Theoretical Survey of Time-Dependent Micropolar Nanoﬂuid Flow over a Linear Curved Stretching Surface

: The heat and mass transfer of the unsteady ﬂow of a micropolar ﬂuid over a curved stretching surface was considered in this study. The Brownian motion and thermophoresis effects were explored in this analysis. The effects of suction/injection cases on the curved surface were discussed. Under ﬂow assumptions, a mathematical model was designed employing boundary layer approximations using partial differential equations. A suitable transformation was developed using the lie symmetry method. Partial differential equations were transformed into ordinary differential equations by suitable transformations. The dimensionless system was elucidated through a numerical technique, namely bvp4c. The involved physical parameters’ inﬂuences are described in the form of graphs as well as numerical results in the form of tables. Our current work is helpful in the engineering and industrial ﬁelds. The unsteadiness parameter increases which Nusselt number at increased but concentrations declined. The thermophoresis parameter increases when increasing the Nusselt number because the small number of nanoparticles enhances the heat transfer rate. The temperature proﬁle declined due to increasing values of unsteadiness parameter for both cases of suction and injection cases.


Introduction
The heat and mass transfer of a nanofluid over a stretched curved surface has been studied. The combination of a base fluid with nanoparticles is known as a nanofluid. Mostly, nanoparticles in a nanofluid are a mixture of oxides, carbides, metals, and carbon nanotube with a base fluid. The heat transfer increases with the thermal conductivity of a nanofluid. The thermal conductivity of the nanofluid plays a vital role in the heat transfer phenomenon. The term nanofluid was first used by Choi and Eastman [1] in 1995. The size of nanoparticles is 1-100 nm and they are dispersed in a base fluid. Masuda et al. [2] observed the variations in the viscosity and thermal conductivity of liquid by the inclusion between base fluid and nanoparticles. Yu and Mittra [3] presented a simple (FDTD) technique which can be used to study curved dielectric surfaces. Nadeem and Lee [4] analytically considered a nanofluid over an exponentially stretching surface. They explored the results of Brownian motion and thermophoresis on a non-linear stretching surface. The based peristaltic hyperbolic tangent fluid flow at the curved channel examined was studied by Nadeem and Maraj [5]. Malvandi et al. [6] elucidated the characteristics of a nanofluid with heat generation over a porous stretching sheet. The effects of heat generation and chemical reaction of a second-order unsteady viscoelastic fluid flow over a rotating cone were studied by Saleem et al. [7]. Nadeem and Abbas [8] analyzed the flow behavior of a modified nanofluid with an exponentially stretching sheet. Hosseinzadeh et al. [9] Symmetry 2022, 14, 1629 2 of 14 highlighted the impact of hybrid nanofluid flow under entropy generation. Recently, a few researchers have discovered the flow of a nanofluid with different assumptions and different geometries, as seen in the literature [10,11].
Micropolar fluids a microstructure and a non-symmetric stress tensor. By defining the nonlinear relationships of shear rate and shear stress, non-viscous liquid can be studied. The model for micropolar fluids is acceptable for biological fluids, exocytic lubricants, colloids, liquid crystals, and animal blood. There are various factors related to non-viscous liquids that can be studied during the studies of physical problems. Moreover, these fluids have a viscosity that is dependent on shear. Micropolar fluids were invented by Eringen [12]. Ahmadi [13] deliberated solutions to micropolar fluid flow considering the effects of microinertia over a semi-infinite surface. A micropolar fluid considering mixed convection toward a vertical sheet along uniform heat flux was obtained by Gorla [14]. The results of a based micropolar fluid at a stretching surface were obtained by Kelson and Desseaux [15]. The results of mixed convection of a base micropolar fluid were obtained by Bhargava et al. [16] in the presence of porous media. Qasim et al. [17] investigated steady micropolar fluid flow by considering a stretching surface along Newtonian heating. The mixed convection of base micropolar fluid under a stagnant region of a vertical sheet was explained by Ishak et al. [18]. Nadeem and Abbas [19] initiated work on hybrid nanofluid flow under slip effects. Nadeem et al. [20] analyzed micropolar fluid flow over a Riga surface. Several investigators (see Refs. [21,22]) highlighted the influence of micropolar fluid flow with various assumptions and flows on various geometries.
The slip wall condition is a case where the viscous impacts at the wall are invisible. Boundary slip is also a proper boundary condition for symmetrical surfaces. In the slip wall condition, the normal velocity is zero, but the tangential velocity is not zero. The boundary-layer theory is very effective for non-Newtonian fluid models and has received great attention in the last few years. Such flows are important in lessening the drag friction forces and enlarging the rate of transfer heat. The study of slip effects play a vital role in micropumps, micronozzles, microvalves, and hard disk drives. The no-slip condition is considered when the relative velocity between the fluid and boundary is zero. Saffman [23] studied the experiments on mass efflux and verified the presence of a non-zero tangential velocity on the surface of a permeable boundary. Chellam et al. [24] reported the impact of mass transfer as well as fluid flow in the occurrence of uniformly porous slip boundaries. Slip effects under a base micropolar fluid in vertical porous sheets were discussed by Chaudhary and Preethi [25]. Sharma [26] discussed the flow of fluctuating mass and thermal diffusion with slip conditions at a vertical plate. The MHD flow of a nanofluid with slip conditions was explored for a permeable sheet by Ibrahim and Shankar [27]. The slip effects on a porous channel with micropolar fluid in a stretching/shrinking porous medium were investigated by Xinhui et al. [28]. Several investigators (see Refs. [29,30]) have been highlighted the effects of a stretching surface.
We discussed the unsteady flow of a micropolar fluid over a curved surface. Mathematical developed under the flow assumptions, the partial differential equations are transformed into ordinary differential equations. A nonlinear system of equations was created by implementing the bvp4c numerical procedure. The effect of the physical parameters is highlighted using graphs and tables. Our current work will be helpful in the engineering and industrial fields. No one has discussed this model before. These results are utilized which helped in the fields of engineering and industry.

