Some Variational Theorems for Non-Newtonian Flow

Johnson, Michelle

doi:10.1063/1.1706150

Cited by 63 publications

(41 citation statements)

References 3 publications

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“…Elements of a variational principle for the Stokes flow of an incompressible non-Newtonian fluid already exist in the literature (Bird, 1960;Johnson, 1960). These have been our starting point for constructing a variational principle suitable for ice-sheet modeling.…”

Section: The Action Principle and The Euler-lagrange Equationsmentioning

confidence: 99%

“…Bird, 1960;Johnson, 1960). Early applications in glaciology of the principle formulated by Johnson (1960) may be found in Fowler (1981) and Oakberg (1981). Variational principles in ice-sheet modeling involving the various approximate models, particularly in the finite-element context, are to be found in Colinge and Rappaz (1999), Glowinski and Rappaz (2003) and Rappaz and Reist (2005) for example, and most recently in the work of Schoof (2006Schoof ( , 2010.…”

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confidence: 99%

“…Bird, 1960;Johnson, 1960). Early applications in glaciology of the principle formulated by Johnson (1960) may be found in Fowler (1981) and Oakberg (1981).…”

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confidence: 99%

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Consistent approximations and boundary conditions for ice-sheet dynamics from a principle of least action

Dukowicz¹,

Price²,

Lipscomb³

2010

J. Glaciol.

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ABSTRACT. The formulation of a physical problem in terms of a variational (or action) principle conveys significant advantages for the analytical formulation and numerical solution of that problem. One such problem is ice-sheet dynamics as described by non-Newtonian Stokes flow, for which the variational principle can be interpreted as stating that a measure of heat dissipation, due to internal deformation and boundary friction, plus the rate of loss of total potential energy is minimized under the constraint of incompressible flow. By carrying out low-aspect-ratio approximations to the Stokes flow problem within this variational principle, we obtain approximate dynamical equations and boundary conditions that are internally consistent and preserve the analytical structure of the full Stokes system. This also allows us to define an action principle for the popular first-order or 'Blatter-Pattyn' shallow-ice approximation that is distinct from the action principle for the Stokes problem yet preserves its most important properties and elucidates various details about this approximation. Further approximations within this new action functional yield the standard zero-order shallow-ice and shallow-shelf approximations, with their own action principles and boundary conditions. We emphasize the specification of boundary conditions, which are problematic to derive and implement consistently in approximate models but whose formulation is greatly simplified in a variational setting.

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Section: The Action Principle and The Euler-lagrange Equationsmentioning

confidence: 99%

mentioning

confidence: 99%

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Consistent approximations and boundary conditions for ice-sheet dynamics from a principle of least action

Dukowicz¹,

Price²,

Lipscomb³

2010

J. Glaciol.

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“…For all remaining solutions, however, all the ruby and steel spheres were used. It is clear from Figure 2 (9) in which C is a constant for a given fluid. From Equation ( vm2/D2) should define straight lines which should intercept the (vm2/D2) = 0 axes at -qs = qo.…”

Section: Extrapolation Procedures For the Zero-shear Vfscosltymentioning

confidence: 93%

An experimental investigation of the flow of aqueous non‐Newtonian high polymer solutions past a sphere

