Analytical study on long wave refraction over a dredge excavation pit

An analytical solution of the mild-slope wave equation is derived to describe long wave propagating over the idealized dredge excavation pit. The pit is assumed to be axisymmetrical and composed of a flat bottom and a convex slope. The convex slope is expressed by a simple power function. The problem is solved in the polar coordination system by the separation of variables. By the obtained solution, the characteristics of the wave refraction and reflection over the dredge excavation pit are discussed. The results show that wave amplitude is attenuated within and in the lee side of the pit and amplified at the rear flank of the pit due to wave refraction. The effects of the incident wave length and the shape of the pit on wave refraction are also discussed.     

Some exact expressions for the temporal evolution of long Rossby waves on a beta-plane

Some exact expressions are derived to describe the temporal evolution of forced Rossby waves in a two-dimensional beta-plane configuration where the background flow has constant zonal-mean velocity. The meridional length scale of the problem is assumed to be small relative to the zonal length scale and so the long-wave limit of zero aspect ratio is taken. In the case where the background flow velocity is zero, an exact solution is obtained in terms of generalized hypergeometric functions. A late-time asymptotic approximation is obtained and it shows that the solution oscillates with time and its amplitude goes to zero in the limit of infinite time. In the case of a non-zero background flow velocity, the solution is evaluated using two different procedures which give two equivalent expressions in terms of different generalized hypergeometric functions. The late-time asymptotic behaviour is investigated and it is found that the solution approaches a steady state in the limit of infinite time.We also derive a solution in the form of an asymptotic series expansion for the more general situation where a Rossby wave packet is generated by a zonally-localized boundary condition comprising a continuous spectrum of wavenumbers or Fourier modes. The exact solutions found here can be used as leading-order solutions in weakly-nonlinear analyses and other studies involving more realistic configurations for time-dependent Rossby waves or wave packets.     

Numerical solutions for impulsively started and decelerated viscous flow past a circular cylinder

A finite difference study of the unsteady two-dimensional flow past a circular cylinder has been conducted using vorticity and streamfunction as the dependent variables. The two cases considered were impulsively started and decelerated flows. The impulsively started problem was considered to validate the method and has yielded results which agree quite closely with existing results from both calculations and experiments. The decelerated flow analysis produced results which can be explained in terms of induced velocity effects from existing wake vortices for both suddenly stopped and uniformly decelerated flows.     

Analytical bending solutions of elastica with one end held while the other end portion slides on a friction support

Summary Treated herein is an elastica under a point load. One end of the elastica is fully restrained against translation, and elastically restrained against rotation, while the other end portion is allowed to slide over a friction support. The considered elastica problem belongs to the class of large-deflection beam problems with variable deformed arc-lengths between the supports. To solve the governing nonlinear differential equation together with the boundary conditions, the elliptic integral method has been used. The method yields closed-form solutions that are expressed in a set of transcendental equations in terms of elliptic integrals. Using an iterative scheme, pertinent bending results are computed for different values of coefficient of friction, elastic rotational spring constant and loading position, so that their effects may be examined. Also, these accurate solutions provide useful reference sources for checking the convergence, accuracy and validity of results obtained from numerical methods and software for large deflection beam analysis. It is interesting to note that this class of elastica problem may have two equilibrium states; a stable one and an unstable one. Received 5 August 1996; accepted for publication 14 February 1997     

Analytical solution of a double moving boundary problem for nonlinear flows in one-dimensional semi-infinite long porous media with low permeability简

Based on Huang＇s accurate tri-sectional nonlin- ear kinematic equation （1997）, a dimensionless simplified mathematical model for nonlinear flow in one-dimensional semi-infinite long porous media with low permeability is presented for the case of a constant flow rate on the inner boundary. This model contains double moving boundaries, including an internal moving boundary and an external mov- ing boundary, which are different from the classical Stefan problem in heat conduction： The velocity of the external moving boundary is proportional to the second derivative of the unknown pressure function with respect to the distance parameter on this boundary. Through a similarity transfor- mation, the nonlinear partial differential equation （PDE） sys- tem is transformed into a linear PDE system. Then an ana- lytical solution is obtained for the dimensionless simplified mathematical model. This solution can be used for strictly checking the validity of numerical methods in solving such nonlinear mathematical models for flows in low-permeable porous media for petroleum engineering applications. Finally, through plotted comparison curves from the exact an- alytical solution, the sensitive effects of three characteristic parameters are discussed. It is concluded that with a decrease in the dimensionless critical pressure gradient, the sensi- tive effects of the dimensionless variable on the dimension- less pressure distribution and dimensionless pressure gradi- ent distribution become more serious; with an increase in the dimensionless pseudo threshold pressure gradient, the sensi- tive effects of the dimensionless variable become more serious; the dimensionless threshold pressure gradient （TPG） has a great effect on the external moving boundary but has little effect on the internal moving boundary.     

Analytical model for thermal resistance due to multiple moving circular contacts on a coated bodyModèle analytique pour la résistance thermique due à des contacts circulaires mobiles multiples à la surface d'un solide revêtu

The heat transfer at the interface of two solids in sliding/rolling contact depends on the constriction phenomenon which occurs at the vicinity of asperities. In order to study this problem, the micro-contacts are represented by multiple moving circular heat sources on the surface of a body. The studied body is constituted of a substrate and a surface coating. The thermal constriction resistance due to those contacts is determined analytically in this paper. The solution is developed by using the integral Fourier transforms, and it is valid regardless of the velocity and the relative contact size values. To cite this article: A. Baïri, C. R. Mecanique 331 (2003).