It is highly unlikely to find an analytical solution of the coupled nonlinear equations; therefore, a numerical solutions for the velocity, microrotation, and temperature along with the anisotropic slip have been obtained using the numerical method discussed in previous section with the purpose of giving the detailed behavior of flow fields and thermal distribution. In this section, the influence of main controlling parameters present in the governing equations is discussed and the graphical representations of entropy generation and Bejan numbers have been presented to observe the effect of pertinent parameters. The considered mathematical problem defined in Eqs. (21)-(26) with the boundary conditions in Eqs. (27)-(28) have been solved by a numerical method (bvp4c) for a suitable range of the values of fluid parameters like the vortex viscosity parameter *C*_{1}, spin gradient viscosity parameter *C*_{2}, microinertia density parameter *C*_{3}, Reynolds number *Re*, Prandtl number Pr, Brinkman number *Br*, radial slip *ζ* and tangential slip *η*. The effects of theses parameters on velocity, microrotation, and temperature fields have been presented graphically through Figures 2-11. All the calculations are taken at high Reynolds number, *i.e. Re* = 2950.

Figure 2 Effect of *C*_{1} on *F* (*Λ*) keeping *ζ* = *η* = 0.2, *C*_{2} = *C*_{3} = 1, Pr = 6.8, *Br* =1 and *Re* = 2950

Figure 3 Effect of *C*_{1} on *G* (*Λ*) keeping *ζ* = *η* = 0.2, *C*_{2} = *C*_{3} = 1, Pr = 6.8, *Br* =1 and *Re* = 2950

Figure 4 Effect of *C*_{1} on −*H* (*Λ*) keeping *ζ* = *η* = 0.2, *C*_{2} = *C*_{3} = 1, Pr = 6.8, *Br* =1 and *Re* = 2950

Figure 5 Effect of *C*_{1} on *M* (*Λ*) keeping *ζ* = *η* = 0.2, *C*_{2} = *C*_{3} = 1, Pr = 6.8, *Br* =1 and *Re* = 2950

Figure 6 Effect of *C*_{1} on −*P* (*Λ*) keeping *ζ* = *η* = 0.2, *C*_{2} = *C*_{3} = 1, Pr = 6.8, *Br* = 1 and *Re* = 2950

Figure 7 Effect of *C*_{1} on *θ* (*Λ*) keeping *ζ* = *η* = 0.2, *C*_{2} = *C*_{3} = 1, Pr = 6.8, *Br* = 1 and *Re* = 2950

Figure 8 Effect of *η* on *F* (*Λ*) keeping *ζ* = 0.2, *C*_{1} = *C*_{2} = *C*_{3} = 20, Pr = 6.8, *Br* = 1 and *Re* = 2950

Figure 9 Effect of *η* on −*H* (*Λ*) keeping *ζ* = 0.2, *C*_{1} = *C*_{2} = *C*_{3} = 20, Pr = 6.8, *Br* = 1 and *Re* = 2950

Figure 10 Effect of *ζ* on *F* (*Λ*) keeping *η* = 0.2, *C*_{1} = *C*_{2} = *C*_{3} = 50, Pr = 6.8, *Br* =1 and *Re* = 2950

Figure 11 Effect of *ζ* on −*H* (*Λ*) keeping *η* = 0.2, *C*_{1} = *C*_{2} = *C*_{3} = 50, Pr = 6.8, *Br* =1 and *Re* = 2950

In Figures 2-7, the effect of the vortex viscosity parameter *C*_{1} on the velocity profiles, microrotation, and temperature distribution have been considered for *ζ* = *η* = 0.2, *C*_{2} = 1, *C*_{3} = 1, Pr = 6.8, *Br* =1 and *Re* = 2950. Figure 2 illustrates the influence of *C*_{1} on the radial velocity *F*(*Λ*). The radial velocity field *F* (*Λ*) increases near the disk, then starts to decrease as fluid moves far from the disk and reduces to zero. Figure 3 indicates that the tangential velocity field *G* (*Λ*) exhibits exponentially decaying behavior as the values of *C*_{1} rise. Figure 4 shows the influence of *C*_{1} on axial velocity −*H* (*Λ*), which illustrates that the axial velocity −*H* (*Λ*) increases when it moves away from the disk by increasing the values of *C*_{1}. From Figure 5, it is observed that the magnitude of microrotation, *M* (*Λ*), increases with *C*_{1} and these profiles decrease near the boundary and increase far from the boundary. The pressure profiles in von Kármán flows rises near the disk and become constant as they move away from the disk. The pressure profiles increase with an increase in the values of *C*_{1} as depicted in Figure 6. The temperature profiles exhibit the increasing behavior with the higher values of *C*_{1} as displayed in Figure 7.

