Study of the Carreau fluid over a linearly curved stretching sheet: A numerical approach

Carreau fluid flow over a linearly curved stretching surface has significant applications in manufacturing industries. This study focuses on the impact of various parameters on the velocity, temperature, pressure skin friction, and heat transfer coefficient of a 2D Carreau flow. Both shear thinning (pseudoplastic) and shear thickening (dilatant) behaviors are considered. The parameters include suction, non-dimensional radius of curvature, Weissenberg number, magnetic parameter, power-law index, Eckert number (viscous dissipation parameter), and Prandtl number. The effects of viscous dissipation, magnetic field, convective boundary conditions, and suction are also taken into account. The study reveals that the flow velocity decreases with increasing magnetic parameter, curvature factor, and suction, but increases with the Weissenberg number and power law index. The heat transfer rate is reduced by the curvature factor, Weissenberg number, suction, power-law index, and Prandtl number, while the magnetic parameter has the opposite effect. The boundary value problem is simplified through appropriate similarity transformation and solved numerically using the shooting method and Matlab’s built-in function bvp4c. The results obtained from these two methods show good agreement. The findings also align well with existing literature in some limiting cases, confirming the validity of the study.


Introduction
Heat transfer flows due to moving boundaries have many important applications in industries.In manufacturing processes the stretching of sheets has significant effects on the quality of finished products.The final products are significantly influenced by stretching and heating/cooling; depending upon the skin friction or frictional drag and the rate of heat transfer.Under various stretching velocities, different real processes take place, viz; extrusion of polymer sheet, manufacturing of plastic films, continuous stretching, hot rolling, metal spinning, rubber sheet, or thin film and many more.
The boundary layer behavior prompted by the motion of a moving surface has taken attraction of researchers because of its applications in the manufacturing process, such as the flow generated in fiber spinning, glass blowing, hot rolling, paper production, and fibers production.Some of the famous applications of stretching process have been given by Fisher 1 and Altan et al. 2 Sakiadis 3 discussed the effects of constant velocity on Newtonian fluid caused by a moving plate.Erickson et al. 4 analyzed the impact of heat mass transfer past a moving plate.Tsou et al. 5 reported the influence of constant velocity over a stretching surface in the boundary layer together with temperature.Crane 6 carried forward the work of Sakiadis and discussed that in polymer industries it is sometimes essential to consider a stretching plastic sheet.The boundary layer analysis due to a stretchable porous plate of the heat and mass transfer is examined by Gupta and Gupta. 7[10][11][12] Carreau fluids are a type of generalized Newtonian fluids whose model was put forth by Carreau in 1698.Power law models are enable to predict the viscosity for very high and low shear rate.Carreau fluid is ideally used to model the phenomena of non-Newtonian fluids like plasma, blood flow, and visco-elastics.This model was put forth by Carreau in 1698.Power law model are enable to predict the viscosity for very high and low shear rate.The Carreau Model is In the above equation, t and _ g represents the extra stress tensor and generalized shear rate, m o is the zero shear rate viscosity, m ' is the infinite shear rate viscosity, G is the material time, n is the power law index.In terms of second invariant tensor P, the generalized shear rate can be expressed as where P = trace rU + (rU ) T À Á , Here we consider the case for which m ' = 0.In the following form the extra stress tensor is rewritten as Hayat et al. 13 investigated the flow of Carreau fluid caused by linear stretching under the effects of Newtonian heating and chemical reactions.Effects of physical parameters on velocity, concentration, and temperature were considered.Khan et al. 14 implemented shooting scheme to study the behavior of Carreau nanofluids due to paraboloid upper horizontal surface.Carreau fluid over a linearly curved stretching surface is an important model in many engineering and industrial applications.][17][18] In numerical analysis, the word shooting for shooting method is originated artillery which describes its working in such a way that it places cannon at initial position and by continuously changing the angle it targets until it reaches and satisfies boundary conditions.Initial value problem is obtain by reducing boundary value problem and by applying appropriate similarity transformation, non-linear ordinary differential equations are obtain from PDE's of differential equations.The equations are then solved by shooting method coupled with iv order Runge-Kutta integration scheme.This scheme is preferable to solve fluid dynamics problems because it takes advantage of speed and adaptivity over other numerical schemes. 19he objectives of this study is concerned with the analysis of boundary driven flow of Carreau fluid over a curved stretching surface.Carreau fluid of both shear thinning (pseudoplastic) and shear thickening (dilatant) nature have been considered.The effects of viscous dissipation, suction and magnetic field are taken into account.Governing equations are reduced to self-similar equations via appropriate similarity transformation.The reduced system of equations are solved numerically through the implementation of shooting method and Matlab built in function bvp4c.The pertinent parameters in this communication are the suctions, nondimensional radius of curvature, Weissenberg number, magnetic parameter, power-law index, Eckert number (viscous dissipation parameter), and Prandtl number.

