2. Oldroyd-B fluids in plane Couette flow

Consider an incompressible viscoelastic fluid flow with velocity $\boldsymbol{u} = u_x \boldsymbol{e}_x + u_y \boldsymbol{e}_y + u_z \boldsymbol{e}_z$ between two parallel plates at $y = 0$ and $y=\ell$ . The bottom plate is stationary and the top plate is travelling at velocity $U$ in the $\boldsymbol{e}_x$ direction. Periodic boundary conditions are imposed on the $\boldsymbol{e}_x$ and $\boldsymbol{e}_z$ directions and no-slip conditions are assumed at the upper and lower boundaries.

In the Oldroyd-B model, polymers dissolved in the fluid solvent are modelled as dumbbells which are advected, stretched and rotated by the flow. The stress tensor $\unicode{x1D64F}$ in the momentum equation

(2.1) \begin{align} \rho \left [\frac {\partial \boldsymbol{u}}{\partial t} + (\boldsymbol{u} \boldsymbol{\cdot } \boldsymbol{\nabla }) \boldsymbol{u}\right ] = \boldsymbol{\nabla }\boldsymbol{\cdot } \unicode{x1D64F} - \boldsymbol{\nabla }p \end{align}

consists of both contributions from the solvent and polymer via

(2.2) \begin{align} \unicode{x1D64F} = \unicode{x1D64F}_p + \eta _s \big [\boldsymbol{\nabla }\boldsymbol{u} + (\boldsymbol{\nabla }\boldsymbol{u})^\top \big ], \end{align}

where $\rho$ is the density, $\eta _s$ is the solvent viscosity, $p$ is the pressure and $\unicode{x1D64F}_p$ is the polymeric contribution to the stress tensor. This is assumed to satisfy the upper-convected Maxwell equation

(2.3) \begin{align} \lambda \left [\frac {\partial \unicode{x1D64F}_p}{\partial t} + \boldsymbol{u} \boldsymbol{\cdot } \boldsymbol{\nabla }\unicode{x1D64F}_p - \unicode{x1D64F}_p \boldsymbol{\cdot } \boldsymbol{\nabla }\boldsymbol{u} - (\boldsymbol{\nabla }\boldsymbol{u})^\top \boldsymbol{\cdot } \unicode{x1D64F}_p\right ] + \unicode{x1D64F}_p = \eta _p \left [\boldsymbol{\nabla }\boldsymbol{u} + (\boldsymbol{\nabla }\boldsymbol{u})^\top \right ], \end{align}

where $\eta _p$ is the polymer viscosity and $\lambda$ is the characteristic relaxation time scale of the dissolved polymers.

The standard non-dimensionalisation of the system is performed using $\ell$ as a characteristic length scale, $U$ as a velocity scale, $\ell / U$ as the associated time scale and by introducing the polymer conformation tensor

(2.4) \begin{align} \unicode{x1D658} = \frac {\lambda }{\eta _p} \unicode{x1D64F}_p + \unicode{x1D644}, \end{align}

where $ \unicode{x1D644}$ is the identity matrix. Denoting the resulting non-dimensional quantities again by $\boldsymbol{u}$ and $p$ , the non-dimensional system reads

(2.5a) \begin{align} \frac {\partial \boldsymbol{u}}{\partial t} + (\boldsymbol{u} \boldsymbol{\cdot } \boldsymbol{\nabla }) \boldsymbol{u} + \boldsymbol{\nabla }p &= \frac {\beta }{Re} {\nabla} ^2 \boldsymbol{u} + \frac {1-\beta }{\textit{Re Wi}} \boldsymbol{\nabla }\boldsymbol{\cdot } \unicode{x1D658},\end{align}

(2.5b) \begin{align} \boldsymbol{\nabla }\boldsymbol{\cdot } \boldsymbol{u} &= 0, \end{align}

(2.5c) \begin{align} \frac {\partial \unicode{x1D658}}{\partial t} + (\boldsymbol{u}\boldsymbol{\cdot } \boldsymbol{\nabla }) \unicode{x1D658} = \unicode{x1D658} \boldsymbol{\cdot } \boldsymbol{\nabla }\boldsymbol{u} &+ (\boldsymbol{\nabla }\boldsymbol{u})^\top \boldsymbol{\cdot } \unicode{x1D658} + \frac {1}{ \textit{Wi}}( \unicode{x1D644} - \unicode{x1D658}), \end{align}

on the domain $\varOmega = [0, \varGamma _x] \times [0, 1] \times [0, \varGamma _z]$ with boundary conditions

(2.6) \begin{align} \begin{aligned} \boldsymbol{u}(x, y, z) &= \boldsymbol{u}(x + \varGamma _x, y, z), & \boldsymbol{u}(x, y, z) &= \boldsymbol{u}(x, y, z + \varGamma _z),\\ \unicode{x1D658}(x, y, z) &= \unicode{x1D658}(x + \varGamma _x, y, z), & \unicode{x1D658}(x, y, z) &= \unicode{x1D658}(x, y, z + \varGamma _z),\\ \boldsymbol{u}(x, 0, z) &= 0, & \boldsymbol{u}(x, 1, z) &= \boldsymbol{e}_x. \end{aligned} \end{align}

The system’s behaviour is governed by three non-dimensional parameters

(2.7) \begin{align} \textit{Re}= \frac {\rho U \ell }{\eta }, \qquad \beta = \frac {\eta _s}{\eta }, \qquad \textit{Wi} = \frac {U \lambda }{\ell }, \end{align}

where the Reynolds number $ \textit{Re}$ is defined in terms of the total viscosity $\eta = \eta _s + \eta _p$ , the Weissenberg number $ \textit{Wi}$ is the ratio between the characteristic time scale of the polymer to that of the fluid and $\beta$ is the ratio of solvent viscosity to total viscosity.

No boundary conditions are needed for $ \unicode{x1D658}$ at the walls. The constitutive equation (2.5c ) is first order in space: the only spatial derivative of $ \unicode{x1D658}$ appears through the transport term $\boldsymbol{u} \boldsymbol{\cdot } \boldsymbol{\nabla }\unicode{x1D658}$ . Since the velocity field satisfies the no-penetration condition $\boldsymbol{u} \boldsymbol{\cdot } \boldsymbol{n} = 0$ at the walls, there is no inflow boundary through which information on the conformation tensor must be supplied. However, it is useful to note that $ \unicode{x1D658}$ can be viewed as the covariance matrix of the end-to-end point vectors of the polymer dumbbells. As a result, it is natural that $ \unicode{x1D658}$ is symmetric and positive definite, which we write in the standard notation $ \unicode{x1D658} = \unicode{x1D658}^\top \succ 0$ . It is well known that if initially