The tangential velocity, microrotation profiles, pressure, and temperature distributions are not greatly affected by the slip parameters which lead us to the decision to consider the effect of these parameters on radial and axial velocity profiles only, as showed in Figures 8-9.

Figure 8 depicts the influence of tangential slip parameter *η* on radial velocity *F* (*Λ*) in the presence of radial slip *ζ* = 0.2 at *C*_{1} = *C*_{2} = *C*_{3} = 20, Pr = 6.8, *Br* = 1, and *Re* = 2950. It is mentioned formerly that different slip lengths in span and stream-wise directions have opposite effect on fluid velocity. From Figures 8-9, it can be observed that radial and axial velocities decrease in both directions as increasing the values of tangential slip parameter *η*.

In order to analyze the effects of radial slip parameter *ζ* on radial velocity *F* (*Λ*) and axial velocity −*H* (*Λ*) profiles, graphs are plotted in Figures 10-11 in the presence of tangential slip *η* = 0.2 and at *C*_{1} = *C*_{2} = *C*_{3} = 50, Pr = 6.8, *Br* =1 and *Re* = 2950. It is clearly seen that both, the radial and axial velocities increase with an increase in the values of radial slip. The reason behind this phenomenon is that the slip at the disk surface increases as the radial slip parameter *ζ* increases.

Figures 12-13 depict the effect of the spin gradient viscosity parameter *C*_{2} for *C*_{1} = *C*_{3} = 200, and the influence of the microinertia density parameter *C*_{3} for *C*_{1} = *C*_{2} = 400, keeping *Br* = 1, Pr = 6.8, and *Re* = 2950. The influences of these micropolar parameters *C*_{2} and *C*_{3} are observed on the pressure distribution −*P* (*Λ*) in the presence of radial and tangential slips as *ζ* = *η* = 0.2. The spin gradient viscosity parameter *C*_{2} has an inverse effect on pressure distribution−*P* (*Λ*) as shown in Figure 12, because pressure decreases by increasing the values of *C*_{2}, whereas pressure increases as raising the values of microinertia density parameter *C*_{3}.

Figure 12 Effect of *C*_{2} on −*P* (*Λ*) keeping *ζ* = *η* = 0.2, *C*_{1} = *C*_{3} = 200, *Br* = 1, Pr = 6.8 and *Re* = 2950

Figure 13 Effect of *C*_{3} on −*P* (*Λ*) keeping *ζ* = *η* = 0.5, *C*_{1} = *C*_{2} = 400, *Br* = 1, Pr = 6.8 and *Re* = 100

The effect of Reynolds number *Re* on radial velocity *F* (*Λ*) profiles and axial velocity −*H* (*Λ*) profiles are exhibited through the graphs shown in Figures 14-15. The graphs are plotted in the presence of radial slip *ζ* = 0.5, tangential slip *η* = 0.2 and at *C*_{1} = *C*_{2} = *C*_{3} = 50, Pr = 6.8, *Br* = 5. It is observed that radial velocity *F*(*Λ*) and axial velocity −*H* (*Λ*) reduce when the Reynolds number *Re* changes from low to high values as displayed in Figures 14-15. Therefore, the parameter *Re* shows the decreasing effect on both the radial and axial velocities.

Figure 14 Effect of *Re* on *F* (*Λ*) keeping *ζ* = 0.5, *η* = 0.2, *C*_{1} = *C*_{2} = *C*_{3} = 50, Pr = 6.8, and *Br* = 5

Figure 15 Effect of *Re* on −*H* (*Λ*) keeping *ζ* = 0.5, *η* = 0.2, *C*_{1} = *C*_{2} = *C*_{3} = 50, Pr = 6.8, and *Br* = 5

Figures 16-17 show the pressure distribution −*P* (*Λ*) for selected values of the radial slip *ζ* and tangential slip *η* parameters.

