Axisymmetric Rayleigh-Benard convection

This thesis considers axisymmetric Rayleigh-Benard convection in an infinite horizontal layer of fluid heated from below major emphasis is placed on a study of the effect of rotation of the layer, where both the stationary and overstable cases are analysed. In Chapter 2, a numerical solution of the linearised equations which govern the non-rotating fluid with rigid boundaries, is presented. In Chapter 3, the non-rotating layer is perturbed by making the elevation of the lower plane a small slowly varying function .1 of the radial coordinate. The modified amplitude equation is found and at the central axis the matching with a local solution in terms of Bessel functions is carried out. In Chapter 4, the effect of rotation is incorporated and the numerical scheme of Chapter 2, is modified to solve the appropriate linearised equations. In Chapter 5, the non-linear amplitude equation is derived for the rotating layer with rigid boundaries in the case when the system is subject to the exchange of stabilities. The matching process with a solution in terms of Bessel functions near the axis of rotation is described in Chapter 6, and is shown to lead to the possibility of'phase-winding' effects associated with variations in the wavelength of convection. 2. In Chapter 7, it is shown that when the rotating layer is subject to overstability a pair of amplitude equations governs the motion away from the axis of rotation. Again one of the main interests lies in how the solution matches with that valid in the neighbourhood of the axis

Homotopy analysis method for solving multi-term linear and nonlinear diffusion–wave equations of fractional order

AbstractIn this paper we have used the homotopy analysis method (HAM) to obtain solutions of multi-term linear and nonlinear diffusion–wave equations of fractional order. The fractional derivative is described in the Caputo sense. Some illustrative examples have been presented

Non-parallel plane Rayleigh Benard convection in cylindrical geometry

This paper considers the effect of a perturbed wall in regard to the classical Benard convection problem in which the lower rigid surface is of the form $z=ε^2 g(s)$, s=ε r, in axisymmetric cylindrical polar coordinates (r,ϕ,z). The boundary conditions at s=0 for the linear amplitude equation are found and it is shown that these conditions are different from those which apply to the nonlinear problem investigated by Brown and Stewartson [1], representing the distribution of convection cells near the center

Generalizing Homotopy Analysis Method to Solve System of Integral Equations

This paper presents the application of the Homotopy Analysis Method (HAM) and Homotopy Perturbation Method (HPM) for solving systems of integral equations. HAM and HPM are two analytical methods to solve linear and nonlinear equations which can be used to obtain the numerical solution. The HAM contains the auxiliary parameter h, provide us with a simple way to adjust and control the convergence region of solution series. The results show that HAM is a very efficient method and that HPM is a special case of HAM

Adomian Decomposition Method for Approximating the Solution of the Parabolic Equations

Abstract In this paper, the Adomian decomposition method for solving the linear and nonlinear parabolic equations is implemented with appropriate initial conditions. In comparison with existing techniques, the decomposition method is highly effective in terms of accuracy and rapid convergence. The numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method. Mathematics Subject Classification: 35K9

Solving a system of nonlinear integral equations by an RBF network

AbstractIn this paper, a novel learning strategy for radial basis function networks (RBFN) is proposed. By adjusting the parameters of the hidden layer, including the RBF centers and widths, the weights of the output layer are adapted by local optimization methods. A new local optimization algorithm based on a combination of the gradient and Newton methods is introduced. The efficiency of some local optimization methods to update the weights of RBFN is studied in solving systems of nonlinear integral equations