The axial velocity field in a tube of arbitrary cross-section geometry is determined for the case in which a viscoelastic fluid obeys a constitutive law of the multiple-integral type. The flow is linearized by means of a perturbation scheme for small amplitud oscillations of the pressure gradient. At the first order for the velocity, a solution is found for cross-section contours of a wide variaty of shapes. In this method the no-slip boundary condition is imposed in contours whose shape depends on an infinite set of arbitrary functions that can be selected according to some guiding rules. In this manner, the resulting cross-sections approach ellipses, triangles, squares and many other regular and non-regular shapes. The analysis shows that very complex flow patterns, as depicted by isovel curves, develop according to the values of the fluid parameters and the solid boundary shapes. These results are necessary for further developments leading to the study of secondary flows. The paper presents the analytical formulations and the flow fields associated to several tube geometries and different values of the fluid parameters.
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