A Theory of Non-linear Waves in Multi-dimensions: with Special Reference to Surface Water Waves

The surface water waves are "modal" waves in which the "physicalspace" (t, x, y, z) is the product of a propagation space (t,x, y) and a cross space, the z-axis in the vertical direction.We have derived a new set of equations for the long waves inshallow water in the propagation space. When the ratio of theamplitude of the disturbance to the depth of the water is small,these equations reduce to the equations derived by Whitham (1967)by the variational principle. Then we have derived a singleequation in (t, x, y)-space which is a generalization of thefourth order Boussinesq equation for one-dimensional waves.In the neighbourhood of a wave froat, this equation reducesto the multidimensional generalization of the KdV equation derivedby Shen & Keller (1973). We have also included a systematicdiscussion of the orders of the various non-dimensional parameters.This is followed by a presentation of a general theory of approximatinga system of quasi-linear equations following one of the modes.When we apply this general method to the surface water waveequations in the propagation space, we get the Shen-Keller equation.     

双金属氰化物在多组分偶联反应中的催化活性(英文)

Several dual metal cyanide catalysts were prepared from potassium ferrocyanide,metal chloride(where metal = Zn2+,Mn2+,Ni2+,Co2+ and Fe2+),t-butanol(complexing agent) and PEG-4000(co-complexing agent).The catalysts were characterized by elemental analysis(CHN and X-ray fluorescence),X-ray diffraction,N2 adsorption-desorption,scanning electron microscopy,Fourier-transform infrared spectroscopy,and UV-Visible spectroscopy.The dual metal cyanide catalysts were used in several acid catalyzed multi-component coup...     

Non-uniform slot suction/injection into mixed convection boundary layer flow over vertical cone

The aim of this work is to study the effect of non-uniform single and double slot suction/injection into a steady mixed convection boundary layer flow over a vertical cone, while the axis of the cone is inline with the flow. The governing boundary layer equations are transformed into a non-dimensional form by a group of non-similar trans- formations. The resulting coupled non-linear partial differential equations are solved nu- merically by employing the quasi-linearization technique and an implicit finite-difference scheme. Numerical computations are performed for different values of the dimensionless parameters to display the velocity and temperature profiles graphically. Also, numerical results are presented for the skin friction and heat transfer coefficients. Results indicate that the skin friction and heat transfer coefficients increase with non-uniform slot suction, but the effect of non-uniform slot injection is just opposite.     

Penalty Methods for Numerical Approximations of Optimal Boundary Flow Control Problems

Abstract

This article is concerned with penalty methods for solving optimal Dirichlet control problems governed by the steady-state and time-dependent Navier-Stokes equations. We present, in two different versions, the penalized methods for solving the steady-slate Dirichlet control problems. These approaches are implemented and compared numerically. We also generalize the penalty methods to the time-dependent case. Scmidiscrete and fully discrete approximations of time-dependent Dirichlet control problems are discussed and implemented. Numerical results for solving both the steady-state and the time dependent Dirichlet control problems are reported.     

Reduced Basis Method for Optimal Control of Unsteady Viscous Flows

In this article we discuss the reduced basis method (RBM) for optimal control of unsteady viscous flows. RBM is a reduction method in which one can achieve the versatility of the finite element method or another for that matter and gain significant reduction in the number of degrees of freedom. The essential idea in this method is to define a reduced order subspace spanned by few basis elements and then obtain the solution via a Galerkin projection. We present several ways to define this subspace. Feasibility of the approach is demonstrated on two boundary control problems in cavity and wall bounded channel flows. Control action is effected through boundary surface movement on part of the solid wall. Application of RBM to the control problems leads to finite dimensional optimal control problems which are solved using Newton's method. Through computational experiments we demonstrate the feasibility and applicability of the reduced basis method for control of unsteady viscous flows.