We construct the Hirota bilinear form of the nonlocal Boussinesq(nlBq) equation with four arbitrary constants for the first time. It is special because one arbitrary constant appears with a bilinear operator together in a product form. A straightforward method is presented to construct quasiperiodic wave solutions of the nl Bq equation in terms of Riemann theta functions. Due to the specific dispersion relation of the nl Bq equation, relations among the characteristic parameters are nonlinear, then the linear method does not work for them. We adopt the perturbation method to solve the nonlinear relations among parameters in the form of series. In fact, the coefficients of the governing equations are also in series form.The quasiperiodic wave solutions and soliton solutions are given. The relations between the periodic wave solutions and the soliton solutions have also been established and the asymptotic behaviors of the quasiperiodic waves are analyzed by a limiting procedure.
Interaction between the injected flow from the porous wall and the main flow can reduce drag effectively.The phenomenon is significant to the flight vehicle design.The intensive flux of injection enhances difficulty of numerical simulation and requires higher demands on the turbulence model.A turbulent boundary layer flow with mass injection through a porous wall governed by Reynolds averaged Navier-Stokers(RANS)equations is solved by using the Wilcox′s k-ωturbulence model and the obtained resistance coefficient agrees well with the experimental data.The results with and without mass injection are compared with other conditions unchanged.Velocity profile,turbulent kinetic energy and turbulent eddy viscosity are studied in these two cases.Results confirm that the boundary layer is blowing up and the turbulence is better developed with the aid of mass injection,which may explain the drag reduction theoretically.This numerical simulation may deepen our comprehension on this complex flow.
The behaviors and dynamics of bubbles,usually described by the Rayleigh-Plesset(RP)equation and its generalizations,have been an interesting area in fluid dynamics.Owing to their simplicity and sufficient accuracy,the RP equation for bubble dynamics and its enhanced models remains widely used in science and engineering,see refs.[1-3].In this paper,we consider the RP equation for N-dimensions spheri-
The investigation of the exact traveling wave solutions to the nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. To understand the mechanisms of those physical phenomena, it is necessary to explore their solutions and properties. The Wronskian technique is a powerful tool to construct multi-soliton solutions for many nonlinear evolution equations possessing Hirota bilinear forms. In the process of utilizing the Wronskian technique,the main difficulty lies in the construction of a system of linear differential conditions, which is not unique. In this paper,we give a universal method to construct a system of linear differential conditions.
