Flow of a class of fluids defined via implicit constitutive equation down an inclined plane: Analysis of the quasi-steady regime

Fusi, Lorenzo; Farína, Angiolo

doi:10.1016/j.euromechflu.2016.11.008

Cited by 4 publications

(6 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Readers interested in analytical and semianalytical solutions to boundary value problems for fluids described by constitutive relations of the type T δ = g(D) or k(T δ , D) = O are referred to Málek et al [8], Le Roux and Rajagopal [7], Srinivasan and Karra [22], Narayan and Rajagopal [23], Fusi and Farina [24], Fusi [25], Housiadas et al [26], Gomez-Constante and Rajagopal [27] and Fetecau and Bridges [28], to name a few. The numerical solution of the corresponding governing equations is investigated in Janečka et al [29], while a rigorous numerical analysis for various models that fall into this class is discussed in Diening et al [30], Stebel [31], Hirn et al [32], Süli and Tscherpel [33], Farrell et al [34], Farrell and Gazca-Orozco [35].…”

Section: Fluidsmentioning

confidence: 99%

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

Průša

Rajagopal

2021

Fluids

View full text Add to dashboard Cite

Viscoelastic fluids are non-Newtonian fluids that exhibit both “viscous” and “elastic” characteristics in virtue of the mechanisms used to store energy and produce entropy. Usually, the energy storage properties of such fluids are modeled using the same concepts as in the classical theory of nonlinear solids. Recently, new models for elastic solids have been successfully developed by appealing to implicit constitutive relations, and these new models offer a different perspective to the old topic of the elastic response of materials. In particular, a sub-class of implicit constitutive relations, namely relations wherein the left Cauchy–Green tensor is expressed as a function of stress, is of interest. We show how to use this new perspective in the development of mathematical models for viscoelastic fluids, and we provide a discussion of the thermodynamic underpinnings of such models. We focus on the use of Gibbs free energy instead of Helmholtz free energy, and using the standard Giesekus/Oldroyd-B models, we show how the alternative approach works in the case of well-known models. The proposed approach is straightforward to generalize to more complex settings wherein the classical approach might be impractical or even inapplicable.

show abstract

Section: Fluidsmentioning

confidence: 99%

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

Průša

Rajagopal

2021

Fluids

View full text Add to dashboard Cite

show abstract

“…The constitutive equation is T * = −p * I + S * where S * is related to the symmetric part of the velocity gradient D * via the implicit relation (7). The balance of mass and linear momentum in cylindrical coordinates yields:…”

Section: Squeeze Flowmentioning

confidence: 99%

“…In particular, they proved that for special values of the constitutive parameters, the relation between the norm of the tensor D * and the norm of the tensor T * can be non-monotone. Simple flows of stress power law fluids were studied in [7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

“…As one can see, the monotonicity is lost for n = −5, −3, −2, −1 for which d n > γ. In this paper, we studied the squeeze flow between parallel circular disks of a fluid whose constitutive equation is of the type of (7) under the assumption that the gap between the squeezing disks is much smaller than their radius (lubrication flow; see Figure 2). Squeeze flows are found in many engineering, biological, and rheological applications.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Squeeze Flow of Stress Power Law Fluids

Fusi¹,

Ballotti²

2021

Fluids

Self Cite

View full text Add to dashboard Cite

In this paper, we studied the squeeze flow between circular disks of a new class of fluids defined by an implicit relation referred to as stress power law fluids. The constitutive response of these fluids was written expressing the symmetric part of the velocity gradient as a tensorial function of the Cauchy stress. We assumed that the aspect ratio between the gap separating the disks and the radius was small so that a lubrication expansion could be adopted. We wrote the general problem and looked for a solution that could be written in terms of the small aspect ratio parameter. We obtained a sequence of problems that could be solved iteratively at each order, and we focused on the leading and first order, deriving explicit expressions for the velocity field, stress, and pressure.

show abstract

“…Flows of fluids with a non-monotone constitutive relation of the type (1.3) have been to our best knowledge investigated only in special geometries, where the corresponding system of governing equations reduces to a system of ordinary differential equations, see for example Málek et al (2010), Le Roux and , Narayan and Rajagopal (2013), Srinivasan and Karra (2015), Mohankumar et al (2015) and Fusi and Farina (2017). However, if one needs to investigate flows in more complex geometries, a suitable numerical scheme for solution of transient flow problems must be developed.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2019

View full text Add to dashboard Cite

We propose a numerical scheme for simulation of transient flows of incompressible non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient (shear rate) and the Cauchy stress tensor (shear stress). The main difficulty in dealing with the governing equations for flows of such fluids is that the nonmonotone constitutive relation allows several values of the stress to be associated with the same value of the symmetric part of the velocity gradient. This issue is handled via a reformulation of the governing equations. The equations are reformulated as a system for the triple pressure-velocity-apparent viscosity, where the apparent viscosity is given by a scalar implicit equation. We prove that the proposed numerical scheme has-on the discrete level-a solution, and using the proposed scheme we numerically solve several flow problems.1 The norm of a tensorial quantity A is defined as the standard Frobenius norm, A = def (Tr (AA ⊺ ))) 1 2 .

show abstract

Flow of a class of fluids defined via implicit constitutive equation down an inclined plane: Analysis of the quasi-steady regime

Cited by 4 publications

References 23 publications

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

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

Squeeze Flow of Stress Power Law Fluids

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

Contact Info

Product

Resources

About