Diffraction by an elastic wedge with stress-free boundary: existence and uniqueness

2005 ◽  
Vol 462 (2065) ◽  
pp. 289-317 ◽  
Author(s):  
Vladimir V Kamotski ◽  
Gilles Lebeau
Keyword(s):  
Free Boundary ◽  
Absolute Continuity ◽  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Elastic Wedge ◽  
Free Boundary Conditions ◽  
Cylindrical Waves ◽  
Radiation Conditions ◽  
Continuity Of The Spectrum ◽  
Stress Free Boundary

Applying the spectral function techniques, developed by Croisille and Lebeau, we prove the existence of solutions to problems of plane and cylindrical waves diffraction by an elastic wedge with stress-free boundary conditions. We also formulate radiation conditions, under which the uniqueness holds. The latter implies absolute continuity of the spectrum of the Lamé operator in a wedge domain with stress-free boundary.

STRESS BEHAVIOR IN THE NEIGHBORHOOD OF SHARP CORNERS

Geophysics ◽  
1964 ◽  
Vol 29 (3) ◽  
pp. 360-369 ◽  
Author(s):  
Frank C. Karal ◽  
Samuel N. Karp
Keyword(s):  
Stress Field ◽  
Boundary Conditions ◽  
Practical Reasons ◽  
Elastic Media ◽  
Elastic Wedge ◽  
Arbitrary Angle ◽  
Free Boundary Conditions ◽  
Stress Behavior ◽  
Sharp Corners ◽  
Stress Free Boundary

Knowledge of the behavior of the stress field in the neighborhood of cracks and sharp corners is important for many theoretical and practical reasons. In an earlier paper the authors examined the behavior of the stress field near the edge of an elastic wedge of arbitrary angle with stress‐free boundary conditions prescribed on both surfaces (Karp and Karal, 1962). Other types of boundary conditions, such as those representing rigid and lubricated boundaries, were not considered. In the present paper these boundary conditions are now studied, and the behavior of the stress field in the neighborhood of the tip is determined. In addition, the stress behavior is examined when two different elastic media meet at a sharp edge.

On Wave Reflection in an Elastic Plate

1974 ◽  
Vol 41 (4) ◽  
pp. 1031-1035
Author(s):  
R. Doby
Keyword(s):  
Free Boundary ◽  
Theoretical Analysis ◽  
Wave Reflection ◽  
Transverse Waves ◽  
Reflected Waves ◽  
Longitudinal Waves ◽  
Free Boundary Conditions ◽  
Transverse Modes ◽  
Parallel Surfaces ◽  
Stress Free Boundary

A precursor boundary concept is introduced which decouples the stress-free boundary conditions at the parallel surfaces of the plate. The resulting orthogonal eigenfunctions are longitudinal and transverse waves. This property is exploited so as to satisfy stress-free conditions at the edge. The theoretical analysis reveals that the reflected waves are reordered so that the longitudinal and transverse modes represent the gradients of the incident transverse and longitudinal waves.

scholarly journals Convectons in a rotating fluid layer

2013 ◽  
Vol 717 ◽  
pp. 417-448 ◽  
Author(s):  
Cédric Beaume ◽  
Alain Bergeon ◽  
Hsien-Ching Kao ◽  
Edgar Knobloch
Keyword(s):  
Free Boundary ◽  
Shear Flow ◽  
Fluid Layer ◽  
Plane Layer ◽  
Two Dimensional ◽  
Rotating Fluid ◽  
Bifurcation Structure ◽  
Free Boundary Conditions ◽  
Localized Structures ◽  
Stress Free Boundary

AbstractTwo-dimensional convection in a plane layer bounded by stress-free perfectly conducting horizontal boundaries and rotating uniformly about the vertical is considered. Time-independent spatially localized structures, called convectons, of even and odd parity are computed. The convectons are embedded within a self-generated shear layer with a compensating shear flow outside the structure. These states are organized within a bifurcation structure called slanted snaking and may be present even when periodic convection sets in supercritically. These interesting properties are traced to the presence of a conserved quantity and hence to the use of stress-free boundary conditions.

scholarly journals Transition to geostrophic convection: the role of the boundary conditions

2016 ◽  
Vol 799 ◽  
pp. 413-432 ◽  
Author(s):  
Rudie P. J. Kunnen ◽  
Rodolfo Ostilla-Mónico ◽  
Erwin P. van der Poel ◽  
Roberto Verzicco ◽  
Detlef Lohse
Keyword(s):  
Boundary Layer ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Large Scale ◽  
Scaling Laws ◽  
Stokes Equations ◽  
Fluid Layer ◽  
Navier Stokes ◽  
Free Boundary Conditions ◽  
Stress Free Boundary

