Abstract

The present study is aimed at assessing the existing results concerning the stability of canonical shear flows in channels and tubes with deformable walls, in light of consistent formulations of the nonlinear solid constitutive model and linearised interface conditions at the fluid–solid interface. We show that a class of unstable shear-wave modes at low Reynolds number, predicted by previous studies for pressure-driven flows through neo-Hookean tubes and channels, is absent upon use of consistent interfacial conditions. Furthermore, we analyse the consequences of the change in solid model on the stability of the canonical shear flows by using both neo-Hookean and Mooney–Rivlin models. We show that the salient features of the stability of the system are adequately captured by a consistent formulation of the neo-Hookean solid model, thus precluding the need to employ more detailed solid models. The stability analysis of planar flows past a neo-Hookean solid subjected to three-dimensional disturbances showed that two-dimensional disturbances are more unstable than the corresponding three-dimensional disturbances within the consistent formulations. We show that prior inconsistent formulations of the solid constitutive equation predict a physically spurious spanwise instability in disagreement with experiments thereby demonstrating their inapplicability to predict instabilities in flow past deformable solid surfaces. Using the consistent formulation, the present work provides an accurate picture, over a range of Reynolds numbers, of the stability of canonical shear flows through deformable channels and tubes. Importantly, it is shown how inconsistencies in either the bulk constitutive relation or in the linearisation of the interface conditions can separately lead to physically spurious instabilities. The predictions of this work are relevant to experimental studies in flow through deformable tubes and channels in the low and moderate Reynolds number regime.

The present study considers a shear-thinning viscoelastic liquid layer sheared by the air and flowing past a deformable-solid layer in the presence of a surfactant at the air–liquid interface to model the airflow in the oral area and airways. The stability analysis reveals the existence of purely elastic and unconditionally unstable ‘liquid elastic’ and ‘solid elastic’ modes. The mechanism responsible for the destabilisation of the solid elastic mode is the shear stresses exerted by the air on the liquid and by the liquid on the deformable solid while for the liquid elastic mode, the mechanism is the first normal stress difference across the air–liquid interface. The liquid and solid elastic modes undergo resonance, resulting in the ‘resonance mode’ of instability. The resonance mode exhibits a much higher growth rate than the liquid and solid elastic modes. The shear-thinning characteristic of the liquid and presence of the surfactant leads to enhancement in the growth rate of the resonance mode. An estimate shows a good correlation between the exhaled fluid particle (i.e. droplets and aerosols) diameters and the wavelength of the perturbations with maximum growth rate. In essence, the present analysis predicts that the airflow in the airways and oral area could lead to an elastic instability arising due to the elastic nature of the saliva, mucus and underlying muscle layers.

We investigate the temporal and spatio-temporal buoyancy instabilities in a horizontal liquid layer supported by a poorly conducting substrate and subjected to an oblique temperature gradient (OTG) with horizontal and vertical components, denoted as HTG and VTG, respectively. General linear stability analysis (GLSA) reveals a strong stabilizing effect of the HTG on the instabilities introduced by the VTG for Prandtl numbers $Pr>1$ via inducing an extra vertical temperature gradient opposing the VTG through energy convection. For $Pr<1$ , a new mode of instability arises as a result of a velocity jump in the liquid layer caused by cellular circulation. A long-wave weakly nonlinear evolution equation governing the spatio-temporal dynamics of the temperature perturbations is derived. Spatio-temporal stability analysis reveals the existence of a convectively unstable long-wave regime due to the HTG. Weakly nonlinear stability analysis reveals the supercritical type of bifurcation changing from pitchfork in the presence of a pure VTG to Hopf in the presence of the OTG. Numerical investigation of the spatio-temporal dynamics of the temperature disturbances in the layer in the weakly nonlinear regime reveals the emergence of travelling wave regimes propagating against the direction of the HTG and whose phase speed depends on $Pr$ . In the case of a small but non-zero Biot number, the wavelength of these travelling waves is larger than that of the fastest-growing mode obtained from GLSA.