High-order asymptotic expansion for the acoustics in viscous gases close to rigid walls

2014 ◽  
Vol 24 (09) ◽  
pp. 1823-1855 ◽  
Author(s):  
Kersten Schmidt ◽  
Anastasia Thöns-Zueva ◽  
Patrick Joly
Keyword(s):  
Asymptotic Expansion ◽  
Normal Component ◽  
Tangential Velocity ◽  
Normal Derivative ◽  
Two Dimensions ◽  
Singularly Perturbed Problem ◽  
Optimal Error Estimates ◽  
Complete Asymptotic Expansion ◽  
Curvature Effects ◽  
Rigid Walls

We derive a complete asymptotic expansion for the singularly perturbed problem of acoustic wave propagation inside gases with small viscosity. This derivation is for the non-resonant case in smooth bounded domains in two dimensions. Close to rigid walls the tangential velocity exhibits a boundary layer of size [Formula: see text] where η is the dynamic viscosity. The asymptotic expansion, which is based on the technique of multiscale expansion is expressed in powers of [Formula: see text] and takes into account curvature effects. The terms of the velocity and pressure expansion are defined independently by partial differential equations, where the normal component of velocities or the normal derivative of the pressure, respectively, are prescribed on the boundary. The asymptotic expansion is rigorously justified with optimal error estimates.

Centrifugal instabilities in finite containers: a periodic model

1980 ◽  
Vol 99 (3) ◽  
pp. 575-596 ◽  
Author(s):  
P. Hall
Keyword(s):  
Model Problem ◽  
Fluid Velocity ◽  
Normal Component ◽  
Tangential Velocity ◽  
Normal Derivative ◽  
Taylor Vortex ◽  
Taylor Vortices ◽  
Periodic Model ◽  
Circulatory Flow ◽  
Imperfect Bifurcation

A simplified model problem has recently been suggested by Schaeffer (1980) in order to explain the results obtained by Benjamin (1978) in his investigation of Taylor vortices in short cylinders. In particular Schaeffer reproduces the results obtained by Benjamin for cylinders so short that only two-cell or four-cell flows are possible. The model given by Schaeffer has artificial conditions imposed on the fluid velocity field at the end walls. These conditions depend on a parameter α and reduce to the no-slip condition when α = 1. If α = 0 the conditions require that the normal component of the velocity and the normal derivative of the tangential velocity vanish at the ends. In this case the onset of Taylor vortex-like motion occurs as a bifurcation from purely circumferential flow. If α is now taken to be small and positive, there is no bifurcation and the circulatory flow develops smoothly. We shall use perturbations method for the case of small α. The imperfect bifurcation problem which we obtain predicts some results consistent with those of Benjamin.

scholarly journals Robust nonconforming virtual element methods for general fourth-order problems with varying coefficients

2021 ◽  
Author(s):  
Andreas Dedner ◽  
Alice Hodson
Keyword(s):  
Fourth Order ◽  
Perturbation Parameter ◽  
Two Dimensions ◽  
Optimal Error Estimates ◽  
Projection Operators ◽  
Varying Coefficients ◽  
Virtual Element Methods ◽  
Virtual Element ◽  
Perturbation Problems ◽  
Fourth Order Problems

Abstract We present a class of nonconforming virtual element methods for general fourth-order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth-order problems with varying coefficients. We also discuss fourth-order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth-order problems.

scholarly journals Superfluid vortex-mediated mutual friction in non-homogeneous neutron star interiors

2020 ◽  
Vol 499 (3) ◽  
pp. 3690-3705
Author(s):  
M Antonelli ◽  
B Haskell
Keyword(s):  
Neutron Star ◽  
Normal Component ◽  
Homogeneous Medium ◽  
Outer Core ◽  
Two Dimensions ◽  
Pinning Force ◽  
Mutual Friction ◽  
Momentum Exchange ◽  
Drag Forces ◽  
Vortex Lines

ABSTRACT Understanding the average motion of a multitude of superfluid vortices in the interior of a neutron star is a key ingredient for most theories of pulsar glitches. In this paper, we propose a kinetic approach to compute the mutual friction force that is responsible for the momentum exchange between the normal and superfluid components in a neutron star, where the mutual friction is extracted from a suitable average over the motion of many vortex lines. As a first step towards a better modelling of the repinning and depinning processes of many vortex lines in a neutron star, we consider here only straight and non-interacting vortices: we adopt a minimal model for the dynamics of an ensemble of point vortices in two dimensions immersed in a non-homogeneous medium that acts as a pinning landscape. Since the degree of disorder in the inner crust or outer core of a neutron star is unknown, we compare the two possible scenarios of periodic and disordered pinscapes. This approach allows us to extract the mutual friction between the superfluid and the normal component in the star when, in addition to the usual Magnus and drag forces acting on vortex lines, also a pinning force is at work. The effect of disorder on the depinning transition is also discussed.