Rotating Rayleigh–Bénard convection, the flow in a rotating fluid layer heated from below and cooled from above, is used to analyse the transition to the geostrophic regime of thermal convection. In the geostrophic regime, which is of direct relevance to most geo- and astrophysical flows, the system is strongly rotating while maintaining a sufficiently large thermal driving to generate turbulence. We directly simulate the Navier–Stokes equations for two values of the thermal forcing, i.e. $Ra=10^{10}$ and $Ra=5\times 10^{10}$, at constant Prandtl number $Pr=1$, and vary the Ekman number in the range $Ek=1.3\times 10^{-7}$ to $Ek=2\times 10^{-6}$, which satisfies both requirements of supercriticality and strong rotation. We focus on the differences between the application of no-slip versus stress-free boundary conditions on the horizontal plates. The transition is found at roughly the same parameter values for both boundary conditions, i.e. at $Ek\approx 9\times 10^{-7}$ for $Ra=1\times 10^{10}$ and at $Ek\approx 3\times 10^{-7}$ for $Ra=5\times 10^{10}$. However, the transition is gradual and it does not exactly coincide in $Ek$ for different flow indicators. In particular, we report the characteristics of the transitions in the heat-transfer scaling laws, the boundary-layer thicknesses, the bulk/boundary-layer distribution of dissipations and the mean temperature gradient in the bulk. The flow phenomenology in the geostrophic regime evolves differently for no-slip and stress-free plates. For stress-free conditions, the formation of a large-scale barotropic vortex with associated inverse energy cascade is apparent. For no-slip plates, a turbulent state without large-scale coherent structures is found; the absence of large-scale structure formation is reflected in the energy transfer in the sense that the inverse cascade, present for stress-free boundary conditions, vanishes.

Non-axisymmetric magnetohydrodynamic shear layers in a rotating spherical shell

2000 ◽  
Vol 408 ◽  
pp. 239-274 ◽  
Author(s):  
ANDREW M. SOWARD ◽  
RAINER HOLLERBACH
Keyword(s):  
Magnetic Field ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Spherical Shell ◽  
Shear Layers ◽  
Numerical Results ◽  
Free Boundary Conditions ◽  
Viscous Forces ◽  
Inner Sphere ◽  
Stress Free Boundary

Constant-density electrically conducting fluid is confined to a rapidly rotating spherical shell and is permeated by an axisymmetric magnetic field. Slow steady non-axisymmetric motion is driven by a prescribed non-axisymmetric body force; both rigid and stress-free boundary conditions are considered. Linear solutions of the governing magnetohydrodynamic equations are derived in the small Ekman number E limit analytically for values of the Elsasser number Λ less than order unity and they are compared with new numerical results. The analytic study focuses on the nature of the various shear layers on the equatorial tangent cylinder attached to the inner sphere. Though the ageostrophic layers correspond to those previously isolated by Kleeorin et al. (1997) for axisymmetric flows, the quasi-geostrophic layers have a new structure resulting from the asymmetry of the motion.In the absence of magnetic field, the inviscid limit exhibits a strong shear singularity on the tangent cylinder only removeable by the addition of viscous forces. With the inclusion of magnetic field, large viscous forces remain whose strength [Zscr ] was measured indirectly by Hollerbach (1994b). For magnetic fields with dipole parity, cf. Kleeorin et al. (1997), [Zscr ] increases throughout the range Λ [Lt ] 1; whereas, for quadrupole parity, cf. Hollerbach (1994b), [Zscr ] only increases for Λ [Lt ] E1/5.The essential difference between the dipole and quadrupole fields is the magnitude of their radial components in the neighbourhood of the equator of the inner sphere. Its finite value for the quadrupole parity causes the internal shear layer – the Hartmann–Stewartson layer stump – to collapse and merge with the equatorial Ekman layer when Λ = O(E1/5). Subsequently the layer becomes an equatorial Hartmann layer, which thins and spreads polewards about the inner sphere surface as Λ increases over the range E1/5 [Lt ] Λ [Lt ] 1. Its structure for the stress-free boundary conditions employed in Hollerbach's (1994b) model is determined through matching with a new magnetogeostrophic solution and the results show that the viscous shear measured by [Zscr ] decreases with increasing Λ. Since [Zscr ] depends sensitively on the detailed boundary layer structure, it provides a sharp diagnostic of new numerical results for Hollerbach's model; the realized [Zscr ]-values compare favourably with the asymptotic theory presented.

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