Visualization of non-Newtonian convective fluid flow with internal heat transfer through a rotating expandable surface impact of chemical reaction

In this work, we studied the mixed convective flow of a non-Newtonian Jeffrey fluid on a stretchable surface. The problem is modeled and then solved numerically using the Runge-Kutta fourth-order method with the shooting technique. The physical model of the problem is shown in Fig. 1. Numerical results are calculated in terms of plots and table. Figures 2 and 3, are discussed in the “Numerical procedure and code validation” section. These plots depict the effects of various variables on the motion and concentration fields, such as the Hartman number, Schmidt number, and chemical reaction parameter. On (f^{prime } {text{ and }}g^{prime })the performance of (Omega_{1}) and (M_{1}) is the same. The thickness of the boundary layer is reduced when the value of (M_{1}) is increased. When Sc increases, the concentration fields (T_{0}) and (T_{1}) decline. The destructive response ((Upsilon) > 0) has the effect of lowering the concentration profiles. For destructor ((Upsilon) > 0) and generative ((Upsilon) (T_{0})and (T_{1}) observed opposite effects. By raising (M_{1}), the surface mass transfer decreases. Statistical results of the heat transfer rate are also documented and reviewed.

Pictorial results for the influence of Deborah numbers (Omega_{1} ,Omega_{2})Hartmann’s number (M) Schmidt number (Se) and the chemical reaction factor (Upsilon) on (f^{prime}{text{ and }}g^{prime})are presented in this section. Figures 4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24 and 25 give examples of such impacts with physical amplifications.

Figure 4

Consequence of (Omega_{1}) on (f^{first }).

Figure 5
number 5

Consequence of (Omega_{1}) on (g^{prime }).

Figure 6
number 6

Consequence of (Omega_{2}) on (f^{first }).

Picture 7
number 7

Consequence of (Omega_{2}) on (g^{prime }).

Figure 8
figure 8

Consequence of (M) on (f^{first }).

Figure 9
number 9

Consequence of (M) on (g^{prime }).

Picture 10
number 10

Consequence of (Omega_{1}) on (T_{0} (eta )).

Picture 11
figure 11

Consequence of (Omega_{1}) on (T_{1} (eta )).

Picture 12
figure 12

Consequence of (Omega_{2}) on (T_{0} (eta )).

Picture 13
figure 13

Consequence of (Omega_{2}) on (T_{1} (eta )).

Picture 14
figure 14

Consequence of (M) on (T_{0} (eta )).

Picture 15
number 15

Consequence of (M) on (T_{1} (eta )).

Picture 16
number 16

Consequence of (Se) on (T_{0} (eta )).

Figure 17
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Consequence of (Se) on (T_{1} (eta )., ).

Picture 18
number 18

Consequence of (Upsilon) on (T_{0} (eta )).

Picture 19
number 19

Consequence of (Upsilon) on (T_{1} (eta )).

Picture 20
number 20

Consequence of (Upsilon) on (T_{0} (eta )).

Picture 21
number 21

Consequence of (Upsilon) on (T_{1} (eta )).

Picture 22
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Consequence of (Sc{text{ and }}Upsilon) on (T_{0}^{first } (0)).

Picture 23
number 23

Consequence of (Sc{text{ and }}Upsilon)on (T_{1}^{first } (0)).

Picture 24
number 24

Consequence of Se and M on (T_{0}^{first } (0)) .

Picture 25
number 25

Consequence of Se and M on (T_{1}^{first } (0)) .

The consequences of (Omega_{1} ,Omega_{2}) and (M) on (f^{prime}{text{ and }}g^{prime}) are examined in Figs. 4, 5, 6 and 7.

Figures 4 and 5 show the flow characteristics for a variable parameter of the Deborah number (Omega_{1})in the presence and absence of Mr. It is detected that, (f^{prime } {text{ and }}g^{prime }) decline through the boundary layer with increasing values ​​of (Omega_{1}) . Speed ​​profiles (f^{prime } {text{ and }}g^{prime })turn out to be decreasing as (Omega_{1}) mounted. When compared to (g^{prime }) however, the resurgence of (f^{first }) is smaller. As Deborah number (Omega_{1}) is improved, the film thickness decreases. In contrast, the Deborah number (Omega_{2}) has an opposite influence on speeds (f^{prime } {text{ and }}g^{prime }) that (Omega_{1}) as shown in Figs. 6 and 7.

