**7. Results and discussion**

The main focus of the present chapter is to study the flow and heat transfer characteristics of a Sisko fluid over a nonlinear radially stretching sheet. To obtain physical insight of the flow and heat transfer, Eqs. (22) and (29) subject to boundary conditions (23) and (30) are solved numerically and the results are illustrated graphically. During the ensuing discussion, the assumption of incompressibility and isotropy of fluid is implicit. The influence of the flow behavior index *n*, the material parameter *A*, stretching parameter *s*, and Eckert number *Ec* on flow and heat transfer is the main interest of the study. Further, the effect of variation of Prandtl number Pr on heat transfer is also analyzed in depth. A comparison amongst the flow and heat transfer aspects of the Bingham, Newtonian, and Sisko fluids is also precisely depicted. Moreover, the flow and heat transfer characteristics are also discussed in terms of the local skin friction coefficient and local Nusselt number.

Figure 4 depicts the influence of the power-law index *n* on the velocity profile for pseudoplastic (*n* <1) and dilatant (*n* >1) regimes for nonlinearly stretching sheet. The lower is the value of *n*, greater is the degree of shear thinning. Figure 4(a) shows that the velocity profile and momentum boundary layer thickness decrease with an increase in the value of the power-law index *n*, owing to increase in apparent viscosity. The shear thickening behavior (*n* >1) is illustrated in figure 4(b). This figure reveals that as the value of the power-law index *n* is progressively incremented, the velocity profile and the corresponding momentum boundary layer thickness decrease due to gradual strengthening of viscous effects.

The heat transfer aspects of the Sisko fluid over a constant surface temperature stretching sheet for shear thinning (*n* <1) and thickening (*n* >1) fluids for different values of the power-law index *n* with nonlinear stretching is illustrated in figure 5. Figure 5(a) depicts that the temperature profile and thermal boundary layer reduce with incrementing the value of the power-law index *n*. The effect on the temperature profile is marginalized when the power-law index approaches unity. Figure 5(b) reveals that the power-law index *n* does not affect the temperature profile strongly for *n* >1. However, a slight decrease in the thermal boundary layer thickness is observed.

The stretching parameter *s* affects the temperature distribution and thermal boundary layer due to the influence of momentum transfer. Its effect on heat transfer is illustrated in figures 6 (a-c). For the power-law index (*n* <1), stretching parameter *s* does not affects the heat transfer in the Sisko fluid very strongly. The thermal boundary layer thickness increases when value of *s* is incremented progressively (figure 6(a)). Although, for *n* =1, the effect of *s* on the temperature profile is significant and a contrary behavior is noticed as figure 6(b) elucidates. Figure 6(c) represents the temperature profiles for *n* =1.5. The temperature profile seems to decrease as value of the stretching parameter is incremented. Moreover, it is also noticed that larger the value of the power-law index *n* the more is decrease in the temperature profile.

**6. Validation of numerical results**

352 Heat Transfer Studies and Applications

confidence to the accuracy of the numerical results.

skin friction coefficient and local Nusselt number.

**7. Results and discussion**

observed.

The validation of present results is essential to check the credibility of the numerical solution methodology. The presently computed results are compared with the exact solutions obtained for some limiting cases of the problem. Figures 2 and 3 compare these results, and an excellent correspondence is seen to exist between the two sets of data. In addition, table 1 shows the comparison values of the local skin friction coefficient with those reported by of Khan and Shahzad [22]. It is seen that the comparison is in very good agreement, and thus gives us

The main focus of the present chapter is to study the flow and heat transfer characteristics of a Sisko fluid over a nonlinear radially stretching sheet. To obtain physical insight of the flow and heat transfer, Eqs. (22) and (29) subject to boundary conditions (23) and (30) are solved numerically and the results are illustrated graphically. During the ensuing discussion, the assumption of incompressibility and isotropy of fluid is implicit. The influence of the flow behavior index *n*, the material parameter *A*, stretching parameter *s*, and Eckert number *Ec* on flow and heat transfer is the main interest of the study. Further, the effect of variation of Prandtl number Pr on heat transfer is also analyzed in depth. A comparison amongst the flow and heat transfer aspects of the Bingham, Newtonian, and Sisko fluids is also precisely depicted. Moreover, the flow and heat transfer characteristics are also discussed in terms of the local

