1,841 research outputs found
A transform method for the biharmonic equation in multiply connected circular domains
No full textA new transform approach for solving mixed boundary value problems for the biharmonic equation in simply and multiply connected circular domains is presented. This work is a sequel to Crowdy (2015, IMA J. Appl. Math., 80, 1902–1931) where new transform techniques were developed for boundary value problems for Laplace’s equation in circular domains. A circular domain is defined to be a domain, which can be simply or multiply connected, having boundaries that are a union of circular arc segments. The method provides a flexible approach to finding quasi-analytical solutions to a wide range of problems in fluid dynamics and plane elasticity. Three example problems involving slow viscous flows are solved in detail to illustrate how to apply the method; these concern flow towards a semicircular ridge, a translating and rotating cylinder near a wall as well as in a channel geometry
Analytical formulas for longitudinal slip lengths over unidirectional superhydrophobic surfaces with curved menisci
No full textThis paper reports new analytical formulas for the longitudinal slip lengths for simple
shear over a superhydrophobic surface, or bubble mattress, comprising a periodic array
of unidirectional circular menisci, or bubbles, protruding into, or out of, the fluid. The
accuracy of the formulas is tested against results from full numerical simulations; they
are found to give small relative errors even at large no-shear fractions. In the dilute limit
the formulas reduce to an earlier result by the author [Phys. Fluids, 22, 121703, (2011)].
They also extend analytical results of Sbragaglia & Prosperetti [Phys Fluids, 19, 043603,
(2007)] beyond a small protrusion angle limit
Effective slip lengths for longitudinal shear flow over partial-slip circular bubble mattresses.
No full textThe problem of longitudinal shear flow over a circular bubble mattress with
partial slip and protrusion angle 90o is solved in a quasi-analytical fashion by a
novel transform scheme recently devised by the author. The general approach
can be readily adapted to other mixed boundary value problems. From the
analysis explicit approximations for the effective slip lengths are found as a
function of the Navier-slip parameter and the area fraction of the surface
covered by protrusions. These new approximation formulas for the slip lengths
both unify and extend those based on empirical polynomial fits to numerical
data given recently by Ng and Wang (2011 Fluid Dyn. Res. 43 065504)
Uniform flow past a periodic array of cylinders
No full textThe problem of uniform flow past a linear periodic array, or grating, of circular cylinders is solved by a new transform technique recently devised by the author. The solutions are expressed as an integral representation dependent on a set of coefficients which are the solutions of an explicit and well-conditioned linear system that is easily solved. We also give new explicit approximation formulas for the so-called blockage coefficients that are accurate well beyond the dilute limit relevant to cylinders that are small relative to their spacing. Finally, we describe an alternative approach to the flow problems through conformal mapping theory and produce a transform representation of a useful conformal mapping from the disc-in-channel geometry to a concentric annulus. The latter result is expected to be useful in many other applications
The effect of core size on the speed of compressible hollow vortex streets
No full textThe effect of weak compressibility on the speed of steadily translating staggered vortex streets of hollow vortices in isentropic subsonic flow is studied. A small-Mach-number perturbation expansion about the incompressible solutions for staggered streets of hollow vortices found recently by Crowdy & Green (Phys. Fluids, 2011, vol. 23, 126602) is carried out; the latter solutions provide a desingularization of the classical point vortex streets of von Kármán. The first-order compressible flow correction is calculated. We employ a novel scheme based on a complex variable formulation of the compressible flow equations (the Imai–Lamla method) combined with conformal mapping theory to track the vortex shape in this free boundary problem. The analysis to find the perturbed streamfunction and compressible vortex shapes is greatly facilitated by exploiting a calculus based on use of the Schottky–Klein prime function of a conformally equivalent parametric annulus. It is found that, for a vortex street of specified aspect ratio comprising vortices of specified circulation, the vortex core size is a key determinant of whether compressibility increases or decreases the steady propagation speed (relative to the incompressible street with the same parameters) and that both eventualities are possible. We focus attention on streets with aspect ratios around 0.28, which is close to the neutrally stable case for incompressible flow, and find that a critical vortex core size exists at which compressibility does not affect the speed of the street at first order in the (squared) Mach number. Streets comprising vortices with core size below the critical value speed up due to compressibility; larger vortices slow down
A constructive method for plane-wave representations of special functions
No full textA general constructive scheme for the derivation of plane-wave representations
of special functions is proposed. Illustrative examples of the construction are
given. As one case study, new integral representations of the elliptic Weierstrass
℘ function are derived; these complement, and generalize, similar new planewave
integral representations of the same function recently found by Dienstfrey
& Huang [J. Math. Anal. Appl., 316, 142–160, (2006)] using other techniques.
Our approach is inspired by recent developments in the so-called Fokas transform
for the solution of boundary value problems for partial differential equations.
Keywords: plane wave representation, elliptic function, transform method,
special function
Slip length for transverse shear flow over a periodic array of weakly curved menisci
No full textBy exploiting the reciprocal theorem of Stokes flow, we find an explicit expression for the first order slip length correction, for small protrusion angles, and for transverse shear over a periodic array of curved menisci. The result is the transverse flow analogue of the longitudinal flow result of Sbragaglia and Prosperetti [“A note on the effective slip properties for microchannel flows with ultrahydrophobic surfaces,” Phys. Fluids 19, 043603 (2007)]. For small protrusion angles, it also generalizes the dilute-limit result of Davis and Lauga [“Geometric transition in friction for flow over a bubble mattress,” Phys. Fluids 21, 011701 (2009)] to arbitrary no-shear fractions. While the leading order slip lengths for transverse and longitudinal flow over flat no-shear slots are well-known to differ by a factor of 2, the first order slip length corrections for weakly protruding menisci in each flow are found to be identical
Flipping and scooping of curved 2D rigid fibers in simple shear: the Jeffery equations
No full textThe dynamical system governing the motion of a curved rigid two-dimensional
circular-arc fiber in simple shear is derived in analytical form. This is achieved by
finding the solution for the associated low-Reynolds-number flow around such a fiber
using the methods of complex analysis. Solutions of the dynamical system display
the “flipping” and “scooping” recently observed in computational studies of threedimensional
fibers using linked rigid rod and bead-shell models [Wang et al, Phys.
Fluids, 24, (2012)]. To complete the Jeffery-type equations for a curved fiber in a
linear flow field we also derive its evolution equations in an extensional flow. It is
expected that the equations derived here also govern the motion of slender, curved,
three-dimensional rigid fibers when they evolve purely in the plane of shear or strain
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