Figure 16 Effect of *ζ* on −*P* (*Λ*) keeping *η* = 0.2, Pr = 6.8, *Br* = 1, *Re* = 2950, *C*_{1} = *C*_{2} = *C*_{3} = 50

Figure 17 Effect of *η* on −*P* (*Λ*) keeping *ζ* = 0.2, Pr = 6.8, *Br* = 1, *Re* = 2950, *C*_{1} = *C*_{2} = *C*_{3} = 20

The influence of the radial slip parameter *ζ* on the pressure distribution −*P* (*Λ*) is displayed in Figure 16, which shows that an increase in the values of radial slip *ζ* results in the fall of pressure. It can be observed from Figure 17, that increasing the values of the tangential slip parameter *η* also decrease the pressure. From Figures 18-19, it is quite evident that there is an increase in temperature *θ* (*Λ*) with increasing the values of the Brinkman number *Br* as well as the microinertia density parameter *C*_{3} in the presence of radial and tangential slips, taking *ζ* = 0.5 and *η* = 0.2.

Figure 18 Effect of *Br* on *θ* (*Λ*) keeping *ζ* = 0.5, *η* = 0.2, *C*_{1} = *C*_{2} = *C*_{3} = 100, Pr = 6.8, *Re* = 100

Figure 19 Effect of *C*_{3} on *θ* (*Λ*) keeping *ζ* = 0.5, *η* = 0.2, *C*_{1} = *C*_{2} = 400, Pr = 6.8, *Br* =1, *Re* = 100

To examine the effects of Prandtl number Pr and Reynolds number *Re* on temperature fields *θ* (*Λ*), the graphs are plotted as shown in Figures 20 and 21 from which, it can be seen that the temperature profiles decrease for an increase in the values of Pr and *Re*.

Figure 20 Effect of Pr on *θ* (*Λ*) keeping *ζ* = 0.5, *η* = 0.2, *C*_{1} = *C*_{2} = *C*_{3} = 400, *Br* = 1, *Re* = 40

Figure 21 Effect of *Re* on *θ* (*Λ*) keeping *ζ* = 0.5, *η* = 0.2, *C*_{1} = *C*_{2} = *C*_{3} = 50, *Br* = 5, Pr = 6.8

Figure 22 gives the influence of the vortex viscosity parameter *C*_{1} on the boundary layer thickness and it is inferred that it increases with increasing the values of vortex viscosity parameter. The circumferential velocity is employed to observe the boundary layer thickness for von Kármán flows of Newtonian fluids, which occurs at the point where the tangential velocity drops to *G* ≈ 0.01 as referred in [24].

Figure 22 Effect of vortex viscosity parameter *C*_{1} on boundary layer thickness *δ* keeping *ζ* = *η* = 0.2, *C*_{2} = *C*_{3} = 10, *Br* =1, Pr = 6.8, Re = 2950

Figures 23-28 exhibit the pronounced variation of entropy generation number *N*_{G} and the Bejan number *Be* associated with the effects of Brinkman number *Br*, vortex viscosity parameter *C*_{1}, and Prandtl number Pr. Figure 23 demonstrates the variation of the entropy generation number, which varies with the increasing values of Brinkman number, which gives its maximum value near the disk and decreases away from the disk. The opposite behavior is shown in Figure 24. The Bejan number decreases as the Brinkman increases.

Figure 23 Effect of *Br* on *N*_{G} keeping *ζ* = *η* = 0.2, *C*_{1} = *C*_{2} = *C*_{3} = 10, *Br* =1, Pr = 6.8, *Re* = 2950, *σ*_{1} = 1, *σ*_{2} = 0.5, *Q*_{0} =0.5

Figure 24 Effect of *Br*_{1} on *Be* keeping *ζ* = *η* = 0.2, *C*_{1} = *C*_{2} = *C*_{3} = 10, *Br* =1, Pr = 6.8, *Re* = 2950, *σ*_{1} = 1, *σ*_{2} = 0.5, *Q*_{0} =0.5

Figure 25 Effect of *C*_{1} on *N*_{G} keeping *ζ* = *η* = 0.2, *C*_{2} = *C*_{3} = 10, *Br* =1, Pr = 1.5, *Re* = 2950, *σ*_{1} = 1, *σ*_{2} = 0.5, *Q*_{0} =0.5

