We consider the existence of solutions for some models of non‐Newtonian fluid. We have found that Poiseuille and Taylor–Couette problems may have no usual solutions in a large class of Oldroyd models of viscoelastic fluids. That is, axisymmetric Couette‐like solutions and unidirectional Poiseuille‐like solutions fail to exist at sufficiently large Deborah numbers. We have observed that some of the flow features become universal and independent of model parameters in the critical case, i.e., on the boundary between existence and nonexistence regions. We have described a duality between unidirectional flows of the 4‐constant Oldroyd fluid and plane potential flows of an ideal fictitious gas. Based on this duality we have interpreted the nonexistence of solutions as a ‘‘block‐up’’ phenomenon, in the language of gas dynamics.

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