Abstract
The purpose of this study is to examine the instabilities of a two-dimensional mixed convection boundary layer flow induced by an impinging ascending flow on a heated horizontal cylinder. A significant amount of works is done in recent years on this problem because of its wide range of applications. However, they did not check the stability of the flow in the face of small disturbances that occur in reality. For this, we adopt the linear stability theory by first solving the steady basic flow and then solving the linear perturbed problem. Thus, the governing equations of the basic flow are reduced to two coupled partial differential equations and solved numerically with the Keller-Box method. The corresponding steady solution is obtained, by varying the position along the cylinder’s surface, for different values of Richardson number (λ) and Prandtl number (Pr), up to, respectively, 3000 and 20. To examine the onset of thermal instabilities, the linear stability analysis is done using the normal mode decomposition with small harmonic disturbances. The Richardson number λ is chosen as the control parameter of these instabilities. The resulting eigenvalue problem is solved numerically by the use of the pseudospectral method based on the Laguerre polynomials. The computed results for neutral and temporal growth curves are depicted and discussed in detail through graphs for various parametric conditions. The critical conditions are illustrated graphically to show from which thermodynamic state, the flow begins to become unstable. As a main result, from ξ = 0 to ξ ≈ π/3, we found that forced and mixed convection flow cases are linearly stable in this region. However, in free convection case (λ > 100), it appears that the stagnation zone is the most unstable one and then the instability decreases along the cylinder’s surface up to the limit of its first third, thus giving the most stable zone of the cylinder. Beyond ξ ≈ 1.2, strong instabilities are noted also for low values of Richardson number, and the flow tends to an unstable state even in the absence of thermal effect, i.e., hydrodynamically unstable Ri = 0, probably due to the occurring of the boundary layer separation.