(2.8) \begin{align} \unicode{x1D658}(\boldsymbol{x}, 0) = \unicode{x1D658}^\top (\boldsymbol{x}, 0)\succ 0, \qquad \boldsymbol{x} \in \varOmega , \end{align}

then this property holds for solutions of (2.5c ) for all subsequent times. Therefore, any such solution of the Oldroyd-B equations (2.5a – c ) at each time instant belongs to the set $\mathcal{S}$ (Binns & Wynn Reference Binns and Wynn2024), defined as

(2.9) \begin{align} \mathcal{S} := \left \{(\boldsymbol{u}, \unicode{x1D658}) : \; \begin{array}{c} \boldsymbol{u} \;\text{satisfies the boundary conditions ( {2.6})}\\ \boldsymbol{\nabla }\boldsymbol{\cdot } \boldsymbol{u} = 0 \;\text{and} \; \unicode{x1D658} = \unicode{x1D658}^\top \succ 0 \end{array}\right \}. \end{align}

2.1. Energy dissipation

Our aim is to understand how the energy dissipation rate of Oldroyd-B fluids in plane Couette flow depends on the three governing parameters ( $\beta$ , $ \textit{Re}$ , $ \textit{Wi}$ ). We follow the widely accepted formulation of total energy of Oldroyd-B fluids (Doering, Eckhardt & Schumacher Reference Doering, Eckhardt and Schumacher2006; Hu & Lelièvre Reference Hu and Lelièvre2007; Boyaval, Lelièvre & Mangoubi Reference Boyaval, Lelièvre and Mangoubi2009; Lukáčová−Medvid’ová et al. Reference Lukáčová−Medvid’ová, Notsu and She2016)

(2.10) \begin{align} E := \frac {1}{2} \left \lVert \boldsymbol{u} \right \rVert ^2 + \frac {1-\beta }{{2 \textit{Re Wi}}} \int _\varOmega \mathrm{tr}( \unicode{x1D658} - \unicode{x1D644}\kern1.5pt) - \log \det \unicode{x1D658}\;\mathrm{d} V, \end{align}

which can be interpreted as the sum of the fluid’s kinetic energy and the potential energy stored in the polymers. This is the standard thermodynamically consistent free-energy formulation for the Oldroyd-B model (Hu & Lelièvre Reference Hu and Lelièvre2007). The important property for the present analysis is that differentiating (2.10) along trajectories yields the balance (2.13), with a non-negative polymer dissipation density $\mathrm{tr} ( \unicode{x1D658} + \unicode{x1D658}^{-1} -2 \unicode{x1D644} \kern1pt)$ . Thus the analysis relies on the free-energy/dissipation structure of the constitutive model, rather than on the particular formulation of the polymer energy. Other constitutive models may have different free energies, but the same strategy is expected to apply whenever an analogous energy balance with non-negative polymer dissipation is available.

Here, and in what follows, we use the following notation for $L^2$ inner product and norm:

(2.11) \begin{align} \left \langle \boldsymbol{u}, \boldsymbol{v}\right \rangle =\int _\varOmega \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{v}\;\mathrm{d} V, \qquad \qquad \left \lVert \boldsymbol{u} \right \rVert ^2=\left \langle \boldsymbol{u}, \boldsymbol{u}\right \rangle . \end{align}

Along trajectories of the flow (2.5), it follows from the Jacobi formula $({\partial }/{\partial t}) \log \det \unicode{x1D658} = \mathrm{tr} ( \unicode{x1D658}^{-1} ({\partial \unicode{x1D658}}/{\partial t}) )$ that

(2.12) \begin{align} \frac {\mathrm{d} E}{\mathrm{d} t} = \int _\varOmega \boldsymbol{u} \boldsymbol{\cdot } \frac {\partial \boldsymbol{u}}{\partial t} \;\mathrm{d} V + \frac {1-\beta }{{2 \textit{Re Wi}}} \int _\varOmega \mathrm{tr}\left [\left ( \unicode{x1D644} - \unicode{x1D658}^{-1}\right ) \frac {\partial \unicode{x1D658}}{\partial t}\right ] \;\mathrm{d} V. \end{align}

Substituting (2.5a ) and (2.5c ) into the above equation, using incompressibility (2.5b ) and boundary conditions (2.6) gives the energy equation

(2.13) \begin{align} \begin{split} \frac {\mathrm{d} E}{\mathrm{d} t} =& -\frac {\beta }{ Re} \left \lVert \boldsymbol{\nabla }\boldsymbol{u} \right \rVert ^2 - \frac {1-\beta }{{2 \textit{Re Wi}}^2} \int _\varOmega \mathrm{tr}\big ( \unicode{x1D658} + \unicode{x1D658}^{-1} - 2 \unicode{x1D644}\kern1.5pt \big) \;\mathrm{d} V \\[5pt] &\qquad \qquad + \frac {\beta }{ Re} \int _{y=1} \frac {\partial u_x}{\partial y} \;\mathrm{d} A + \frac {1-\beta }{\textit{Re Wi}} \int _{y=1} c_{xy} \;\mathrm{d} A. \end{split} \end{align}

The first two terms in (2.13) are necessarily negative, while the final two boundary integrals are associated with the power required to enforce the boundary conditions on the upper surface. We therefore interpret the first two terms as the energy dissipation rate

(2.14) \begin{align} \mathcal{E} = \frac {\beta }{ Re} \left \lVert \boldsymbol{\nabla }\boldsymbol{u} \right \rVert ^2 + \frac {1-\beta }{{2 \textit{Re Wi}}^2} \int _\varOmega \mathrm{tr}\big ( \unicode{x1D658} + \unicode{x1D658}^{-1} - 2 \unicode{x1D644} \kern1pt \big ) \;\mathrm{d} V \end{align}

and the last two as production

(2.15) \begin{align} \mathcal{P} = \frac {\beta }{ Re} \int _{y=1} \frac {\partial u_x}{\partial y} \;\mathrm{d} A + \frac {1-\beta }{\textit{Re Wi}} \int _{y=1} c_{xy} \;\mathrm{d} A. \end{align}

If we assume that $E(t)$ is uniformly bounded along trajectories of the system then, taking the time average of (2.13) gives

(2.16) \begin{align} \overline {\mathcal{E}} = \overline {\mathcal{P}}, \end{align}

where $\overline {\mathcal{E}} = \limsup _{T\to \infty }( {1}/{T})\int _0^T \mathcal{E}(\boldsymbol{u}(t), \unicode{x1D658}(t)) \;\mathrm{d} t$ .

The aim of this paper is to prove – for a range of parameter values $(\beta , \textit{Re}, \textit{Wi})$ – rigorous upper bounds on the magnitude of the averaged energy dissipation rate $\overline {\mathcal{E}}$ which must hold for any possible realisation of the flow. To do this, we will use the auxiliary-functional method (Chernyshenko et al. Reference Chernyshenko, Goulart, Huang and Papachristodoulou2014).

