In the formulation of stability of fluid flow through channels and tubes with deformable walls, while the fluid is naturally treated in an Eulerian framework, the solid can be treated either in a Lagrangian or Eulerian framework. A consistent formulation, then, should yield results that are independent of the chosen framework. Previous studies have demonstrated this consistency for the stability of plane Couette flow past a deformable solid layer modelled as a neo-Hookean solid, in the creeping-flow limit. However, a similar exercise carried out in the creeping-flow limit for the stability of pressure-driven flow in a neo-Hookean tube shows that while the flow is stable in the Lagrangian formulation, it is unstable in the existing Eulerian formulation. The present work resolves this discrepancy by presenting consistent Lagrangian and Eulerian frameworks for performing stability analyses in flow through deformable tubes and channels. The resolution is achieved by making important modifications to the Lagrangian formulation to make it fundamentally consistent, as well as by proposing a proper formulation for the neo-Hookean constitutive relation in the Eulerian framework. In the neo-Hookean model, the Cauchy stress tensor in the solid is proportional to the Finger tensor. We demonstrate that the neo-Hookean constitutive model within the Eulerian formulation used in the previous studies is a special case of the Mooney–Rivlin solid, with the Cauchy stress tensor being proportional to the inverse of the Finger tensor unlike in a true neo-Hookean solid. Remarkably, for plane Couette flow subjected to two-dimensional perturbations, there is perfect agreement between the results obtained using earlier Eulerian and Lagrangian formulations despite the crucial difference in the constitutive relation owing to the rather simple kinematics of the base state. However, the consequences are drastic for pressure-driven flow in a tube even for axisymmetric disturbances. We propose a consistent neo-Hookean constitutive relation in the Eulerian framework, which yields results that are in perfect agreement with the results from the Lagrangian formulation for both plane Couette and tube flows at arbitrary Reynolds number. The present study thus provides an unambiguous formulation for carrying out stability analyses in flow through deformable channels and tubes. We further show that unlike plane Couette flow and Hagen–Poiseuille flow in rigid-walled conduits where there is a remarkable similarity in the linear stability characteristics between these two flows, the stability behaviour for these two flows is very different when the walls are deformable. The instability of plane Couette flow past a deformable wall is very robust and is not sensitive to the constitutive nature of the solid, but the stability of pressure-driven flow in a deformable tube is rather sensitive to the constitutive nature of the deformable solid, especially at low Reynolds number.

The instability of the flow of a viscous fluid past a soft, two-layered gel is probed using experiments, and the observations are compared with results from a linear stability analysis. The experimental system consists of the rotating top plate of a rheometer and its stationary bottom plate on which the two-layer gel is placed. When the flow between the top plate and the two-layer gel is viscometric (i.e., laminar), the viscosity obtained from the rheometer is a measure of the material property of the fluid. However, after a critical shear stress, there is a sudden increase in apparent viscosity, indicating that the flow has undergone an instability due to the deformable nature of the two-layer gel. Experiments are carried out to quantify how the critical value of fluid shear stress required to destabilize the flow varies as a function of ratio of solid to fluid layer thickness, and the ratio of the shear moduli of the two gels. A linear stability analysis is carried out for plane Couette flow of a Newtonian fluid past the two-layered gel, by assuming the two solid layers to be elastic neo-Hookean materials. In order to compare the experimental and theoretical results, the effective shear modulus (Geff, defined by H/Geff=H1/G1+H2/G2) of the two-layer gel is found to be useful, where H=H1+H2. Here, Hi and Gi (i=1,2), respectively, denote the thickness and shear modulus of each layer. Results for the nondimensional parameter Γeff=ηV/(dGeff) (V is the velocity of the top plate; η is fluid viscosity, d is the fluid thickness) as a function of solid to fluid thickness H/d obtained from the stability analysis agree well with experimental observations, without any fitting parameters. In general, we find that the flow is more unstable if the softer gel is adjacent to the fluid flow compared to the case when it is not. This suggests that the instability is more interfacial in nature and is crucially dependent on the relative placement of the two layers, and not just on the effective modulus of the two-layer gel. We further show that the theoretical and experimental data for two-layer gels can be suitably collapsed onto the results obtained for a single-gel layer.

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The linear stability of plane Couette flow of a power-law fluid past a deformable solid is analyzed at arbitrary Reynolds number (Re). For flow of a Newtonian fluid past a deformable solid, at high Re, there are two different modes of instability: (i) “wall modes” (Γ∝Re−1∕3) and (ii) “inviscid modes” (Γ∝Re−1) where Γ=VμfGR is the non-dimensional shear-rate in the fluid (V, μf, G, and R denote the top-plate velocity, fluid viscosity, shear modulus of the solid, and fluid thickness, respectively). In this work, we consider the power-law model for the fluid to elucidate the effect of shear-thickening/shear-thinning behaviour on the modes of instability present in the flow, especially at moderate and high Re. At high Re, our numerical results show that wall modes exhibit different scalings in Γ (VηfGR, where ηf is Newtonian-like constant viscosity) vs Re for different values of the power-law index (n), and the scaling exponents are different from that for a Newtonian fluid. This drastic modification in the scaling of wall modes is not observed in viscoelastic (modelled as upper-convected Maxwell or Oldroyd-B fluids) plane Couette flow past a deformable solid. We show that the difference in scaling exponents can be explained by postulating that the wall modes in a power-law fluid are determined by the actual viscosity corresponding to the shear rate of the laminar flow denoted by ηapp. A non-dimensional shear rate based on this viscosity Γapp=VηappGR can be defined, and we show that the postulate Γapp∼Re−1∕3 (motivated by the wall-mode scaling in a Newtonian fluid) captures all the numerically observed scalings for Γ vs Re for different values of n>0.3, which is found to be Γ∼Re−12n+1. Further, we numerically evaluated the wall layer thickness and this agreed with the theoretical scaling of δ∼Re−n2n+1. Interestingly, the theoretical and numerical prediction of wall modes is found to be valid for power-law index, n≥ 0.3. For n≤ 0.3, there is a marked departure from the wall-mode scalings, and our results show a scaling of Γ ∼ Re−1 corresponding to inviscid modes. The variation of the power-law index (n) can stabilise/destabilise the inviscid mode when compared with Newtonian fluid, and this result is observed only in the power-law model and is not seen in the flow of viscoelastic fluid past deformable surfaces. Thus, the present study shows that the shear-rate dependence of viscosity has a significant impact on both the qualitative and quantitative aspects of stability of non-Newtonian fluid flow past deformable surfaces.

Edible films can be used for food packaging to extend the shelf life of foods and protect them from exposure to moisture, microorganisms, pathogens, and bacteria. In the present study, the effect of plasticization and cross‐linking, which are modification approaches for upgrading the properties of starch, are investigated. Glycerol and sorbitol are used as plasticizing agents, and succinic acid is used as a cross‐linking agent. The results demonstrate that the addition of glycerol and sorbitol to the film‐forming solutions reduces brittleness and increases peeling ability and flexibility upon drying. Furthermore, the film matrix supports improvements in water vapor permeability (WVP), swelling index (SI), and mechanical properties. The glycerol‐based finger millet starch (FMS) films exhibits better thermal stability than the sorbitol‐based FMS film. The microscopic images of the films show that the use of glycerol in the film improves compactness and homogeneity compared to sorbitol‐based FMS films.