Stability of the flow of a fluid through a flexible tube at intermediate Reynolds number

Abstract

The stability of the flow of a fluid in a flexible tube is analysed over a range of Reynolds numbers 1<Re<104 using a linear stability analysis. The system consists of a Hagen–Poiseuille flow of a Newtonian fluid of density ρ, viscosity η and maximum velocity V through a tube of radius R which is surrounded by an incompressible viscoelastic solid of density ρ, shear modulus G and viscosity ηs in the region R<r<HR. In the intermediate Reynolds number regime, the stability depends on the Reynolds number Re=ρVR/η, a dimensionless parameter [sum ]=ρGR2/η2, the ratio of viscosities ηr= ηs/η, the ratio of radii H and the wavenumber of the perturbations k. The neutral stability curves are obtained by numerical continuation using the analytical solutions obtained in the zero Reynolds number limit as the starting guess. For ηr=0, the flow becomes unstable when the Reynolds number exceeds a critical value Rec, and the critical Reynolds number increases with an increase in [sum ]. In the limit of high Reynolds number, it is found that Rec∝[sum ]α, where α varies between 0.7 and 0.75 for H between 1.1 and 10.0. An analysis of the flow structure indicates that the viscous stresses are confined to a boundary layer of thickness Re−1/3 for Re[Gt ]1, and the shear stress, scaled by ηV/R, increases as Re1/3. However, no simple scaling law is observed for the normal stress even at 103<Re<105, and consequently the critical Reynolds number also does not follow a simple scaling relation. The effect of variation of ηr on the stability is analysed, and it is found that a variation in ηr could qualitatively alter the stability characteristics. At relatively low values of [sum ] (about 102), the system could become unstable at all values of ηr, but at relatively high values of [sum ] (greater than about 104), an instability is observed only when the viscosity ratio is below a maximum value η*rm.

Observations of instability, hysteresis, and oscillation in low-Reynolds-number flow past polymer gels

“…By assuming that interfacial tension was absent and obtaining estimates of the gel viscosity from linear viscoelasticity data, KM found good quantitative agreement between the theoretical and experimental values of Γ for the range of gel-fluid thickness ratios (ratio 5) they investigated. Linear stability analyses have also been performed for finite Reynolds numbers and for pressure-driven flows through tubes lined with a gel layer [6][7][8][9]. While a weakly nonlinear analysis has been carried out which shows that the instability is subcritical [10], we are not aware of theoretical work exploring the strongly nonlinear regime of the instability.…”

Temporal Stability of Flow Through Viscoelastic Tubes

“…This trend is reversed for Re'50. Consequently, it is di$cult to predict from this result the shape of the neutral-stability curves for large values of n where there may exist other kinds of instabilities as was shown in the inviscid calculations of Kumaran (1996). Figure 11 shows the neutral-stability curves in the plane (k, Re) for di!erent values of .…”

“…At high Re, the rate of transport of energy due to the deformation work at the interface is the negative of the rate of transport of energy in the wall layer due to the Reynolds stress v P v V . Kumaran (1996) employed a di!erent model for the compliant wall, considered an inviscid #ow and developed a stability condition for both axisymmetric and azimuthally varying disturbances with high azimuthal wavenumbers. The application of this condition to the Hagen}Poiseuille #ow showed that the axisymmetric disturbance is stable while the azimuthally varying mode with high wavenumber may be unstable.…”

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

“…The lacunae involved in using a linearized elastic model were first pointed out by Gkanis & Kumar (2003) in the context of plane Couette flow past a deformable solid layer. Much of the earlier work in this subject were based on the linearized elastic model (Yeo & Dowling 1987;Yeo 1988;Yeo et al 1994;Lucey & Peake 2003;Kumaran 1995Kumaran , 1998aShankar & Kumaran 1999, and it is important to examine whether the predictions obtained using the linearized elastic model would be modified by using a more accurate nonlinear model. We will specifically address this issue at various points in this review.…”