3. The auxiliary-functional method

The key step in proving the energy balance (2.16) was to observe that

(3.1) \begin{align} \overline {\frac {\mathrm{d} }{\mathrm{d} t}E[\boldsymbol{u}(t), \unicode{x1D658}(t)]} = 0 \end{align}

holds if energy is bounded along trajectories of the system. The auxiliary-functional method uses a similar observation but with energy replaced by an arbitrary ‘auxiliary’ functional in order to prove bounds for time averages.

In particular, for any functional $V : (\boldsymbol{u}, \unicode{x1D658}) \to \mathbb{R}$ , let $\mathcal{L}_V$ be the functional representing the Lie derivative of $V$ , meaning that

(3.2) \begin{align} \frac {\mathrm{d} }{\mathrm{d} t} V[\boldsymbol{u}(t), \unicode{x1D658}(t)] = \mathcal{L}_V [\boldsymbol{u}(t), \unicode{x1D658}(t)] \end{align}

for any solution $(\boldsymbol{u}(t), \unicode{x1D658}(t))$ of (2.5a – c ). If $V[\boldsymbol{u}(t), \unicode{x1D658}(t)]$ is bounded along trajectories of the system, it then follows that

(3.3) \begin{align} \overline {\mathcal{L}_V [\boldsymbol{u}(t), \unicode{x1D658}(t)]} = 0. \end{align}

As an example, the energy equation (2.13) implies the functional equality $\mathcal{L}_E = -\mathcal{E} + \mathcal{P}$ , and its time average along trajectories of the system gives (2.16).

This idea can be generalised since (3.3) implies that every acceptable $V$ satisfies the equality

(3.4) \begin{align} \overline {\mathcal{E}[\boldsymbol{u}(t), \unicode{x1D658}(t)]} = \overline {(\mathcal{E} + \mathcal{L}_V) [\boldsymbol{u}(t), \unicode{x1D658}(t)]} \end{align}

along trajectories of the system. This gives many different ways to express the time average of energy dissipation. To exploit them, we search for a functional $V$ which satisfies an upper bound of the form

(3.5) \begin{align} \left (\mathcal{E} + \mathcal{L}_V\right )[\boldsymbol{u}, \unicode{x1D658}] \leq \kappa \qquad \forall \, (\boldsymbol{u}, \unicode{x1D658}) \in \mathcal{S}, \end{align}

where $\kappa \in \mathbb{R}$ is a constant and $\mathcal{S}$ is the admissible, forward-invariant set (2.9). Since we know all solutions to the Oldroyd-B model belong to $\mathcal{S}$ if they do so initially, it follows from time-averaging (3.5) that $\overline {\mathcal{E}} \leq \kappa$ for any such trajectory. This observation is a special case of the following weaker, but sufficient for our case, version of the general result (Chernyshenko et al. Reference Chernyshenko, Goulart, Huang and Papachristodoulou2014; Chernyshenko Reference Chernyshenko2022, Theorem 2.2).

The key advantage of using the auxiliary-functional method is that inequality (3.5) can be checked without solving the partial differential equations. Given a candidate functional $V$ , one only needs to verify that $\mathcal{E} + \mathcal{L}_V$ is bounded on the admissible set $\mathcal{S}$ . Since $\mathcal{S}$ is relatively simple, this can be easier than directly computing the average dissipation $\overline {\mathcal{E}}$ . While auxiliary functionals provide arbitrarily sharp bounds on time averages (Tobasco, Goluskin & Doering Reference Tobasco, Goluskin and Doering2018; Rosa & Temam Reference Rosa and Temam2020), this might require constructing more ‘intricate’ auxiliary functionals $V$ , which, essentially, seek to mathematically extract the correct physics.

To highlight the difficulties involved, suppose we take $V = E$ . Then (3.5) becomes

(3.6) \begin{align} (\mathcal{E} + \mathcal{L}_V) [\boldsymbol{u}, \unicode{x1D658}] = \mathcal{P}[\boldsymbol{u}, \unicode{x1D658}] = \frac {\beta }{ Re} \int _{y=1} \frac {\partial u_x}{\partial y} \;\mathrm{d} A + \frac {1-\beta }{\textit{Re Wi}} \int _{y=1} c_{xy} \;\mathrm{d} A. \end{align}

However, since flow and polymer fields in $\mathcal{S}$ can become arbitrarily large, there is no constant $\kappa$ for which the inequality (3.5) can hold in this case. That is, while we can assert that $\overline {\mathcal{E}} = \overline {\mathcal{P}}$ , we cannot rigorously say anything about the size of either of these quantities.

The classical way to obtain bounds for time averages is the background-flow method (Doering & Constantin Reference Doering and Constantin1992, Reference Doering and Constantin1994, Reference Doering and Constantin1995). This is a special case of the auxiliary-functional method in which one seeks an auxiliary functional in the form of the energy of flow perturbations from a suitably selected ‘background flow’, multiplied by a constant $\varLambda \gt 0$ called a ‘balancing parameter’. The background-flow method has found wide application for Newtonian fluids ( $\beta =1$ , $ \textit{Re} \neq 0$ , $ \textit{Wi} = 0$ ) and, in this setting, corresponds to studying auxiliary functionals $V$ of the form

(3.7) \begin{align} V[\boldsymbol{u}] =\text{Const}\,\left \lVert \boldsymbol{u} - \boldsymbol{b} \right \rVert ^2, \end{align}

where $\boldsymbol{b}$ is the background flow which can be chosen to shape $V$ .

For Newtonian Couette flow, a typical choice is to take $\boldsymbol{b} = b(y) \,\boldsymbol{e}_x$ as a pure shear profile. At sufficiently low Reynolds numbers, the choice of $\boldsymbol{b}$ which gives the best bound (i.e. the lowest possible $\kappa$ in (3.5)) is the steady solution and the corresponding bound is $\kappa = \mathcal{E}_{\textit{steady}} = 1/ \textit{Re}$ . For higher Reynolds numbers, the optimal choice of $\boldsymbol{b}$ is known to develop a boundary-layer-like structure indicated in figure 1 (Doering & Constantin Reference Doering and Constantin1992), and the exact optimal shape of $b(y)$ must be determined numerically (Plasting & Kerswell Reference Plasting and Kerswell2003). (Also see Rajkotia-Zaheer & Goluskin Reference Rajkotia-Zaheer and Goluskin2026 for more rigorous justification of these numerical results.) Via (3.5), this optimisation gives an $ \textit{Re}$ -dependent upper bound $\overline {\mathcal{E}} \leq \kappa (\textit{Re})$ . At low Reynolds numbers this bound coincides with the laminar dissipation $1/ \textit{Re}$ , while at high Reynolds numbers the best available background-method bounds approach an $\mathcal{O}(1)$ constant. In this sense, the method captures the transition from the steady Couette scaling to a Reynolds-number-independent upper bound relevant to turbulent Couette flow.

Figure 1.

