To explain the phenomena in nature, physics are still using many models. In most cases, natural phenomena are simplified assuming that they can be mathematically reduced to relatively simple equations involving continuous and differentiable functions. The reality is much more complex, and these simplifications cannot explain all the aspects of an observation. The paper presents a generalization of the movement of a solid particle in a complex fluid, considering that the fluid particles dynamics occurs on continuous, but non-differentiable curves, a situation much closer to reality. Movement analysis has taken into account the consequences of non-differentiability and of scale covariance principle for continuous but non-differentiable curves. The equation of motion was derived showing that the behavior of the fluid is of viscoelastic or of hysteretic type. The analysis of the lift force was made for both rotational and irrotational movements. The numerical simulation shows that the simultaneous presence of solitons and anti-solitons packages from the force fields specify the transition from a differentiable flow (along the same stream line) to a non-differentiable flow (the jump from one stream line to another). We demonstrated that, in any type of motion (rotational or irrotational), the differentiable part of the velocity field generates circulation, while the non-differentiable part of the same field induces the lift force. (C) 2012 Elsevier B.V. All rights reserved.