Materials and Methods
We discussed base micropolar fluids with the Buongiorno model at a permeable curved surface. In the flow direction, the arc length is s, and the normal to tangent vector is r which is shown in Figure 1a,b. Figure 1a,b shows the geometry of the stretching. The surface is stretched, where v w is in the S-direction and c is constant, while v w occurs because of the porous surface, which characterizes two cases: v w < 0 and v w > 0, which are related to injection and suction, respectively.

Materials and Methods
We discussed base micropolar fluids with the Buongiorno model at a permeable curved surface. In the flow direction, the arc length is , and the normal to tangent vector is which is shown in Figure 1a,b. Figure 1a,b shows the geometry of the stretching. The surface is stretched, where v is in the -direction and c is constant, while v occurs because of the porous surface, which characterizes two cases: v < 0 and v > 0, which are related to injection and suction, respectively. With the boundary layer approximations as well as the above assumptions, the governing equations for the flow problem and mathematical model were developed as follows (see Refs. [31,32]): The respective boundary conditions are: The similarity variables are presented as (see Refs. [33][34][35]): With the boundary layer approximations as well as the above assumptions, the governing equations for the flow problem and mathematical model were developed as follows (see Refs. [31,32] ∂T ∂t The respective boundary conditions are: The similarity variables are presented as (see Refs. [33][34][35]): Equation (8), for dimensionless terms, was utilized in Equations (1)- (7), which then become dimensionless, as follows: 1 The pressure profile can be computed as Dismissing the pressure term, we obtain 1 1 with boundary conditions of The physical interest is defined as where the quantities (τ rs and q w ) are calculated as: Using Equation (19) in Equation (20), then we obtain where Re s is the local Reynold number.

