The steady flow of viscoelastic fluid in a toroidal pipe of circular cross-section is analytically solved with the White-Metzner constitutive equation by a perturbation method. The results are as follows:
(1) The flow rate of a secondary flow in the cross-sectional plane increases with decreasing the viscosity index n, and with increasing the elasticity index s, the Reynolds number Re and the Weissenberg number We.
(2) The position of the maximum axial flow velocity shifts on the central plane more outward from the pipe center for smaller n and larger Re in the case of Power law fluid, and for larger s in the case of White-Metzner fluid.
(3) Owing to the centrifugal and elastic forces applied on the fluid, the stream line starting at a pipe center shifts outward on the central plane. The larger the values of s, Re and We are, the larger the amount of the outward shift is.
Hence, the effects of Re and We on the flow are shown to be similar to an available experimental result as well as earlier theoretical results. Also, it is found that the values n and s, which represent the degrees of decrement of the fluid viscosity and the first normal stress coefficient with the shear-rate respectively, have significant effects on the flow at high shear-rate to which the earlier analyses could not be applied.

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Constitutive equations applicable to viscoelastic materials with/without initial elastic responses are derived by using internal variables, and applied to viscoelastic composite materials. Stress analyses are carried out by the finite element method, using linear and non-linear viscoelastic constitutive queations with internal variables.

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