Figure 4 depicts the influence of the power-law index *n* on the velocity profile for pseudoplastic (*n* <1) and dilatant (*n* >1) regimes for nonlinearly stretching sheet. The lower is the value of *n*, greater is the degree of shear thinning. Figure 4(a) shows that the velocity profile and momentum boundary layer thickness decrease with an increase in the value of the power-law index *n*, owing to increase in apparent viscosity. The shear thickening behavior (*n* >1) is illustrated in figure 4(b). This figure reveals that as the value of the power-law index *n* is progressively incremented, the velocity profile and the corresponding momentum boundary

The heat transfer aspects of the Sisko fluid over a constant surface temperature stretching sheet for shear thinning (*n* <1) and thickening (*n* >1) fluids for different values of the power-law index *n* with nonlinear stretching is illustrated in figure 5. Figure 5(a) depicts that the temperature profile and thermal boundary layer reduce with incrementing the value of the power-law index *n*. The effect on the temperature profile is marginalized when the power-law index approaches unity. Figure 5(b) reveals that the power-law index *n* does not affect the temperature profile strongly for *n* >1. However, a slight decrease in the thermal boundary layer thickness is

The stretching parameter *s* affects the temperature distribution and thermal boundary layer due to the influence of momentum transfer. Its effect on heat transfer is illustrated in figures

layer thickness decrease due to gradual strengthening of viscous effects.

The effect of the material parameter *A* on the temperature profile for nonlinear stretching is presented through figures 7(a-c). These figures also make a comparison amongst the temper‐ ature profiles of the Newtonian fluid (*A*=0 and *n* =1) and the power-law fluid (*A*=0 and *n* ≠1) with those of the Sisko fluid (*A*≠0 and *n* ≠0). Qualitatively, figures 7(a-c) reveal that the temperature profile and the corresponding thermal boundary layer thickness reduce in each case with increasing value of the material parameter *A*.

The Prandtl number Pr of a fluid plays a significant role in forced convective heat transfer. Figures 8(a,b) present its effect on heat transfer to Sisko fluid for pseudo-plastic (*n* <1) and dilatant (*n* >1) regimes. These figures depict that the Pr affects the heat transfer process strongly by thinning the thermal boundary layer thickness. It in turn augments the heat transfer at the wall. The augmentation can be ascribed to the enhanced momentum diffusivity for larger Prandtl number. The temperature profile is slightly lower for fluids with shear thickening behavior than that of the shear thinning, for the same Prandtl number.

Eckert number *Ec* measures the transformation of kinetic energy into heat by viscous dissipa‐ tion. The variation of temperature with increasing *Ec* is given in figures 9(a,b). Figure 9(a) describes the effect of increasing *Ec* on highly shear thinning (*n* =0.2) and moderately shear thinning (*n* =0.6) Sisko fluids. These figures clearly elucidate that the temperature profile increases with increasing *Ec*. Further the strongly shear thinning fluids are dominantly affected by *Ec* as compared to that of the moderately shear thinning regime. Figure 9(b) describes the same phenomenon for shear thickening fluid (*n* >1), but here the effects are less prominent.

Figures 10(a,b) compare the velocity and temperature profiles of the Bingham (*n* =0 and *A*≠0) and Newtonian (*n* =1 and *A*=0) fluids with those of the Sisko fluid (*n* ≠0 and *A*≠0). While sketching figure 10(a), the value of material parameter *A* is adjusted at 1.5. This figure clearly shows that for the particular value of variable *η*, the velocity profile of the Bingham fluid substantially higher than those of Newtonian and Sisko fluids. This figure also delineates that the velocity variation of the Bingham fluid approaches free stream velocity for larger values of *η* as compared to those the Newtonian and Sisko fluids. The temperature profiles of three different fluids are demonstrated in figure 10(b). The Bingham fluid shows larger thermal boundary layer thickness, whilst that of the Sisko fluid least, resulting better heat transfer. A wider variation in the concomitant heat transfer slopes at wall is observed in this figure, with least slop for the Bingham fluid, showing minimum heat transfer at the wall.

