Effects of Radial Temperature Gradient on MHD Stability of Couette Flow between Conducting Cylinders - A Wide Gap Problem

Abstract

Stability of Couette flow under different conditions and assuming a narrow gap annulus has received great attention in the past. Notable among these are studies by Taylor (1923), Chandrasekhar (1953, 1954, 1961), Edwards (1958), Becker and Kaye (1962), Lai (1962), Kurzweg (1963), Harris and Reid (1964), Krueger, Gross and DiPrima (1966), Hassard, Cheng and Ludford (1972), Bahl (1972), Soundalgekar, Takhar, Smith (1981), Takhar, Smith and Soundalgekar (1985). Takhar, Ali and Soundalgekar (to be published) presented the study of MHD stability of Couette flow on taking into account the presence of radial temperature gradient and the axial magnetic field, in a narrow-gap annulus. The corresponding stability of Couette flow in a wide-gap annulus has been studied by very few researchers because of its complex nature. Notable among these are studies by Chandrasekhar (1958). Chandrasekhar and Elbert (1962), Walowit, Tsao and DiPrima (1964), Sparrow, Munro and Jonsson (1964), Astill and Chung (1976), Takhar, Ali and Soundalekar (1988), Chandrasehkar (1958) derived results for \(\eta = \frac{1}{2}\) = (i.e. \({R_1} = \frac{1}{2}{R_2}\)) whereas in other papers, Chandrasekhar and Elbert (1962) simplified the numerical procedure by considering the corresponding adjoint eigenvalue problem. Walowit et al. (1964) simplified the method of solution of an eigenvalue problem by giving an algebraic series solution instead of Chandrasekhar’s trigonometric series solution. Sparrow et al., Takhar et al. solved the eigenvalue problem numerically using the Runge-Kutta method, whereas Asti11 and Chung solved it by a finite-difference method. The only paper which deals with MHD stability of wide-gap problem is the one by Chang and Sartory (1967). It deals with the stability of the flow of an electrically conducting fluid in a wide-gap of permeable, perfectly conducting cylinders. In the narrow gap case, there are many papers on the MHD stability of Taylor flows with both conducting and non-conducting walls of the two concentric cylinders. So Ali, Soundalgekar and Takhar (to be published) solved this MHD stability of Taylor flow for both conducting and non-conducting impermeable cylinders separated by a wide-gap. This eigenvalue problem was solved numerically following the method of Harris and Reid (1964) and Sparrow et al. (1964). In Chang and Sartory’s (1967) paper, the basic velocity was assumed to be A/r, where A is a constant. We have assumed the basic velocity of the form Ar + B/r to solve this eigenvalue problem. Hence a comparison is not possible between our results and those of Chang and Sartory (1967).