Magnetohydrodynamic Impact on Carreau Thin Film Couple Stress Nanofluid Flow over an Unsteady Stretching Sheet

A mathematical model of time-dependent thin-ﬁlm ﬂow of Carreau liquid over a stretching surface is studied in this investigation in the presence of couple stress and uniform magnetic ﬁeld. To explain the properties of heat and mass transport phenomena, the inﬂuence of both thermophoresis and Brownian motion variables is used. For the conversion of the model framework (mo-mentum, heat, and concentration expression with boundary conditions) into a set of ordinary diﬀerential equations, the ap-propriate transformation technique is followed. By using analytical tool, HomotopyAnalysis Method (HAM), the transformed model expressions are solved. For diﬀerent estimations of the aﬀected physical factors, the numerical results involving skin-friction coeﬃcient, Nusselt number, Sherwood number, ﬂuid velocity proﬁle, thermal proﬁle, and concentration proﬁle are displayed graphically. Besides, the ﬁndings for skin-friction coeﬃcient, Nu x , and ϕ ( η ) are given in the table format. In raising the ﬂuid temperature, the eﬀect of thermophoresis and magnetic parameters is beneﬁcial. With the Brownian motion and Schmidt number, the solute concentration is found to reduce.


Introduction
e thin-film liquid has numerous applications in various sectors such as coating industries, oil recovery, colloid suspension, soap bubbles, and interface sciences. e analysis of the stretching phenomena with heat and mass transfer together with the thin liquid film has dynamic usages. In mechanics, there are some nonlinear issues open towards complete mathematical learning. Heat transfer remains an essential portion of individual parts such as chemical production, atomic energy, and lubrication. Non-Newtonian fluids regarding the thin film are mainly used in wire coating, paintings and fiber coatings, removal of foodstuff objects, numerous freezing subjects, container fluidization, and sheet coatings. To see the provisions used on behalf of all these applications, many researchers focused on studying the flow of thin liquid film over rigid and stretched surfaces.
Andersson et al. [1] have examined the Power-law fluid thin film flow affected through the time-dependent extending medium using similarity alteration. Myers [2] studied Power-law fluid flow considering small shear forces. e key features of his study are to investigate the influence of the modeled parameters over the thin film of the Power-law liquid when the shear rate approaches zero. Ali et al. [3] inspected the flow of the liquid film, considering the micropolar fluid flow using the extending surface. Similarly, Abbas et al. [4] examined the liquid film motion of the viscoelastic liquid using the extending surface. Siddiqui et al. [5] considered the liquid film flow using the two non-Newtonian fluids over a vertical surface.
ey used the analytical technique to obtain the results of the model problem. Li et al. [6] examined the nanofluid streaming on a thin layer liquid including thermal radiations. Dandapat et al. [7] examined the film flow considering the Marangoni convection and the biviscosity terminology. In addition to nonlinear free surface boundary conditions, complete expression of Navier-Stokes and continuity are analytically solved. Gul et al. [8] discussed the model of effective Pr number for thin film flow of nanofluids over stretching sheet.
Rudraiah et al. [9] are the pioneer to introduce the Marangoni convection concept in the fluid flow considering the open cavity. Chen et al. [10] have explored Marangoni convection's effect considering the Power-law model liquid film flow past an unstable surface. ey observed the influence of the various parameters in the existence of the surface tension. Wang [11], Narayana and Sibanda [12], and Qasim et al [13] have examined the liquid film flow over a stretched surface. ey have used the unsteady flow using the same similarity transformations. Gul et al. [14] have extended the idea of the liquid film flow by adding the Marangoni convection influence over the liquid film flow. Vajravelu et al. [15] explored the mathematical model for the thin layer streaming of the Power-law liquid over an unstable extending and permeable surface. Noor et al. [16] have analyzed thin-film liquid properties in thermos capillary and magnetic fields. Similarly, Shah et al. [17] investigated the fluid's convective flow over an extending surface.
It is important to remember that, in the case of non-Newtonian fluids, the interaction between the deformation rate and shear forces is not linear. e Carreau fluid's extreme shear ratio functions as the Power-law fluid, and the Carreau fluid behaves as Newtonian as the modest shear ratio. To find the mathematical results of the problem, they used the bvp4c scheme. e thin-film flow of the Carreau fluid over an inclined surface was scrutinized by Tshehla [18]. Akbar et al. [19] explored the model of Carreau fluid flow across a porous surface. Using a curved channel, Abbasi et al. [20] examined the peristaltic transport of the Carreau fluid. Khellaf and Lauriat [21] have examined the same fluid flow among the double concentric cylinders. Myers et al. [2] studied the liquid film flow considering Power-law, Ellis, and Carreau models. e related and interesting work can also be seen in the existing literature [22,23]. e general Newtonian Carreau liquid remained was first offered by Carreau [24]. e model turns sensibly fit through the interruptions of polymers' performance in numerous stream positions. Mostly, the Carreau model is well-matched for certain diluted polymer results and meltdown. It determines the behavior of many non-Newtonian liquids to incorporate shear thinning and shear thickening. e Carreau fluid model was subsequently determined by numerous investigators, as its uses in many natural and then scientific innovations. Akbar et al. [19] found the double resolutions of magneto-hydrodynamics stagnation point stream of Carreau liquid near a porous stretching surface. Abbasi et al. [20], Khan et al. [25], and Iqbal et al. [26] discussed the thermodynamics of a Carreau fluid tinny film using different geometries. Kohilavani et al. [27] measured the flow of thin layer plus energy transmission in the Carreau liquid over a porous extending surface. Similarly, Sarada et al. [28], Gowda et al. [29], and Makinde et al. [30] examined the simultaneous convective flow of Carreau fluid, the behavior of non-Newtonian fluids, and ferromagnetic nanofluid flow over stretching surfaces along with MHD and thermal radiation effects.
In view of the above interesting debate, the contribution of this analysis is to extend the existing study of thin layer flow of the Carreau fluid [25] including the following steps.
(i) e couple stress fluid is included to extend the published work [25] (ii) e previous work [25] is extended with the addition of the magnetic field (iii) also includes the influence of Marangoni convection

