Perspectives on using implicit type constitutive relations in the modelling of the behaviour of non-Newtonian fluids

Implicit Type Constitutive Relations for Elastic Solids and Their Use in the Development of Mathematical Models for Viscoelastic Fluids

“…introduced in Le Roux and Rajagopal [7] (for a through discussion of a simpler model with δ = 0, see Málek et al [8]). With properly tuned parameter values, this constitutive relation leads to the characteristic S-shaped curve in the strain rate-stress diagram; thus, the model (4) can serve as simple phenomenological model for flows of substances that exhibit such behaviorsee for example Boltenhagen et al [9], Grob et al [10] and Mari et al [11] to name a few (further references and a thorough discussion can be found in Perlácová and Průša [12] and Janečka and Průša [13]). Note that if the constitutive relation leads to the S-shaped curve in the strain rate-stress diagram, then (4) is not invertible, and it can not be rewritten in the classical form T δ = g(D).…”

“…For further discussion of generalized models of type k(T δ , D) = O, we also refer the reader to Perlácová and Průša [12]. Note that, in all cases discussed above, the constitutive relation between the stress and the symmetric part of the velocity gradient has been related to entropy production mechanisms (see for example Janečka and Průša [13] for a through discussion).…”

“…which allows us to seamlessly deal with incompressible materials as well (if the material is homogeneous and incompressible, we can simply remove ρ from (13) since the density is in this case constant. Similarly, if the response of the elastic component is modeled as a response of an incompressible material, then h κ p(t) is a constant and we can simply remove the h κ p(t) form (13). However, we need to introduce the corresponding Lagrange multipliers that enforce the incompressibility constraint).…”

Numerical scheme for simulation of transient flows of non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient and the Cauchy stress tensor

“…Moreover, one can also think about general implicit relations of the type h(D, T δ ) = 0, (1.4) where h is a tensorial function. One of the very first observations of the fluid response that could be characterised by (1.3) is due to Boltenhagen et al (1997), and the amount of experimental or theoretical works concerning the non-monotonous response of the type (1.3) has been growing since then, see for example Perlácová and Průša (2015), Janečka and Průša (2015), Rajagopal and Saccomandi (2016) or Janečka and Pavelka (2018) for further references.…”