## Abstract

The stability of plane Couette flow of an upper-convected Maxwell (UCM) fluid of thickness R, viscosity η and relaxation time τ_{R} past a deformable wall (modeled here as a linear viscoelastic solid fixed to a rigid plate) of thickness HR, shear modulus G and viscosity η_{w} is determined using a temporal linear stability analysis in the creeping-flow regime where the inertia of the fluid and the wall is negligible. The effect of wall elasticity on the stable modes of Gorodtsov and Leonov [J. Appl. Math. Mech. 31 (1967) 310] for Couette flow of a UCM fluid past a rigid wall, and the effect of fluid elasticity on the unstable modes of Kumaran et al. [J. Phys. II (Fr.) 4 (1994) 893] for Couette flow of a Newtonian fluid past a deformable wall are analyzed. Results of our analysis show that there is only one unstable mode at finite values of the Weissenberg number, W=τ_{R}V/R (where V is the velocity of the top plate) and nondimensional wall elasticity, Γ=Vη/(GR). In the rigid wall limit, Γ≪1 and at finite W this mode becomes stable and reduces to the stable mode of Gorodtsov and Leonov. In the Newtonian fluid limit, W→0 and at finite Γ this mode reduces to the unstable mode of Kumaran et al. The variation of the critical velocity, Γ_{c}, required for this instability as a function of W̄=τ_{R}G/η (a modified Weissenberg number) shows that the instability exists in a finite region in the Γ _{c}-W̄ plane when Γ_{c}>Γ _{c,Newt} and W̄<W̄_{max}, where Γ_{c,Newt} is the value of the critical velocity for a Newtonian fluid. The variation of Γ_{c} with W̄ for various values of H are shown to collapse onto a single master curve when plotted as Γ_{c}H versus W̄/H, for H≫1. The effect of wall viscosity is analyzed and is shown to have a stabilizing effect.

Original language | English (US) |
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Pages (from-to) | 371-393 |

Number of pages | 23 |

Journal | Journal of Non-Newtonian Fluid Mechanics |

Volume | 116 |

Issue number | 2-3 |

DOIs | |

State | Published - Jan 10 2004 |

### Bibliographical note

Funding Information:SK thanks the Shell Oil Company Foundation for support through its Faculty Career Initiation Funds program, and 3M for a Nontenured Faculty Award. Also, acknowledgment is made to the Donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research.

## Keywords

- Creeping flow
- Deformable solids
- Interfacial instability
- Linear stability analysis
- Viscoelastic fluids