Problem formulation
Carreau fluid flow over a curved surface is investigated with incompressible steady property.The surface is curved with radius R and curvilinear natural coordinates are s and r.R describes the distance from origin to the stretched surface.It defines the shape of the surface, for larger values of R the surface goes from curved to flat.Linear stretching velocity is responsible for fluid flow that is U w = cs along sÀ direction.Therefore, the stream forms boundary layer regime through rÀ direction and temperature distribution of the sheet is assumed T w = T o s in which c and T o show reference velocity and temperature.The geometry of flow is shown in Figure 1 The governing equations for the flow are given in vector form as follows see Sajid et al. 20 :

Continuity equation
Momentum equation Energy equation where D Dt denotes the material derivative, U = (u, v) and F are the velocity vector and the body force, respectively.p, c p , and k o are the pressure, specific heat and thermal conductivity.The function f is the viscous dissipation.In the presence of viscous dissipation the governing equations takes the following form:

Mass conservation equation
Momentum equation Energy equation boundary conditions for the above defined problems are (see Rosca and Pop 21 ): To formulate the system dimensionless, we utilized the following similarity variables Governing equations 7-10 with boundary conditions 11-12 are renewed into non-dimensional form as follows: and the pressure term from (17) refers to as: by utilizing equations ( 16) into (17), one can eliminate the pressure term boundary conditions are: Where S = À

Physical quantities
The quantity of physical interest in this problem is the skin friction coefficient C fs and heat transfer coefficient Nu s which is defined as where using equations ( 13)-( 15) in (35), we obtain; The Reynolds number is Re s = cs 2 n .

Numerical scheme
Solutions to boundary value problems related to fluid flow can be determined using appropriate numerical techniques.The shooting method, one of the oldest and most efficient numerical methods, is often used.This method initially transforms the boundary value problem (BVP) into a system of first-order initial value problems (IVP).Then, the missing initial conditions at the starting point are estimated.This transformed IVP is solved using a highly efficient sixth-order Runge-Kutta method.The accuracy of the resulting solution is verified by comparing it with the given values at the end point.If the solution's accuracy meets the desired level, it is accepted.Otherwise, the estimated values are iteratively refined using the Newton-Raphson method until the boundary conditions are satisfied within a specified tolerance limit at infinity.

Shooting method
The non-linear ODE for heat and momentum equations that is, subject to the boundary conditions are solved numerically using sixth order Runge-Kutta method along with shooting technique.
The boundary conditions are defined as:

