V. Shankar

Viscoelasticity of dilute solutions of semiflexible polymers

We show using Brownian dynamics simulations and theory how the shear relaxation modulus G(t) of dilute solutions of relatively stiff semiflexible polymers differs qualitatively from that of rigid rods. For chains shorter than their persistence length, G(t) exhibits three time regimes: At very early times, when the longitudinal deformation is affine, G(t) approximately t(-3/4). Over a broad intermediate regime, during which the chain length relaxes, G(t) approximately t(-5/4). At long times, G(t) mimics that of rigid rods. A model of the polymer as an effectively extensible rod with a frequency dependent elastic modulus B(omega) approximately ((i)omega)(3/4) quantitatively describes G(t) throughout the first two regimes.

Stability of wall modes in fluid flow past a flexible surface

The stability of wall modes in fluid flow past a flexible surface is analyzed using asymptotic and numerical methods. The fluid is Newtonian, while two different models are used to represent the flexible wall. In the first model, the flexible wall is modeled as a spring-backed, plate-membrane-type wall, while in the second model the flexible wall is considered to be an incompressible viscoelastic solid of finite thickness. In the limit of high Reynolds number (Re), the vorticity of the wall modes is confined to a region of thickness

Stability of fluid flow through deformable neo-Hookean tubes

O(Re ^{-1/3})

Theory of linear viscoelasticity of semiflexible rods in dilute solution

in the fluid near the wall of the channel. An asymptotic analysis is carried out in the limit of high Reynolds number for Couette flow past a flexible surface, and the results show that wall modes are always stable in this limit if the plate-membrane wall executes motion purely normal to the surface. However, the flow is shown to be unstable in the limit of high Reynolds number when the wall can deform in the tangential direction. The asymptotic results for this case are in good agreement with the numerical solution of the complete governing stability equations. It is further shown using a scaling analysis that the high Reynolds number wall mode instability is independent of the details of the base flow velocity profile within the channel, and is dependent only on the velocity gradient of the base flow at the wall. A similar asymptotic analysis for flow past a viscoelastic medium of finite thickness indicates that the wall modes are unstable in the limit of high Reynolds number, thus showing that the wall mode instability is independent of the wall model used to represent the flexible wall. The asymptotic results for this case are in excellent agreement with a previous numerical study of Srivatsan and Kumaran.

Weakly nonlinear stability of viscous flow past a flexible surface

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 high-wavenumber 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.

Stability of pressure-driven flow in a deformable neo-Hookean channel

We present a theory of the linear viscoelasticity of dilute solutions of freely draining, inextensible, semiflexible rods. The theory is developed expanding the polymer contour about a rigid rod reference state, in a manner that respects the inextensibility of the chain, and is asymptotically exact in the rodlike limit where the polymer length L is much less than its persistence length Lp. In this limit, the relaxation modulus G(t) exhibits three time regimes: At very early times, less than a time τ∥∝L8/Lp5 required for the end-to-end length of a chain to relax significantly after a deformation, the average tension induced in each chain and G(t) both decay as t−3/4. Over a broad range of intermediate times, τ∥≪t≪τ⊥, where τ⊥∝L4/Lp is the longest relaxation time for the transverse bending modes, the end-to-end length decays as t−1/4, while the residual tension required to drive this relaxation and G(t) both decay as t−5/4. As later times, the stress is dominated by an entropic orientational stress, giving ...

Electric-field– and contact-force–induced tunable patterns in slipping soft elastic films

The weakly nonlinear stability of viscous fluid flow past a flexible surface is analysed in the limit of zero Reynolds number. The system consists of a Couette flow of a Newtonian fluid past a viscoelastic medium of non-dimensional thickness H (the ratio of wall thickness to the fluid thickness), and viscosity ratio

Experimental study of the instability of laminar flow in a tube with deformable walls

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Stability of gravity-driven free-surface flow past a deformable solid at zero and finite Reynolds number

(ratio of the viscosities of wall and fluid media). The wall medium is bounded by the fluid at one surface and two different types of boundary conditions are considered at the other surface of the wall medium – for ‘grafted’ gels zero displacement conditions are applied while for ‘adsorbed’ gels the displacement normal to the surface is zero but the surface is permitted to move in the lateral direction. The linear stability analysis reveals that for grafted gels the most unstable modes have \alpha \sim O(1), while for adsorbed gels the most unstable modes have \alpha \rightarrow 0, where \alpha is the wavenumber of the perturbations. The results from the weakly nonlinear analysis indicate that the nature of the bifurcation at the linear instability is qualitatively very different for grafted and absorbed gels. The bifurcation is always subcritical for the case of flow past grafted gels. It is found, however, that relatively weak but finite-amplitude disturbances do not significantly reduce the critical velocity required to destabilize the flow from the critical velocity predicted by the linear stability theory. For the case of adsorbed gels, it is found that a supercritical equilibrium state could exist in the limit of small wavenumber for a wide range of parameters

Stability of two-layer Newtonian plane Couette flow past a deformable solid layer

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