Michal Bathory

1. Loading publications...

A typical system of equations governing the motion of rate-type viscoelastic fluids can take the form \[\begin{aligned}&\partial_t\ve+\ve\cdot\nabla\ve-\nu\Delta\ve+\nabla p=\di\S\\[0.1in] &\partial_t\S+\ve\cdot\nabla\S+\S\W-\W\S+\PP(\S)=\mathcal{F}(\S)\D\end{aligned}\] where \[\nabla\ve=\D+\W\] is the decomposition of the velocity gradient into the symmetric and skew-symmetric parts, respectively. Whilst the 2nd-order tensor \(\PP(\S)\) is always monotone in the extra stress \(\S\) and models an elastic dissipation, the choice of the 4th-order tensor \(\mathcal{F}(\S)\) is more delicate and corresponds to the type of tensor objective derivative being considered.

There even seems to be a tradeoff: While the simplest case of \[\mathcal{F}(\S)=0\] is often too restrictive in physical applications, the more realistic choices, such as \[\mathcal{F}(\S)\D=\S\D+\D\S\] are very difficult to treat mathematically, often requiring additional assumtpions on the function \(\PP\) or some other modelling relaxations (see arXiv:2002.11224), in order to have just a global existence result, for example. However, it turns out, there is a middle ground, a curious example being \[\mathcal{F}(\S)\D=\ad_{\S}\coth\ad_{\S}\D,\] where \(\ad_{\S}\) is the matrix commutator operator. With this choice one can both exploit the mathematical benefits of corotational derivatives and, at the same time, link it to the upper-convected rates, that are characteristic for the famous Oldroyd-B and Giesekus models.

For details, see arXiv:2505.07987 and arXiv:2504.15692.

Having two solutions \((\ve,\S)\) and \((\we,\Z)\) (of the Navier-Stokes-Giesekus system, let us say) emanating from different initial data and such that \(\S\), \(\Z\) are positive definite, a natural candidate for measuring their distance is \[\mathcal{L}=\int|\ve-\we|^2+\operatorname{tr}(\S-\Z)-\log\det(\S\Z^{-1}).\] The goal is to show that if one of those solutions, say \((\we,\Z)\), is uniformly sufficiently small, then \(\mathcal{L}\) decays to zero with time (preferably at an exponential rate), i.e., that \(\mathcal{L}\) is actually the Lyapunov function for the system, no matter the initial perturbation of \((\ve,\S)\).

This behaviour is indeed observed experimentally, even for viscoelastic fluids, for example in the Taylor-Couette flow, where the "small solution" (which in practice can be surprisingly large) represents a steady state, induced by the movement of one (or both) of the cylinders enclosing the fluid.

However, confirming these observations rigorously and in a general setting (not just cylinders) is not as straightfoward. The problem is that the only solutions that are (currently) known to exist globally-in-time are weak solutions. Unfortunately, the available regularity of those solutions seems insufficient to express \(\frac{d}{dt}\mathcal{L}\).

A possible resolution is to construct weak solutions with their stability already in mind, leading to the concept of relative energy inequality, which for viscoelastic fluids takes the form \[\begin{aligned}&\frac{d}{dt}\mathcal{L}+\int|\nabla(\ve-\we)|^2+\int\big(|\Z^{-1}\S^{\frac12}-\S^{-\frac12}|^2+|\Z^{-\frac12}(\S-\Z)|^2\big)\nonumber\\ &\quad\leq-\int(\ve-\we)\cdot\nabla\we\cdot(\ve-\we)-\int(\partial_t\we+\we\cdot\nabla\we-\nu\Delta\we-\di\Z)\cdot(\ve-\we)\nonumber\\ &\quad\quad+\int\big(\nabla(\ve-\we)\cdot(\Z^{-1}-\I)(\S-\Z)+(\ve-\we)\cdot\nabla(\Z^{-1}-\I)\cdot(\S-\Z)\big)\nonumber\\ &\quad\quad+\int(\partial_t\Z+\we\cdot\nabla\Z+\PP(\Z)-\nabla\we\Z-\Z(\nabla\we)^T)\cdot(\Z^{-1}-\Z^{-1}\S\Z^{-1})\end{aligned}\] If this inequality is enforced for all smooth vector fields \(\we\) and smooth positive definite tensor fields \(\Z\), then it represents a generalization of the standard energy inequality, which actually also holds the complete information about the solution of the given system.

Rather than tracking the evolution of the temperature \(\theta\) directly, it is more natural to impose the balance of entropy \(\eta\) instead: \[\partial_t\eta+\ve\cdot\nabla\eta-\di\big(\kappa(\theta)\nabla\log\theta\big)\ge\xi.\] This inequality is mathematically well-structured as it implicitly yields control over the (average of) right hand side, which is the non-negative entropy production \[\xi\coloneqq\frac{2\nu(\theta)}{\theta}|\D|^2+\kappa(\theta)|\nabla\log\theta|^2+\frac1{\theta}\PP(\theta,\S)\cdot\partial_{\S}\psi(\theta,\S),\] whose control yields essential apriori estimates of a solution.

The function \(\psi\) models here the Helmholtz free energy that characterizes the energy storage mechanism of the fluid and is linked to the entropy via the 1st law of thermodynamics: \[\partial_{\theta}\psi(\theta,\S)=-\eta.\] For example, the case of the linear shear modulus, that is \[\psi(\theta,\S)=-c_v\theta(\log\theta-1)+\mu\theta(\operatorname{tr}(\S-\I)-\log\det\S),\quad c_v,\mu>0,\] is treated in arXiv:2308.04570. One of the difficulties is to ensure that \(\theta>0\) and that the extra stress \(\S\) is positive definite, so that the above expression makes sense. More general choices of \(\psi\) lead to more complicated dependence of the internal energy on temperature and the extra stress \(\S\).

In any case, the usual intuitive link between the viscous dissipation and thermal heating does not apply for viscoelastic fluids. Instead, the apriori estimates for the velocity \(\ve\) can only be based on \[\sup_{(0,T)}\int_{\Omega}|\ve|^2+\int_{0}^{T}\int_{\Omega}\frac{2\nu(\theta)}{\theta}|\D|^2<\infty\] thanks to the energy conservation and entropy inequality. To proceed further and obtain an estimate for the symmetric velocity gradient \(\D\) only, it is clear that one has to derive estimates also for the temperature. This is done by carefully taking into consideration also the elastic dissipation mechanism and closing the circle of estimates.

Including a full-fledged description of the temperature evolution would be a major step towards real-world applicability of viscoelastic fluid models.