Adiabatic Eckert number *Eca* is a measure of direction of heat flux *qx* ; the heat flows from heated sheet to fluid (*qx* >0), when *Ec* <*Eca*, and vice versa. Figures 11(a,b) shows a series of adiabatic Eckert numbers evaluated numerically for shear thinning (*n* =0.5) and shear thick‐

ening (*n* =1.5) Sisko fluids. It is clearly noticed from these figures that the adiabatic Eckert number increases at an accelerated pace for smaller values of Prandtl number. Further inspection of these figures reveal that the values of *Eca* decrease at a rapid rate for shear thinning fluid as compared to that of the shear thickening fluid. Adiabatic Eckert number *Eca* is a measure of direction of heat flux *<sup>x</sup> q* ; the heat flows from heated sheet to fluid ( 0) *<sup>x</sup> q* , when *Ec Ec <sup>a</sup>* , and vice versa. Figures 11(a,b) shows a series of adiabatic Eckert numbers evaluated numerically for shear

11

Table 2 summarizes the overall trends of the skin friction coefficient for shear thinning and thickening fluids when the material parameter *A*, stretching parameter *s* and Eckert number *Ec* are varied. This table reveals that the value of the skin friction increases with each increment in the value of the material parameter *A* for linear as well as nonlinear stretch‐ ing of the sheet, which results in increased drag to the Sisko fluid. The drag is slightly lower for the higher value of the power-law index. Further, this table also depicts the variation in the local Nusselt number with the increasing value of *A* for linear and nonlinear stretching. The Nusselt number shows improvement with each increment of *A*. It is also clear that the improvement is better for the power-law index *n* >1. Moreover, a decrease in the local Nusselt number is observed with an increase in *Ec*. It is further noticed that the decrease in heat transfer from wall to fluid is about 24*%* for shear thinning and 13*%* for shear thickening fluids. Table 3 demonstrates the effects of the Prandtl number Pr on the local Nusselt number for different values of the power-law index *n*. An increase in the Prandtl number augments the Nusselt number at the sheet. Moreover, an increase in the Nusselt number is larger for fluids with medium power-law index *n*. thinning ( 0.5) *n* and shear thickening ( 1.5) *n* Sisko fluids. It is clearly noticed from these figures that the adiabatic Eckert number increases at an accelerated pace for smaller values of Prandtl number. Further inspection of these figures reveal that the values of *Eca* decrease at a rapid rate for shear thinning fluid as compared to that of the shear thickening fluid. Table 2 summarizes the overall trends of the skin friction coefficient for shear thinning and thickening fluids when the material parameter *A* , stretching parameter*s* and Eckert number *Ec* are varied. This table reveals that the value of the skin friction increases with each increment in the value of the material parameter *A* for linear as well as nonlinear stretching of the sheet, which results in increased drag to the Sisko fluid. The drag is slightly lower for the higher value of the power-law index. Further, this table also depicts the variation in the local Nusselt number with the increasing value of *A* for linear and nonlinear stretching. The Nusselt number shows improvement with each increment of *A*. It is also clear that the improvement is better for the power-law index *n* 1. Moreover, a decrease in the local Nusselt number is observed with an increase in *Ec* . It is further noticed that the decrease in heat transfer from wall to fluid is about 24% for shear thinning and 13% for shear thickening fluids. Table 3 demonstrates the effects of the Prandtl number Pr on the local Nusselt number for different values of the

The stream function appearing in Eq. (20) is plotted in figure 12 for several values of *ψ* and different values of the power-law index *n*. The streamlines are symmetrical about the midway vertical axis, owing to the fact that sheet is being stretched radially by applying equal force in each direction. Moreover, the flow seems nearly straight down at large values of *η* (away from the sheet) and tends to horizontal at small values of *η* (vicinity of the stretching sheet). This figure further describes that the flow become identical for each value of the power-law index *n* close to the sheet. power-law index *n*. An increase in the Prandtl number augments the Nusselt number at the sheet. Moreover, an increase in the Nusselt number is larger for fluids with medium power-law index *n*. The stream function appearing in Eq. (20) is plotted in figure 12 for several values of and different values of the power-law index *n* . The streamlines are symmetrical about the midway vertical axis, owing to the fact that sheet is being stretched radially by applying equal force in each direction. Moreover, the flow seems nearly straight down at large values of (away from the sheet) and tends to horizontal at small values of (vicinity of the stretching sheet). This figure further describes that the flow

become identical for each value of the power-law index *n* close to the sheet.

