear theory, $\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}$ , the perturbations are most strongly shearing above the critical layer at $y\approx -0.75$ , whereas for the nonlinear correction, $\unicode[STIX]{x1D700}^{2}\boldsymbol{{\mathcal{G}}}_{(2)}$ , the shearing is focused closer to the critical and localized in $x$ . This nonlinear correction with significant localized shear explains the observation previously noted regarding figure 11: there is a region, localized in $x$ and close to the critical layer, that shows the most significant geodesic deviation but locally only weakly changes the LVR. These two factors indicate a shearing deformation, which is consistent with a large $xy$ component of the higher-order correction, $\unicode[STIX]{x1D700}^{2}\boldsymbol{{\mathcal{G}}}_{(2)}$ . The ease with which the localized shearing deformation, whose effects appear prominently in the saturated state, was deduced simply from a consideration of a slight deviation from linear theory demonstrates the effectiveness of using physically relevant polynomial expansions.

Figure 14 shows $\unicode[STIX]{x1D700}\boldsymbol{{\mathcal{G}}}_{(1)}$ and $\unicode[STIX]{x1D700}^{2}\boldsymbol{{\mathcal{G}}}_{(2)}$ for $W6.67$ . The isocontours of both quantities similarly show that regions of expansion/compression are concentrated near the critical layer and are dominated by the $k_{x}=1$ Fourier mode. This behaviour echoes the findings in figure 12. Moreover, in that figure we noted the spreading of stretched/compressed regions near the critical layer towards the centreline. The source of this spreading is clear in figure 14: the regions of largest stretch and compression are located right above the critical layer.