**5. Concluding remarks**

**Figure 11.** Plots of isovels for n=3, ε=0.3849 and N=0.5

354 Advances in Bioengineering

**Figure 12.** Plots of isovels for n=3, ε=0.3849 and N=1.0

more energy by supplied in order to keep constant the rate of flow.

a factor that determines stagnant areas close to such corners.

The plug zones are defined through the following joint conditions

One significant feature of plastic flows are plug (or solid) and stagnant zones, which, as a general rule, tend to decrease the rate of flow for a given pressure gradient, or require that

Figure 13 shows that energy dissipation is greatly affected by the tube geometry. The maximum energy dissipation in the case of an equilateral triangle appears in the middle of the sides, where the axial shear stress is maximum. Also the shear stress is zero at the corners, which is Some analytical results concerning the effects of fluid elasticity and plasticity, coupled with tube cross-sectional geometry variation have been presented in this chapter.

The analysis is unified through some concepts that make it possible to explore, in a rather general fashion, the mechanisms of transversal transport that arise with the coupling of viscoelasticity and non-circular shape.

The circular shape is the most energy-efficient shape when only longitudinal mass transfer is considered. However, this not the case when transversal motion becomes important, as in heatexchange processes, or in cases when also particle distribution is relevant.

General results indicate that viscoelasticity tends to increase transversal transport, which can be demonstrated by analytical means. These in particular, show that starting at third order in *Wi* , secondary flows appear when the fluid exhibits non-linear viscoelasticity, as prescribed by the fluid models of Giesekus and Phan-Tien-Tanner. Such secondary, or transversal motions, when coupled with the temperature, through the energy equation, determine a temperature field that improve heat-transfer between the fluid and the tube wall.

Plastic effects, on another perspective, do not induce transversal motion if the cross-section is not circular. Rather plasticity increases energy consumption when maintaining a given rate of flow is a priority. Shapes that include sharp corners lead to stagnant zones of fluids in the vicinity of those corners, thus decreasing the flow. The analytical method herein applied can be used for determining plug-zones of limiting values of the yield parameter *N* for which such zones exist properly.

The application of these findings to biological flows have been commented in previous sections of the chapter.