Discussion
The current study examines the behavior of Carreau fluid over a stretching sheet.It explores how various physical parameters, such as the radius of curvature, Weissenberg number, magnetic parameter, and suction parameter, influence the velocity, temperature, and pressure of the fluid.This is considered for both shear thinning (m\1) and shear thickening (m.1) scenarios.The impact of an increasing radius of curvature on velocity is depicted in Figures 2 and 3. It's observed that for both shear thinning and shear thickening cases, the velocity profile decreases.This is because an increase in the radius of curvature reduces the centripetal force responsible for circular motion, leading to a decrease in velocity.The Weissenberg number, a viscoelastic parameter of the fluid, signifies the degree of deformation.
As the Weissenberg number increases, the fluid's thickness also increases, resulting in a decline in the velocity profile.This is because the Weissenberg number measures the fluid's relaxation time.Figures 4 and 5 illustrate the effect of the Weissenberg number for shear thinning and shear thickening cases.In the shear thickening case, an increase in the Weissenberg number leads to an increase in the particles' relaxation time, which in turn increases the resistance for molecules, resulting in a decrease in velocity.However, the opposite behavior is observed in the shear thinning case.A magnetic field applied perpendicular to the particle's velocity directly affects the particle's motion direction.An increase in the magnetic force reduces the radius of the circular path, which in turn decreases the particle's velocity.Figures 6 and 7 demonstrate the decrease in the velocity profile with an increase in the magnetic field parameter M for both (m\1) and (m.1) cases.Suction parameter represents the amount of fluid being removed from the system.It means that suction parameter reduces the amount of fluid from system which results in decrease of velocity profile of fluid.Figures 8 and 9 deliberates the effect of suction parameter by decrease in velocity profile for both M for (m\1) and (m.1).Increase in Weissenberg number is directly related to increase in turbulence of fluid.The turbulence in return causes the increase in heat transfer.Whereas increase in magnetic field hinders the motion of particles of fluid which results in decrease of motion of particles.With increase in Weissenberg number the thermal field shows up surd for smaller value of magnetic field for (M = 0:5)Figures 12 and 13.However, an opposite behavior is noticed for greater value of magnetic parameter with (M = 1:5).Figures 14 and  15 have been provided to describe the features of temperature against the magnetic parameter M for (m\1) and (m.1).
Suction parameter refers to the amount of fluid being removed from system.When the amount of fluid reduces, the internal energy of the particles also reduces which effects the temperature of system.Figures 16 and  17 portrays the effect of increase in suction parameter on temperature profile which shows decline with increases in suction parameter.Increases in Eckert number produces heat in fluid due to frictional heating.Eckert number is fraction of kinetic energy to specific heat enthalpy.So when the Eckert number increases, it works against fluid stresses to transform kinetic energy into internal energy.Figures 20 and 21 display the increase in temperature profile in both cases by increasing Eckert number Ec.
When the radius of curvature keep increasing the surface become flat hence causes a decrease in pressure of fluid.The effect of increasing radius of curvature and Magnetic parameter on the pressure is plotted in Figures 22 and 23.For large values of d the velocity decreases, while the pressure tends to zero for the flat surface.Figure 23 shows the effect of radius of curvature on pressure.Figure 24 displays the behavior of powerlaw index on pressure profile.It is observed that pressure decreases for large values of m.With increases    in power law index parameter, the fluid become less viscous, and it can flow more conveniently.So, fluid will require less pressure to flow.Therefore, in Figure 24 we noticed a decrease in pressure with increases in value of power law index parameter m.
Table 1 is presented to show the validity of the present results with the previous published work.The results for all values of d are align with previous published works.The distinction in the skin friction coefficient and the heat transfer rate owing to the alteration of We, M, d, Ec, and Pr are computed to tabulate the numerical values for shear thinning (m\1) and shear thickening (m.1) case in Tables 2 and 3.Both the skin friction coefficient and local heat transfer rate experience a decrease in magnitude with an increase in We for shear thinning (m\1) and it increases for shear thickening (m.1) case.The skin friction coefficient increases with an increase in Magnetic parameter M and suction S, but decreases with an increase in radius of curvature parameter d.An increase in M, d and Ec results in a decrease in local heat transfer rate, while an increase in Pr leads to an increase in local heat transfer rate.The above promising results gives future directions for this model to utilize for further industrial approaches by considering the effect of entropy generation.Moreover, for other non-Newtonian fluids this work can be carry forward.

Conclusions
Numerical investigations have been carried out on the flow of the Carreau fluid along with a linear stretchable curved wall.On the basis of our investigation, the key points of the pre-sent analysis are given below.
For shear thinning fluids, increases in Weissenberg number intensify the velocity of fluid while it mitigates the velocity for shear thickening fluids.An increase in the magnetic parameter, curvature factor and suction parameter results in a decrease in the flow velocity.On the other hand,         fluid velocity increases with an increase in the power law index.For shear thinning fluids, the temperature profile increases while it decreases for shear thickening fluids when the Weissenberg number is large.The heat transfer rate decreases with an increase in the suction parameter, curvature factor, powerlaw index and Prandtl number.Conversely, the opposite behavior is observed for the magnetic parameter.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

2 o
v w ffiffi c p n S.0 shows suction, We = G 2 c 2 Re s is the Weissenberg number, d = ffiffi c n p R is the dimensionless radius of curvature and M = sB cr is the magnetic parameter.

