Computations of Numerical Solutions in Polymer Flows Using Giesekus Constitutive Model in the<i>hpk</i>Framework with Variationally Consistent Integral Forms

Surana, Karan S.; Deshpande, Kedar; Romkes, Albert; Reddy, J. N.

doi:10.1080/15502280903106465

Cited by 4 publications

(10 citation statements)

References 12 publications

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“…(ii) Converged solution obtained using lower convected (LC) and Jaumann (Jm) rate equations are then compared with those from the UC case to determine their validity: (a) for progressively increasing flow rate that correspond to the progressively increasing strain rates; (b) and for progressively increasing Deborah number (De) for a fixed ∂p ∂x . (iii) Surana et al [37] have shown that for this model problem a coarse uniform discretization of ten elements (half domain i.e. H ) with p-levels 5 − 11 for k = 2 (solution of class C 1 for u, p and τ p ) produces converged solutions that satisfy GDEs quite well.…”

Section: Remarksmentioning

confidence: 86%

“…Thus minimally conforming k for p and τ p is two but k = 3 is needed for the velocity u if the integrals are to be in the Riemann sense. Surana et al [37] have shown that choice of k = 2 for velocity u suffices as well due to smoothness of the solution but for this choice the term corresponding to ∂ 2 u ∂y 2 in the integrals are in Lebesgue sense. However, upon convergence the desired smoothness in the numerical solution is achieved [37].…”

Section: Remarksmentioning

confidence: 99%

“…Surana et al [37] have shown that choice of k = 2 for velocity u suffices as well due to smoothness of the solution but for this choice the term corresponding to ∂ 2 u ∂y 2 in the integrals are in Lebesgue sense. However, upon convergence the desired smoothness in the numerical solution is achieved [37]. In summary, for all numerical solution we choose a ten element uniform discretization with k = 2 for u, p and τ p .…”

Section: Remarksmentioning

confidence: 99%

“…In summary, for all numerical solution we choose a ten element uniform discretization with k = 2 for u, p and τ p . Based on reference [37] p-levels beyond 5 yield good accuracy of I . With p-levels of 11, I of the order of O(10 −8 ) or lower is achieved indicating that the GDEs are satisfied accurately by the computed solutions.…”

Section: Remarksmentioning

confidence: 99%

“…local approximations of class C 1 (¯ e ) for u, p as well as τ p . Based on reference [37] we choose p = 11 for all dependent variables. Thus the local approximations for all dependent variables is of the same order and same degree.…”

Section: (A) Numerical Studies For a Fixed Deborah Number With Progrementioning

confidence: 99%

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The Rate Constitutive Equations and Their Validity for Progressively Increasing Deformation

Surana

Romkes

et al. 2010

Mechanics of Advanced Materials and Structures

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The rate constitutive equations based on upper convected, lower convected, Jaumann, Truesdell and Green-Naghdi stress rates, etc.[1-10] have been used for the solid matter and polymeric liquids when the mathematical models are derived employing conservation laws in the Eulerian description. The rate constitutive equations provide relationship between convected time derivatives of the stress tensor and the convected time derivatives of strain tensor through the constitution of the matter. It is well known that if one assumes constant properties of the matter, then the use of various rate constitutive equations yield different responses [9] when the strains and strain rates are not infinitesimal (finite deformation) even though the rate constitutive equations are objective or frame invariant. This has been rationalized by the argument that in order for the different rate constitutive equations to produce the same response, the material tensor must be different for each case [9]. Our point of view is rather pragmatic in the sense that if all rate constitutive models describe the behavior of the same matter in which material constants do not change during deformation, then surely there must be some inherent assumptions in their derivations that are responsible for this anomaly.The covariant and contravariant convected bases in the current configuration of the deforming matter provide two possible alternate means of defining the convected time derivatives of the contravariant and covariant Cauchy stress tensors as well as the strain tensors. The relationship between the convected time derivatives of the stress tensor, material tensor and convected time derivative of the strain tensor result in the rate constitutive equations. Thus there are at least two obvious approaches for deriving rate constitutive equations: one based on covariant description referred to as lower convected rate constitutive equations and the other based on a contravariant description referred to as upper convected rate constitutive equations. It can be shown that the other rate constitutive equations available (in the literature) can also be derived using these two basic descriptions by modifications that are either justified based on the physics of the deforming matter or mathematical manipulations. When the strains and the strain rates are small (closer to infinitesimal assumption), there is isomorphism (or equivalence) between the covariant and contravariant descriptions, hence in this case the two descriptions will yield identical results even though the explicit forms of the rate equation expressions in the two descriptions are different. It is shown that when the strains and the strain rates are finite (finite deformation), the isomorphism or equivalence between the two descriptions is lost. The paper demonstrates that with progressively increasing deformation leading to finite deformation only the contravariant description has physical basis, hence the rate constitutive equations derived using contravariant description remain valid whereas...