Figure 26 Effect of *C*_{1} on *Be* keeping *ζ* = *η* = 0.2, *C*_{2} = *C*_{3} = 10, *Br* =1, Pr = 6.8, *Re* = 2950, *σ*_{1} = 1, *σ*_{2} = 0.5, *Q*_{0} =0.5

Figure 27 Effect of Pr on *N*_{G} keeping *ζ* = *η* = 0.2, *C*_{1} = *C*_{2} = *C*_{3} = 10, Pr = 6.8, *Re* = 2950, *σ*_{1} = 1, *σ*_{2} = 0.5, *Q*_{0} =0.5

Figure 28 Effect of Pr on *Be* keeping *ζ* = *η* = 0.2, *C*_{1} = *C*_{2} = *C*_{3} = 10, *Br* =1, *Re* = 2950, *σ*_{1} = 1, *σ*_{2} = 0.5, *Q*_{0} =0.5

It is illustrated by Figures 25-26 that the entropy generation and the Bejan numbers decrease near the disk as *C*_{1} increases. The influence of the Prandtl number on the entropy generation number and the Bejan number are presented in Figure 27-28. The entropy generation number increases with Prandtl number as depicted in Figure 27. A rise in the values of Prandtl number increases the Bejan number but it becomes minimum as fluid moves away from the disk as shown in Figure 28.

presents numerical values for local skin friction coefficients *C*_{f} and *C*_{g}, and local Nusselt number *N*_{u} for different pertinent parameters. The vortex viscosity parameter *C*_{1} decreases both the skin friction coefficients and Nusselt number, which increases the fluid flow over disk and heat transfer increases as a decrement in the Nusselt number. The increase in the values of spin gradient viscosity parameter *C*_{2} increases the skin friction coefficient *C*_{f}, which reduces the fluid flow in the radial direction and also increases the Nusselt number, while skin friction coefficient *C*_{g} diminishes allowing more fluid to flow in a tangential direction. The radial skin friction coefficient and Nusselt number decrease as the values of the microinertia density parameter *C*_{3} increase, whereas skin friction coefficient increases in the tangential direction. The skin friction coefficient decreases for increasing values of radial slip parameter *ζ*, which allows more fluid to flow in a radial direction. On the other hand, the tangential skin friction and Nusselt number increase, which reduces the flow in a tangential direction. The increasing effect in the values of tangential slip parameter *η* decreases both the physical quantities like skin friction coefficient and Nusselt number. Therefore, fluid velocity increases with increase in the tangential slip parameter. Finally, it is observed that variations in the Prandtl number Pr change the Nusselt number only, which reduces heat transfer as the Nusselt number increases.

Table 1 Numerical values of local skin friction coefficients *C*_{f} and *C*_{g}, local Nusselt number *N*_{u} for physical parameters *C*_{1}, *C*_{2}, *C*_{3}, *ζ*, *η* and Pr while keeping *Re* = 2950 and *Br* = 1

For the reliability of numerical outcomes, comparison of the present results with [24] is presented in . It can be seen that the results are in good agreement at *C*_{1} = *C*_{2} = *C*_{3} = *ζ* = *η* = 0, which validates the results and computational technique. This table also shows the effects of four other parameters by varying the value of one parameter while the others are kept at zero. It is observed that the vortex viscosity parameter *C*_{1} has a decreasing effect on radial and tangential skin frictions while it increases the volumetric flow rate, pressure, and bound ary layer thickness for *C*_{1} = 5. On the other hand, when the spin gradient viscosity parameter is fixed at *C*_{2} = 10, it gives the same results as obtained in [24] but it changes the volumetric flow rate from 0.8838 to 0.8844 and the pressure from 0.3906 to 0.3911. The radial skin friction, pressure, and boundary layer thickness are found decreasing for *ζ* = 0.2, while it enhances the tangential skin friction and volumetric flow rate. The tangential slip parameter illustrates the decreasing behavior on all the quantities except boundary layer thickness, which slightly increases from 5.4 to 5.55 at *η* = 0.2. It can be seen that radial and tangential skin friction reduce for *C*_{1} = *C*_{2} = *C*_{3} = 10, and *ζ* = *η* = 0.2, while it increases the volumetric flow rate, pressure, and boundary layer thickness.

Table 2 Numerical comparison of some physical quantities with Ref. [24] obtained at *Re* = 2950

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