To the best of our knowledge, the effect of a uniform vertical throughflow on the hydrodynamic stability of PCF against small disturbances has not been investigated so far, and the objective of this investigation is to furnish the missing information. It is well known that the PCF is always stable for all the values of Reynolds number so far as small disturbances are concerned with no throughflow. At first glance, it is natural to assume that the PCF in the presence of throughflow will also be stable against small disturbances for all the values of Reynolds number. However, the present investigation has revealed that this conjecture is not always true because the basic velocity profile is distorted by the effect of throughflow and permits the unstable mode of disturbance. The stability of fluid flow is analyzed by solving the stability eigenvalue problem numerically using the Chebyshev collocation method.
drazin reid hydrodynamic stability pdf 31
The change in the hydrodynamic stability of plane Couette flow (PCF) due to the presence of a uniform vertical throughflow has been investigated. A modal stability analysis of small-amplitude disturbances has been performed and the resulting stability eigenvalue problem is solved numerically using the Chebyshev collocation method. Some interesting and unexpected results are unveiled on the stability characteristics of the fluid flow. The numerical approach has shown that that three-dimensional disturbances are always more stable than two-dimensional ones. Suitable symmetries in the stability eigenvalue problem is found allowing one to gather information on the regime \(\left R_T \right 0\). It is found that PCF is always stable in the absence of throughflow and also for values of \(\left R_T \right \le 3.352\) and beyond this value it becomes unstable as the growth rate turns out to be positive from negative. In the parameter space \(3.353 \le \left R_T \right \le 4.899\), the flow gets destabilized manifesting itself as a minimum in the \((R_T ,R_c )\)-plane and stabilized reversely for values of \(\left R_T \right > 4.899\) and attains an asymptotic value 27,189.31 \(\left R_T \right\) for sufficiently large values of \(\left R_T \right\). Besides, the instability is found to set in always via travelling-wave mode and PCF has a more stabilizing effect than PPF in the presence of vertical throughflow. The size of convection cells decrease with increasing \(\left R_T \right\) and the cells may move up (\(c_c 0)\) the layer depending on the direction of throughflow. 2ff7e9595c
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