The linear stability of fully developed Poiseuille flow of a Newtonian fluid in a deformable neo-Hookean tube is analysed to illustrate the shortcomings of extrapolating the linear elastic model for the tube wall outside its domain of validity of small strains in the solid. We show using asymptotic analyses and numerical solutions that a neo-Hookean description of the solid dramatically alters the stability behaviour of flow in a deformable tube. The flow-induced instability predicted to exist in the creeping-flow limit based on the linear elastic approximation is absent in the neo-Hookean model. In contrast, a new low-wavenumber (denoted by k) instability is predicted in the limit of very low Reynolds number (Re ≪ 1) with k ∝ Re1/2 for purely elastic (with ratio of solid to fluid viscosities ηr = 0) neo-Hookean tubes. The first normal stress discontinuity in the neo-Hookean solid gives rise to a high-wavenumber interfacial instability, which is stabilized by interfacial tension at the fluid–wall interface. Inclusion of dissipation (ηr ≠ 0) in the solid has a stabilizing effect on the low-k instability at low Re, and the critical Re for instability is a sensitive function of ηr. For Re ≫ 1, both the linear elastic extrapolation and the neo-Hookean model agree qualitatively for the most unstable mode, but show disagreement for other unstable modes in the system. Interestingly, for plane-Couette flow past a deformable solid, the results from the extrapolated linear elastic model and the neo-Hookean model agree very well at any Reynolds number for the most unstable mode when the wall thickness is not small. The results of this study have important implications for experimental investigations aimed at probing instabilities in flow through deformable tubes.
Non-linear elastic model incorporating temperature effects
