Algorithm for analysis of peristaltic annular flows. (2nd April 2017) - Moradi, H.V.; Zandi, S.; Floryan, J.M.
Highlights: A spectrally accurate algorithm is proposed for the analysis of peristaltic flows in annular geometries. The Galileo transformation is used to convert the unsteady physical problem into a steady problem. The Stokes stream function is employed to reduce the number of field equations to a single fourth-order PDE. Numerical discretization uses Fourier and Chebyshev expansions in the streamwise and radial directions, respectively. Difficulties associated with the irregularities of the boundaries are overcome using the immersed boundary conditions method. Abstract: A spectrally accurate algorithm suitable for the analysis of peristaltic flows in annular geometries has been developed. The bounding surfaces have cylindrical forms modified by axisymmetric waves of arbitrary form traveling in the axial direction. The problem is expressed as a superposition of the flow in a smooth annulus and modifications associated with the surface waves. The Galileo transformation has been used to convert the unsteady physical problem into a steady problem in the frame of reference moving with the wave. The Stokes stream function has been used to reduce the number of field equations to a single fourth-order partial differential equation. Numerical discretization uses Fourier expansions in the streamwise direction and Chebyshev expansions in the radial direction. Difficulties associated with the irregularities of the boundaries have been overcome using the immersed boundary conditions method (IBC). Various numerical tests have been carried out in order to verify the spectral accuracy of the proposed algorithm. Analytical solutions for the long wavelength waves as well as for the waves with small amplitudes have been used to verify the algorithm. It has been shown that the leading-order approximation for the long wavelength waves provides sufficiently accurate results for wavelengths longer than 10 π . It has also been demonstrated that the changes in the mean axial pressure gradient vary proportionally to the second power of the wave amplitude S 2 for waves with small enough amplitudes.