Perturbation Series Method Research Articles

Magnetohydrodynamic mixed convection in a vertical channel

The problem of combined free and forced convective magnetohydrodynamic flow in a vertical channel is analysed by taking into account the effect of viscous and ohmic dissipations. The channel walls are maintained at equal or at different constant temperatures. The velocity field and the temperature field are obtained analytically by perturbation series method and numerically by finite difference technique. The results are presented for various values of the Brinkman number and the ratio of Grashof number to the Reynolds number for both equal and different wall temperatures. Nusselt number at the walls is determined. It is found that the viscous dissipation enhances the flow reversal in the case of downward flow while it counters the flow in the case of upward flow. It is also found that the analytical and numerical solutions agree very well for small values of ε .

Evolution of Small Periodic Disturbances into Roll Waves in Channel Flow with Internal Dissipation*

This paper concerns the asymptotic behavior of small disturbances subject to periodic boundary conditions as they evolve in time to produce the quasi‐steady pattern of roll waves. The mathematical problem is rather interesting as solutions of the linearized equations are unstable. We use a combination of perturbation and Fourier series methods to derive evolution equations for asymptotic solution, which corresponds to a single mode initially. Due to the nonlinear wave and wave interactions higher modes are generated at higher order. We show that by including the interactions between the first and second modes, these evolution equations turn out to be a system of one first‐order real equation and two second‐order complex equations. We present asymptotic and numerical results to show that our theory predicts the solution accurately for both transient and quasi‐steady traveling wave phases.

Magnetic Saturation Analysis of Cylindrical Synchronous Machine by Perturbation Series Expansion and a Method for Steady State Characteristics Calculation

Power system dynamic performance is very much affected by synchronous machines model. In particular, magnetic saturation gives substantial effects and its physical interpretation is usually complicated and difficult. Therefore saturation phenomenon has been studied long in the field of electric power system and machine analysis. Recently numerical magnetic field analysis such as FEM are widely used to consider this effect, but it needs heavy computation of large dimension matrix and necessary computing time is too long for us to apply it to multi machine power system analysis. From the viewpoint of power system analysis more sophisticated and compact model is highly required. This paper proposes a new method to take into account effects of magnetic saturation in a synchronous machine with a cylindrical rotor. It gives the additional calculation which represent the effects of magnetic saturation using so called perturbation series method applied to the nonlinearity in the magnetic permiability. Using the model generator it is confirmed that the proposed method shows good accuracy even in heavily loaded condition, in which linear analysis failed.

A mathematical study of peristaltic transport of a casson fluid

In this paper, the peristaltic flow of rheologically complex physiological fluids when modelled by a non-Newtonian Casson fluid in a two-dimensional channel is considered. A perturbation series method of solution of the stream function for zeroth and first order in amplitude ratio is sought. Of interest is the difference between peristaltic transport of Newtonian and non-Newtonian fluids. It is found that Newtonian fluid is an important subclass of non-Newtonian fluids that may adequately represent some physiological phenomena. Analytical and numerical solutions are found for the zeroth and first order in stream function and compared to well-documented research in the literature. It is shown that for a Casson fluid, when certain approximations are made in the most generalized form of constitutive equation, the fluid may be adequately represented as an improvement of a Newtonian fluid.

Effect of viscous dissipation on mixed convection in a vertical channel with boundary conditions of the third kind

The effect of viscous dissipation on fully-developed mixed convection is analysed for the laminar flow in a parallel-plate vertical channel whose walls exchange heat with an external fluid. Both conditions of equal and of different reference temperatures of the external fluid are considered. First, the simpler cases of either negligible Brinkman number or negligible Grashof number are solved analytically. Then, the combined effects of buoyancy forces and of viscous dissipation are analysed by a perturbation series method. In the examined cases, the velocity field, the temperature field and the Nusselt numbers are evaluated.

Laminar mixed convection with viscous dissipation in a vertical channel

Combined free and forced convection flow in a parallel-plate vertical channel is analysed in the fully developed region by taking into account the effect of viscous dissipation. The two boundaries are considered as isothermal and kept either at equal or at different temperatures. The velocity field, the temperature field and the Nusselt numbers are obtained by a perturbation series method which employs a perturbation parameter proportional to the Brinkman number. Dimensionless coefficients which allow the evaluation of the dimensionless mean velocity, of the dimensionless bulk temperature and of the Nusselt numbers are determined.

The effect of buoyancy forces on laminar convection with viscous dissipation in a vertical channel

The interaction of the viscous dissipation effect with the presence of buoyancy forces is investigated for laminar-flow heat transfer in a parallel-plate vertical channel. One of the channel walls is considered as isothermal with a prescribed temperature, while the other wall is considered as insulated. The velocity field is assumed to be parallel. The velocity field, the temperature field and the Nusselt number are obtained by a perturbation series method which employs the ratio between the Grashof number and the Reynolds number as the perturbation parameter. The radius of convergence of the perturbation series is estimated.

A note to the solution of cheng-minkowycz equation arising in free convection flow in porous media

In this paper we reconsider the problem of boundary layer free convection flow along a vertical flat plate with a power law surface temperature and embedded in a fluid-saturated porous medium. The resulting ordinary differential equation is solved numerically using the method of perturbation series in combination with Shanks transformation. The accuracy of the present solution is discussed and is found to be in excellent agreement with theoretical predictions of Banks[7].

Analytical solutions for diffusion-controlled adsorption kinetics with non-linear adsorption isotherms

A regular perturbation series method provides a solution to the diffusion equation when the boundary condition is a non-linear adsorption isotherm. For adsorption at the interface the Freundlich and Langmuir isotherms yield power series in the square root of time. Convergence of the power series solutions is improved by applying the Shanks transformation. The solutions are compared to limiting cases and to published numerical solutions. The results are most accurate for small time where the numerical finite difference method is least reliable.

Piecewise Analytical Perturbation Series Solutions of the Radial Schrödinger Equation: One-Dimensional Case

We develop a piecewise analytical perturbation series method (PAPSM) for solving the radial Schrodinger equation. The method centers around seeking a perturbation series solution when the coefficient function of the radial Schrodinger equation is approximated by piecewise constant polynomials.We perform a number of numerical experiments designed to evaluate the accuracy and efficiency of PAPSM as compared to Gordon's piecewise analytical solution method. The calculations are performed for both scattering and bound state problems.

Some comments on quasi-bound-state calculations

Quasi-bound-state energy calculations are made for a perturbed oscillator and a perturbed hydrogen atom. The theory of the least-squares method is analysed. A simple numerical variant of the well-known stabilization method is tested, and the results are compared with those from the least-squares and perturbation-series methods.

A fast, approximation method for LEED intensity calculations

A perturbation series method for obtaining LEED intensities from the transfer matrix method is reported. This approach is based on simple physical considerations and is adaptable to a variety of surface structures. The intensities obtained from this approximation are of comparable accuracy with those obtained by the complete procedure but the computation times are considerably reduced. The method holds promise for intensity calculations on crystals with surface layers of adsorbed atoms.