A viscoelastic flow model of Maxwell-type with a symmetric-hyperbolic formulation

Maxwell models for viscoelastic flows are famous for their potential to unify elastic motions of solids with viscous motions of liquids in the continuum mechanics perspective. But the usual Maxwell models allow one to define well motions mostly for one-dimensional flows only. To define unequivocal multi-dimensional viscoelastic flows (as solutions to well-posed initial-value problems) we advocated in [ESAIM:M2AN 55 (2021) 807-831] an upper-convected Maxwell model for compressible flows with a symmetrichyperbolic formulation. Here, that model is derived again, with new details.


Elastic and viscous motions in the continuum perspective
First, let us recall seminal systems of PDEs that unequivocally model the motions φ t : B →⊂ R 3 of continuum bodies B on a time range t ∈ [0, T ).PDEs governing elastic flows are a starting point for all continuum bodies.PDEs governing viscoelastic flows, for liquid bodies in particular, shall come next in Section 2.
In the sequel, assuming ρ ∈ R + * constant, we recall how one standardly defines e(F ) for solids and fluids dynamics, on considering the determinant |F i α | of the deformation gradient (also denoted |F | hereafter) and the cofactor matrix C i α of F i α (C in tensor notation) as variables independent of F .Next, in Section 2, we recall with much details the function e(F ) that we proposed in [2] so as to properly define a viscoelastic dynamics of Maxwell type that unifies solids and fluids.
) enters the framework of symmetric-hyperbolic systems.In particular, a unique time-continuous solution can be built in . The latter solution, associated with a unique mapping φ t , is equivalently defined by [17] where ∀α .Indeed, with the Eulerian description (7-10) of the body motions (i.e. in spatial coordinates, as opposed to the Lagrangian description (1-5) in material coordinates) where C T is the dual (matrix transpose) of C, and with Piola's identity (11) (16) div(ρF T ) = 0 = ∇ × (ρC T ) , one can show that, when e(F ) is polyconvex, the symmetric-hyperbolic framework applies to (12)(13)(14)(15)(16) insofar as smooth solutions also satisfy the conservation law for ρ 2 |u|2 + ρe, a functional convex in a set of independent conserved variables [7].
A first example of a physically-meaningful internal energy is the neo-Hookean with c 2 1 > 0.Then, the quasilinear system (1-3) is symmetric-hyperbolic insofar as smooth solutions additionally satisfy a conservation law for |u| 2 /2+e strictly convex in (u, F ). Unequivocal motions can be defined1 , equivalently by (12)(13).The latter neo-Hookean model satisfyingly predicts the small motions of some solids.
However, (18) is oversimplistic : it does not model the deformations that are often observed orthogonally to a stress applied unidirectionally, see e.g.[16] regarding rubber.Many observations are better fitted when the Cauchy stress σ contains an additional spheric term −pI, with a pressure p(ρ) function of volume changes.
Next, instead of (18), one can rather assume a compressible neo-Hookean energy The functional (19) is polyconvex as soon as γ > 1 [7].Thus, using either (1-5) or (7-10) one can define unequivocal smooth motions with where an additional pressure term arises2 in comparison with (18).Precisely, one can build unique solutions to a symmetric reformulation of a system of conservation laws for conserved variables They combine with (20) using Ξ(U ) T = pu y −pu x to yield a symmetric system after premultiplication by The symmetric formulation allows one to establish the key energy estimates in the existence proof of smooth solutions [7], as well as weak self-similar solutions to the 1D Riemann problem using generalized eigenvectors R solutions to For application to real materials3 , one important question remains: how to choose c 2 1 and d 2 1 .
In most real applications of elastrodynamics, the material parameters c 2 1 and d 2 1 should vary, as functions of F e.g., but also as functions of an additional temperature variable so as to take into account microscopic processes not described by the macroscopic elastodynamics system.For instance, the deformations endured by stressed elastic solids increase with temperature, until the materials become viscous liquids.Then, one natural question arises: could (19) remain useful for liquids which are mostly incompressible (i.e.div u ≈ 0 holds) and much less elastic than solids ?In Sec.1.2, we recall the limit case when the volumic term dominates the internal energy, and p = C 0 ρ γ dominates σ, which coincides with seminal PDEs for perfect fluids (fluids without viscosity).In Section 2, we next consider how to rigorously connect fluids like liquids to solids using an enriched elastodynamics system.1.2.Fluid dynamics.Consider the general Eulerian description (12-15) for continuum body motions.It is noteworthy that given u, each kinematic equation ( 10), ( 8) and ( 9) is autonomous.As a consequence, in spatial coordinates, motions can be defined by reduced versions of the full Eulerian description (7-10), with an internal energy e strictly convex in ρ but not in F !One famous case is the polytropic law with C 0 > 0.Then, one obtains Euler's system for perfect (inviscid) fluids (31) with a pressure p := −∂ ρ −1 e = C 0 ρ γ characterizing spheric stresses: (32) The system (31) is symmetric-hyperbolic.It is useful to define unequivocal timeevolutions of Eulerian fields (on finite time ranges) [7], although multi-dimensional solutions are then not equivalently described by one well-posed Lagrangian description [8].In fact, for application to real fluids, the