Determination of the critical velocity of the fluid flowing in a single-walled carbon nanotubes embedded in a polymer matrix

Since 90’s carbonic nanotubes are broadly used in nanophysics, nanobiology and nanomechanics in nanofluidic devices, nanocontainers for gas storage and nanopipes conveying fluid. They have a perfect hollow cylindrical geometry and superior mechanical strength. The flowing fluid can be water, oil, dynamic flow of methane, ethane and ethylene molecules. The problem of the fluid-structure interaction could be considered in the case of nanoscale. However, the experiments at the nanoscale are difficult and expensive. That is why the continuum elastic models have been used to study the fluid-structure interaction. The carbon nanotubes are considered with Euler- and Timoshenko-beam models. In this paper the dynamic stability of a single-walled carbon nanotube is investigated on the basis of the Euler-beam model and with the employment of the Generalized Differential Quadrature Method. The tube under investigation is assumed hinged at its both ends and is embedded in a polymer matrix. To study the influence of the surrounding elastic medium (for example, a polymer) on the stability of the pipe, an elastic base of Pasternak is introduced. A differential equation is presented that describes the transverse vibrations of a nanotube embedded in a polymer matrix. Dimensionless parameters are introduced. The scheme of Chebycheva-Gauss-Lobato is used for sampling. The coefficients are calculated using Lagrange interpolation functions. A system of homogeneous equations is written in the matrix form. The obtained numerical results are for flowing fluids with different densities. In order to study the effect of the surrounding elastic medium (such as polymer) on the stability of the pipe the Pasternak elastic foundation is introduced. The critical velocities of each type of fluid are determined for different stiffnesses of this matrix. A decrease in the critical speed with the increasing mass ratio has been established.