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We build a general formalism to describe thin viscous jets as one-dimensionalobjects with an internal structure. We present in full generality the stepsneeded to describe the viscous jets around their central line, and we arguethat the Taylor expansion of all fields around that line is convenientlyexpressed in terms of symmetric trace-free tensors living in the two dimensionsof the fiber sections. We recover the standard results of axisymmetric jets andwe report the first and second corrections to the lowest order description,also allowing for a rotational component around the axis of symmetry. Whenapplied to generally curved fibers, the lowest order description corresponds toa viscous string model with circular sections. However, when including thefirst corrections we find that curved jets generically develop ellipticsections. Several subtle effects imply that the first corrections cannot bedescribed by a rod model, since it amounts to selectively discard somecorrections. For completeness, we also recover the constitutive relations forforces and torques in rod models and exhibit a missing term in the lowest orderexpression of viscous torque. Given that our method is based on tensors, thecomplexity of all computations has been beaten down by using an appropriatetensor algebra package such as {\it xAct}, allowing us to obtain aone-dimensional description of curved viscous jets with all the first ordercorrections consistently included. Finally, we find a description for straightfibers with elliptic sections as a special case of these results, and recoverthat ellipticity is dynamically damped by surface tension. An application totoroidal viscous fibers is presented in the companion paper [Pitrou, Phys. Rev.E 97, 043116 (2018)].

One-dimensional reduction of viscous jets. I. Theory