From these figures, it is clear that the influence of (Omega_{1}) and (Omega_{2}) on (f^{prime } {text{ and }}g^{prime }) is higher in the absence of M. Figures 8 and 9 show that the inspiration of the Hartman number M on the velocities (f^{prime } {text{ and }}g^{prime }) in the presence and in the absence of the Deborah number. Figure 8 shows the influence of M on the velocity component (f^{first }) which is a function of (eta) . It is observed that the speed decreases as M increases. Also, from Fig. 9, it is quite obvious that the speed (g^{prime })decrease significantly as the M increases. The effect of M on (f^{prime}{text{ and }}g^{prime}) is analogous to that of (Omega_{1}) . From this analysis, it is also investigated that the effect of M for (Omega_{2} = 0.4)is higher than (Omega_{2} = 0.3.)

Figures 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 and 21 show the fluctuations of entrepreneurs on the concentration components (T_{0}) and (T_{1}) . Figures 10 and 11 show the impact of Deborah’s number (Omega_{1}) on (T_{0}) and (T_{1}) in the event of a destructive chemical change ((Upsilon) > 0). Each time the value of (Omega_{1})is higher, the amplitude of the concentration profile (T_{0}) increases, while the scale of (T_{1})drops. Note that for high values ​​of (Omega_{1}) the variance of (T_{1}) is higher than that of (T_{0}) .

In the event of a destructive chemical reaction ((Upsilon) > 0), fig. 12 and 13 explore the fluctuations of (Omega_{2}) on concentration fields (T_{0}) and (T_{1}) for two different values ​​of M = 0 and M = 0.3. When linking Figs. 10 and 11, it can be seen that Figs. 12 and 13 have opposite qualitative consequences. Moreover, for M = 0, the concentration fields decrease rapidly compared to M = 0.3.

The consequence of Hartmann’s number (M) on (T_{0}) and (T_{1}) is seen in Figs. 14 and 15 respectively in the presence of the numbers Deborah (Omega_{1} = 0.3{text{ and }}Omega_{2} = 0.4.)As (M) is reinforced, the concentration fields (T_{0})and (T_{1}) are found to increase. In both cases, the improvement is more significant for a large value of the Deborah number, i.e. for (Omega_{2} = 0.4) .

Figures 16 and 17 show how the Schmidt number Sc varies with concentration profiles (T_{0}) and (T_{1}). When Sc increases, respectively,(T_{0}) and (T_{1}) drop. Figures 18 and 19 illustrate the impact of the interfering chemical reaction factor ((Upsilon)> 0) on profiles (T_{0}) and (T_{1}) . The concentration profile (T_{0}) and (T_{1}) turn out to be a decreasing function of (Upsilon) . It is also evident that, as (Upsilon) improves, the strength of (T_{0}) and (T_{1}) reduce.

Figures 20 and 21 show how the concentration profiles (T_{0}) and (T_{1}) change as a generating chemical reaction ((Upsilon) (T_{0}) increases for large generating chemical reaction parameters, as shown in Fig. 18. The amplitude of (T_{1}) grows like (Upsilon) ((Upsilon)

The effect of Sc and (Upsilon) on mass transfer (T_{0}^{first } (0)) and (T_{1}^{first } (0)) is shown in Figs. 22 and 23. It is revealed that surface mass transfer (T_{0}) improves while (T_{1}) decreases as Sc and (Upsilon) increase. Figures 24 and 25 show the influence of (M) and Sc on the (T_{0}^{first } (0))and (T_{1}^{first } (0).)From these figures, it is clear from this figure that, as (M)increases, the transfer of surface mass (T_{0})decreases and (T_{1}) improved.

Table 1 shows the variation in surface mass transfer (- T_{0}^{first } (0))and (- T_{1}^{first } (0))for some values ​​of (Se) , (Upsilon), (M) and (Omega_{1} = Omega_{2}). From this table, it is observed that the mass transfer values (- T_{0}^{first } (0))increases for (Se) , (Upsilon)and (Omega_{1} = Omega_{2}) while decreases to increase (M). Likewise, the amount of (- T_{1}^{first } (0))increases with increasing values ​​of (Se), (M)and (Omega_{1} = Omega_{2})and is reduced for large values ​​of (Upsilon). For confirmation of the calculation of the current works, the existing works are also compared with the published works reported by Liao21 for mass transfer (- T_{0}^{first } (0))and (- T_{1}^{first } (0))and excellent agreement is found as shown in the table below.

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