Stability of the flow of a viscoelastic fluid past a deformable surface in the low Reynolds number limit

Chokshi, Paresh; Kumaran, V.

doi:10.1063/1.2798069

Cited by 19 publications

(22 citation statements)

References 23 publications

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“…In most studies, the viscoelastic fluid is described either by the upper convected Maxwell (UCM) model or the Oldroyd-B model. For the creeping flow over a flexible surface, the fluid elasticity tends to either stabilize (suppression of instability) or delay the flow instability depending upon the values of viscoelasticity parameters [20][21][22][23]. Thus, introducing fluid elasticity is found to have a stabilizing influence on the Newtonian instability mode for Re -> 0.…”

Section: Introductionmentioning

confidence: 80%

Wall-mode instability in plane shear flow of viscoelastic fluid over a deformable solid

2015

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The linear stability analysis of a plane Couette flow of an Oldroyd-B viscoelastic fluid past a flexible solid medium is carried out to investigate the role of polymer addition in the stability behavior. The system consists of a viscoelastic fluid layer of thickness R, density ρ, viscosity η, relaxation time λ, and retardation time βλ flowing past a linear elastic solid medium of thickness HR, density ρ, and shear modulus G. The emphasis is on the high-Reynolds-number wall-mode instability, which has recently been shown in experiments to destabilize the laminar flow of Newtonian fluids in soft-walled tubes and channels at a significantly lower Reynolds number than that for flows in rigid conduits. For Newtonian fluids, the linear stability studies have shown that the wall modes become unstable when flow Reynolds number exceeds a certain critical value Re(c) which scales as Σ(3/4), where Reynolds number Re=ρVR/η,V is the top-plate velocity, and dimensionless parameter Σ=ρGR(2)/η(2) characterizes the fluid-solid system. For high-Reynolds-number flow, the addition of polymer tends to decrease the critical Reynolds number in comparison to that for the Newtonian fluid, indicating a destabilizing role for fluid viscoelasticity. Numerical calculations show that the critical Reynolds number could be decreased by up to a factor of 10 by the addition of small amount of polymer. The critical Reynolds number follows the same scaling Re(c)∼Σ(3/4) as the wall modes for a Newtonian fluid for very high Reynolds number. However, for moderate Reynolds number, there exists a narrow region in β-H parametric space, corresponding to very dilute polymer solution (0.9≲β<1) and thin solids (H≲1.1), in which the addition of polymer tends to increase the critical Reynolds number in comparison to the Newtonian fluid. Thus, Reynolds number and polymer properties can be tailored to either increase or decrease the critical Reynolds number for unstable modes, thus providing an additional degree of control over the laminar-turbulent transition.

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Section: Introductionmentioning

confidence: 80%

Wall-mode instability in plane shear flow of viscoelastic fluid over a deformable solid

2015

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“…To simplify our calculations, we assume a frequency-independent viscosity to describe the dissipative effects in solid medium. Chokshi & Kumaran (2007) have argued that since the velocity of the gel in the base state is zero, it is possible to replace (within a linear stability analysis) the constant viscosity by a frequency-dependent viscosity. Real soft solid materials often exhibit frequency-dependent viscosity (Muralikrishnan & Kumaran 2002;Eggert & Kumar 2004), and the neutral stability curves for such cases can be obtained from a calculation that assumes frequency-independent viscosity by following an iterative procedure described in Muralikrishnan & Kumaran (2002).…”

Section: Governing Equationsmentioning

confidence: 99%

“…We also discuss the stability characteristics of plane-Couette flow past a neo-Hookean elastic solid at arbitrary Reynolds number in order to compare and contrast the differences in the two geometries. The governing equations for plane-Couette flow past a neo-Hookean solid in the creeping-flow limit are given in Gkanis & Kumar (2003) and Chokshi & Kumaran (2007). These equations are generalized to include fluid and solid inertia in this work, and are not displayed here in the interests of brevity.…”

Section: Governing Equationsmentioning

confidence: 99%

Stability of fluid flow through deformable neo-Hookean tubes

Gaurav

Shankar

2009

J. Fluid Mech.

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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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“…In most of the earlier studies, the fluid was assumed to be Newtonian, and the same assumption will be made here as well. The governing equations for an incompressible Newtonian fluid are the Navier-Stokes mass and momentum conservation equations given by However, it must be mentioned here that there have been some studies (Shankar & Kumar 2004;Kumar & Shankar 2005;Choskshi & Kumaran 2007;Neelamegam et al 2013), wherein the role of viscoleastic effects in the fluid on the stability has been analyzed.…”

Section: Systems and Modelsmentioning

confidence: 99%

Stability of fluid flow through deformable tubes and channels: An overview

Shankar

2015

Sadhana

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The aim of this paper is to provide a systematic overview of the study of instabilities in flow past deformable solid surfaces, with particular emphasis on internal flows through tubes and channels. The subject is certainly more than five decades old, and arguably began with Kramer's pioneering experiments on drag reduction by compliant surfaces. This was immediately followed by the theoretical studies of Benjamin and Landhal in the early 1960s. Most earlier theoretical studies were focused on stability of external flows such as boundary layers, and used relatively simple wall models composed of spring-backed plates. There has been a resurgence in the field since the mid-1980s, and more attention was focused on internal flows through deformable tubes and channels. The wall deformation was described by both phenomenological spring-backed plate models and continuum linear viscoelastic solid models. All these studies predict several types of instabilities in flow past deformable surfaces. This paper will attempt to place the various theoretical results in perspective, and to classify the instabilities predicted by various studies. Recent studies have also emphasized the importance of using a frame-invariant constitutive model, such as the neo-Hookean model, for the solid deformation. Until recently, however, the field has been dominated by theoretical and numerical studies, with very little experimental observations to corroborate the theoretical predictions. Recent experiments in flow through deformable tubes and channels indeed show instability at Reynolds number much lower than their rigid counterparts, and the experimental observations are in qualitative agreement with some of the theoretical predictions. There have also been a few studies on the non-linear aspects of the instability using the weakly non-linear formulation to determine the nature of the bifurcation at the linear instability. A brief discussion on weakly nonlinear analyses is also provided in this paper.

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Stability of the flow of a viscoelastic fluid past a deformable surface in the low Reynolds number limit

Cited by 19 publications

References 23 publications

Wall-mode instability in plane shear flow of viscoelastic fluid over a deformable solid

Wall-mode instability in plane shear flow of viscoelastic fluid over a deformable solid

Stability of fluid flow through deformable neo-Hookean tubes

Stability of fluid flow through deformable tubes and channels: An overview

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