Stabilization of periodic Stokesian Hele–Shaw flows of ferrofluids

Abstract

We consider an incompressible ferrofluid in a vertical Hele–Shaw cell and develop a proper analytic framework for the free interface and the velocity potential of the fluid in a periodic geometry. The flow is assumed to obey a non-Newtonian Darcy law. The forces influencing the fluid are gravity, surface tension and the response to a magnetic field induced by a current. In addition, the flow is stabilized at the lower boundary component by an external source b. We prove a well-posedness result for the flow near flat solutions. Moreover, we find conditions on the parameters and on the slope of b for the exponential stability and instability of flat interfaces. Furthermore, we identify values for the current's intensity ι where critical bifurcation of nontrivial finger-shaped solutions from the branch of trivial (flat) solutions takes place.

Keywords:

• quasilinear elliptic equation
• parabolic evolution equation
• non-Newtonian fluid
• stability of equilibria
• local bifurcation

AMS Subject Classifications::

• 35K90
• 35Q35
• 70K50

Acknowledgements

We thank the anonymous referees for carefully reading the manuscript. The corresponding author is grateful for the support within IRTG 1627 granted by the Deutsche Forschungsgemeinschaft.