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Instability of high-frequency modes in viscoelastic plane Couette flow past a deformable wall at low and finite Reynolds number. (English) Zbl 1187.76656
Summary: The linear stability of plane Couette flow of an upper convected Maxwell (UCM) fluid of viscosity $$\eta$$, density $$\rho$$, relaxation time $$\tau _{R}$$ and thickness $$R$$ flowing past a linear viscoelastic solid of shear modulus $$G$$ and thickness $$HR$$ is analyzed using a combination of asymptotic analysis (at low Reynolds number) and numerical solution (at finite Reynolds number). The asymptotic analysis is used to analyze the effect of wall deformability on a class of high frequency modes in the UCM fluid with $$c \sim O(Re^{-1/2})$$ at $$Re \ll 1$$ first studied by Gorodtsov and Leonov [J. Appl. Math. Mech. 31 (1967) 310-319; abbreviated GL here], who showed that these modes are stable in a rigid channel. Here, $$c$$ is the wavespeed of perturbations (nondimensionalised by $$V), Re=\rho VR/\eta$$ is the Reynolds number, $$V$$ is the velocity of the moving plate. Our asymptotic results show that all the high frequency-low $$Re$$ modes of GL are rendered unstable by solid wall deformability. The variation of the growth rate with the nondimensional solid elasticity parameter $$\Gamma =V\eta /(GR)$$ shows an oscillatory behavior alternating between stable and unstable regions, and the variation for upstream traveling waves is completely antiphase with that for downstream waves. The parameter $$\Gamma$$ is shown to be proportional to $$Re^{1/4}$$ at $$Re \ll 1$$ for neutrally stable downstream waves. Numerical continuation shows that the instability at low $$Re$$ continues to finite $$Re$$, and typically the instability ceases to exist above a critical $$Re$$. This critical $$Re$$ increases with an increase in the Weissenberg number $$W=\tau _{R}V/R$$. In some cases, however, the instability continues to very high $$Re$$, and $$\Gamma \propto Re^{-1}$$ in that limit for neutrally stable modes. Neutral stability curves in the $$\Gamma -W$$ plane show that the predicted instability of the high frequency GL modes exists only at finite and large $$W$$, and is absent in the Newtonian fluid $$(W \rightarrow 0)$$ limit. The asymptotic analysis for the Oldroyd-B model shows that the ratio of solvent viscosity to total viscosity of the solution should be $$O(Re^{1/2})$$ in order for the instability to exist at low $$Re$$, meaning the instability is absent for realistic values of solvent viscosity at low $$Re$$. However, numerical results show that the instability of the high-frequency GL modes is present at finite $$Re$$ in an Oldroyd-B fluid, and increasing solvent viscosity ratio has a stabilizing effect on the instability. Similarly, the solid to fluid viscosity ratio $$\eta _{r}$$ also has a stabilizing effect at finite $$Re$$, but in the limit of low $$Re, \eta _{r} \sim O(Re^{1/2})$$ in order for the instability to exist. Our study thus demonstrates the presence of an instability in viscoelastic plane Couette flow past a deformable wall which is primarily due to the viscoelastic nature of the fluid, and is absent in Newtonian fluids.

MSC:
 76E05 Parallel shear flows in hydrodynamic stability 76E17 Interfacial stability and instability in hydrodynamic stability 76A10 Viscoelastic fluids
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