*Figure 2.* A comparison of the exact and numerical results (solid line exact results and open circles numerical results) when *n s* 0 and 1 are fixed. **Figure 2.** A comparison of the exact and numerical results (solid line exact results and open circles numerical results) when *n* =0 and *s* =1 are fixed.

ening (*n* =1.5) Sisko fluids. It is clearly noticed from these figures that the adiabatic Eckert number increases at an accelerated pace for smaller values of Prandtl number. Further inspection of these figures reveal that the values of *Eca* decrease at a rapid rate for shear

 Adiabatic Eckert number *Eca* is a measure of direction of heat flux *<sup>x</sup> q* ; the heat flows from heated sheet to fluid ( 0) *<sup>x</sup> q* , when *Ec Ec <sup>a</sup>* , and vice versa. Figures 11(a,b) shows a series of adiabatic Eckert numbers evaluated numerically for shear thinning ( 0.5) *n* and shear thickening ( 1.5) *n* Sisko fluids. It is clearly noticed from these figures that the adiabatic Eckert number increases at an accelerated pace for smaller values of Prandtl number. Further inspection of these figures reveal that

11

Table 2 summarizes the overall trends of the skin friction coefficient for shear thinning and thickening fluids when the material parameter *A*, stretching parameter *s* and Eckert number *Ec* are varied. This table reveals that the value of the skin friction increases with each increment in the value of the material parameter *A* for linear as well as nonlinear stretch‐ ing of the sheet, which results in increased drag to the Sisko fluid. The drag is slightly lower for the higher value of the power-law index. Further, this table also depicts the variation in the local Nusselt number with the increasing value of *A* for linear and nonlinear stretching. The Nusselt number shows improvement with each increment of *A*. It is also clear that the improvement is better for the power-law index *n* >1. Moreover, a decrease in the local Nusselt number is observed with an increase in *Ec*. It is further noticed that the decrease in heat transfer from wall to fluid is about 24*%* for shear thinning and 13*%* for shear thickening fluids. Table 3 demonstrates the effects of the Prandtl number Pr on the local Nusselt number for different values of the power-law index *n*. An increase in the Prandtl number augments the Nusselt number at the sheet. Moreover, an increase in the

 Table 2 summarizes the overall trends of the skin friction coefficient for shear thinning and thickening fluids when the material parameter *A* , stretching parameter*s* and Eckert number *Ec* are varied. This table reveals that the value of the skin friction increases with each increment in the value of the material parameter *A* for linear as well as nonlinear stretching of the sheet, which results in increased drag to the Sisko fluid. The drag is slightly lower for the higher value of the power-law index. Further, this table also depicts the variation in the local Nusselt number with the increasing value of *A* for linear and nonlinear stretching. The Nusselt number shows improvement with each increment of *A*. It is also clear that the improvement is better for the power-law index *n* 1. Moreover, a decrease in the local Nusselt number is observed with an increase in *Ec* . It is further noticed that the decrease in heat transfer from wall to fluid is about 24% for shear thinning and 13% for shear thickening fluids. Table 3 demonstrates the effects of the Prandtl number Pr on the local Nusselt number for different values of the power-law index *n*. An increase in the Prandtl number augments the Nusselt number at the sheet. Moreover, an increase in the

the values of *Eca* decrease at a rapid rate for shear thinning fluid as compared to that of the shear thickening fluid.

The stream function appearing in Eq. (20) is plotted in figure 12 for several values of *ψ* and different values of the power-law index *n*. The streamlines are symmetrical about the midway vertical axis, owing to the fact that sheet is being stretched radially by applying equal force in each direction. Moreover, the flow seems nearly straight down at large values of *η* (away from the sheet) and tends to horizontal at small values of *η* (vicinity of the stretching sheet). This figure further describes that the flow become identical for each value of the power-law index

power-law index *n* . The streamlines are symmetrical about the midway vertical axis, owing to the fact that sheet is being stretched

*Figure 2.* A comparison of the exact and numerical results (solid line exact results and open circles numerical results) when *n s* 0 and 1 are fixed. **Figure 2.** A comparison of the exact and numerical results (solid line exact results and open circles numerical results)

radially by applying equal force in each direction. Moreover, the flow seems nearly straight down at large values of

(vicinity of the stretching sheet). This figure further describes that the flow

and different values of the

(away from

thinning fluid as compared to that of the shear thickening fluid.

