## 3 Numerical Algorithms and Computational Methodologies
### Response of Flow Variables in the Longitudinal Direction  
All the graphical results thus far are presented in the radial direction, across the inner pipe diameter. For this reason, the results have been symmetric about the pipe centerline. It is important to also illustrate the development of the solutions in the longitudinal direction (i.e., the flow direction) in response to variations in flow parameters. Figure 29 illustrates the behaviour of the core-fluid temperature in the longitudinal direction, in response to variations in the nanoparticle volume-fraction, \(\varphi\). The results of Figure 29 show, as expected, that the core-fluid temperature decreases from the initial high values at the inlet to lower values at the outlet. This is the hallmark of a heat exchanger design and function--that a hot fluid is cooled as it flows and interacts with a colder fluid in a connected channel even when the two connected channels containing the hot and cold fluids, respectively, are separated by a solid wall. Additionally, Figure 29 illustrates, as also shown in the corresponding Figure 6, that the core fluid temperature decreases with increasing nanoparticle volume-fraction, \(\varphi\). As already explained under Figure 6, this clearly indicates that an increase in the nanoparticles volume-fraction, \(\varphi\), correspondingly increases the thermal conductivity of the shell-nanofluid, and hence, also increases the heat-exchange characteristics from the hot core-fluid to the colder shell-nanofluid. Figure 28: Effects of Deborah number, \(De\), on the second normal stress difference, \(N_{2}\). Figure 30 illustrates the behaviour of the shell-fluid temperature in the longitudinal direction, in response to variations in the nanoparticle volume-fraction, \(\varphi\). The results of Figure 30 show, as expected, that the shell-fluid temperature will increase from the initial low values at the inlet to higher values at the outlet. This is again the hallmark of a heat exchanger design and function--that a coolant fluid is heated as it flows and interacts with a hotter fluid in a connected channel even when the two connected channels, respectively, containing the coolant and hot fluids, are separated by a solid wall. Figure 31 illustrates the behaviour of the connecting wall temperature in the longitudinal direction, in response to variations in the nanoparticle volume-fraction, \(\varphi\). The connecting wall temperature would roughly average the core-fluid and shell-fluid temperatures, and hence, the behaviour of the connecting wall temperature largely mirrors the average behavior of the core-fluid and shell-fluid temperatures. Of specific note is that the connecting wall temperature decreases with increasing nanoparticle volume-fraction, \(\varphi  
Figure 31: Effects of nanoparticles volume-fraction, \(\varphi\), on the connecting wall temperature. Figure 30: Effects of nanoparticles volume-fraction \(\varphi\) on shell-fluid temperature. Fluid moving from inlet at \(z=4\) to outlet at \(z=0\), i.e., flow direction is from right to left (\(\longleftarrow\)). Figure 32: Effects of Prandtl number, \(Pr\), on core-fluid temperature. Fluid moving from inlet at \(z=0\) to outlet at \(z=4\), i.e., flow direction is from left to right (\(\longleftrightarrow\)). Figure 33: Effects of Prandtl number, \(Pr\), on shell-fluid temperature. Fluid moving from inlet at \(z=4\) to outlet at \(z=0\), i.e., flow direction is from right to left (\(\longleftrightarrow\)). Figure 34: Effects of Reynolds number, \(Re\), on core-fluid temperature. Fluid moving from inlet at \(z=0\) to outlet at \(z=4\), i.e., flow direction is from left to right (\(\longleftrightarrow\)). Figure 35: Effects of Reynolds number, \(Re\), on the shell-fluid temperature. Fluid moving from inlet at \(z=4\) to outlet at \(z=0\), i.e., flow direction is from right to left (\(\longleftrightarrow\)). Figure 36 illustrates the behaviour of the core-fluid temperature, in the longitudinal direction, in response to variations in the Reynolds number, \(Re\).