Steady Couette solution and an example background-flow profile. Here $\delta$ denotes the boundary-layer thickness of the background-flow profile.

To date, neither the auxiliary nor the background-flow method has been applied to viscoelastic flows. In Newtonian flow, a quadratic kinetic-energy functional is natural since the advection term does not contribute to the energy balance: $\langle \boldsymbol{u}, \boldsymbol{u} \boldsymbol{\cdot } \boldsymbol{\nabla }\boldsymbol{u}\rangle = 0$ . The same cancellation remains available for the velocity field in the Oldroyd-B equations, which is why the velocity-dependent part of the auxiliary functional below remains quadratic. The additional difficulty is that the constitutive equation contains the stretching terms $ \unicode{x1D658} \boldsymbol{\cdot } \boldsymbol{\nabla }\boldsymbol{u} + (\boldsymbol{\nabla }\boldsymbol{u})^\top \boldsymbol{\cdot } \unicode{x1D658}$ . The trace/logarithmic structure of the Oldroyd-B free energy is what gives the cancellation underlying the energy equation (2.13): the terms produced by polymer stretching cancel with a polymer stress-work term in the kinetic-energy balance, while polymer relaxation gives a sign-definite dissipation. As a result, the natural polymer-dependent part of the auxiliary functional is non-polynomial in $ \unicode{x1D658}$ . In the Newtonian case, the search for the best $V$ can be viewed as searching over auxiliary functionals obtained by adding to the quadratic energy a term that is linear in the state, together with a constant, since the associated background-flow functional in (3.7) can be written as $V[\boldsymbol{u}] = \text{const}\, (\left \lVert \boldsymbol{u} \right \rVert ^2 -2 \langle \boldsymbol{b}, \boldsymbol{u}\rangle + \left \lVert \boldsymbol{b} \right \rVert ^2 )$ . Polymeric perturbation energies derived from (2.10) are non-polynomial, and this significantly complicates the subsequent analysis.

4. Bounds on energy dissipation for Oldroyd-B fluids in plane Couette flow

Throughout, we will use the following decomposition of the fluid and polymer fields

(4.1) \begin{align} \boldsymbol{u} = \boldsymbol{b} + \boldsymbol{v}, \qquad \unicode{x1D658} = \unicode{x1D658}_b + \unicode{x1D659}, \end{align}

where $\boldsymbol{b}$ and $ \unicode{x1D658}_b$ are independent of time, satisfy the boundary conditions (2.6) and $\boldsymbol{\nabla }\boldsymbol{\cdot } \boldsymbol{b}=0$ . These choices imply that the velocity perturbation $\boldsymbol{v}$ satisfies homogeneous boundary conditions and that $\boldsymbol{\nabla }\boldsymbol{\cdot } \boldsymbol{v}=0$ . Moreover, as it is often done in the background-flow method, we will choose $\boldsymbol{b}$ to satisfy the Euler equations, so that $\boldsymbol{b}\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{b}$ is a gradient of some scalar. Consequently, $\left \langle \boldsymbol{v}, \boldsymbol{b}\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{b}\right \rangle =0$ , which simplifies certain formulae. We will refer to $\boldsymbol{v}$ and $ \unicode{x1D659}$ as perturbations, but it should be emphasised that they are perturbations from the background field $(\boldsymbol{b}, \unicode{x1D658}_b)$ . Unlike in stability studies, $(\boldsymbol{b}, \unicode{x1D658}_b)$ is not required to be a solution of the governing equations.

The form of the auxiliary functional is motivated by the perturbation entropy functional introduced by Binns & Wynn (Reference Binns and Wynn2024) for the global stability problem. In their setting, the reference state is the steady Couette solution, and the functional is used as a Lyapunov functional to prove decay towards that steady state. In the present work, we use the same structure, but replace the steady reference state by the general background pair ( $\boldsymbol{b}, \unicode{x1D658}_b$ ). Thus the functional below may be viewed as the background-flow analogue of the Binns & Wynn (Reference Binns and Wynn2024) perturbation entropy: the background fields are not required to solve the governing equations, but are instead chosen to obtain the best possible dissipation bound.

Accordingly, we define the functional

(4.2) \begin{align} V[\boldsymbol{u}, \unicode{x1D658}] = \frac {1}{2} \|\boldsymbol{v}\|^2 + \frac {1-\beta }{2\alpha \textit{Re Wi}} \int _\varOmega \mathrm{tr}( \unicode{x1D644} + \alpha \unicode{x1D659}\kern1.5pt) -3 - \log \det ( \unicode{x1D644} + \alpha \unicode{x1D659}\kern1.5pt) \;\mathrm{d} V, \end{align}

where $\boldsymbol{v} = \boldsymbol{u} - \boldsymbol{b}$ , $ \unicode{x1D659} = \unicode{x1D658} - \unicode{x1D658}_b$ . The time-independent background fields $\boldsymbol{b}, \unicode{x1D658}_b$ , together with the constant $\alpha \gt 0$ , must all be specified to define $V$ . The parameter $\alpha$ is used to ensure that $ \unicode{x1D644} + \alpha \unicode{x1D659} \succ 0$ , so that the logarithmic term is well defined. In the special case $ \unicode{x1D658}_b = \unicode{x1D644}$ and $\alpha = 1$ , the polymeric term in (4.2) coincides with the polymeric contribution to the total energy (2.10). For more general choices of $\boldsymbol{b}$ and $ \unicode{x1D658}_b$ , the functional (4.2) can be viewed as the sum of the kinetic energy of the velocity perturbation and a rescaled polymeric perturbation energy relative to the chosen background pair ( $\boldsymbol{b}, \unicode{x1D658}_b$ ).

To emphasise the ‘energy like’ properties of $V$ , we now collect some of its important mathematical properties used in the subsequent analysis. First, the requirement that $ \unicode{x1D644} + \alpha \unicode{x1D659} \succ 0$ ensures $V$ is non-negative and $V[\boldsymbol{u}, \unicode{x1D658}]=0$ if and only if $(\boldsymbol{v}, \unicode{x1D659}\kern1.5pt) = (\boldsymbol{0},\boldsymbol{\mathsf{0}})$ (Binns & Wynn Reference Binns and Wynn2024). Furthermore, $V$ is bounded from above by dissipative terms in the perturbation variables via

(4.3) \begin{align} 0 \leq V[\boldsymbol{u}, \unicode{x1D658}] \leq C_p \frac 12 \|\boldsymbol{\nabla }\boldsymbol{v}\|^2 + \frac {1-\beta }{2\alpha \textit{Re Wi}} \|\alpha \unicode{x1D659}\|_{\alpha \unicode{x1D659}}^2 \qquad \forall \,(\boldsymbol{u}, \unicode{x1D658}) \in \mathcal{S}. \end{align}

