The stability of fluid flow in a flexible tube to non-axisymmetric perturbations is
analysed in this paper. In the first part of the paper, the equivalents of classical
theorems of hydrodynamic stability are derived for inviscid flow in a flexible tube
subjected to arbitrary non-axisymmetric disturbances. Perturbations of the form
vi = v˜i exp
[ik(x − ct) + inθ] are imposed on a
steady axisymmetric mean flow U(r) in a flexible
tube, and the stability of mean flow velocity profiles and bounds for
the phase velocity of the unstable modes are determined for arbitrary values of
azimuthal wavenumber n. Here r, θ and x are
respectively the radial, azimuthal and axial coordinates, and
k and c are the axial wavenumber and phase velocity
of disturbances. The flexible wall is represented by a standard constitutive relation
which contains inertial, elastic and dissipative terms. The general results indicate
that the fluid flow in a flexible tube is stable in the inviscid limit if the quantity
Ud[Gscr ]/dr [ges ] 0, and could be unstable for
Ud[Gscr ]/dr < 0, where [Gscr ] ≡
rU′/(n2 + k2r2).
For the case of Hagen–Poiseuille flow, the general result implies that the flow is stable
to axisymmetric disturbances (n = 0), but could be unstable to non-axisymmetric
disturbances with any non-zero azimuthal wavenumber (n ≠ 0). This is in marked
contrast to plane parallel flows where two-dimensional disturbances are always more
unstable than three-dimensional ones (Squire theorem). Some new bounds are derived
which place restrictions on the real and imaginary parts of the phase velocity for
arbitrary non-axisymmetric disturbances.In the second part of this paper, the stability of the Hagen–Poiseuille flow in
a flexible tube to non-axisymmetric disturbances is analysed in the high Reynolds
number regime. An asymptotic analysis reveals that the Hagen–Poiseuille flow in a
flexible tube is unstable to non-axisymmetric disturbances even in the inviscid limit,
and this agrees with the general results derived in this paper. The asymptotic results
are extended numerically to the moderate Reynolds number regime. The numerical results reveal
that the critical Reynolds number obtained for inviscid instability to non-axisymmetric
disturbances is much lower than the critical Reynolds numbers obtained in the previous studies
for viscous instability to axisymmetric disturbances when the dimensionless parameter
Σ = ρGR2/η2 is large.
Here G is the shear modulus of the elastic medium,
ρ is the density of the fluid, R is the radius of the tube and
η is the viscosity of the fluid. The viscosity of the wall medium is found to have a
stabilizing effect on this instability.
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