Nusselt number is larger for fluids with medium power-law index *n*.

The stream function appearing in Eq. (20) is plotted in figure 12 for several values of

Nusselt number is larger for fluids with medium power-law index *n*.

become identical for each value of the power-law index *n* close to the sheet.

*n* close to the sheet.

354 Heat Transfer Studies and Applications

when *n* =0 and *s* =1 are fixed.

the sheet) and tends to horizontal at small values of

*Figure 3.* A comparison of the exact and numerical results (solid line exact results and open circles numerical results) when *n s* 1 and 3 are fixed. **Figure 3.** A comparison of the exact and numerical results (solid line exact results and open circles numerical results) when *n* =1 and *s* =3 are fixed. *Figure 3.* A comparison of the exact and numerical results (solid line exact results and open circles numerical results) when *n s* 1 and 3 are fixed. *Figure 3.* A comparison of the exact and numerical results (solid line exact results and open circles numerical results) when *n s* 1 and 3 are fixed.

fixed. fixed. **Figure 4.** The velocity profile *f* ′ (*η*) for different values of the power-law index *n* when *s* =1.5 and *A*=1.5 are fixed. fixed.

for different values of the power-law index *n* when *s* 1.5 and *A* 1.5 are

for different values of the power-law index *n* when *s* 1.5 and *A* 1.5 are

*Figure 4.* The velocity profile *f* ( )

*Figure 5.* The temperature profile for different values of the power-law index *n* when *s* 1.5 , *Figure 5.* The temperature profile for different values of the power-law index *n* when *s* 1.5 , *Figure 5.* The temperature profile for different values of the power-law index *n* when *<sup>s</sup>* 1.5 , **Figure 5.** The temperature profile *θ*(*η*) for different values of the power-law index *n* when *<sup>s</sup>* =1.5, *A*=1.5, Pr=2.0, *Ec* =0.1 and *Br* =0.1 are fixed.

A = = 1, Pr 2.0 , Ec = 0.1 and Br = 0.1 are fixed.

13

Figure 6. The temperature profileθ η( ) for different values of the stretching parameter s when A = = 1.5, Pr 2.0 , Ec = 0.1 and Br = 0.1 are fixed. **Figure 6.** The temperature profile *θ*(*η*) for different values of the stretching parameter *s* when *A*=1.5, Pr=2.0, *Ec* =0.1 and *Br* =0.1 are fixed.

13

A = = 1, Pr 2.0 , Ec = 0.1 and Br = 0.1 are fixed.

Figure 6. The temperature profileθ η( ) for different values of the stretching parameter s when A = = 1.5, Pr 2.0 , Ec = 0.1 and Br = 0.1 are fixed. **Figure 6.** The temperature profile *θ*(*η*) for different values of the stretching parameter *s* when *A*=1.5, Pr=2.0,

356 Heat Transfer Studies and Applications

*Ec* =0.1 and *Br* =0.1 are fixed.

Figure 7. The temperature profile θ η( ) for different values of the material parameter A when s = = 1.5, Pr 2.0 , **Figure 7.** The temperature profile *θ*(*η*) for different values of the material parameter *A* when *s* =1.5, Pr=2.0, *Ec* =0.1 and *Br* =0.1 are fixed.

Ec = 0.1 and Br = 0.1 are fixed.

*Figure 9.* Temperature profiles

 

*Figure 7.* The temperature profile

 

*Ec* 0.1 and *Br* 0.1 are fixed.

for different values of the material parameter *A* when *s* 1.5, Pr 2.0 ,

14

*Figure 8.* Temperature profile ( ) for different values of the Prandtl number Pr when *s A* 1.5 , and *Ec Br* 0.1 are fixed. **Figure 8.** Temperature profile *θ*(*η*) for different values of the Prandtl number Pr when *s* = *A*=1.5, and *Ec* = *Br* =0.1 are fixed. <sup>15</sup> 15

and Pr 2.0 are fixed. **Figure 9.** Temperature profiles *θ*(*η*) for different values of the Eckert number *Ec* when *s* =1.5, *A*=1.5, *Br* =0.1, and Pr=2.0 are fixed. *Figure 9.* Temperature profiles ( ) for different values of the Eckert number *Ec* when *<sup>s</sup>* 1.5, *<sup>A</sup>* 1.5, *Br* 0.1 , and Pr 2.0 are fixed.

