## 1 Introduction  
It is very common to find many examples of complex fluids in our daily lives. From food, health care products or materials for construction to human biofluids. These complex fluids are characterized by their intermediate behavior between a solid and a liquid. In the special case of the human biofluids, blood is a clear example of a complex fluid whose behavior is given by a complex mixture of its components: red blood cells, platelets, white blood cells and other elements in plasma [1; 2]. The red blood cells are the most abundant component and also responsible for the rheological behavior of whole blood. When the blood is at rest for several seconds, the red cells begin to form rouleaux, leading to an interconnected network extending throughout the blood, thus increasing its resistance to deformation and imparting also an elastic response to blood. These rouleaux could be broken up by imposed shearing causing a relative decrease in the viscosity, and, therefore, a shear-thinning behaviour [1; 2; 3; 4]. Another characteristic of this complex behavior is the viscoelasticity given by the relaxation time, which measures the time taken by the internal fluid structures to relax internal stresses imposed by the flow [6]. At the microscale, flows of viscoelastic fluids can be significantly different from those of their Newtonian counterparts. The most common dimensionless numbers that characterize these kinds of viscoelastic fluid flows are the Reynolds number (_Re_), defined as the ratio between inertial and viscous forces, _Re_= \(\rho VL/\eta\), and the Weissenberg number (_Wi_), which is the ratio between elastic and viscous forces, _Wi_= \(\lambda V/L\)[7]. \(V\) represents the average velocity, \(L\) the characteristic length-scale and \(\rho\), \(\lambda\) and \(\eta\) are the density, relaxation time and viscosity of the fluid, respectively. Viscoelastic fluids are of great importance in some nanotechnological applications, for instance, a complete rheological characterization and understanding of the flow behavior of these fluids is crucial for the assessment of the movement of microbots immersed in complex fluids. One of the most impressive inventions of nanotechnology is the creation of nanobots and microbots, which are mechanical or electromechanical devices that are able to perform specific functions at the nano- and micro-scale, respectively [8]. Nanobots (artificial machines sized down below 10 microns) have become very popular, although nobody has yet built artificial non-biological nanorobots, and they remain a hypothetical concept. However, microbots (sized between 10 and 1000 microns) are a reality [9] and they can be considered as _\"training wheels\"_ for nanotechnology. Microbots have been developed for several purposes like industrial applications or biomedicine [10; 11]. The area of biomedicine is currently one of the most challenging and offers a huge number of potential applications for the microbots to perform precise and delicate tasks inside the human body, i.e., providing a mobile viewing platform enhancing a surgeon's view, or for protecting and treating the human body against pathogens. Microbots can provide a new method for unblocking blood clots, performing eye surgery and delivering drugs [12; 13; 14]. This latter application is especially interesting for the case of treatments to fight cancer, offering an alternative to the traditional chemotherapy, increasing its effectiveness while diminishing side-effects. In this field, some research has been directed to the fabrication method of the microbot prototypes [15; 16], to their control and propulsion [17; 18; 19; 20; 21; 22], and to the study of their communication mechanisms [23; 24; 25]. It is well known that at a low-Reynolds number regime, non-reciprocal motion is required for a self-propelling device. [PERSON] [26] proposed that traveling waves propagating along a tail was the mechanism by which spermatozoa propel themselves. Later on, [PERSON] and [PERSON] [27] discovered that some bacteria, such as _E.