Here, $C_p$ is a constant arising from the Poincaré inequality $\|\boldsymbol{v}\|^2 \leq C_p \|\boldsymbol{\nabla }\boldsymbol{v}\|^2$ , which holds since $\boldsymbol{v}$ has homogeneous boundary conditions. We have also introduced notation for the weighted inner product and norm

(4.4) \begin{align} \left \langle \unicode{x1D63C}, \unicode{x1D63D} \right \rangle _{\unicode{x1D648}} := \int _\varOmega \mathrm{tr}\left [ \unicode{x1D63C}^\top \left ( \unicode{x1D644}+ \unicode{x1D648}\right )^{-1} \unicode{x1D63D} \right ] \;\mathrm{d} V, \qquad \left \lVert \unicode{x1D63C} \right \rVert ^2_{ \unicode{x1D648}} = \left \langle \unicode{x1D63C}, \unicode{x1D63C} \right \rangle _{ \unicode{x1D648}}, \end{align}

defined for any vectors or matrices $ \unicode{x1D63C}$ , $ \unicode{x1D63D}$ and any matrix satisfying $ \unicode{x1D648} \succ - \unicode{x1D644}$ , all of which may be spatially varying. With this notation, the standard $L^2$ norm defined in (2.11) can be written as $\|\boldsymbol{\cdot } \|_{\boldsymbol{\mathsf{0}}} = \|\boldsymbol{\cdot } \|$ , although we will continue to use the standard notation $\|\boldsymbol{\cdot } \|$ throughout.

The system’s total energy dissipation rate (2.14) can now be written as

(4.5) \begin{align} \mathcal{E} = \frac {\beta }{ Re} \|\boldsymbol{\nabla }\boldsymbol{u}\|^2 + \frac {1-\beta }{{2 \textit{Re Wi}}^2} \| \unicode{x1D658} - \unicode{x1D644}\|_{ \unicode{x1D658} - \unicode{x1D644}}^2, \end{align}

highlighting the nature of $ \unicode{x1D658}= \unicode{x1D644}$ as the un-deformed polymer states. An interesting feature of this polymeric term

(4.6) \begin{align} \| \unicode{x1D658} - \unicode{x1D644}\|_{ \unicode{x1D658} - \unicode{x1D644}}^2 = \int _\varOmega \mathrm{tr}{ \left ( ( \unicode{x1D658} - \unicode{x1D644}\kern1.5pt) \unicode{x1D658}^{-1} ( \unicode{x1D658} - \unicode{x1D644}\kern1.5pt) \right )} \;\mathrm{d} V \end{align}

is that close to the un-deformed configuration $ \unicode{x1D658} \approx \unicode{x1D644}$ the norm behaves quadratically, yet for large deformations of the conformation tensor it scales linearly due to the presence of the inverse matrix in (4.6).

4.1. Stability and dissipation of steady viscoelastic channel flow

Before investigating the unsteady behaviour of the flow, we recall the known results on its stability. Binns & Wynn (Reference Binns and Wynn2024) studied global stability of Oldroyd-B fluids using perturbation energies of the form (4.2) with

(4.7) \begin{align} \boldsymbol{b} = \boldsymbol{b}_{\textit{steady}} = y\boldsymbol{e}_x, \qquad \qquad \unicode{x1D658}_b = \unicode{x1D658}_{\textit{steady}} = \begin{bmatrix} 1+2 \textit{Wi}^2 & \textit{Wi} & 0 \\ \textit{Wi} & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \end{align}

taken as the laminar steady solution to the governing equations (2.5a – c ). The key step in the analysis of Binns & Wynn (Reference Binns and Wynn2024) is to show that if the non-dimensional parameters $(\beta , \textit{Re}, \textit{Wi})$ are sufficiently small, then $\alpha$ can be chosen in the definition of $V$ (4.2) such that

(4.8) \begin{align} \frac {\mathrm{d} V}{\mathrm{d} t} \leq -c \, V \end{align}

holds along trajectories of (2.5a – c ) for some constant $c\gt 0$ . Consequently, $V \rightarrow 0$ as $t \rightarrow \infty$ and one can conclude that the system is globally stable. That is, $(\boldsymbol{u}, \unicode{x1D658}) \rightarrow (\boldsymbol{b}_{\textit{steady}}, \unicode{x1D658}_{\textit{steady}})$ as $t \rightarrow \infty$ from any arbitrary initial condition in $\mathcal{S}$ .

The specific parameter range in which this argument applies has the form

(4.9) \begin{align} c_1 \left (\frac {1-\beta }{1 - \textit{Wi}} \right ) \textit{Wi} + c_2 \textit{Re}\lt \beta , \qquad 0 \lt \textit{Wi} \lt 1, \end{align}

where $c_1, c_2\gt 0$ are positive constants. Within this parameter range the dissipation of the steady flow can be computed by substituting $\boldsymbol{b}_{\textit{steady}}$ and $ \unicode{x1D658}_{\textit{steady}}$ into (2.14) to give $ \overline {\mathcal{E}} = \mathcal{E}_{\textit{steady}} = 1/ Re.$ This highlights an interesting feature of the Oldroyd-B model in the Couette geometry, in that the laminar viscoelastic dissipation rate is always equal to the corresponding Newtonian value.

4.2. Perturbation energy and dissipation in unsteady viscoelastic channel flows

Our aim now is to extend the parameter range (4.9) over which a quantifiable estimate on the size of $\overline {\mathcal{E}}$ can be proven. Since $V$ is non-negative and vanishes only when $(\boldsymbol{v}, \unicode{x1D659}\kern1.5pt) = (\boldsymbol{0}, \boldsymbol{\mathsf{0}})$ , a Lyapunov decay estimate of the form (4.8) would imply $V(t) \to 0$ and hence convergence to the chosen reference state $(\boldsymbol{b}, \unicode{x1D658}_b)$ ; see for example Binns & Wynn (Reference Binns and Wynn2024). Since we aim to make statements about the unsteady solutions of (2.5a – c ), possibly far from the chosen background fields, we cannot in general expect such an estimate to hold. For the purpose of bounding time-averaged dissipation, it is sufficient instead to prove the weaker inequality

(4.10) \begin{align} \frac {\mathrm{d} V}{\mathrm{d} t} \leq -c\, V + \gamma \end{align}

for some constants $c,\gamma \gt 0$ . This is important since it implies that $V$ is bounded along trajectories of the system, meaning that condition (i) of Lemma1 holds. Furthermore, the negative coefficient on the right-hand side of (4.10) gives some hope that the bounding condition (ii) of Lemma1 can also hold.

