The Navier–Stokes equations are important for science and engineering, since they describe the motion of fluids.  However, these equations can not describe the physical responses of fluids with a complex microstructure.

Michal Bathory, Miroslav Bulíček, and Josef Málek, Charles University, Czech Republic, have developed a robust mathematical theory for viscoelastic fluids. Which could serve as an analytical framework, to quantify errors between exact and computed solutions for these models.

Read some of their latest work here: https://doi.org/10.1515/anona-2020-0144

Transcript:

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The Navier–Stokes equations are important for science and engineering, since they describe the motion of fluids. For instance, they are used to model blood flow within arteries, an ocean’s currents, and dynamics of air around aeroplanes. A century ago, theoretical physicists and mathematicians made advances in the use of Navier–Stokes equations. However, Navier–Stokes equations cannot describe the physical responses of fluids with a complex microstructure. Michal Bathory, Miroslav Bulíček, and Josef Málek at Charles University in the Czech Republic have developed a robust mathematical theory for viscoelastic fluids.

Since the first half of the nineteenth century, the Navier–Stokes equations have been successfully used to describe flows of simple fluids, such as water or oil, under standard conditions. To predict the flows of a Navier–Stokes fluid in a vessel (think of water in a garden hose), we also need to know the initial state of the fluid and its behaviour on the boundary (think of the interaction of water with the inner layer of the garden hose). If we have all pieces of information (that is, the equations in the bulk, the equations on the boundary, and the initial conditions), we can find the explicit formulae for the velocity and the pressure, and solve the Navier–Stokes equations explicitly, but only in special geometrical settings and under restricted type of admissible flows. The only other way to obtain information about the character of the flow in the container is either by performing experiments (but these are again done in specific geometries) or by using scientific computing methods and tools.

Today, the latter approach is increasingly used as it is much more flexible, the underlying numerical methods are becoming ever-more efficient, and computational software and hardware allows researchers to compute three-dimensional flows of the time-dependent Navier–Stokes equations in complex geometries. The question then is: What is the precise meaning, or definition, of the solution that we are trying to approximate using these advanced computational methods?

Although scientific computing was in its infancy before the Second World War, theoretical physicist Carl Wilhelm Oseen and mathematician Jean Leray asked the question: Can we establish the existence of a solution to the Navier–Stokes equations which corresponds to a state of the velocity given arbitrarily at an initial instant? Leray succeeded in answering this question.

Paying attention to connections between mathematical approaches and the physical underpinnings of the problems, Leray introduced the concept of a solution reflecting available energetic information, proved its existence for any initial state of the velocity with bounded kinetic energy, and investigated its further qualitative properties.

However, despite their unquestionable success and broad applications, the Navier–Stokes equations cannot adequately describe the features exhibited by fluids like water and oil in extreme regimes, or describe the behaviour of many liquids and fluid-like materials. This is because there are various phenomena and rather counterintuitive observations that cannot be captured by the linear relation between the shear-stress and the shear-rate, which is the relation that characterises the Navier–Stokes fluid equations.

Non-Newtonian fluids are fluid-like materials that cannot be described by the Navier–Stokes equations. Non-Newtonian phenomena can be split into two groups. The first one is characterised by a nonlinear relationship between the shear-stress and the shear-rate inclusive of those connected with the presence of activation criteria. This group includes power-law fluids and their various generalisations, Bingham, Herschel–Bulkley, or activated Euler fluids. Most of these fluids are covered successfully by the framework of implicit constitutive equations that relate to Cauchy stress and the symmetric part of the velocity gradient.

The second group of non-Newtonian characteristics includes fascinating phenomena connected with viscoelastic fluids, such as the presence of normal stress differences in a simple shear flow, stress relaxation, nonlinear creep, shear, and vorticity banding. The governing equations for both groups of fluids are more complicated than the Navier–Stokes equation. Each of these groups include hundreds of models designed by engineers and physicists to describe the behaviour of various polymers or geo- and biomaterials such as glaciers, the Earth’s mantle, asphalt binder, synovial liquid, creams, paints and gels, the eye sclera, blood, or food products such as ketchup, honey, and mayonnaise.

From the mathematical perspective, it is natural and fundamental to ask whether we can establish the mathematical foundations, in the sense of Leray’s programme, for the models belonging to these groups. Michal Bathory and his mentors have focused on the development of such a theory for a class of models belonging to the second group, namely the viscoelastic rate-type fluid models with stress diffusion.

Models of viscoelastic rate-type fluids with stress diffusion have been around for more than thirty years. They are able to accurately describe the majority of non-Newtonian phenomena and, thanks to the presence of the stress diffusion, they are attractive both from the mathematical and physical points of view.

From a physics perspective, the presence of the stress diffusion coefficient is linked with proper finite thickness of the shear or vorticity bands. From a mathematical perspective, the presence of a nice linear diffusion operator in the governing equation for the elastic part of the Cauchy stress should significantly help in the analysis. Interestingly, despite the efforts of various mathematical groups, a theory with Leray’s concept in mind is not in place. In addition, there’s ambiguity regarding the proper choice of the objective time derivative for this (elastic) part of the Cauchy stress. Moreover, these derivatives include additional nonlinear terms that complicates the analysis in an essential manner.

How did Michal Bathory and his colleagues overcome these difficulties?

There are three crucial ingredients. Firstly, Bathory’s study builds on several papers concerning a general thermodynamic methodology that determines the constitutive equations involving part of the Cauchy stress from two pieces of information: namely how the material stores energy and how entropy is produced. This requires us to specify the equations for two scalar quantities. If they are known, then thermodynamics provides the remaining pieces of information for tensorial quantities and their evolution. There is yet another benefit of this approach, this time on the mathematical side. These two pieces of information are essential for the complete thermodynamic development but also determine the function spaces in which Leray’s programme should be conducted. With this understanding, a second ingredient enters. Bathory and colleagues modify the mechanism how the material stores energy in such a way that it is not distinguishable from standard models provided that the elastic response is small. However, it provides a new piece of information that improves the mathematical qualities of the elastic part of the Cauchy stress. The third ingredient is a novel method, which in the analysis of tensorial nonlinear equations guarantees that the tensorial quantity is positive definite. This issue is linked to the development of a suitable approximation system where nonlinear terms in the equations vanish if the spectrum of the elastic part of the Cauchy stress reaches zero.

With these three ingredients, Bathory and colleagues have developed a robust mathematical theory for unsteady flows of viscoelastic rate-type fluids with stress diffusion.

The theory covers any choice of objective time derivative of the elastic tensor and is the same type as Leray’s theory for the Navier–Stokes equations, even though the system treated by Bathory and colleagues is much more complicated. The components of the velocity and of the elastic part of the Cauchy stress belong to the same function space. The essential point in establishing this result is a detailed understanding of the derivation of the fluid models using the tools of continuum thermodynamics. Without it, we are blind in the forest; a vast array of viscoelastic models are used in different areas, different contexts, and developed on different (often rather intuitive) bases. The overall proof is constructive and starts with a proper finite-dimensional approximation. As such, it can serve numerical analysts as the essential tool in the analysis of convergence of suitably discretized finite-element schemes and for estimating and guaranteeing the error between their solution and the solution computed by carefully designed numerical solvers. Bathory and colleagues are now testing their newly designed class of models in computational simulations.

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