Does shear flow stabilize an immersed thread?

The stability of one liquid thread immersed in a fluid in a shear field is considered by linear stability analysis. A constant shear stress is imposed far away from the thread. The shear flow tends to deform and elongate the thread. The stability of the thread is characterized by the growth rate of a random perturbation. The equation for the growth rate leads to an eigenvalue problem with the wave number, the ratio of viscosities and the capillary number as parameters. Using Hurwitz's criterion, we determine the range of the ratio of viscosities for which the shear stabilizes the thread. A critical capillary number above which the thread is always stable is found. Special attention is paid to the special case of thread and fluid having equal viscosity. Then, the critical capillary number can be calculated analytically.