6. Discussion about graphical results

The purpose of this study is to enhance the heat and mass diffusion by choosing a thin-layer spray of the Casson nanofluid over a stretching cylinder. The physical configuration of the problem is shown in Figure 1. The solution of the problem has been obtained using the homotopy approach, and the main features for the convergence (h-curves) of homotopy analysis method (HAM) have been shown in Figures 2 and 3. These figures demonstrate the h-curves for velocity, temperature, and concentration fields, respectively. The impact of buoyancy parameter λ and buoyancy ratio Nr on velocity field is prescribed in Figure 4. Velocity grows with the rising values of λ because the natural convection parameter λ and momentum boundary layer are in direct relation. The similar effect for the rising values of Nr can be seen in Figure 4. The effect of thickness parameter β and Casson fluid parameter β<sup>1</sup> versus velocity field is shown in Figure 5. Increasing values of β generate friction force and decline the velocity field because the thicker flow creates hurdles in fluid motion, while the thin layer is comparatively fast flowing. The Heat and Mass Transfer Analysis During Bunch Coating of a Stretching Cylinder by Casson Fluid 49 http://dx.doi.org/10.5772/intechopen.79772

Figure 1. Physical geometry of the problem.

Figure 2. h-curve for velocity profile.

Figure 3. Combined h-curve for temperature and concentration fields.

Figure 4. Variation of velocity with Nr and λ.

Therefore, larger amount of β declines the flow motion. The similar effect of the larger amount of the Casson fluid parameter β<sup>1</sup> is shown in Figure 5. The rising values of the parameter β<sup>1</sup> imply a decline in the yield stress of the Casson fluid. In Figure 6, the behavior of the thermophoretic parameter Nt and Reynolds number Re is observed over the field of temperature. The Heat and Mass Transfer Analysis During Bunch Coating of a Stretching Cylinder by Casson Fluid 51 http://dx.doi.org/10.5772/intechopen.79772

Figure 5. Variation of velocity with β and β1.

Figure 6. Variation of Nt and Re.

The larger amount of thermophoresis parameter Nt depreciates temperature profile because rising values of Nt enhance the concentration profile due to its direct relation and its product in the model equation increases the cooling effect to reduce the temperature field. The larger quantity of Re reduces temperature field. Rising values of Reynolds number Re enhance the

Figure 7. Variation of Nt and Nb.

inertial forces. The powerful inertial forces kept the fluid particles tightly closed, and more heat energy is required to break down the bonds among these atoms. In other words the inertial forces raise the boiling point of the fluid, and more heat energy is required to enhance the temperature. Figure 7 shows the influences of thermophoretic parameter Nt and Brownian motion parameter Nb in concentration field. The larger amount of Nb displays a falling performance against concentration field. The parameter Nb is owing to the thinning of boundary layer because the random flow of liquid particles makes the decline in the concentration. The rising values of thermophoresis parameter Nt enhance the concentration field. The reason behind this is that Nt is in direct relation with concentration pitch, while the Nb is in inverse relation to the concentration field.

Figure 8 represents the behavior of the concentration field with respect to Reynolds number Re and Lewis number Le. The larger amount of Re improves the concentration field. The reason is that larger values of Re generate the enhancement in the inertial forces to rise concentration field. The concentration boundary layer is falling with the rising value of Lewis number Le.

Figure 9 shows the relationship between pressure distribution over the stretching surface versus Reynolds number Re and Casson fluid parameter β1. The larger amount of β<sup>1</sup> increases the viscous forces, and more pressure are required at the surface. Thus the larger amount of β<sup>1</sup> decreases the pressure distribution. The larger amount of the Reynolds number Re decreases the pressure distribution. The strong inertial effects packed the fluid particle tightly, and as a result the pressure distribution decreases.

Table 1 shows the numerical values of the skin friction coefficient, local Nusselt number, and Sherwood number of various physical parameters. The skin friction coefficient rises with the

The Heat and Mass Transfer Analysis During Bunch Coating of a Stretching Cylinder by Casson Fluid 53 http://dx.doi.org/10.5772/intechopen.79772

Figure 8. Variation of Re and Le.

Figure 9. Variation of pressure term.

growth of thickness parameter β. The thick boundary layer increases friction force and improves the cooling effect. Therefore, the Nussselt and Sherwood numbers are increased. The Reynolds number Re decreases the fluid flow due to inertial forces. Due to this reason, the larger quantity of Re enhances the f 00ð Þ1 , Θ<sup>0</sup> ð Þ1 and ϕ<sup>0</sup> ð Þ1 . Similar effect for rising values of the


Table 1. Numerical values for skin friction coefficient, local Nusselt number, and Sherwood number for various physical parameters when h ¼ �0:5, Pr ¼ 0:5, β<sup>1</sup> ¼ 1:2, β ¼ 1:5, Nt ¼ 0:5, Nb ¼ 1, Nr ¼ 0:6, Re ¼ 0:8, Le ¼ 0:5.


Table 2. Values of f 00ð Þ1 for various Reynolds numbers when h ¼ �0:5, Pr ¼ 0:5, β<sup>1</sup> ¼ 1:2, β ¼ 1:1, Nt ¼ 0:5, Nb ¼ 1, Nr ¼ 0:6, Le ¼ 0:5.


Table 3. Values of �θ<sup>0</sup> ð Þ1 for various Prandtl numbers when h ¼ �0:5, β<sup>1</sup> ¼ 1:2, β ¼ 1:1, Nt ¼ 0:5, Nb ¼ 1, Nr ¼ 0:6, Re ¼ 0:8, Le ¼ 0:5.

Casson parameter β<sup>1</sup> has been shown in Table 1. The reason is that the viscous forces become dominant with the larger amount of β<sup>1</sup> to enhance the f 00ð Þ1 , Θ<sup>0</sup> ð Þ1 and ϕ<sup>0</sup> ð Þ1 . The comparison of present work and published work has been shown in Tables 2 and 3, and closed agreement for f 00ð Þ1 , Θ<sup>0</sup> ð Þ1 and ϕ<sup>0</sup> ð Þ1 has been achieved.