To show how the inequality (4.10) may be proven, consider the following expression for $\mathrm{d} V/\mathrm{d} t$ :

(4.11) \begin{align} \frac {\mathrm{d} V}{\mathrm{d} t}& = -\overbrace {\frac {\beta }{2 Re} \left ( \left \lVert \boldsymbol{\nabla }\boldsymbol{v} \right \rVert ^2 + \|\boldsymbol{\nabla }\boldsymbol{u}\|^2\right )}^{\text{fluid `dissipation'}} - \overbrace {\frac {\alpha (1-\beta )}{{2 \textit{Re Wi}}^2} \left \lVert \unicode{x1D659} \right \rVert ^2_{\alpha \unicode{x1D659}} }^{\text{polymer `dissipation'}}-\overbrace { \left \langle \boldsymbol{v}, \boldsymbol{v} \boldsymbol{\cdot } \boldsymbol{\nabla }\boldsymbol{b}\right \rangle + \frac {1-\beta }{\textit{Re Wi}} \left \langle \boldsymbol{v}, \boldsymbol{\nabla }\boldsymbol{\cdot } \unicode{x1D658}_b\right \rangle }^{\text{fluid/background coupling}} \nonumber \\ & \qquad +\; \overbrace {\frac {\alpha (1-\beta )}{\textit{Re Wi}} \left \langle \unicode{x1D659}( \unicode{x1D644} + \unicode{x1D659}\kern1.5pt), \boldsymbol{\nabla }\boldsymbol{b} \right \rangle _{\alpha \unicode{x1D659}}}^{\text{polymer/background coupling}} \;-\;\overbrace {\frac {1-\beta }{\textit{Re Wi}} \left \langle \unicode{x1D659}, ( \unicode{x1D644} - \alpha \unicode{x1D658}_b)\boldsymbol{\nabla }\boldsymbol{v} + \frac {\alpha }{2} \boldsymbol{v} \boldsymbol{\cdot } \boldsymbol{\nabla }\unicode{x1D658}_b \right \rangle _{\alpha \unicode{x1D659}} }^{\text{polymer/fluid coupling}}\nonumber \\ & \qquad + \overbrace {\frac {\alpha (1-\beta )}{{2 \textit{Re Wi}}} \left \langle \unicode{x1D659}, -\boldsymbol{b} \boldsymbol{\cdot } \boldsymbol{\nabla }\unicode{x1D658}_b + 2( \unicode{x1D658}_b - \unicode{x1D644}\kern1.5pt)\boldsymbol{\nabla }\boldsymbol{b} + \frac {1}{ \textit{Wi}} ( \unicode{x1D644} - \unicode{x1D658}_b)\right \rangle _{\alpha \unicode{x1D659}} + \frac {\beta }{2 Re} \left \lVert \boldsymbol{\nabla }\boldsymbol{b} \right \rVert ^2 }^{\text{background `constants'}} . \end{align}

For completeness, (4.11) is proven in Appendix A, but we refer the reader to Binns & Wynn (Reference Binns and Wynn2024) for further details of the derivation.

The first two, dissipation-like, terms in (4.11) are negative. Whether (4.10) holds then depends on whether the other terms can be appropriately balanced by the ‘dissipation’ terms. We now discuss each of these terms, in order to give a high-level guide to the subsequent mathematical analysis in § 4.3.

Fluid/background coupling:

The fluid/background coupling terms can be controlled from above by the ‘perturbation dissipation’ terms $\|\boldsymbol{\nabla }\boldsymbol{v}\|^2$ , provided that $\boldsymbol{\nabla }\boldsymbol{b}$ and $ \unicode{x1D658}_b$ are sufficiently small. For the first term, estimates of the form

(4.12) \begin{align} \left \langle \boldsymbol{v}, \boldsymbol{v} \boldsymbol{\cdot } \boldsymbol{\nabla }\boldsymbol{b}\right \rangle \leq \frac {\text{const}}{ Re} \| \boldsymbol{\nabla }\boldsymbol{v}\|^2 \end{align}

follow from the Poincaré inequality and are at the heart of the background-flow method for Newtonian fluids (see e.g. Doering & Constantin Reference Doering and Constantin1992). At low Reynolds numbers, the inequality can be satisfied for $\| \boldsymbol{\nabla }b \| \approx \mathcal{O}(1)$ , and it is important to note that $\|\boldsymbol{\nabla }\boldsymbol{b}\| \geq 1$ must always hold, due to the assumption that $\boldsymbol{b}$ satisfies the Couette boundary conditions. However, at higher Reynolds numbers, the estimate (4.12) can only be enforced if $\boldsymbol{\nabla }\boldsymbol{b}$ is concentrated in appropriately thin boundary layers, as indicated in figure 1. This exploits the homogeneous boundary conditions on $\boldsymbol{v}$ at the walls to control the left-hand side of (4.12). As will be detailed in § 4.3, this forces the choice

(4.13) \begin{align} \| \boldsymbol{\nabla }\boldsymbol{b}\| \geq \mathcal{O}(\textit{Re}) \end{align}

in the selection of the fluid-dependent part of the auxiliary functional $V$ .

Polymer/background coupling:

Due to its dependence on $ \unicode{x1D659}$ , for (4.10) to hold this term must be controlled by the ‘polymer dissipation’. This is possible if $\boldsymbol{\nabla }\boldsymbol{b}$ is chosen to be small with respect to the magnitude of the polymer dissipation. However, given the constraint (4.13), this corresponds to requiring that the system parameters satisfy $ \textit{Re} \textit{Wi} \leq \mathcal{O}(1)$ . This places a crucial restriction on our analysis: while arbitrarily large Reynolds numbers can be considered, rigorous bounds are only guaranteed for weakly elastic flows satisfying $ \textit{Wi} \lesssim Re^{-1}$ . This restriction arises due to the assumption of only linear spring damping in the Oldroyd-B model, and is unlikely to be overcome without considering alternative constitutive models. Possible solutions to this barrier will be discussed later.

Polymer/fluid coupling:

This term is bilinear in $ \unicode{x1D659}$ and $\boldsymbol{\nabla }\boldsymbol{v}$ and can therefore be upper bounded by a combination of the fluid and polymeric ‘dissipations’ if $ \unicode{x1D644}-\alpha \unicode{x1D658}_b$ and $\boldsymbol{\nabla }\unicode{x1D658}_b$ are chosen to be sufficiently small. Again, there is no barrier to making such choices, other than modifying the value of the background constant terms, discussed above.

Background ‘constants’:

Although there is a dependence on the polymer perturbation $ \unicode{x1D659}$ , the first term in this expression has the form $\text{tr}( \unicode{x1D659}(\unicode{x1D644}+\alpha \unicode{x1D659}\kern1.5pt)^{-1} f(\boldsymbol{b}, \unicode{x1D658}_b))$ and can therefore be upper bounded by a constant depending only on the background fields $ \unicode{x1D657}$ and $ \unicode{x1D658}_b$ . Overall, the two terms considered here contribute only to the constant $\gamma$ in (4.10).

