scholarly journals Asymptotic formulae for flow in superhydrophobic channels with longitudinal ridges and protruding menisci

2018 ◽  
Vol 839 ◽  
Author(s):  
Toby L. Kirk
Keyword(s):  
Asymptotic Expansion ◽  
Shear Flow ◽  
Numerical Solutions ◽  
Large Range ◽  
Slip Length ◽  
Matched Asymptotic Expansion ◽  
Asymptotic Formulae ◽  
Relative Errors

This paper presents new asymptotic formulae for flow in a channel with one or both walls patterned with a longitudinal array of ridges and arbitrarily protruding menisci. Derived from a matched asymptotic expansion, they extend results by Crowdy (J. Fluid Mech., vol. 791, 2016, R7) for shear flow, and thus make no restriction on the protrusion into or out of the liquid. The slip length formula is compared against full numerical solutions and, despite the assumption of small ridge period in its derivation, is found to have a very large range of validity; relative errors are small even for periods large enough for the protruding menisci to degrade the flow and touch the opposing wall.

scholarly journals Asymptotic expansions of Lambert series and related q-series

2017 ◽  
Vol 13 (08) ◽  
pp. 2097-2113 ◽  
Author(s):  
Shubho Banerjee ◽  
Blake Wilkerson
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Theta Functions ◽  
Lambert Series ◽  
Pochhammer Symbol ◽  
Complete Asymptotic Expansion ◽  
Polygamma Functions ◽  
Jacobi Theta Functions ◽  
Relative Errors ◽  
Asymptotic Forms

We study the Lambert series [Formula: see text], for all [Formula: see text]. We obtain the complete asymptotic expansion of [Formula: see text] near [Formula: see text]. Our analysis of the Lambert series yields the asymptotic forms for several related [Formula: see text]-series: the [Formula: see text]-gamma and [Formula: see text]-polygamma functions, the [Formula: see text]-Pochhammer symbol and the Jacobi theta functions. Some typical results include [Formula: see text] and [Formula: see text], with relative errors of order [Formula: see text] and [Formula: see text] respectively.

Asymptotics of Sturm-Liouville eigenvalues for problems on a finite interval with one limit-circle singularity, I

1984 ◽  
Vol 99 (1-2) ◽  
pp. 51-70 ◽  
Author(s):  
F. V. Atkinson ◽  
C. T. Fulton
Keyword(s):  
Asymptotic Expansion ◽  
Eigenvalue Problem ◽  
Hypergeometric Function ◽  
Finite Interval ◽  
Confluent Hypergeometric Function ◽  
Limit Circle ◽  
Asymptotic Formulae ◽  
Sturm Liouville

SynopsisAsymptotic formulae for the positive eigenvalues of a limit-circle eigenvalue problem for –y” + qy = λy on the finite interval (0, b] are obtained for potentials q which are limit circle and non-oscillatory at x = 0, under the assumption xq(x)∈L1(0,6). Potentials of the form q(x) = C/xk, 0<fc<2, are included. In the case where k = 1, an independent check based on the limit-circle theory of Fulton and an asymptotic expansion of the confluent hypergeometric function, M(a, b; z), verifies the main result.

scholarly journals Mechanical Quadrature Methods and Extrapolation for Solving Nonlinear Problems in Elasticity

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Pan Cheng ◽  
Ling Zhang
Keyword(s):  
Asymptotic Expansion ◽  
Nonlinear Boundary ◽  
Numerical Solutions ◽  
Boundary Integral Equations ◽  
Logarithmic Singularity ◽  
Nonlinear Problems ◽  
Boundary Integral ◽  
Convergence Theory ◽  
Mechanical Quadrature ◽  
Quadrature Methods

This paper will study the high accuracy numerical solutions for elastic equations with nonlinear boundary value conditions. The equations will be converted into nonlinear boundary integral equations by the potential theory, in which logarithmic singularity and Cauchy singularity are calculated simultaneously. Mechanical quadrature methods (MQMs) are presented to solve the nonlinear equations where the accuracy of the solutions is of three orders. According to the asymptotical compact convergence theory, the errors with odd powers asymptotic expansion are obtained. Following the asymptotic expansion, the accuracy of the solutions can be improved to five orders with the Richardson extrapolation. Some results are shown regarding these approximations for problems by the numerical example.

scholarly journals Nusselt numbers for Poiseuille flow over isoflux parallel ridges accounting for meniscus curvature

2016 ◽  
Vol 811 ◽  
pp. 315-349 ◽  
Author(s):  
Toby L. Kirk ◽  
Marc Hodes ◽  
Demetrios T. Papageorgiou
Keyword(s):  
Nusselt Number ◽  
Poiseuille Flow ◽  
Numerical Solutions ◽  
Surface Deformation ◽  
Slip Length ◽  
Constant Heat Flux ◽  
Flux Boundary Condition ◽  
Nusselt Numbers ◽  
Dual Series Equations ◽  
Hydrodynamic Slip

We investigate forced convection in a parallel-plate-geometry microchannel with superhydrophobic walls consisting of a periodic array of ridges aligned parallel to the direction of a Poiseuille flow. In the dewetted (Cassie) state, the liquid contacts the channel walls only at the tips of the ridges, where we apply a constant-heat-flux boundary condition. The subsequent hydrodynamic and thermal problems within the liquid are then analysed accounting for curvature of the liquid–gas interface (meniscus) using boundary perturbation, assuming a small deflection from flat. The effects of this surface deformation on both the effective hydrodynamic slip length and the Nusselt number are computed analytically in the form of eigenfunction expansions, reducing the problem to a set of dual series equations for the expansion coefficients which must, in general, be solved numerically. The Nusselt number quantifies the convective heat transfer, the results for which are completely captured in a single figure, presented as a function of channel geometry at each order in the perturbation. Asymptotic solutions for channel heights large compared with the ridge period are compared with numerical solutions of the dual series equations. The asymptotic slip length expressions are shown to consist of only two terms, with all other terms exponentially small. As a result, these expressions are accurate even for heights as low as half the ridge period, and hence are useful for engineering applications.