( ) for different values of the Eckert number *Ec* when *s* 1.5, *A* 1.5, *Br* 0.1 ,

*Br* 0.1, and Pr 2.0 are fixed. **Figure 10.** A comparison among the velocity and temperature profiles for different fluids when *s* =1.5, *Ec* =0.1, *Br* =0.1, and Pr=2.0 are fixed.

*Figure 11.* Adiabatic Eckert number *Eca* with variation of Prandtl number Pr when *s* 1.5, *Br* 0.1 , and *A* 1.0 are fixed.

*Figure 11.* Adiabatic Eckert number *Eca* with variation of Prandtl number Pr when *s* 1.5, *Br* 0.1 , and *A* 1.0 are fixed.

*Br* 0.1, and Pr 2.0 are fixed.

*Figure 10.* A comparison among the velocity and temperature profiles for different fluids when *s* 1.5, *Ec* 0.1 ,

( ) for different values of the Eckert number *Ec* when *s* 1.5, *A* 1.5, *Br* 0.1 ,

15

*Figure 10.* A comparison among the velocity and temperature profiles for different fluids when *s* 1.5, *Ec* 0.1 , *Br* 0.1, and Pr 2.0 are fixed.

and Pr 2.0 are fixed.

*Figure 9.* Temperature profiles

 

14

15

*Figure 7.* The temperature profile

358 Heat Transfer Studies and Applications

*Figure 8.* Temperature profile

*Figure 9.* Temperature profiles

*Figure 9.* Temperature profiles

*Br* =0.1, and Pr=2.0 are fixed.

and Pr=2.0 are fixed.

 

   

 

for different values of the material parameter *A* when *s* 1.5, Pr 2.0 ,

( ) for different values of the Prandtl number Pr when *s A* 1.5 , and

( ) for different values of the Eckert number *Ec* when *s* 1.5, *A* 1.5, *Br* 0.1 ,

( ) for different values of the Eckert number *Ec* when *<sup>s</sup>* 1.5, *<sup>A</sup>* 1.5, *Br* 0.1 , and Pr 2.0 are fixed.

*Ec* 0.1 and *Br* 0.1 are fixed.

*Ec Br* 0.1 are fixed. **Figure 8.** Temperature profile *θ*(*η*) for different values of the Prandtl number Pr when *s* = *A*=1.5, and *Ec* = *Br* =0.1 are fixed. <sup>15</sup>

and Pr 2.0 are fixed.

**Figure 9.** Temperature profiles *θ*(*η*) for different values of the Eckert number *Ec* when *s* =1.5, *A*=1.5, *Br* =0.1,

*Figure 10.* A comparison among the velocity and temperature profiles for different fluids when *s* 1.5, *Ec* 0.1 , *Br* 0.1, and Pr 2.0 are fixed.

*Figure 10.* A comparison among the velocity and temperature profiles for different fluids when *s* 1.5, *Ec* 0.1 , *Br* 0.1, and Pr 2.0 are fixed.

**Figure 10.** A comparison among the velocity and temperature profiles for different fluids when *s* =1.5, *Ec* =0.1,

*Figure 11.* Adiabatic Eckert number *Eca* with variation of Prandtl number Pr when *s* 1.5, *Br* 0.1 , and *A* 1.0 are fixed.

*Figure 11.* Adiabatic Eckert number *Eca* with variation of Prandtl number Pr when *s* 1.5, *Br* 0.1 , and *A* 1.0 are fixed.

*Figure 11.* Adiabatic Eckert number *Eca* with variation of Prandtl number Pr when *s* 1.5, *Br* 0.1 , and *A* 1.0 are fixed. **Figure 11.** Adiabatic Eckert number *Eca* with variation of Prandtl number Pr when *<sup>s</sup>* =1.5, *Br* =0.1, and *A*=1.0 are fixed.

**Figure 12.** The streamlines for different values of the power-law index *n* when *s* =1.5 and *A*=1.5 are fixed.


**Table 1.** A tabulation of the local skin friction coefficient in terms of the comparison between the present results and the HAM results (ref. [22])


**Table 2.** A tabulation of the local skin friction and the local Nusselt number.


**Table 3.** A tabulation of the local Nusselt number for different values of Prandtl number when *s* =1.5 and *A*=1.5 are fixed.