Results and Discussion
The present problem was solved numerically by using the bvp4c/shooting procedure. The effect of distinct parameters is graphically portrayed along the temperature and concentration profiles. Figures 2-5 depict the variation in temperature profiles along the distinct values of flow parameters. Figure 2 shows that by enlarging the value of the unsteadiness parameter β 0 , the temperature profile declines for in the cases of both suction and injection. A physical increase in β 0 causes an increase in the thickness of the thermal layer. In Figure 3, the variation in the temperature plot is displayed against the thermophoresis parameter. Figure 3 shows the impacts of N T on the temperature profile. The temperature profile reduced due to increasing values of N T . Physically, N T is directly proportional to the heat thermal coefficient which arises with fluid; hence, the heat transfer rate at the surface diminishes. In Figure 4, the variation in the temperature plot is displayed against the thermal slip parameter. Figure 4 reveals that the value of δ increases when the temperature plot is reduced. Physically, δ is directly proportional to the heat thermal coefficient which arises with fluid; hence, the heat transfer rate at the surface diminishes. In Figure 5, it is shown that θ(ζ) decreases with greater values of K 0 in the cases of both suction and injection. Figures 6 and 7 illustrate the variation in the concentration field for altered values of physical parameters. Figure 6 shows the diversity in the concentration profile for different values of the unsteadiness parameter. It is clarified that the concentration field decreases for the parameter of β 0 in the cases of both suction and injection. Figure 7 shows that the concentration field is reduced due to enhancement in the values of N B . centration profiles. Figures 2-5 depict the variation in temperature profiles along the distinct values of flow parameters. Figure 2 shows that by enlarging the value of the unsteadiness parameter , the temperature profile declines for in the cases of both suction and injection. A physical increase in causes an increase in the thickness of the thermal layer. In Figure 3, the variation in the temperature plot is displayed against the thermophoresis parameter. Figure 3 shows the impacts of on the temperature profile. The temperature profile reduced due to increasing values of . Physically, is directly proportional to the heat thermal coefficient which arises with fluid; hence, the heat transfer rate at the surface diminishes. In Figure 4, the variation in the temperature plot is displayed against the thermal slip parameter. Figure 4 reveals that the value of increases when the temperature plot is reduced. Physically, is directly proportional to the heat thermal coefficient which arises with fluid; hence, the heat transfer rate at the surface diminishes. In Figure 5, it is shown that ( ) decreases with greater values of in the cases of both suction and injection. Figures 6 and 7 illustrate the variation in the concentration field for altered values of physical parameters. Figure 6 shows the diversity in the concentration profile for different values of the unsteadiness parameter. It is clarified that the concentration field decreases for the parameter of in the cases of both suction and injection. Figure 7 shows that the concentration field is reduced due to enhancement in the values of .            Tables 1 and 2 present the influence of several physical parameters on the Re 1/2 s C f , Re 1/2 s N u s , and Re 1/2 s C m for micropolar nanofluid. We also acknowledge the influence of strong and weak concentrations. The effects of δ on Re 1/2 s N u s are shown. It is noted that δ increases as the value of Re 1/2 s N u s decreases, because the thermal slip reduces the heat transfer rate on the curved surface. The value of Re 1/2 s N u s declines in the case of a weak concentration correlated to a strong concentration. Thermal slip does not influence Re 1/2 s C f or Re 1/2 s C m . In terms of the impacts of Π on Re 1/2 s C f , Re 1/2 s N u s , and Re 1/2 s C m , it is observed that the value of Π increases as the values of Re 1/2 s C f , Re 1/2 s N u s , and Re 1/2 s C m increase, because the slip velocity increases, which increased in heat transfer, Re 1/2 s C m , or skin frictions because the motion of the fluid flow increases. The Re 1/2 s C f , Re 1/2 s N u s , and Re 1/2 s C m values decline when a weak concentration is correlated to a strong concentration. In terms of the influence of λ on the Re 1/2 s C f , Re 1/2 s N u s , and Re 1/2 s C m values, it is observed that the value of λ increases as the Re 1/2 s C f , Re 1/2 s N u s , and Re 1/2 s C m values increase which increase in heat transfer, Re 1/2 s C m , and skin frictions due to the increase in the motion of the fluid flow. In terms of the influence of ω on the Re 1/2 s C f , Re 1/2 s N u s , and Re 1/2 s C m , it is seen that the values of Re 1/2 s C f and Re 1/2 s C m decrease with an increase in ω due to the suction flow, which prevents any decreases in the skin friction and Re 1/2 s C m . Meanwhile, the Re 1/2 s N u s value increases as the ω heat transfer rate increases. The Re 1/2 s C f , Re 1/2 s N u s , and Re 1/2 s C m values decrease in the case of n = 0.5 compared to when n = 0. In terms of the impact of K 0 on the Re 1/2 s C f , Re 1/2 s N u s , and Re 1/2 s C m , it is seen that the Re 1/2 s C f and Re 1/2 s C m values decrease with an increase in K 0 due to the increased values of curvature, which force a reduction in the skin friction and Re 1/2 s C m values. Meanwhile, the Re 1/2 s N u s value increases with larger values of K 0 because the increases in the values of the curvature parameter prevent the heat transfer rate at the surface from increasing. The Re 1/2 s C f , Re 1/2 s N u s , and Re 1/2 s C m values decrease in the case of a weak concentration as compared to that of a strong concentration. In terms of the effects of K 1 on Re 1/2 s C f , Re 1/2 s N u s , and Re 1/2 s C m , it is seen that the Re 1/2 s C f and Re 1/2 s C m values increase with an increase in K 1 because the values of the micropolar parameter are increased, which prevent any decreases in skin friction and Re 1/2 s C m . Meanwhile, the Re 1/2 s N u s value decreases as the value of K 1 increases, because decreasing the values of the micropolar parameter forces reductions to the heat transfer rate at the surface to occur. In terms of the influence of T 0 on Re 1/2 s C f , Re 1/2 s N u s , and Re 1/2 s C m , it is seen that the Re 1/2 s C f and Re 1/2 s C m values increase with an increase in T 0 because the skin friction enhances due to the heat, while Re 1/2 s N u s declines with as the values of T 0 increase because radiation increases, which reduces the heat transfer rate at the surface. In terms of the impact of N T on Re 1/2 s N u s , it is noted that N T increases as Re 1/2 s N u s increases because the small number of nanoparticles enhances the heat transfer rate of the wall. There are no effects of thermal slip on Re 1/2 s C f and Re 1/2 s C m . The impact of N B on Re 1/2 s N u s is shown. It is noted that N B increases with decreased Re 1/2 s N u s due to the collision of fluid and nanoparticles, which reduce the heat transfer rate at the surface. No impacts of δ on Re 1/2 s C f or Re 1/2 s C m were found. In terms of the effects of β 0 on Re 1/2 s C f , Re 1/2 s N u s , and Re 1/2 s C m , it is seen that the Re 1/2 s C f and Re 1/2 s C m values increase with an increase in β 0 , while Re 1/2 s N u s decreases with larger values of β 0 , because the decreasing values of unsteadiness force a decline in the heat transfer rate at the surface. Table 3 compares the present analysis with Nogrehabadi et al. [36] and Sahoo and Do [37] for Re 1/2 s C f with diverse values of the suction parameter.

