Circle theorems for steady Stokes flow

Sen, S. K.

doi:10.1007/bf00945316

Cited by 12 publications

(10 citation statements)

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“…In solving the present problem with circular stirring rods, 4 we previously used a "circle theorem" for the biharmonic equations 11,12 to modify the form of to ensure that the no-slip boundary condition on the vat wall is satisfied automatically; essentially the modification involves appending to appropriate images in this wall. However, the use of the circle theorem, while mathematically elegant, adds considerably to the algebraic complexity of , and here we adopt instead the simpler device of including in an additional series in positive powers of z ͓cf.…”

Section: Methods Of Solutionmentioning

confidence: 99%

Two-dimensional Stokes flow driven by elliptical paddles

Cox

Finn

2007

Physics of Fluids

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A fast and accurate numerical technique is developed for solving the biharmonic equation in a multiply connected domain, in two dimensions. We apply the technique to the computation of slow viscous flow ͑Stokes flow͒ driven by multiple stirring rods. Previously, the technique has been restricted to stirring rods of circular cross section; we show here how the prior method fails for noncircular rods and how it may be adapted to accommodate general rod cross sections, provided only that for each there exists a conformal mapping to a circle. Corresponding simulations of the flow are described, and their stirring properties and energy requirements are discussed briefly. In particular the method allows an accurate calculation of the flow when flat paddles are used to stir a fluid chaotically.

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Section: Methods Of Solutionmentioning

confidence: 99%

Two-dimensional Stokes flow driven by elliptical paddles

Cox

Finn

2007

Physics of Fluids

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“…The theorem can also be used for non-circular boundaries if one can find conformal transformation that maps the given boundary to a circle. The circle theorem of Milne-Thomson has also its analogue in electrostatics [29], Stokes flows [1,35] and in isotropic elasticity [20]. Furthermore, the circle theorem has also been extended to include surface singularity distributions [33,2].…”

Section: Two-dimensional Inviscid Flowmentioning

confidence: 99%

“…The earlier sphere theorems for axisymmetric slow viscous flows [8,9,13] also used the inversion theorem implicitly. The circle theorems for Stokes flows [1,35] further exploited the use of the inversion theorem for biharmonic functions. It is worth citing the notable extensions of circle and sphere theorems for isotropic elastic media [20,5].…”

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confidence: 99%

Generalized Circle and Sphere Theorems for Inviscid and Viscous Flows

Daripa¹,

Palaniappan²

2001

SIAM J. Appl. Math.

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The circle and sphere theorems in classical hydrodynamics are generalized to a composite double body. The double body is composed of two overlapping circles/spheres of arbitrary radii intersecting at a vertex angle π/n, n an integer. The Kelvin's transformation is used successively to obtain closed form expressions for several flow problems. The problems considered here include two-dimensional and axisymmetric three-dimensional inviscid and slow viscous flows. The general results are presented as theorems followed by simple proofs. The two-dimensional results are obtained using complex function theory while the three-dimensional formulas are obtained using Stokes stream function.The solutions for several flows in the presence of the composite geometry are derived by the use of these theorems. These solutions are in singularity forms and the image singularities are interpreted in each case. In the case of three-dimensional axisymmetric viscous flows, a Faxen relation for the force acting on the composite bubble is derived.

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“…In the case of two-dimensional slow flow theory there is a complex variable circle theorem [1] for the solutions of Stokes flows due to singularities outside a circular cylinder which corresponds to Milne-Thomson's circle theorem [2,3] for potential flow outside the same cylinder, in the inviscid flow theory. Again it is notable that the same complex variable circle theorem can solve, in particular, some particular Stokes flow problems which cannot be done by the application of the real variable circle theorem [4,5]. Moreover, the "condition for zero perturbation velocity" referred to in the former theorem may suggest, in many cases, relatively easily the strengths or positions or both, of the singularities of the basic flow so that the viscous flows outside the circular boundary exist.…”

Section: Introductionmentioning

confidence: 99%