Lamb's Solution of Stokes's Equations: A Sphere Theorem

Palaniappan, D.; Nigam, S. D.; Amaranath, T.; Usha, R.

doi:10.1093/qjmam/45.1.47

Cited by 56 publications

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“…Differential 4,6,7,9,16,18,19,20,21] for the displacement field in elasticity are based on the action of the gradient and the rotation operator on harmonic or biharmonic functions. The same is true for the representation of the velocity and the pressure fields in Stokes flow where similar representations have been proposed [10,12,17,22,23]. For electromagnetism, extended use of polyadic fields and their representations can be found in [13], while in [11,14] differential representations for angular momentum with direct physical interpretation are provided.…”

Section: Introductionmentioning

confidence: 79%

On the Helmholtz decomposition for polyadics

Dassios¹,

Lindell²

2001

Quart. Appl. Math.

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Abstract.A polyadic field of rank n is the tensor product of n vector fields. Helmholtz showed that a vector field, which is a polyadic field of rank 1, is nonuniquely decomposable into the gradient of a scalar function plus the rotation of a vector function. We show here that a polyadic field of rank n is, again nonuniquely, decomposable into a term consisting of n successive applications of the gradient to a scalar function, plus a term that consists of (n -1) successive applications of the gradient to the rotation of a vector function, plus a term that consists of (n -2) successive applications of the gradient to the rotation of the rotation of a dyadic function and so on, until the last (n + l)th term, which consists of n successive applications of the rotation operator to a polyadic function of rank n. Obviously, the n = 1 case recovers the Helmholtz decomposition theorem. For dyadic fields a more symmetric representation is provided and formulae that provide the potential representation functions are given. The special cases of symmetric and antisymmetric dyadics are discussed in detail. Finally, the multidivergence type relations, which reduce the number of independent scalar representation functions to n2, are presented.

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Section: Introductionmentioning

confidence: 79%

On the Helmholtz decomposition for polyadics

Dassios¹,

Lindell²

2001

Quart. Appl. Math.

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“…Based on this method, we examined the Papkovich-Neuber (PN) [8,10] and the Palaniappan et al (PNAU) [9] differential representations, which offer solutions for such flow problems in spherical geometry. The important physical flow fields (velocity, total pressure) are presented in terms of vector spherical harmonics.…”

Section: Discussionmentioning

confidence: 99%

“…On the other hand, Palaniappan et al [9] assumed another 3D differential representation for the solutions of Stokes equations as a function of the harmonic and biharmonic potentials A(r) and B(r), respectively:…”

Section: Fundamentals Of Stokes Flowmentioning

confidence: 99%

“…The introduction of differential representations of the solutions of Stokes equations [1,8,9,10] serves to unify our own approach on all 3D incompressible fluid motions. Based on the previous formulation of cell models, the problem is now focused on the use of the appropriate representation that coincides with the physical problem.…”

Section: Introductionmentioning

confidence: 99%

“…The major advantage of the differential representations is that they provide us with the flow fields for Stokes flow in terms of harmonic potentials. The most famous general spatial solutions are the PN solution [8,10], the Boussinesq-Galerkin solution [1,10], and the PNAU solution [9]. Recently, a method of connecting 3D differential representations has been developed [2], where the PN and the Boussinesq-Galerkin differential representations were interrelated and connection formulae between the corresponding spherical harmonic and biharmonic potentials were developed.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Comparison of differential representations for radially symmetricStokes flow

Dassios

Vafeas

2004

Abstract and Applied Analysis

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Papkovich and Neuber (PN), and Palaniappan, Nigam, Amaranath, and Usha (PNAU) proposed two different representations of the velocity and the pressure fields in Stokes flow, in terms of harmonic and biharmonic functions, which form a practical tool for many important physical applications. One is the particle-in-cell model for Stokes flow through a swarm of particles. Most of the analytical models in this realm consider spherical particles since for many interior and exterior flow problems involving small particles, spherical geometry provides a very good approximation. In the interest of producing ready-to-use basic functions for Stokes flow, we calculate the PNAU and the PN eigensolutions generated by the appropriate eigenfunctions, and the full series expansion is provided. We obtain connection formulae by which we can transform any solution of the Stokes system from the PN to the PNAU eigenform. This procedure shows that any PNAU eigenform corresponds to a combination of PN eigenfunctions, a fact that reflects the flexibility of the second representation. Hence, the advantage of the PN representation as it compares to the PNAU solution is obvious. An application is included, which solves the problem of the flow in a fluid cell filling the space between two concentric spherical surfaces with Kuwabara-type boundary conditions.

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Lorentz’s theorem on the Stokes equation

Hasimoto

1996

The Centenary of a Paper on Slow Viscous Flow by the Physicist H.A. Lorentz

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Lamb's Solution of Stokes's Equations: A Sphere Theorem

Cited by 56 publications

References 0 publications

On the Helmholtz decomposition for polyadics

On the Helmholtz decomposition for polyadics

Comparison of differential representations for radially symmetricStokes flow

Lorentz’s theorem on the Stokes equation

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