Problem Formulation
e mathematical model has been focused on the two-dimensional thin film flow considering Carreau fluid. e liquid film's width h(t) is uniform, and the surface stretches across the x-axis, as illustrated in Figure 1.
e y-axis is considered vertical to the x-axis and in the direction of the liquid film's thickness. e thin film is stretched with the U w (x, t) � bx(1 − αt) − 1 (uniform velocity), where α and b are constants and b > 0 shows the stretching rate. Here, t is the time and α is the constant used in the unsteadiness parameter. V w � ����������� (υb/(1 − αt)) is the suction/injection velocity such that V w � 0, V w > 0, and V w < 0 belong to impermeability, suction, and injection cases, respectively. e surface temperature is symbolized by T w , and thin film thickness is h(t). Since, in the above mathematical modeling, we considered the Carreau fluid flowing past an extending hot plate, the expression of τ (Cauchy stress tensor) for the said fluid is known as [27] Here, Here, ε demonstrates the apparent viscosity, ε ∞ , p, τ, and ε 0 denote the infinite-shear-rate viscosity, the pressure term, the Cauchy stress tensor, and zero-shear rate viscosity, respectively, I used to show identity tensor, λ is the material time constant, and n is the power-law index. e shear rate is defined as and τ is defined as [27] 2 Mathematical Problems in Engineering e power-law index falls between 0 < n < 1, and the Carreau model shows the shear-thinning or pseudoplastic properties. e dilatants' behavior accurses when n > 1. e magnetic field is considered in the transverse direction in the unsteady form such as B(t) � B 0 (1 − at) − 0.5 . About the Carreau fluid model, the liquid film flow can be compiled as Heat/energy equation is where the element of velocity (u, v) varies along the x and y direction, λ represents the material time constant, ρ denotes density, κ shows thermal conductivity, c p is the capacity specific heat, and T is temperature of fluid.
e surface tension varies in a linear way, depending on the temperature. e other is made unchanged by the other properties of the fluid. Surface tension is described as follows [7,[11][12][13][14]: According to the abovementioned assumptions, the physical conditions are e wall temperature (T w ) is defined as where T 0 and C 0 and T ref and C ref indicate the slit and reference temperature and concentration, respectively.
T ref and C ref can be treated as a constant or differences of a uniform temperature and concentration. e concept of T w impersonates the situation in which the temperature of the sheet reduces in proportion to x 2 from T 0 at the slit; also, the quantity of thermal drop along the surface of the sheet improves over time.
Next, we introduce the similarity factors as in [16]: By using equations (13) and (5), it is satisfied identically but equations (6)- (8) and (11) take the following form: where where M, S, Pr, N T , w 2 e , N B , Sc, τ, K, and Re x are the magnetic field parameter, unsteady factor, Prandtl number, thermophoresis factor, local Weissenberg number, Brownian motion factor, Schmidt number, Cauchy stress tensor, couple stress parameter of the fluid, and local Reynolds number. δ is the suction\injection parameter.
In this model problem, the physical quantities of practical importance are C fx , Nu x , and Sh x as follows [7,15]: where τ w , q w , and q m are represented by the below equations:

