Instability of high-frequency modes in viscoelastic plane Couette flow past a deformable wall at low and finite Reynolds number.

*(English)*Zbl 1187.76656Summary: 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.