Summary of the perturbation energy analysis:

Given the discussion of the non-dissipative terms in the perturbation energy equation (4.11), it is possible to prove that $V$ is bounded along trajectories of the system, at least for viscoelastic flows with $ \textit{Re} \textit{Wi} \leq \mathcal{O}(1)$ . Consequently, $\varLambda V$ , and equivalently $V$ , satisfies condition (i) of Lemma1 and can therefore be used as a candidate for an auxiliary functional with which to satisfy the bounding condition (ii) of Lemma1. To satisfy the bounding inequality, we will therefore search for constants $\varLambda$ and $\kappa$ such that

(4.14) \begin{align} \mathcal{E} + \mathcal{L}_{\varLambda V} \leq \kappa \end{align}

holds on all flow/polymer fields in $\mathcal{S}$ .

4.3. An analytical bound on dissipation

The perturbation energy equation (4.11) can be significantly simplified by making the choices $\alpha =1, \unicode{x1D658}_b= \unicode{x1D644}$ , which eliminate all $ \unicode{x1D658}_b$ -dependent contributions to the terms on the right-hand side of (4.11). We also choose $\boldsymbol{b} = b(y)\, \boldsymbol{e}_x$ with

(4.15) \begin{align} b(y) = \left \{ \begin{array}{rcl} \frac {1}{2\delta }y, && 0 \leq y \lt \delta ,\\ \frac {1}{2}, & &\delta \leq y \leq 1- \delta ,\\ \frac {1}{2} + \frac {1}{2\delta }(y - (1 - \delta )), && 1-\delta \lt y \leq 1. \end{array} \right . \end{align}

This corresponds to the ‘background profile’ used in the classical analysis of dissipation bounds for Newtonian Couette flow, indicated in figure 1. While there may be potential for optimisation, such choices will allow us to present the main result more clearly. In particular, (4.11) reduces to

(4.16) \begin{align} \begin{split} \frac {\mathrm{d} V}{\mathrm{d} t} =& -\frac {\beta }{2 Re} \left (\left \lVert \boldsymbol{\nabla }\boldsymbol{v} \right \rVert ^2 + \left \lVert \boldsymbol{\nabla }\boldsymbol{u} \right \rVert ^2 \right ) - \frac {1-\beta }{{2 \textit{Re Wi}}^2} \left \lVert \unicode{x1D659} \right \rVert ^2_{ \unicode{x1D659}} \\[5pt] &\qquad + \int _\varOmega b'v_xv_y\;\mathrm{d} V + \frac {1-\beta }{\textit{Re Wi}} \left \langle \unicode{x1D659}( \unicode{x1D644}+ \unicode{x1D659}\kern1.5pt), \boldsymbol{\nabla }\boldsymbol{b} \right \rangle _{ \unicode{x1D659}} + \frac {\beta }{2 Re} \left \lVert \boldsymbol{\nabla }\boldsymbol{b} \right \rVert _2^2, \end{split} \end{align}

where $b' = \mathrm{d} b(y) /\mathrm{d} y$ denotes the derivative of the scalar background profile $b(y)$ in $\boldsymbol{b} = b(y)\,\boldsymbol{e}_x$ with respect to the wall-normal coordinate $y$ . Now recall from (4.5) that the expression for dissipation can be written as

(4.17) \begin{align} \mathcal{E} = \frac {\beta }{ Re} \| \boldsymbol{\nabla }\boldsymbol{u}\|^2 + \frac {1-\beta }{{2 \textit{Re Wi}}^2} \| \unicode{x1D658}- \unicode{x1D644}\|_{ \unicode{x1D658} - \unicode{x1D644}}^2. \end{align}

This can be compared with the dissipative terms in $\mathrm{d} V/\mathrm{d} t$ by noticing that, with the choice $ \unicode{x1D658} = \unicode{x1D644} + \unicode{x1D659}$ , we have that

(4.18) \begin{align} \left \| \unicode{x1D659} \right \|^2_{ \unicode{x1D659}} =\| \unicode{x1D658}- \unicode{x1D644}\|_{ \unicode{x1D658} - \unicode{x1D644}}^2 . \end{align}

We now seek to show that (4.14) holds for some constants $\varLambda ,\kappa$ . Using (4.16)

(4.19) \begin{align} \begin{split} \mathcal{E} + \mathcal{L}_{\varLambda V} &= \mathcal{E} + \varLambda \frac {\mathrm{d} V}{\mathrm{d} t} \\ &= \varLambda \left (-\frac {\beta }{2 Re} \| \boldsymbol{\nabla }\boldsymbol{v} \|^2 - \int _\varOmega b' v_x v_y \; \mathrm{d} V + \frac {\beta }{2 Re} \| \boldsymbol{\nabla }\boldsymbol{b}\|^2 \right ) \\ & \qquad + \frac {(2-\varLambda )\beta }{2 Re} \| \boldsymbol{\nabla }\boldsymbol{u}\|^2\\ & \qquad +\varLambda \frac {1-\beta }{\textit{Re Wi}} \int _\varOmega b' c_{xy}\,\mathrm{d} V - \frac {(\varLambda -1)(1-\beta )}{{2 \textit{Re Wi}}^2} \| \unicode{x1D658}- \unicode{x1D644}\|_{ \unicode{x1D658} - \unicode{x1D644}}^2 . \end{split} \end{align}

To begin, we upper bound the first bracketed term in (4.19). Since $b'$ is supported only in boundary layers of width $\delta$ , no-slip boundary conditions, Young’s inequality and incompressibility imply that (Doering & Constantin Reference Doering and Constantin1992)

(4.20) \begin{align} \left | \int _\varOmega b'v_x v_y \;\mathrm{d} V \right | \leq & \frac {1}{2\delta } \left (\left | \int _0^{\varGamma _z} \!\!\!\int _0^{\delta } \!\!\int _0^{\varGamma _x} v_x v_y \;\mathrm{d} x \,\mathrm{d} y \,\mathrm{d} z \right | + \left | \int _0^{\varGamma _z} \!\!\!\int _{1-\delta }^{1} \!\int _0^{\varGamma _x} v_x v_y \;\mathrm{d} x \,\mathrm{d} y \,\mathrm{d} z\right |\right ) \nonumber\\[3pt] \leq & \frac {\delta }{4} \left \| \frac {\partial v_x}{\partial y} \right \| \left \|\frac {\partial v_y}{\partial y} \right \| \nonumber\\[3pt] \leq & \frac {\delta }{4} \left (\frac {1}{2\sqrt {2}} \left \lVert \frac {\partial v_x}{\partial y} \right \rVert ^2 + \frac {\sqrt {2}}{2} \left \lVert \frac {\partial v_y}{\partial y} \right \rVert ^2 \right ) \leq \frac {\delta }{8\sqrt {2}} \left \lVert \boldsymbol{\nabla }\boldsymbol{v} \right \rVert ^2. \end{align}

Consequently, $\int _\varOmega b' v_x v_y \;\mathrm{d} V$ can be upper bounded by the ‘perturbation dissipation’ if $\delta$ is chosen to be sufficiently small. Namely, if

