Details
| Original language | English |
|---|---|
| Pages (from-to) | 1474-1494 |
| Number of pages | 21 |
| Journal | Applicable Analysis |
| Volume | 92 |
| Issue number | 7 |
| Publication status | Published - 23 May 2012 |
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
- local bifurcation, non-Newtonian fluid, parabolic evolution equation, quasilinear elliptic equation, stability of equilibria
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Applied Mathematics
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In: Applicable Analysis, Vol. 92, No. 7, 23.05.2012, p. 1474-1494.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Stabilization of periodic Stokesian Hele-Shaw flows of ferrofluids
AU - Escher, Joachim
AU - Wenzel, Michael
N1 - Funding information: 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.
PY - 2012/5/23
Y1 - 2012/5/23
N2 - 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.
AB - 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.
KW - local bifurcation
KW - non-Newtonian fluid
KW - parabolic evolution equation
KW - quasilinear elliptic equation
KW - stability of equilibria
UR - http://www.scopus.com/inward/record.url?scp=84880391600&partnerID=8YFLogxK
U2 - 10.1080/00036811.2012.683788
DO - 10.1080/00036811.2012.683788
M3 - Article
AN - SCOPUS:84880391600
VL - 92
SP - 1474
EP - 1494
JO - Applicable Analysis
JF - Applicable Analysis
SN - 0003-6811
IS - 7
ER -