Abstract
In this paper, the stability of Dean’s problem in the presence of a radial temperature gradient is studied for narrow gap case. The analytical solution of the eigen value problem is obtained by using the Galerkin’s method. The critical values of parameters and Λ are computed, where  is wave number and Λ is a parameter determining the onset of stability from the obtained analytical expressions for the first, second and third approximations. It is found that the difference between the numerical values of critical Λ corresponding to the second and third approximations is very small as compared to the difference between first and second approximations. The critical values of Λ obtained by the third approximation agree very well with the earlier results computed numerically by using the finite difference method. This clearly indicates that for the better result one should obtain the numerical values by taking more terms in approximation. Also, the amplitude of the radial velocity and the cell-patterns are shown on the graphs for different values of the parameter M, which depends on difference of temperatures of outer cylinder to the inner one i.e. on (), where is the temperature of inner cylinder and  is the temperature of outer cylinder.
Highlights
The stability of flow phenomenon of a viscous incompressible fluid between two concentric rotating cylinders with one or both cylinder rotating in the same or opposite directions, was first studied by Taylor [1]
To determine Tac, different methods were given by Taylor [1], Chandrasekhar [2], DiPrima [3], Duty and Reid [4], Harris and Reid [5].This problem was later studied by many researchers because of its practical importance in engineering applications, and is known as the Taylor stability problem
If the two concentric cylinders are assumed to be stationary, and the flow is caused by a pressure gradient acting round the curved channel, the effect of small disturbances on the stability of such a motion, was first studied by Dean [6], and is known as the Dean problem
Summary
The stability of flow phenomenon of a viscous incompressible fluid between two concentric rotating cylinders with one or both cylinder rotating in the same or opposite directions, was first studied by Taylor [1]. The effects of a radial temperature gradient on the Dean-problem between the narrow-gap annular flow under a pressure gradient acting round the cylinders was studied by Ali et al [15] using finite difference method. Our aim is to study the narrow-gap Dean-stability problem in the case when the flow is due to the pressure gradient acting round the cylinders. We have solved this problem by using the Galerkin’s method and the results are compared with those obtained by Ali et al [15]. 4asinh(a) K, Eq (12) is the result which is obtained by Chandrashekhar [20]
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