Slip length for longitudinal shear flow over an arbitrary-protrusion-angle bubble mattress: the small-solid-fraction singularity

2017 ◽  
Vol 820 ◽  
pp. 580-603 ◽  
Author(s):  
Ory Schnitzer
Keyword(s):  
Shear Flow ◽  
Asymptotic Analysis ◽  
Solid Fraction ◽  
Asymptotic Expansions ◽  
Slip Length ◽  
Longitudinal Shear ◽  
Unidirectional Flow ◽  
Effective Slip Length ◽  
Effective Slip ◽  
Scaling Arguments

We study the effective slip length for unidirectional flow over a superhydrophobic mattress of bubbles in the small-solid-fraction limit $\unicode[STIX]{x1D716}\ll 1$. Using scaling arguments and utilising an ideal-flow analogy we elucidate the singularity of the slip length as $\unicode[STIX]{x1D716}\rightarrow 0$: relative to the periodicity it scales as $\log (1/\unicode[STIX]{x1D716})$ for protrusion angles $0\leqslant \unicode[STIX]{x1D6FC}<\unicode[STIX]{x03C0}/2$ and as $\unicode[STIX]{x1D716}^{-1/2}$ for $0<\unicode[STIX]{x03C0}/2-\unicode[STIX]{x1D6FC}=O(\unicode[STIX]{x1D716}^{1/2})$. We continue with a detailed asymptotic analysis using the method of matched asymptotic expansions, where ‘inner’ solutions valid close to the solid segments are matched with ‘outer’ solutions valid on the scale of the periodicity, where the bubbles protruding from the solid grooves appear to touch. The analysis yields asymptotic expansions for the effective slip length in each of the protrusion-angle regimes. These expansions overlap for intermediate protrusion angles, which allows us to form a uniformly valid approximation for arbitrary protrusion angles $0\leqslant \unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x03C0}/2$. We thereby explicitly describe the transition with increasing protrusion angle from a logarithmic to an algebraic small-solid-fraction slip-length singularity.

Thermal stability in a rotating spherical fluid shell with non-uniform heating

1977 ◽  
Vol 353 (1674) ◽  
pp. 349-362 ◽  
Keyword(s):  
Shear Flow ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Rapid Change ◽  
Polar Angle ◽  
Quiescent State ◽  
Uniform Heating ◽  
Thermally Induced ◽  
Spherical Fluid ◽  
Geophysical Flow

The theory is developed for the convective stability of a rotating spherical shell of fluid upon which is initially imposed a stable thermally induced shear flow. The fluid shell contains heating sources which are distributed proportional to the sine of the polar angle squared. Thus the analysis has a number of similarities to some geophysical flow situations. It is found that the properties of the solution are strongly dependent on the initial conditions. Thus to obtain further insight concerning the stability of the system numerical solutions are obtained a t two shell thicknesses. The critical values of the Taylor number ( T ) and the Rayleigh number ( C ) are generally similar to those found in previous studies of rotating fluid shells. How ever the effect of the initial shear flow is to reduce the critical value of C for a given T , below that found for uniform heating and an initially quiescent state. The flows obtained at the onset of instability are toroidal cells which vary in number dependent on T and C .A maximum of six cells are found at large values of T . A significant effect of the initial shear flow is the occurrence of a rapid change in stability when the number of toroidal cells changes.

LATTICE BOLTZMANN SIMULATIONS OF DROP–DROP INTERACTIONS IN TWO-PHASE FLOWS

2005 ◽  
Vol 16 (01) ◽  
pp. 25-44 ◽  
Author(s):  
KANNAN N. PREMNATH ◽  
JOHN ABRAHAM
Keyword(s):  
Shear Flow ◽  
Lattice Boltzmann ◽  
Flow Model ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Drop Deformation ◽  
Simple Shear Flow ◽  
Two Phase ◽  
Lattice Boltzmann Simulations ◽  
Boltzmann Method

In this paper, three-dimensional computations of drop–drop interactions using the lattice Boltzmann method (LBM) are reported. The LBM multiphase flow model employed is evaluated for single drop problems and binary drop interactions. These include the verification of Laplace–Young relation for static drops, drop oscillations, and drop deformation and breakup in simple shear flow. The results are compared with experimental data, analytical solutions and numerical solutions based on other computational methods, as applicable. Satisfactory agreement is shown. Initial studies of drop–drop interactions involving the head-on collisions of drops in quiescent medium and off-center collision of drops in the presence of ambient shear flow are considered. As expected, coalescence outcome is observed for the range of parameters studied.

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