system (31) is better understood as the limit of a kinetic model based on Boltzmann's statistical description of molecules [9], and the model indeed describes gaseous fluids better than condensed fluids (liquids).In any case, the fluid model (31) still lacks viscosity.One classical approach adds viscous stresses as an extra-stress term τ in (32) i.e. ( The extra-stress is required symmetric (to preserve angular momentum), objective (for the sake of Galilean invariance), and "dissipative" (to satisfy thermodynamics principles) [5].Precisely, introducing the entropy η as an additional state variable for heat exchanges at temperature θ = ∂ s e > 0, thermodynamics requires with a dissipation term D ≥ 0. Usually, denoting D(u) ij := 1 2 ∂ i u j + ∂ j u i , one then postulates a Newtonian extra-stress with two constant parameters ℓ, μ > 0 (34) . The Newtonian model allows for the definition of causal motions through the resulting Navier-Stokes equations.But it is not obviously unified with elastodynamics; and letting alone that (34) is far from some real "non-Newtonian" materials, it implies that shear waves propagate infinitelyfast, an idealization that is also a difficulty for the unification with elastodynamics.
By contrast, Maxwell's viscoelastic fluid models for τ possess well-defined shear waves of finite-speed, and they can be connected with elastodynamics with a view to unifying solids and fluids (liquids) in a single continuum description.
2.1.Viscoelastic 1D shear waves for solids and fluids.Some particular solutions to ( 12)-( 14)-( 33)-( 35)-(36) unequivocally model viscoelastic flows, and rigorously link solids to fluids.Shear waves e.g. for a 2D body moving along e x ≡ e x 1 following b = y ≡ x 2 , a = x − X(t, y), X(0, y) = 0 are well-defined by (7) 38) are well defined given initial conditions plus possibly boundary conditions when the body has finite dimension along e y ≡ e x 2 , such as y ≡ x 2 > 0 in Stokes first problem see e.g.[15].Moreover, the latter 1D shear waves rigorously unify solids and fluids insofar as they are structurally stable [14,4] yy u like elastic solids, and when λ → 0, they satisfy yy u like viscous liquids.So the 1D shear waves illustrate well the structural capability of Maxwell's model to unify solid and Newtonian fluid motions.
But a problem arises with multi-dimensional motions: solutions to ( 12)-( 14)-( 33)-( 35)-( 36) are not well-defined in general. 4Other objective derivatives than UC can be used, which also allow symmetric-hyperbolic reformulations.They will not be considered here for the sake of simplicity.
Proof.We will show (3) in material coordinates (the Lagrangian description).On one hand, computing ∂ t |u| 2 = 2u • ∂ t u is straightforward.One the other hand, using Interestingly, notice that our free energy (41) is not useful for well-posedness: it is not strictly convex in conserved variables.Morover, our formulation ( 12)-( 13)-( 14)-(39) for a sound Maxwell model admits the 1D shear waves examined in Sec.2.1 as solutions, so it preserves some well-established interesting properties of the standard (incompressible) formulation of Maxwell model.
Let us finally detail the symmetric structure of our hyperbolic formulation for (compressible) viscoelastic flows of Maxwell-type, with Lagrangian description To that aim, we consider a 2D system when λ → ∞: Rewriting ∂ t U + ∂ α G α (U ) = 0 the system above, involutions M α ∂ α U = 0 hold with M a = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 M b = 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 and A symmetric formulation is obtained for our quasilinear formulation of Maxwell (compressible) viscoelastic flows similarly to the standard compressible elastodynamics case: on premultiplying the system (48-55) by D 2 η(U ), insofar as the matrix (D 2 η(U )DG α (U ) + DΞ(U ) T M α )ν α is symmetric given a unit vector ν = (ν a , ν b ) ∈ R 2 .We do not detail the symmetric matrix (D 2 η(U )DG α (U ) + DΞ(U ) T M α )ν α here: its upper-left block coincides with (28), but the other blocks are complicate and depend on the choice of the variable Y = A − 1 2 (key to exhibit the symmetrichyperbolic structure using a fundamental convexity result from [10] -Theorem 2 p.276 with r = 1 2 and p = 0) a choice which is not unique (ours may not be optimal).In any case, the symmetric structure yields a key energy estimate for the construction of unique smooth solutions, and it also allows one to construct 1D waves similarly from (29) when λ → ∞ (otherwise one has to take into account the source term of relaxation-type).

Conclusion and Perpsectives
Our symmetric-hyperbolic formulation of viscoelastic flows of Maxwell type [2] allows one to rigorously establish multidimensional motions, within the same continuum perspective as elastodynamics and Newtonian fluid models.It remains to exploit that mathematically sound framework, e.g. to establish the structural stability of the model and rigorously unify (liquid) fluid and solid motions through parameter variations in our model: see [4] regarding the nonsingular limit toward elastodynamics.Another step in that direction is to drive the transition between (liquid) fluid and solid motions more physically, e.g. on taking into account heat transfers: see [3] for a model of Cattaneo-type for the heat flux, which preserves the symmetric-hyperbolic structure.Last, one may want to add physical effects for particular applications: the purely Hookean internal energy in (41) can be modified to include finite-extensibility effects as in FENE-P or Gent models, or to use another measure of strain, with lower-convected time-rate for instance, see [3].