On the behaviour of small disturbances in plane Couette flow with a temperature gradient

Abstract

The stability of plane Couette flow with a heated lower plate is considered with respect to a two-dimensional infinitesimal disturbance. The eigenvalues are found with the aid of a digital computer as the latent roots of a matrix. Neutral stability curves for various Prandtl numbers at Reynolds numbers up to 150 are obtained by a second method. It is found that the principle of the exchange of stabilities does not hold for this problem. With the aid of Squire’s transformation the conclusion is drawn that all fluids will become unstable at the same value of the Rayleigh number irrespective of whether shear is present or not.

Bioconvection under uniform shear: linear stability analysis

“…In figure 23, we show a neutral stability curve computed by artificially suppressing the advection transport term of L C | β=0 in (5.7) at S = 8, and compare it with the original one given in figure 8 (c). The inhibition of the advection term results in a significant amount of destabilisation at α < 10, suggesting that the two-dimensional mode tends to be stabilised by the advection, similarly to RayleighBénard convection in a shear flow (Gallagher & Mercer 1965). However, we note that the Reynolds number in this case is about Re = 3.35.…”

“…We note that if G 1 = G 2 = 0, the form of the linearised equation is identical to that for Rayleigh-Bénard convection with a cross flow (e.g. Gallagher & Mercer 1965;Kelly 1992;Jerome et al 2012). Computation for the linear instability by excluding the operators for…”

“…In Rayleigh-Bénard convection, introducing shear has been shown to stabilise the gravitational overturning instability at non-zero streamwise wavenumbers (Gallagher & Mercer 1965;Kelly 1992;Jerome et al 2012). The same stabilisation mechanism is also found to act in the present study.…”

Stability of a Plane Parallel Flow with Variable Vertical Shear and Unstable Stratification

“…Since Jeffrey's (1928) suggestion, a number of studies have been made both experimentally (Terada 1928, Graham 1934, Chandra 1938 and theoretically (Kuo 1963, Asai 1964and 1970, Deardorff 1965, Gallagher and Mercer 1965, Ingersoll 1966 from the point of view of thermal convection in a shear flow. These studies concluded that a constant shear flow exerts a suppressing influence on convective motion in a vertical plane parallel to the basic flow in which the convection is imbedded and this is responsible for the formation of a longitudinal roll.…”

Three-Dimensional Features of Thermal Convection in a Plane Couette Flow

“…Recently, Deardorff (1965), Gallagher andMercer (1965), Ingersoll (1966) and Asai and Nakasuji (1968) analyzed the full sixth-order stability problem numerically instead of Kuo's incomplete treatment on diffusion terms. Then they showed neutral stability curve for various ranges of some physical parameters such as Rayleigh number, Reynolds number, Prandtl number and horizontal wavenumber of a perturbation.…”