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Section: Remarksmentioning

confidence: 86%

Section: Remarksmentioning

confidence: 99%

Section: Remarksmentioning

confidence: 99%

Section: Remarksmentioning

confidence: 99%

Section: (A) Numerical Studies For a Fixed Deborah Number With Progrementioning

confidence: 99%

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The Rate Constitutive Equations and Their Validity for Progressively Increasing Deformation

Surana

Romkes

et al. 2010

Mechanics of Advanced Materials and Structures

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3D modeling of generalized Newtonian fluid flow with data assimilation using the least-squares finite element method

Averweg

Schwarz

et al. 2022

Computer Methods in Applied Mechanics and Engineering

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Alternative least‐squares finite element models of Navier‐Stokes equations for power‐law fluids

Vallala

Reddy²,

Surana³

2011

Engineering Computations

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Purpose -Most studies of power-law fluids are carried out using stress-based system of Navier-Stokes equations; and least-squares finite element models for vorticity-based equations of power-law fluids have not been explored yet. Also, there has been no study of the weak-form Galerkin formulation using the reduced integration penalty method (RIP) for power-law fluids. Based on these observations, the purpose of this paper is to fulfill the two-fold objective of formulating the least-squares finite element model for power-law fluids, and the weak-form RIP Galerkin model of power-law fluids, and compare it with the least-squares finite element model. Design/methodology/approach -For least-squares finite element model, the original governing partial differential equations are transformed into an equivalent first-order system by introducing additional independent variables, and then formulating the least-squares model based on the lower-order system. For RIP Galerkin model, the penalty function method is used to reformulate the original problem as a variational problem subjected to a constraint that is satisfied in a least-squares (i.e. approximate) sense. The advantage of the constrained problem is that the pressure variable does not appear in the formulation. Findings -The non-Newtonian fluids require higher-order polynomial approximation functions and higher-order Gaussian quadrature compared to Newtonian fluids. There is some tangible effect of linearization before and after minimization on the accuracy of the solution, which is more pronounced for lower power-law indices compared to higher power-law indices. The case of linearization before minimization converges at a faster rate compared to the case of linearization after minimization. There is slight locking that causes the matrices to be ill-conditioned especially for lower values of power-law indices. Also, the results obtained with RIP penalty model are equally good at higher values of penalty parameters. Originality/value -Vorticity-based least-squares finite element models are developed for power-law fluids and effects of linearizations are explored. Also, the weak-form RIP Galerkin model is developed.

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Computations of Numerical Solutions in Polymer Flows Using Giesekus Constitutive Model in thehpkFramework with Variationally Consistent Integral Forms

Cited by 4 publications

References 12 publications

The Rate Constitutive Equations and Their Validity for Progressively Increasing Deformation

The Rate Constitutive Equations and Their Validity for Progressively Increasing Deformation

3D modeling of generalized Newtonian fluid flow with data assimilation using the least-squares finite element method

Alternative least‐squares finite element models of Navier‐Stokes equations for power‐law fluids

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