HAM Solution
e analytical approach has been used to examine the modeled problem. is technique includes the latest version BVPh 2.0 package of the Homotopy Analysis Method (HAM). BVPh 2.0 package tends to converge rapidly, and more iterations are possible in this method in a short time.
e recent solution has been achieved using the package mentioned above up to the 30 order estimations. e trail solution for the modeled problem is obtained as Mathematical Problems in Engineering where A n (n � 1, 2, . . . , 8) are the constants of the integration. e residual errors considering average estimations up to the k th order is defined as e sum of the total residual errors ε t m has been attained from the fluid velocity, temperature, and concentration profiles.  Figure 2. e velocity profile f ′ (η) has shown the decline behavior by increasing the magnitude of the magnetic parameter M. With increasing strength of the magnetic field, we will encounter a resistive force known as Lorentz force. e rate of Lorentz force increases with a greater value of magnetic parameter. is force behaves as an agent of resistance to fluid motion. Due to this fact, the speed of the liquid film declined. e effect of S(unsteady factor) on f ′ (η) (velocity field) is presented in Figure 3. We witnessed from the plot a decrease in velocity fields for rising magnitude of S. Obviously, for the greater degree of the S unsteadiness factor, the density of the thin film rises, and hence, the fluid velocity declines with maximum value of S. Figure 4 indicates another major effect of we 2 on thin layer liquid f ′ (η) velocity. We observed a drop in f ′ (η) from the plot for the increasing amount of we 2 . In this graph, we can see contradictory patterns occurring with various values of we 2 on fluid velocity. In case of shear-thinning, the velocity of the liquid thin film reduces, whenever we strengthen its value we 2 . In comparison, the behavior of we 2 is opposite for shear thickening fluid. is has been confirmed that the boundary film width reduces for 0 < n < 1, although it improves for n > 1. Figure 5 illustrates the importance of K(couple stress factor) on the f ′ (η)(velocity profile). Here, as the magnitude of the couple stress factor increases, the f ′ (η) velocity profile is reduced. It is interesting that the liquid becomes a Newtonian fluid when the couple stress factor tends to zero (K ⟶ 0). Figure 6 highlights influence of β (layer thickness factor) on the f ′ (η) velocity profiles of the fluid film. A rise in the value of f ′ (η) depreciation in the fluid flow velocity profiles has been observed. Physically, the viscosity of the momentum boundary film is enlarged by the growth of the value of β. For this purpose, we observed a depreciation in fluid velocity. e influence of N T (thermophoresis factor) is captured in Figures 7 and 8 on θ(η) (temperature field) and ϕ(η)(concentration field).

Result and Discussion
erefore, the conclusion is taken from Figure 7 that θ(η) is growing with rising values N T . Figure 8 illustrates the influence of N T on ϕ(η), where an increasing behavior is observed in ϕ(η) with the growing quantity of N T . e impact of N B on θ(η) (thermal profile) and ϕ(η) (concentration field) is shown in Figures 9 and 10. Figure 9 indicates that θ(η) grows with the rising valuesN B . Brownian motion particles spontaneously travel across the fluid, and therefore, increase in N B produces an improvement in the fluid objects' random motion and as a result more heat can be transported. But at the other side, in the case of ϕ(η) Figure 10, the opposite behavior is noticed. Figure 11 demonstrates the influence of Pr on the θ(η) (temperature filed) of thin film fluid. It is shown from this graph that, with the rising values of Pr, θ(η) indicates a decrease. Physically, with the increasing values of Pr, decline the temperature field for its larger values. Figure 12 examines the influence of Sc on the ϕ(η) field. It is clearly found from the graph that ϕ(η) and its corresponding width of the boundary layer are decreasing function of Sc. Also, Sc is inversely related to the coefficient of diffusion. erefore, a rise in Sc leads to a lower coefficient of diffusion. So, a 6 Mathematical Problems in Engineering decrease in the ϕ(η) profile is caused by such a lower diffusion coefficient. e skin friction of the Carreau liquid film increasing with the larger magnitude of the parameters M, n, S, k, and we is shown in Table 1. e resistance force enlarges due the increasing magnitude of these parameters, and consequently, the skin friction boots up. Similarly, the thermophoretic parameter and Brownian motion parameter improve the heat transfer rate for the larger values of these parameters, as shown in Table 2, while the increasing value of the unsteady parameter reduces the heat transfer rate.  Nt, Nb, and S augmenting the Sherwood number for the increasing amount of these parameters is displayed in Table 3, but the parameter Sc reduces the mass transfer rate for its larger values. e comparison of the present study and published work [31] is shown in Table 4.

Conclusions
In this analysis, by taking into account the influence of Marangoni convection, couple stress, and MHD, we investigated the flow, heat transport and concentration properties of thin film fluid flow of the Carreau fluid model over an unsteady stretching sheet. e Buongiorno model was also used to develop the effects of thermophoresis and Brownian diffusion. e key finding of the present analysis is taken in the following way.

Data Availability
All the data exist in the manuscript.

Conflicts of Interest
e authors declare that they have no conflicts of interest.