**4. Plastic flow**

The velocity field and related variables for the flow of a Bingham fluid in non-circular tubes is next found using a similar technique. The axial velocity is defined as

Energy Transfer and Dissipation in non-Newtonian Flows in non-Circular Tubes http://dx.doi.org/10.5772/59907 353

$$
w = w\_0 + \varepsilon w\_1 + \varepsilon \, ^2 \overline{w}\_2 + \dots \tag{59}$$

The constitutive equations (29, 30) can be expanded in series around the parameter ε by using (58), where from [24]

$$\text{tr}\_{rz} = \text{N} - \frac{\partial w\_0}{\partial r} - \varepsilon \frac{\partial w\_1}{\partial r} + \text{O}\{\varepsilon^2\} \tag{60}$$

$$\pi\_{\partial z} = \text{ - } \frac{\varepsilon}{r} (\frac{\partial w\_0}{\partial r} - \text{N}) \underbrace{\frac{\partial w\_1}{\partial \theta}}\_{\overset{\partial w\_0}{\partial r}} + \text{0} \{\varepsilon^2\} \tag{61}$$

From the above the governing equation for *w*1 are found, i.e.

$$r\_T \left( r - \frac{N}{2} \right) \left( \frac{\partial^2 w\_1}{\partial r^2} + \frac{1}{r} \frac{\partial w\_1}{\partial r} \right) + \frac{\partial^2 w\_1}{\partial \theta^2} = 0 \tag{62}$$

which can be solved by separation of variables.

**Figure 9.** Characteristics plots of isothermal curves for n=3 and ε=0.2

also *Nu* increases for a given value of *Wi*.

to wall elasticity and peristaltic motion.

normal physiological states.

augment transport capacity.

**4. Plastic flow**

352 Advances in Bioengineering

These results, as initially presented and discussed in [13, 25, 26] show that the Nusselt number, i.e., the heat-transfer between fluid and wall, increases as the viscoelastic parameter *Wi* increases, with an asymptotic trend, for a given value of the Reynolds number. As *Re* increases,

The relevance of these findings for biological flows may be related to vessel deformation due

The preceding analysis shows that a very small deviation of the cross-section contour from the circular geometry, as represented by the value of the parameter *ε*, may induce secondary flows, and so increase the transversal transport capacity of biological flows. It seems to be an open research area the study of physiological and morphological changes that, in this context, may induce pathologies that accelerate heat transfer inside the human body. Such changes may create conditions that improve or worsen transport processes that should lead to restore

Previous results associated to the flow of viscoelastic fluid in channels of axially-varying crosssection [27] show also that axial change of geometry, as found in biological vessels, also

The velocity field and related variables for the flow of a Bingham fluid in non-circular tubes

is next found using a similar technique. The axial velocity is defined as

The dissipation function, in terms of the normal axial shear stress, is given in equation (35) and can be computed, as an indicator of energy dissipation due to friction.

Typical results for the velocity field and dissipation function are shown in the following figures.

**Figure 10.** Plots of isovels for n=3, ε=0.3849 and N=0.2

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

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

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

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 a factor that determines stagnant areas close to such corners.

The plug zones are defined through the following joint conditions

**Figure 13.** Plots of the dissipation function (35) for n=3, ε=0.3849 and N=0.5

$$\begin{array}{ccc}\frac{dw}{dt} = 0 & ; & \text{ $w = const$ } \end{array} \tag{63}$$

which can be met up to certain value of *N* according to the cross-section shape.

Abnormal vessel geometries may occur in biological flow arising from many sources. Espe‐ cially in the case of stenosis, or geometry change due to solid deposition in artery walls, plastic effects may lead to blood clotting in corners, when the arteries are small.
