The stability of pressure-driven flow in a rectangular channel with deformable neo-Hookean viscoelastic solid walls is analysed for a wide range of Reynolds numbers (from Re ≪ 1 to Re ≫ 1) by considering both sinuous and varicose modes for the perturbations. Pseudospectral numerical and asymptotic methods are employed to uncover the various unstable modes, and their stability boundaries are determined in terms of the solid elasticity parameter Γ = Vη/(ER) and the Reynolds number Re = RV ρ/η; here V is the maximum velocity of the laminar flow, R is the channel half-width, η and ρ are respectively the viscosity and density of the fluid and E is the shear modulus of the solid layer. We show that for small departures from a rigid solid, wall deformability could have a destabilizing or stabilizing effect on the Tollmien–Schlichting (TS) instability (a sinuous mode) depending on the solid-layer thickness. Upon further increase in solid deformability, the TS mode coalesces with another unstable mode (absent in rigid channels) giving rise to a single unstable mode which extends to very low Reynolds number (<1) for highly deformable walls. There are other types of instabilities that exist only due to wall deformability. In the absence of inertia (Re = 0), there is a short-wave instability of both sinuous and varicose modes arising due to the discontinuity of the first normal stress difference across the fluid–solid interface. For both sinuous and varicose modes, it is shown that inclusion of inertia is important even for Re ≪ 1, wherein a new class of long-wavelength unstable modes are predicted which are absent at Re = 0. These unstable modes are a type of shear waves in an elastic solid which are destabilized by the flow. These long- and short-wave instabilities are absent if a simple linear elastic model is used for the solid. At intermediate and high Re, upstream and downstream travelling waves of both sinuous and varicose modes become unstable. We show that sinuous and varicose modes become critical in different parameter regimes, thereby demonstrating the importance of capturing all the unstable modes. Inclusion of dissipative effects in the neo-Hookean model is generally shown to play a stabilizing role on the instabilities due to both sinuous and varicose modes. The predicted instabilities will be important for the flow of liquids (with viscosity ≥ 10−3 Pa s) in deformable channels of width ≤1 mm, and with shear modulus ≤ 105 Pa.

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 ∝ Re 1/2 for purely elastic (with ratio of solid to fluid viscosities η r = 0) neo-Hookean tubes. The first normal stress discontinuity in the neo-Hookean solid gives rise to a highwavenumber interfacial instability, which is stabilized by interfacial tension at the fluid-wall interface. Inclusion of dissipation (η r = 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 η r . 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.

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A general formulation is presented for the analysis of arching in soils as an extension of the conventional shear plane approach, in which the slip surfaces within the soil mass are considered as vertical planes. Such an assumption results in a low estimate of the design load on the yielding buried structure because of a considerable separation between the assumed vertical slip surfaces and the actual curved slip surfaces. The arching theory presented in this paper overcomes this limitation of the conventional theory by assuming inclined slip surfaces close to the actual slip surfaces. Based on this concept, an analytical expression for the vertical stress on the yielding structure is derived; a special case of this general expression reduces to the conventional expression. The variation of the vertical stress on the yielding structure with depth has been presented graphically along with a comparison with the conventional vertical shear plane approach. It is observed that the amount of arching increases with increase in the ratio of depth to width of the yielding structure as reported in the literature

The linear stability of Newtonian liquid flow down an inclined plane lined with a deformable elastic solid layer is analyzed at zero and finite Reynolds number. There are two qualitatively different interfacial modes in this composite system: the free-surface or gas-liquid ͑"GL"͒ mode which becomes unstable at low wave numbers and nonzero Reynolds number in flow down a rigid plane, and the liquid-solid ͑"LS"͒ mode which could become unstable even in the absence of inertia at finite wave numbers when the solid layer is deformable. The objectives of this work are to examine the effect of solid layer deformability on the GL and LS modes at zero and finite inertia, and to critically assess prior predictions concerning GL mode instability suppression at finite inertia obtained using the linear elastic model by comparison with the more rigorous neo-Hookean model for the solid. In the creeping-flow limit where the GL mode instability is absent in a rigid incline, we show that for both solid models, the GL and LS modes become unstable at finite wavelengths when the solid layer becomes sufficiently soft. At finite wavelengths, the labeling of the two interfacial modes as GL and LS becomes arbitrary because these two modes get "switched" when the solid layer becomes sufficiently deformable. The critical strain required for instability becomes independent of the solid thickness ͑at high enough values of thickness͒ for both GL and LS modes in the linear elastic solid, while it decreases with the thickness of the neo-Hookean solid. At finite Reynolds number, it is shown for both the solid models that the free-surface instability in flow down a rigid plane can be suppressed at all wavelengths by the deformability of the solid layer. The neutral curves associated with this instability suppression are identical for both linear elastic and neo-Hookean models. When the solid becomes even more deformable, both the GL and LS modes become unstable for finite wave numbers at nonzero inertia, but the corresponding neutral curves obtained from the two solid models differ significantly in detail. At finite inertia, for both the solid models, there is a significant window in the shear modulus of the solid for moderate values of solid thickness where both the GL and LS modes are stable at all wave numbers. Thus, using the neo-Hookean model, the present study reaffirms the prediction that soft elastomeric coatings offer a passive route to suppress and control interfacial instabilities.

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