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The perturbation method is used to solve the control equations of a three-dimensional annular flow inside a small gap. The nonlinear equations are separated into zeroth-order and first-order perturbation equations. The velocity and pressure distributions are solved successively by different numerical methods with the zeroth-order and first-order equation. Agreement in results is found with the present method and software ANSYS-CFX, which illustrates the applicability of perturbation method in solving complicated flow field inside small gaps.

The annular flow inside small gaps between rotors and stators can be found in many fluid circumstances such as sliding bearings, radial dynamic pressure seals, submersible pumps and nuclear pumps. The study of dynamical effects related to gap fluid field has been one of the research hotspots of fluid mechanics. Fritz [

In this study, the perturbation method is used to solve three-dimensional control equations of an annular fluid flow inside a small gap that separates a rotating shaft and a fixed stator. The equations are expanded into zeroth- order and first-order perturbation equations of small eccentricity. The velocity and pressure distribution for the flow domain are solved successively with the zeroth-order and first-order equations by difference methods.

Consider a small-sized control volume, CV, shown in

where

The control equations can be expressed based on overall flow theory and Moody’s wall friction coefficient equations [

For nonlinear ordinary and partial differential equations, perturbation methods can be used to quantify the change in solution with respect to unperturbed linear systems due to tiny disturbance to parameters. These methods have been used and developed in various fields with different backgrounds [

The eccentricity of the rotor shaft is defined as the small disturbance in this study. The thickness of the gap, the pressure and velocity of the annular flow are assumed to be:

where

Equations (3) through (5) are separated in terms of order of

The first-order perturbation equations are:

where

The difference method is used to discretize the zeroth-order perturbation equations. The nonlinear partial differential equations are transformed into algebraic equations. The zeroth-order solution is substituted into the first-order perturbation equations to obtain the solution of the first order. The results of Equations (3) through (5) are the addition of zeroth-order perturbation solutions to Equations (7) through (9) and the first-order perturbation solutions to Equations (10) through (12).

Numerical simulations using software ANSYS-CFX [

The results displayed in

equations also agree well with the results through ANSYS-CFX qualitatively which will not be discussed any further.

The solution to a three-dimensional annular flow inside a small gap between rotor and stator is obtained using zeroth-order and first-order perturbed control equations. The perturbed solutions are compared with the numerical results through ANSYS-CFX and they are found to agree qualitatively. Consequently, it is applicable to solve three-dimensional nonlinear control equations of the small-gap annular fluid when the eccentricity of axis is small enough compared with the average clearance between the rotor and the stator.

This paper is sponsored by the National Basic Research Program of China (Grant 2015CB057300), the Key Specific Projects of Liaoning Scientific Innovation (201303002), Liaoning Provincial Science and Technology Programs(2014010499-301), Program of Cultivated Key Project of Dalian University of Technology, and the Free Exploration Project of State Key Laboratory of Structural Analysis for Industrial Equipment (S14204).

Xiaojian Cao,Yuefang Wang, (2015) Perturbation Solutions for Annular Flow of Small Gap. Journal of Applied Mathematics and Physics,03,761-765. doi: 10.4236/jamp.2015.37092