TY - JOUR
T1 - Hopf-Hopf and steady-Hopf mode interactions in Taylor-Couette flow of an upper convected Maxwell liquid
AU - Renardy, M.
AU - Renardy, Y.
AU - Sureshkumar, R.
AU - Beris, A. N.
N1 - Funding Information:
The work of Y. Renardy is supported by the National Science Foundation under Grant No. CTS-9307238 and the Office of Naval Research under Grant No. N00014-92-J-1664. The work of M. Renardy is supported by the National Science Foundation under Grant No. DMS-9306635 and by the Office of Naval Research under Grant No. N00014-92-J-1664. Antony N. Beris and R. Sureshkumar acknowledge support by the NSF Division of Fluid Mechanics and Particulate Flows under Grant No. CTS-9114508.
PY - 1996/3
Y1 - 1996/3
N2 - In the viscoelastic Taylor-Couette flow, it is known that at certain parameter values, the onset condition can involve more than one mode becoming unstable. We focus on the upper convected Maxwell liquid, and on the situation where two Hopf modes, or a Hopf mode and a steady mode, are simultaneously at criticality. The interaction of such modes is investigated at critical situations. Weakly non-linear amplitude equations are derived for the interaction of these modes, based on a center manifold reduction scheme. The two modes generate a two-parameter bifurcation. The coefficients involved in the equations are determined numerically, based on the physical parameters of the system at criticality. For the cases investigated, it is found that all the bifurcated branches of solutions are unstable. Moreover, direct numerical simulation of the amplitude equations does not lead to bounded solutions.
AB - In the viscoelastic Taylor-Couette flow, it is known that at certain parameter values, the onset condition can involve more than one mode becoming unstable. We focus on the upper convected Maxwell liquid, and on the situation where two Hopf modes, or a Hopf mode and a steady mode, are simultaneously at criticality. The interaction of such modes is investigated at critical situations. Weakly non-linear amplitude equations are derived for the interaction of these modes, based on a center manifold reduction scheme. The two modes generate a two-parameter bifurcation. The coefficients involved in the equations are determined numerically, based on the physical parameters of the system at criticality. For the cases investigated, it is found that all the bifurcated branches of solutions are unstable. Moreover, direct numerical simulation of the amplitude equations does not lead to bounded solutions.
KW - Hopf-Hopf mode
KW - Steady-Hopf mode
KW - Taylor-Couette flow
KW - Upper convected Maxwell liquid
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U2 - 10.1016/0377-0257(95)01415-2
DO - 10.1016/0377-0257(95)01415-2
M3 - Article
AN - SCOPUS:0030088274
SN - 0377-0257
VL - 63
SP - 1
EP - 31
JO - Journal of Non-Newtonian Fluid Mechanics
JF - Journal of Non-Newtonian Fluid Mechanics
IS - 1
ER -