Final Remarks
We analyzed micropolar fluid flow over a stretched curved surface. The results of the dimensionless system and physical parameters that appeared under the flow assumptions were elaborated upon through graphs and tables. The main achievements of the current analysis are highlighted below:

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In terms of the values of Re 1/2 s C f , Re 1/2 s N u s , and Re 1/2 s C m , a weak concentration (n = 0.5) enables greater values in comparison to a strong concentration (n = 0.0).

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The temperature profile achieves greater values near the surface in the case of ω > 0 as compared to ω < 0. • Surprisingly, the concentration profile achieves greater values near the surface in the case of ω > 0 as compared to ω < 0 due to increments in the β 0 and N B .

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The unsteadiness parameter increases, which resists increases in the Nusselt number in a strong concentration (n = 0.0) but declines in a weak concentration (n = 0.5).

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The thermophoresis parameter increases as the Nusselt number increases because the small number of nanoparticles enhances the heat transfer rate.

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The higher value of the unsteadiness parameter β 0 , the lower the temperature profile for the cases of both suction and injection. K 0 (1) Curvature parameter u (m/s) Velocity vector r-direction Sh s (1) Sherwood number N T (1) Thermophoresis parameter n (1) Microgyration ω (1) Stretching parameter α m 2 /s Thermal diffusivity T (K) Temperature T w (K) Wall temperature λ (1) Stretching parameter K 1 (1) Micropolar parameter k Ns/m 2 Vertex viscosity c p f (J/kg K) Heat capacity of fluid s (m) Arc length N B (1) Brownian motion parameter τ rs (pa) Wall shear stress T w (K) Wall temperature Pr (1) Prandtl number R (m) Radius of curvature ρ kg/m 3 Fluid density T ∞ (K) Ambient temperature µ Ns/m 2 Dynamic viscosity T ∞ (K) Ambient temperature Π (1) Velocity slip