## 3 Results and Discussion  
The velocity profiles obtained could be characterized in the following way. For the lowest flow rates, when the wall shear stress was below the yield stress \(\tau_{y}\) = 3.4 Pa of the MFC suspension [10], the velocity profile corresponded to a pure plug flow in the whole pipe, excluding a yielding marginal wall layer of a few microns in thickness. When the wall shear stress exceeded the yield stress, the velocity profiles consisted of three distinctive parts (see Figure 3). In the outer (bulk) region, at the distances greater than 20 \(\upmu\)m, the shear rate was small. In the region of 2-20 \(\upmu\)m the velocity profile was rather steep and approached zero towards the wall. In the immediate vicinity of the wall (distance 0-2 \(\upmu\)m), the velocity dropped abruptly to zero. The most natural explanation for the observed behavior of the velocity profile was a development of a consistency profile in the pipe, caused by wall depletion [26], when the distance from the wall was smaller than 20 \(\upmu\)m.  
The measured velocity profiles were fitted by the empirical formula  
\[u(y)=\dot{\gamma}_{w}^{a}y+u_{s}^{a}\Big{(}1-e^{-\frac{y}{\dot{\gamma}_{w}}} \Big{)}+u_{s}\;, \tag{1}\]  
where \(y\) is the distance from the wall and \(\dot{\gamma}_{w}^{a}\), \(u_{s}^{a}\), \(u_{s}\) and \(\lambda_{w}\) are free parameters. Parameter \(\dot{\gamma}_{w}^{a}\) is the apparent shear rate at wall, \(u_{s}^{a}\) and \(u_{s}\) are the apparent slip velocities, and \(\lambda_{w}\) is the characteristic thickness of the (apparent) slip layer. The local viscosity of the suspension could then be calculated from the formula  
\[\mu(y)=\frac{\tau(y)}{\dot{\gamma}(y)}, \tag{2}\]  
where \(y\) is the distance from the pipe wall, \(\tau(y)=\tau_{w}(D-2y)/D\), and \(\dot{\gamma}(y)=\mathrm{d}u/\mathrm{d}y\) is the local shear rate. Outside the wall depletion layer (\(y>20\)\(\upmu\)m), but still close to the wall (\(y<200\)\(\upmu\)m), the shear rate and shear stress are approximately \(\dot{\gamma}(y)\)\(\dot{\gamma}_{w}^{a}\) and \(\tau(y)\)\(\tau_{w}\), respectively. Figure 4 shows the viscosity \(\mu\)  
\(\tau_{w}\) / \(\dot{\gamma}_{w}^{a}\) of the MFC suspension in the bulk (\(y>20\)\(\upmu\)m). Above its yield stress, the well-known shear  
Figure 3: An example of a measured flow velocity profile (dots) together with the corresponding OCT amplitude signal (solid line) at the flow rate of 158 mL/min. The peak in the amplitude signal due to the tube wall is marked with an arrow. thinning (power-law) behaviour of MFC suspensions is evident in Figure 4 (5-15 Pa). This behaviour is also typical for fibre suspensions and is believed to be due to adhesive contacts between the fibres that are broken by the shear forces when the shear rate increases [27; 28]. As a result, the (floc) structure of the MFC suspension changes [29; 30]. Notice, that close to and below the yield stress the viscosity values are large, and there are strong fluctuation in their values. Large values are due to the values of \(\dot{\gamma}_{w}^{a}\) being close to zero when the suspension is non-yielded. The strong variation in the viscosity values is due to continuous break-up and recovery of the local network structure in the flow. Such fluctuations may be caused by small variations in shear history and a non-homogeneous floc structure of the sample suspension. This effect could be minimized by using longer measurement times. Figure 4 shows that there is a narrow transition region between two power laws, which takes place when the shear stress is ca. 10 Pa. To our knowledge, this kind of viscosity behavior has not been reported earlier for MFC suspensions. Such abrupt change in the rheological behavior of the MFC suspension is likely to be related with a sudden structural change in the suspension, for example, in fibre orientation [31] and/or flocculation [28]. For the floc size analysis, 50 images consisting of 1000 successive A-scans were analysed for the amplitude signals.