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

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

Finite amplitude axisymmetric convection between rigid rotating planes

AbstractThis paper considers the nature of stationary axisymmetric convection in a rotating layer of fluid heated from below. The nonlinear amplitude equation for the case of rigid boundaries is derived, and it is found that for certain ranges of the speed of rotation and Prandtl number subcritical instability is possible. The outer solution is matched with the inner solution (which can be expressed in terms of Bessel functions) and is similar to that described by Brown and Stewartson (1978)

NON-PARALLEL PLANE RAYLEIGH BENARD CONVECTION IN CYLINDRICAL GEOMETRY

&#x0D; &#x0D; &#x0D; This paper considers the effect of a perturbed wall in regard to the classical Benard convection problem in which the lower rigid sur­ face is of the form $z =\varepsilon^2 g (s)$, in axisymmetric cylindrical polar coordinates, $(r,\phi, z)$. The boundary conditions at $s =0$ for the linear amplitude equation is found and it is shown that these conditions are different from those which apply to the nonlinear problem investigated by Stewartson (1978) [2], representing a distribution of convection cells near the centre. &#x0D; &#x0D; &#x0D; </jats:p

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

A variational iteration method for solving parabolic partial differential equations

AbstractIn this paper, He’s variational iteration method is employed successfully for solving parabolic partial differential equations with Dirichlet boundary conditions. In this method, the solution is calculated in the form of a convergent series with an easily computable component. This approach does not need linearization, weak nonlinearity assumptions or perturbation theory. The results reveal that the method is very effective and convenient