## 4 Fluid Dynamic Performance
### Factors
#### 4.1.2 Complex Fluid Nature, Viscosity and Elasticity  
Viscoelasticity makes a big difference in the fluid flow around the prototypes. They behave in a different way depending on the flow conditions. For instance, [PERSON] and [PERSON] [64; 65; 66] studied at the macro scale the influence of the elasticity in the critical Reynolds number for different flow regimes. The presence of elasticity can increase the drag reduction at low Reynolds numbers. In the shear-layer transition regime, the increase of the wake angle and pressure rise coefficient reduced the drag coefficient and narrowed the near wake. At the micro scale, the relevance of fluid elasticity is enhanced and elastic instabilities can be observed easily even at low or negligible \(Re\)[7; 67]. [PERSON] et al. [54] showed that, for a viscoelastic fluid (Boger fluid), symmetric vortices started to form upstream of a hyperbolic contraction at a critical \(Re\) of 0.82; meanwhile, for the same micro channel and flow conditions, for a Newtonian fluid, the critical \(Re\) at which symmetric vortices start to form downstream of the hyperbolic contraction is higher (\(Re=49.3\)) due to inertial effects (Figure 2).  
It has also been reported that, in viscoelastic fluids, both enhancement and inhibition of swimming of some microorganisms' speed occurs depending on the swimming strategy and the rheological characteristics of the media. For instance, helical bacteria such as _Leptospira_ and _B. burgdorferi_ swims faster in a viscoelastic fluid than in a Newtonian fluid of the same viscosity [68; 69], whereas _C. elegans_, which undulates its body in a planar wave, swims at a slower pace [70]. [PERSON] et al. [71] studied numerically the locomotion of spherical swimmers with tangential squirming motion in a viscoelastic fluid. They reported that the swimming speed of the viscoelastic fluid decreased in relation to the Newtonian fluid in all the cases analyzed, and also the required power for swimming is smaller for the viscoelastic fluid in addition to the hydrodynamic efficiency being higher than for the Newtonian case. Experiments carried out by [PERSON] et al. [72] to measure the speed of a swimmer immersed in Newtonian and viscoelastic fluids revealed differences between a viscoelastic fluid that has a viscosity independent of shear rate (Boger fluid) and more complex shear-thinning viscoelastic fluids that have multiple relaxation time scales. In the latter case, the speed was slower than in the Boger-like fluid.  
Figure 2: Flow patterns for distilled water (\(Re\) = 49.3) and 400 ppm polyacrylamide (PAA) solution with 1% NaCl (\(Re\) = 0.82). The flow direction is from left to right. Images adapted from [PERSON] et al. [54]. Copyright Journal of Non-Newtonian Fluid Mechanics, 2011.  
This is a controversial issue, and there is not yet a consensus with experimental studies leading to clarify whether viscoelasticity increases or decreases swimming velocity. [PERSON] and [PERSON] [73] considered the general problem of swimming using small-amplitude periodic waves in a viscoelastic fluid described by the classical Oldroyd-B constitutive relationship, and they showed that if all travelling waves move in the same direction, the locomotion speed of the organism is systematically decreased. However, if the waves travel in two opposite directions, this can lead to enhancement of the swimming speed. Nevertheless, more investigation is still needed to check if the theoretical models are in agreement with biological tests.