Modelling the wall deformation of fluid conveying elastic-walled tubes

We investigate the small-amplitude deformations of a long, thin-walled elastic tube that is initially axially uniform with an arbitrary cross-sectional shape. The tube is deformed by a (possibly non-uniform) transmural pressure. For an initially elliptical tube, the leading-order deformations are shown to be governed by a single partial differential equation (PDE) for the azimuthal displacement as a function of the axial and azimuthal co-ordinates and time. Previous authors have obtained solutions of this PDE by making ad-hoc approximations based on truncating an approximate Fourier representation. In this new work, we present a generalised governing PDE, which permits arbitrary initial cross-sectional shapes, and describe a new solution method in which we instead write the azimuthal displacement as a sum over the azimuthal eigenfunctions of a generalised eigenvalue problem. We show that we are able to derive an uncoupled system of linear PDEs with constant coefficients for the amplitude of the azimuthal modes as a function of the axial co-ordinate and time. This results in a formal series solution of the whole system being found as a sum over the azimuthal modes. We show that the nth mode's contribution to the tube's relative area change is governed by a simplified second-order PDE. For the case in which wall deformation is driven by a uniform transmural pressure, we determine a family of initial cross-sectional shapes that have the property that only the first azimuthal mode is excited, which results in a semi-analytical solution of this three-dimensional problem.