(4.21) \begin{align} 0 \lt \delta \leq \min \left \{\frac {1}{2}, \frac {4\sqrt {2} \beta }{ Re}\right \} ,\end{align}

then

(4.22) \begin{align} -\frac {\beta }{2 \textit{Re}} \left \lVert \boldsymbol{\nabla }\boldsymbol{v} \right \rVert ^2 - \int _\varOmega b'v_x v_y \;\mathrm{d} V \leq 0. \end{align}

If we make the additional assumption that

(4.23) \begin{align} \varLambda \geq 2, \end{align}

so that the term involving $\|\boldsymbol{\nabla }\boldsymbol{u}\|^2$ in (4.19) can be upper bounded by zero, and use that $\left \lVert \boldsymbol{\nabla }\boldsymbol{b} \right \rVert ^2 = 1/(2\delta )$ , it follows from (4.19) and (4.22) that

(4.24) \begin{align} \mathcal{E} + \mathcal{L}_{\varLambda V} \leq \frac {\varLambda \beta }{4 \delta \textit{Re}} + \varLambda \frac {1-\beta }{\textit{Re Wi}} \int _\varOmega b'c_{xy} \;\mathrm{d} V - \frac {(\varLambda -1) (1-\beta )}{{2 \textit{Re Wi}}^2} \| \unicode{x1D658}- \unicode{x1D644}\|_{ \unicode{x1D658} - \unicode{x1D644}}^2 . \end{align}

To complete the proof of the bound, i.e. to upper bound (4.24) by a constant $\kappa$ , it is necessary to estimate the sign-indefinite term $\int _\varOmega b' c_{xy} \, \mathrm{d} V$ . This corresponds to the production of energy via polymer stretching induced by the ‘background’ shear, which must be appropriately controlled by the polymeric energy dissipation arising from the spring damping in the Oldroyd-B model. To see why this is possible, the following fact, which acts similarly to a Young-type inequality for off-diagonal elements of positive-definite matrices, is required.

Lemma 2. Let $ \unicode{x1D658}$ be a $3\times 3$ symmetric positive-definite matrix. Then

(4.25) \begin{align} \left | c_{xy} \right | \leq \frac {a}{2} \| \unicode{x1D658}- \unicode{x1D644}\|^2_{ \unicode{x1D658}- \unicode{x1D644}} + \frac {2-\sqrt {2}}{a}, \end{align}

holds for any $a \geq 1$ .

The proof of this lemma is given in Appendix B.

In terms of the physical interpretation of the conformation tensor, this lemma implies that the cross-correlation of polymer stretching in the $\boldsymbol{e}_x$ and $\boldsymbol{e}_y$ directions cannot exceed the sum of the variances of the polymer deformations in each direction.

To make use of Lemma2, we let $\varOmega _\delta := \{\varOmega :\,y\in [0,\delta ]\cup [1-\delta , 1]\}$ and again use that $b'$ is supported in boundary layers of width $\delta$ to show that

(4.26) \begin{align} \begin{split} \varLambda \frac {1-\beta }{\textit{Re Wi}}\int _\varOmega b' c_{xy} \;\mathrm{d} V &\leq \frac {\varLambda (1-\beta )}{2\delta \textit{Re Wi}} \int _{\varOmega _\delta } \left | c_{xy} \right | \;\mathrm{d} V\\ &\leq \frac {\varLambda (1-\beta )}{2\delta \textit{Re Wi}} \left [\frac {a}{2}\| \unicode{x1D658}- \unicode{x1D644}\|_{ \unicode{x1D658} - \unicode{x1D644}}^2 + \frac {2\delta \left (2-\sqrt 2\right )}{a} \right ], \end{split} \end{align}

which holds for any $a \geq 1$ . This freedom is useful, since if $a$ is chosen to satisfy

(4.27) \begin{align} 1 \leq a \leq \frac {2\delta (1-\varLambda ^{-1})}{ \textit{Wi}} \end{align}

then the first term on the right-hand side of (4.26) can be upper bounded by the final negative term in (4.24). In summary, we have proven that if the three constraints (4.21), (4.23) and (4.27) hold on the parameters $\delta ,\varLambda$ and $a$ then we have the upper bound

(4.28) \begin{align} \mathcal{E} + \mathcal{L}_{\varLambda V} \leq \frac {\varLambda \beta }{4\delta Re} + \left (2- \sqrt 2 \right ) \frac {\varLambda (1-\beta )}{\textit{a {Re Wi}}}. \end{align}

This shows that condition (ii) of Lemma1 holds for the auxiliary functional $\varLambda V$ .

One possible choice for the parameters is given by

(4.29) \begin{align} \delta = \min \left \{\frac {1}{2}, \frac {4\sqrt 2\beta }{ Re}\right \}, \qquad \varLambda = \frac {2}{1- \textit{Wi}}, \qquad a = \frac {2\delta }{ \textit{Wi}}\left (1- \frac {1}{\varLambda } \right ). \end{align}

The choice of $\varLambda$ is made so that the polymer-production term $b'c_{xy}$ can be controlled by the dissipation in (4.24), and already requires $0 \lt \textit{Wi} \lt 1$ . These parameters satisfy the required constraints (4.21), (4.23) and (4.27) provided $a \geq 1$ . This final condition is equivalent to

(4.30) \begin{align} a = \delta \left (1 + \textit{Wi}^{-1}\right ) \geq 1 \end{align}

which places an inherent restriction on the range of system parameters ( $ \textit{Re}, \textit{Wi}, \beta$ ) to which the result can apply. Since $\delta \leq 1/2$ , the condition $a \geq 1$ can only hold if $ \textit{Wi} \lt 1$ . At larger Reynolds numbers when $\delta$ is small, we have the extra restriction $ \textit{Re} \lt 4\sqrt 2 \beta ( 1 + \textit{Wi}^{-1})$ . We collect these observations in the main analytical result of the paper.

Theorem 1. Let $0 \lt \textit{Wi} \lt 1$ and $0 \lt \beta \lt 1$ . If

(4.31) \begin{align} 0\lt \textit{Re} \lt 4\sqrt 2 \beta \left (1 + \textit{Wi}^{-1} \right ) ,\end{align}

then the upper bound on the infinite-time-averaged energy dissipation rate

(4.32) \begin{align} \overline {\mathcal{E}} \leq \left (\frac {1}{1- \textit{Wi}}\right ) \left ( \frac {1}{8\sqrt 2} +\frac {(\sqrt {2}-1)(1-\beta )}{2\beta (1+ \textit{Wi})} \right )\max \left \{ 1, \frac {8 \sqrt {2}\beta }{ Re} \right \} \end{align}

must hold for any solution to the Oldroyd-B equations ( 2.5 ) whose initial condition belongs to the admissible set $\mathcal{S}$ given by ( 2.9 ).

Proof. This requires checking the technical conditions of Lemma1, as detailed in Appendix C.