STABILITY OF WALL MODES IN A FLEXIBLE TUBE

The asymptotic results (Kumaran 1998b) obtained for Lambda similar to 1 for the flow in a flexible tube are extended to the limit Lambda muc h less than 1 using a numerical scheme, where Lambda is the dimensionl ess parameter Re-1/3(G/rho V-2), Re = (rho VR/nu) is the Reynolds numb er, rho and nu are the density and viscosity of the fluid, R is the tu be radius and G is the shear modulus of the wall material. The results of this calculation indicate that the least-damped mode becomes unsta ble when Lambda decreases below a transition value at a fixed Reynolds number, or when the Reynolds number increases beyond a transition val ue at a fixed Lambda. The Reynolds number at which there is a transiti on from stable to unstable perturbations for this mode is determined a s a function of the parameter Sigma = (pGR(2)/nu(2)), the scaled waven umber of the perturbations kR, the ratio of radii of the wall and flui d H and the ratio of viscosities of the wall material and the fluid nu (r). For nu(r) = 0, the Reynolds number at which there is a transition from stable to unstable perturbations decreases proportional to Sigma (1/2) in the limit Sigma much less than 1, and the neutral stability c urves have a rather complex behaviour in the intermediate regime with the possibility of turning points and isolated domains of instability. In the limit Sigma much greater than 1, the Reynolds number at which there is a transition from stable to unstable perturbations increases proportional to Sigma(alpha), where alpha is between 0.7 and 0.75. An increase in the ratio of viscosities nu(r), has a complex effect on th e Reynolds number for neutrally stable modes, and it is observed that there is a maximum ratio of viscosities at specified values of H at wh ich neutrally stable modes exist; when the ratio of viscosities is gre ater than this maximum value, perturbations are always stable.