**5. Discussion**

To study the influence of many embedded parameters on temperature and velocity, the graphs are plotted by using the Mathcad-15 software. Figure 1 shows the physical sketch of the problem. Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 are prepared to highlight the influence of time-fractional parameter 0 <α< 1 on fluid flow and temperature. The time-fractional parameter α controls the velocity as well as the temperature profile. Sodium alginate (SA–NaAlg) is taken as a specific example of Casson fluid. The physical sketch of the problem is provided in Figure 1.

**Figure 1.** Physical sketch of the problem with the coordinates system.

Figure 2 is plotted for the influence of Casson parameter γ on velocity, which shows that if the value of the Casson parameter is increased, the fluid velocity increases. This is because of the circumstance that with a large value of γ, the yield stress falls through, and the boundary layer thickness reduces. Figure 3 investigates the impact of Grashof number *Gr* on velocity. The greater value *Gr* leads to an increase in the velocity of the fluid. Physically, the increase of *Gr* leads to an increase in the bouncy force, and as a result, the velocity of the fluid increases.

**Figure 2.** Plots of velocity for four different γ when *K* = 0.5, Pr = 7.2, *M* = 1, *Gr* = 10, *N* = 0.5 , and *t* = 10.

**Figure 3.** Plots of velocity for four different *Gr* when *K* = 0.5, Pr = 7.2, γ = 0.3, *M* = 1, *N* = 0.5 , and *t* = 10.

The effect of porosity parameter *K* against the velocity profile is investigated in Figure 4. To increase the value of the porosity parameter *K*, first we need to decrease the flow of fluid. Physically, the resistance of the porous medium is depressed, which raises the momentum development of the flow regime, and finally accelerates the velocity of the fluid. Figure 5 shows the influence of *M* on flow of the fluid; the rise in *M* decreases the flow of fluid. It is physically true because the increase of *M* means to increase the frictional force (Lorentz force), which leads to a decrease in the velocity of the fluid.

**Figure 4.** Plots of velocity for four different *K* when Pr = 7.2, γ = 0.3, *M* = 1, *Gr* = 10.*N* = 0.5 , and *t* = 10.

**Figure 5.** Plots of velocity for four different *M* values when *K* = 0.5, Pr = 7.2, γ = 0.3, *Gr* = 10, *N* = 0.5 , and *t* = 10.

The influence of radiation parameter *N* is highlighted in Figure 6. For the higher value of *N*, the fluid velocity decreases. Physically, the increase in radiation parameter means the release of heat energy from the flow region, and so the fluid temperature decreases as the thermal boundary layer thickness become thinner. Figures 7 and 8 illustrate the influence of Prandtl number Pr on velocity and temperature respectively, the increase of Pr causing a decrease in the temperature and as a result a decrease in the fluid velocity. The small degree of thermal diffusion causes expanding in velocity boundary layer width. Pr controls the comparative thickness of the momentum and thermal boundary layers in the heat transfer problems. Subsequently, Pr can be applied to develop the percentage of cooling.

**Figure 6.** Plots of velocity for four different*N* values when*K* = 0.5, Pr = 7.2, γ = 0.3, *M* = 1, *Gr* = 10, and *t* = 10.

**Figure 7.** Plots of velocity for four different Pr values when *K* = 0.5, γ = 0.3, *M* = 1, *Gr* = 10, *N* = 0.5 , and *t* = 10.

**Figure 8.** Plots of temperature for four different Pr values when *N* = 0.5, α = 0.1 and *t* = 10.

Figures 9 and 10 illustrate the outcome of time *t* on temperature and velocity profiles. Figure 11 investigates the effect of time fraction derivative parameter α on temperature. It is investigated that the time-fractional derivative parameter controls the temperature profile.

**Figure 9.** Plots of velocity for four different *t* values when *K* = 0.5, Pr = 7.2, γ = 0.3, *M* = 1, *Gr* = 10, and *N* = 0.5.

**Figure 10.** Plots of temperature for four different *t* values when Pr = 7.2, α = 0.1, and *N* = 0.5 .

**Figure 11.** Plots of temperature for four different α values when Pr = 7.2, *N* = 0.5, and *t* = 10 .

A comparison of the Atangana–Baleanu fractional model with an ordinary model is investigated in Figures 12 and 13 for velocity and temperature, respectively. For both cases, it is detected that the temperature and velocity profile for the Atangana–Baleanu fractional model is less than that of the ordinary model.

Note that all these graphs of velocity are plotted for the phase angle ω*t* equal to 90 degrees, and this value cosine is zero; therefore, all the graphs of velocity (Figures 2–7, Figure 9, Figure 12) have a unique pattern of velocity. That is, at the plate surface y = 0, the fluid is at rest or there is no motion in the fluid, and for large values of the independent variable y, the fluid velocity decays, and as y approaches infinity (further bigger values of y), the fluid motion disappears and velocity tends to zero. This physical pattern of the graphs of velocity agrees with the imposed condition on velocity given in Equation (6). Similarly, the unique style of all the graphs of temperature, that is temperature at y = 0, is 1, and far away from the plate surface; that is, for larger values of y, the temperature decays and tends to zero as y tends to infinity.

**Figure 12.** Comparison of fractional SA fluid and ordinary SA fluid (velocity) when *K* = 0.5, Pr = 7.2, γ = 0.3, *M* = 1, *Gr* = 10 *N* = 0.5 , and *t* = 10.

**Figure 13.** Comparison of fractional SA fluid and ordinary SA fluid (temperature) when Pr = 7.2, α = 0.1, *N* = 0.5 , and *t* = 10.

Tables 1 and 2 are plotted to show the variation of different parameters on Nusselt number and skin fraction. Table 1 clarified that the Nusselt number is increased when Pr, and *N* and are increased, while an increase in fractional derivative parameter α and *t* decreased the Nusselt number. The behavior of the present results is identical with the published results of Khan et al. [49] and Ali et al. [50]. Table 2 shows the impact of the deferent parameter on skin fraction. It is observed that *t*, α, *Gr*, and Pr have a positive (increasing) impact on the skin fraction, while *M*, *N*, γ, and *K* and show a negative (decreasing) impact on the skin fraction. This behavior of skin fraction against different parameters is identical with the published results of Mackolil and Mahanthesh [51].


**Table 1.** Variation of Nusselt number.


**Table 2.** Variation of skin-friction.