Turian

1967

AIChE Journal

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The terminal velocities of spheres falling in aqueous solutions of hydraxyethyl cellulose and polyethylene oxide were determined with ruby and steel spheres. For each sphere the terminal velocity was obtained in seven cylinder sizes covering a range of sphere-to-cylinder diameter ratio from 0.0067 to 0.18. The data were used in an extrapolation to correct for the cylindrical wall effect. Several alternative extrapolation procedures for estimating the zero shear viscosity were attempted and the results were compared. An empirical correlation for the drag coefficient was developed in terms of the Ellis parameters of the fluids, and the Faxen wall correction for Newtonian flow. The correlation gives the drag coefficient within 4% for all cases in which the quantity qovt/Dtl/a is less thon 0.3. N E W T O N I A N F L O W PAST A SPHEREThe drag force exerted on a sphere of radius R resulting from the flow past it of an unbounded incompressible Newtonian fluid of viscosity To and approach velocity v, is given by Stokes' law ( 1 2 ) :Equation (1) Oseen's approximation is represented by taking the terms up to and including N R e in Equation (2), whereas Goldstein's solution, purportedly an extension of Oseen's approximation, does not include the logarithmic term. These solutions are classified as the low Reynolds number flow approximations to the Navier-Stokes equations, and they possess certain interesting features, some of which have been discussed by Chester ( 4 ) (with regard to the Oseen approximation) and also by Proudman and Pearson (1 7). to (D/D,) = 0.32. Ladenburg (11) has also presented a correction for the effect of the bottom of the cylinder. This correction, based on the solution for a sphere falling in an infinite fluid bounded at the bottom by an infinite flat plate, appears to overestimate the bottom correction. More recently, Tanner (24) carried out a numerical calculation in which the effects of both the cylindrical wall and the bottom of the container were taken into account. He found that the top and bottom corrections are negligible, provided the sphere is at least one cylinder radius away from either end. These conclusions are corroborated by experimental evidence and, consequently, the need for a top and bottom correction can be eliminated by appropriate design of the falling-sphere system.Many other studies on the flow of Newtonian fluids past a sphere have been reported. These have been reviewed (8, 13) and will not be discussed inasmuch as the results are not employed in the present study. N O N -N E W T O N t A N F L O W PAST A SPHEREReported studies on non-Newtonian flow past a sphere include solutions based on perturbation procedures, solutions based on variational schemes, and empirical correlations and associated experimental studies. Solutions Bored o n Perturbation ProceduresA perturbation approximation for the flow of a ReinerRivlin-Prager fluid ast a sphere was obtained by Rathna ity are constant. The rheological model subject to these restrictions is inadequate in describing non-Newton...

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“…Most treatments of the problem have considered the deformation and the basal slip separately, but a variational principle for stresses presented by Johnson (1960) can be used to investigate the interaction between the two modes of flow.…”

Section: Flow Of Ice In a Cylindrical Channel Of Parabolic Cross-sectionmentioning

confidence: 99%

Variational Methods for Glacier Mechanics Problems

Oakberg¹

1981

J. Glaciol.

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ABSTRACT. The object of the research is to determine whether direct methods from the calculus of variations can provide convenient approximate solutions of complex problems in glacier mechanics. The Ritz technique is used to minimize an appropriate functional. Coordinate functions obtained from a finiteelement model are combined with a coordinate function that is the solution of a related problem. The finite-element coordinate functions make localized adjustments to the related solution. Solutions of two sample problems are presented. An analysis of the closure of an intergranular vein in ice at the melting point is based upon a variational principle for velocities. An analysis of the flow of ice in a cylindrical channel is based upon a variational principle for stresses. RESUME. Les methodes variationnelles pour les probLemes de mecanique des glaciers. L'objet de la recherche est de determiner si des methodes directes issues du calcul des variations peut fournir d es solutions approchees convenables de problemes complexes de mecanique des glaciers. La technique d e Ritz est utilisee pour minimiser un fonctionnel approprie. L es fonctions coordonnees obtenues it partir d'un modele aux elements finis sont combinees avec une fonction coordonnee qui est la solution d'un probleme particulier. Les fon ctions coordonnees issues du modele aux elements finis permettent d es ajustements locaux de la solution particuliere. On presente les solutions de deux problemes types. Une analyse d e la fermeture d'un pore intergranulaire dans de la glace it la temperature de fusion est basee sur un principe variationnel pour les vitesses. Une analyse de l'ecoulement de la glace dans un chenal cylindrique est fondee sur le principe variationnel pour les efforts.ZUSAMMENFASSUNG

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Some Variational Theorems for Non-Newtonian Flow

Cited by 63 publications

References 3 publications

Consistent approximations and boundary conditions for ice-sheet dynamics from a principle of least action

Consistent approximations and boundary conditions for ice-sheet dynamics from a principle of least action

An experimental investigation of the flow of aqueous non‐Newtonian high polymer solutions past a sphere

Variational Methods for Glacier Mechanics Problems

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