Stability of the viscous flow of a fluid through a flexible tube

Abstract

The stability of Hagen-Poiseuille flow of a Newtonian fluid of viscosity η in a tube of radius R surrounded by a viscoelastic medium of elasticity G and viscosity ηs occupying the annulus R < r < HR is determined using a linear stability analysis. The inertia of the fluid and the medium are neglected, and the mass and momentum conservation equations for the fluid and wall are linear. The only coupling between the mean flow and fluctuations enters via an additional term in the boundary condition for the tangential velocity at the interface, due to the discontinuity in the strain rate in the mean flow at the surface. This additional term is responsible for destabilizing the surface when the mean velocity increases beyond a transition value, and the physical mechanism driving the instability is the transfer of energy from the mean flow to the fluctuations due to the work done by the mean flow at the interface.The transition velocity Γt for the presence of surface instabilities depends on the wavenumber k and three dimensionless parameters: the ratio of the solid and fluid viscosities ηr = (ηs/η), the capillary number λ = (T/GR) and the ratio of radii H, where T is the surface tension of the interface. For ηr = 0 and λ = 0, the transition velocity Γt diverges in the limits k [Lt ] 1 and k [Gt ] 1, and has a minimum for finite k. The qualitative behaviour of the transition velocity is the same for λ < 0 and ηr = 0, though there is an increase in λt in the limit k > 1. When the viscosity of the surface is non-zero (ηr < 0), however, there is a qualitative change in the λtvs. k curves. For ηr < 1, the transition velocity λt is finite only when k is greater than a minimum value kmin, while perturbations with wavenumber k < kmin are stable even for λ → ∞. For ηr > 1, λt is finite only for kmin < k < kmax, while perturbations with wavenumber k < kmin or k > kmax are stable in the limit λ → ∞. As H decreases or ηr increases, the difference kmax − kmin decreases. At a minimum value H = Hmin which is a function of ηr, the difference kmax − kmin = 0, and for H < Hmin, perturbations of all wavenumbers are stable even in the limit λ → ∞. The calculations indicate that Hmin shows a strong divergence proportional to exp (0.0832nr2) for ηr [Gt ] 1.

Temporal Stability of Flow Through Viscoelastic Tubes

“…A comparison with the solution obtained by Kumaran (1995a) for the Hagen}Poiseuille #ow in a viscoelastic tube has been made in Hamadiche & Al-Farkh (1997). The agreement between the results of the present numerical approach and the results of Kumaran was found to be of the same order as the algorithm round-o!…”

“…Furthermore, the continuation method was used to "nd the eigenvalues for new values of the four dimensionless control parameters. In order to validate our numerical method, we compared the present results with the results of Kumaran (1995a) and the results of Salwen & Grosch (1972) and Gill (1973). The agreement between ours and their results is very good.…”

“…The stability of Hagen}Poiseuille #ow in a compliant circular pipe has been studied by Evrensel et al (1993) and Kumaran (1995a), both of whom considered only low-Reynoldsnumber #ows and therefore neglected inertia. Kumaran has shown that the coupled system is unstable to axisymmetric perturbations.…”

Reversal of the Bernoulli Effect and Channel Flutter

“…In yet another separate development, Kumaran et al (1994) and Kumaran (1995a) studied the destabilizing mechanism in Stokes flow through flexible tubes lined with a gel-like viscoelastic medium. The coupling between the fluid shear stress and the tangential motion of the fluid-solid interface is shown to be responsible for the interface instabilities.…”

Experimental studies on the flow through soft tubes and channels

“…However, there have been a series of experiments carried out by Kumaran & Muralikrishnan (2000); Muralikrishnan & Kumaran (2002) and Eggert & Kumar (2004) in a commercial rheometer which is used for measuring fluid viscosity. Though this is an open flow, where the fluid is sheared between two parallel plates, the experiments were motivated by theoretical predictions of an instability in channel and tube flows at low Reynolds number (Kumaran et al 1994;Kumaran 1995).…”