Roll instability in shear flow

To understand these curves let us examine the curve for a given eff (fig-3.11.3). Let us suppose that the frequency of shear is kept constant at 0.8 Hz and the voltage is increased. At F = 0, the instability is of the Y type at the instability disappears at the instability reappears 

The principal mechanism for the onset of RI is shown schematically in fig. 3.11.4. A spatially periodic j fluctuation (or fluctuation) of the form cosq x results in a secondary velocity that is also periodic in x. This effect has been referred to as hydrodynamic focussing. Under appropriate conditions, can have a destabilizing effect and generate convective rolls. 

For a given 4 there is the torque S due to the primary velocity (as in HI) as well as a torque due to the secondary velocity gradient and together the total viscous torque F, brings about a distortion 0. In turn, a 0 distortion gives rise to a torque that increases f . (We have already seen in the case of HI that T = —p S0, but this value is now modified slightly by a torque due to v. ) 

The complete calculation of the threshold for RI involves a solution of the form expi( x+ z) with boundary conditions for 0, v,  

It is seen at once that (3.11.5) and (3.11.6) bear a close similarity to (3.10.14) and (3.10.18) for electrohydrodynamic instability ny, and S correspond to the charge, curvature, and electric field, and the Y and Z regimes correspond to the conduction and dielectric regimes, respectively. 

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