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.

Curvature effects and radial homoclinic snaking

2021 ◽  
Author(s):  
Damià Gomila ◽  
Edgar Knobloch
Keyword(s):  
Scaling Law ◽  
Spatial Dynamics ◽  
Two Dimensions ◽  
Localized States ◽  
Homogeneous State ◽  
Curvature Effects ◽  
Localized Structures ◽  
Homoclinic Snaking ◽  
Curvature Driven ◽  
Axisymmetric Structures

Abstract In this work, we revisit some general results on the dynamics of circular fronts between homogeneous states and the formation of localized structures in two dimensions (2D). We show how the bifurcation diagram of axisymmetric structures localized in radius fits within the framework of collapsed homoclinic snaking. In 2D, owing to curvature effects, the collapse of the snaking structure follows a different scaling that is determined by the so-called nucleation radius. Moreover, in the case of fronts between two symmetry-related states, the precise point in parameter space to which radial snaking collapses is not a ‘Maxwell’ point but is determined by the curvature-driven dynamics only. In this case, the snaking collapses to a ‘zero surface tension’ point. Near this point, the breaking of symmetry between the homogeneous states tilts the snaking diagram. A different scaling law is found for the collapse of the snaking curve in each case. Curvature effects on axisymmetric localized states with internal structure are also discussed, as are cellular structures separated from a homogeneous state by a circular front. While some of these results are well understood in terms of curvature-driven dynamics and front interactions, a proper mathematical description in terms of homoclinic trajectories in a radial spatial dynamics description is lacking.

scholarly journals Data Processing Method of Well Testing

2021 ◽  
Vol 134 (3) ◽  
pp. 35-38
Author(s):  
A. M. Svalov ◽  
Keyword(s):  
Boundary Condition ◽  
Asymptotic Expansion ◽  
Test Data ◽  
Traditional Method ◽  
Processing Method ◽  
Well Testing ◽  
Well Test ◽  
Conductivity Equation ◽  
Complete Asymptotic Expansion ◽  
Data Processing Method

Horner’s traditional method of processing well test data can be improved by a special transformation of the pressure curves, which reduces the time the converted curves reach the asymptotic regimes necessary for processing these data. In this case, to take into account the action of the «skin factor» and the effect of the wellbore, it is necessary to use a more complete asymptotic expansion of the exact solution of the conductivity equation at large values of time. At the same time, this method does not allow to completely eliminate the influence of the wellbore, since the used asymptotic expansion of the solution for small values of time is limited by the existence of a singular point, in the vicinity of which the asymptotic expansion ceases to be valid. To solve this problem, a new method of processing well test data is proposed, which allows completely eliminating the influence of the wellbore. The method is based on the introduction of a modified inflow function to the well, which includes a component of the boundary condition corresponding to the influence of the wellbore.

Landau constants and their q-analogues

2015 ◽  
Vol 13 (02) ◽  
pp. 217-231 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
Xin Li ◽  
M. Rahman
Keyword(s):  
Asymptotic Expansion ◽  
Large Degree ◽  
Complete Asymptotic Expansion

We derive inequalities and a complete asymptotic expansion for the Landau constants Gn, as n → ∞ using the asymptotic sequence n!/(n + k)!. We also introduce a q-analogue of the Landau constants and calculate their large degree asymptotics. In the process, we also establish q-analogues of identities due to Ramanujan and Bailey.

scholarly journals A REMARK ON LONG RANGE SCATTERING FOR THE NONLINEAR KLEIN–GORDON EQUATION

2005 ◽  
Vol 02 (01) ◽  
pp. 77-89 ◽  
Author(s):  
HANS LINDBLAD ◽  
AVY SOFFER
Keyword(s):  
Asymptotic Expansion ◽  
Initial Data ◽  
Long Range ◽  
Gordon Equation ◽  
Scattering Problem ◽  
One Dimension ◽  
Complete Asymptotic Expansion ◽  
Arbitrary Size ◽  
Klein Gordon ◽  
Klein Gordon Equation

We consider the scattering problem for the nonlinear Klein–Gordon Equation with long range nonlinearity in one dimension. We prove that for all prescribed asymptotic solutions there is a solution of the equation with such behavior, for some choice of initial data. In the case the nonlinearity has the good sign (repulsive) the result hold for arbitrary size asymptotic data. The method of proof is based on reducing the long range phase effects to an ODE; this is done via an appropriate ansatz. We also find the complete asymptotic expansion of the solutions.

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