Hydrodynamic forces on two moving discs

Burton, David A.; Gratus, Jonathan; Tucker, Robin

doi:10.2298/tam0402153b

Cited by 23 publications

(16 citation statements)

References 8 publications

(18 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently, it was solved by several methods. One evident approach is related with conformal mapping of the annulus to a rectangle [12,16] and uses the image method for the last one. It produces a double periodic lattice of images, described in terms of elliptic functions [2].…”

Section: Introduction: Methods Of Images and Circle Theoremmentioning

confidence: 99%

Vortex images andq-elementary functions

Pashaev¹,

Yılmaz²

2008

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

In the present paper, the problem of vortex images in the annular domain between two coaxial cylinders is solved by the q-elementary functions. We show that all images are determined completely as poles of the q-logarithmic function, and are located at sites of the q-lattice, where a dimensionless parameter q = r 2 2 r 2 1 is given by the square ratio of the cylinder radii. The resulting solution for the complex potential is represented in terms of the Jackson q-exponential function. Our approach in this paper provides an efficient path to rediscover known solutions for the vortex-cylinder pair problem and yields new solutions as well. By composing pairs of q-exponents to the first Jacobi theta function and conformal mapping to a rectangular domain we show that our solution coincides with the known one, obtained before by elliptic functions. The Schottky-Klein prime function for the annular domain is factorized explicitly in terms of q-exponents. The Hamiltonian, the KirchhoffRouth and the Green functions are constructed. As a new application of our approach, the uniformly rotating exact N-vortex polygon solutions with the rotation frequency expressed in terms of q-logarithms at Nth roots of unity are found. In particular, we show that a single vortex orbits the cylinders with constant angular velocity, given as the q-harmonic series. Vortex images in two particular geometries with only one cylinder as the q → ∞ limit are studied in detail.

show abstract

Section: Introduction: Methods Of Images and Circle Theoremmentioning

confidence: 99%

Vortex images andq-elementary functions

Pashaev¹,

Yılmaz²

2008

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…There exists a velocity potential φ such that u = ∇φ in D. Let ∂ D j denote the boundary of the object D j while the outer boundary of D will be denoted ∂ D 0 . The flow must satisfy the boundary conditions u.n = U j .n, on ∂D j (2) where, for j = 0, 1, . .…”

Section: The Problem Of Fluid Stirrersmentioning

confidence: 99%

“…In what follows, if a = (a x , a y ) denotes a vector then the notation a will be used to denote its natural complex analogue a = a x + ia y . It is wellknown that the solution of the mathematical problem just stated is equivalent to finding a complex potential w(z), analytic in D, satisfying the boundary conditions (2) where dw/dz = u − iv. Since we have additionally imposed that there is no circulation around any of the objects {D j | j = 1, .…”

Section: The Problem Of Fluid Stirrersmentioning

confidence: 99%

“…D ζ is a triply connected circular domain In the case of just two stirrers (so that the fluid region is doubly connected) considered by Wang [1] and Burton et al [2], the problem admits a celebrated integral-formula solution known as the Villat formula [6]-a fact recently pointed out by Crowdy et al [7]. The Villat formula is only relevant to doubly connected domains.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Explicit solution for the potential flow due to an assembly of stirrers in an inviscid fluid

Crowdy

2008

J Eng Math

View full text Add to dashboard Cite

The motion of an irrotational incompressible fluid driven by an assembly of stirrers, of arbitrary shape, moving at specified velocities in the fluid is considered. The problem is shown to be equivalent to a standard mathematical problem in potential theory known as the modified Schwarz problem. It turns out that the solution to this problem can be written down, in closed form, as an explicit integral depending on the conformal mapping to the fluid region from a canonical pre-image region and a kernel function expressed in terms of a transcendental function called the Schottky-Klein prime function. In this way, an explicit integral solution, up to conformal mapping, for the complex potential of the flow generated by an arbitrary assembly of stirrers can be written down.

show abstract

“…It admits an exact analytic solution in terms of elliptic functions and has been studied by Johnson and McDonald [6], Burton et al [7] and Crowdy and Marshall [8]. Analysis of the same problem by the method of images in terms of the q-calculus has been examined by the present authors in [9].…”

Section: Introductionmentioning

confidence: 90%

Power-series solution for the two-dimensional inviscid flow with a vortex and multiple cylinders

Pashaev

Yılmaz

2009

J Eng Math

View full text Add to dashboard Cite

The problem of a point vortex and N fixed cylinders in a two-dimensional inviscid fluid is studied and an analytical-numerical solution in the form of an infinite power series for the velocity field is obtained using complex analysis. The velocity distribution for the case of two cylinders is compared with the existing results of the problem of a vortex in an annular region which is conformally mapped onto the exterior of two cylinders. Limiting cases of N cylinders and the vortex, being far away from each other are studied. In these cases, "the dipole approximation" or "the point-island approximation" is derived, and its region of validity is established by numerical tests. The velocity distribution for a geometry of four cylinders placed at the vertices of a square and a vortex is presented. The problem of vortex motion with N cylinders addressed in the paper attracted attention recently owing to its importance in many applications. However, existing solutions using Abelian function theory are sophisticated and the theory is not one of the standard techniques used by applied mathematicians and engineers. Moreover, in the N ≥ 3 cylinder problem, the infinite product involved in the presentation of the Schottky-Klein prime function must also be truncated. So, the approach used in the paper is simple and an alternative to existing methods. This is the main motivation for this study.

show abstract

Hydrodynamic forces on two moving discs

Cited by 23 publications

References 8 publications

Vortex images andq-elementary functions

Vortex images andq-elementary functions

Explicit solution for the potential flow due to an assembly of stirrers in an inviscid fluid

Power-series solution for the two-dimensional inviscid flow with a vortex and multiple cylinders

Contact Info

Product

Resources

About