Figure 2 .
Figure 2. Radius of curvature d impact on velocity profile.

Figure 3 .
Figure 3. Radius of curvature d impact on velocity profile.

Figure 4 .
Figure 4. Weissenberg number We impact on velocity profile.

Figures 10 and 11
Figures 10 and 11 are sketch to know the behavior of radius of curvature d on velocity profile for shear thinning and shear thickening fluid.It is seen in both cases that velocity decreases for increasing values of d.Increase in Weissenberg number is directly related to increase in turbulence of fluid.The turbulence in return causes the increase in heat transfer.Whereas increase in magnetic field hinders the motion of particles of fluid which results in decrease of motion of particles.With increase in Weissenberg number the thermal field shows up surd for smaller value of magnetic field for (M = 0:5)Figures 12 and 13.However,

Figure 6 .
Figure 6.Magnetic parameter M impact on velocity profile.Figure 8. Suction parameter S impact on velocity profile.

Figure 8 .
Figure 6.Magnetic parameter M impact on velocity profile.Figure 8. Suction parameter S impact on velocity profile.

Figure 7 .
Figure 7. Magnetic parameter M impact on velocity profile.Figure 5. Weissenberg number We impact on velocity profile.

Figure 5 .
Figure 7. Magnetic parameter M impact on velocity profile.Figure 5. Weissenberg number We impact on velocity profile.

Figures 18 and 19
Figures 18 and 19 interpret the influence of Prandtl number on the thermal field.Both cases showed decline in temperature profile.Increases in Eckert number produces heat in fluid due to frictional heating.Eckert number is fraction of kinetic energy to specific heat enthalpy.So when the Eckert number increases, it works against fluid stresses to transform kinetic energy into internal energy.Figures 20 and 21 display the increase in temperature profile in both cases by increasing Eckert number Ec.

Figure 9 .
Figure 9. Suction parameter S impact on velocity profile.

Figure 10 .
Figure 10.Radius of curvature d impact on temperature profile with n = 0.5.

Figure 12 .
Figure 12.Weissenberg number We impact on temperature profile with m = 0.5.

Figure 11 .
Figure 11.Radius of curvature d impact on temperature profile with m = 0.5.

Figure 14 .
Figure 14.Magnetic parameter M impact on temperature profile with m 0.5.

Figure 16 .
Figure 16.Suction parameter S impact on temperature profile with m = 0.5.

Figure 15 .
Figure 15.Magnetic parameter M impact on temperature profile with m = 1.5.Figure 13.Weissenberg number We impact on temperature profile with m = 1.5.

Figure 13 .
Figure 15.Magnetic parameter M impact on temperature profile with m = 1.5.Figure 13.Weissenberg number We impact on temperature profile with m = 1.5.

Figure 18 .
Figure 18.Prandtl number Pr impact on temperature profile with m = 0.5.

Figure 20 .
Figure 20.Eckert number Ec impact on temperature profile with m = 0.5.

Figure 19 .
Figure 19.Prandtl number Pr impact on temperature with m = 1.5.

Figure 17 .
Figure 17.Suction parameter S impact on temperature profile with m = 1.5.

Figure 21 .
Figure 21.Eckert number Ec impact on temperature profile with m = 1.5.

Figure 22 .
Figure 22.Radius of curvature d impact on pressure profile.Figure24.Power-law index m impact on temperature profile.

Figure 24 .
Figure 22.Radius of curvature d impact on pressure profile.Figure24.Power-law index m impact on temperature profile.

Figure 23 .
Figure 23.Magnetic parameter M impact on temperature profile.

Table 1 .
Comparison of present values of the local friction coefficient ÀRe C fs with Rosca and Pop 21 for several values of d. s

Table 2 .
Numerical values of the local friction coefficient ÀRe 1 2 s C fs for different physical parameters.

Table 3 .
Numerical values of local heat transfer coefficient for different physical parameters.