Heat transfer analysis of generalized second-grade fluid with exponential heating and thermal heat flux

The aim of present work is to apply the Caputo–Fabrizio fractional derivative in the constitutive equations of heat transfer. Natural convection flow of an unsteady second grade fluid over a vertical plate with exponential heating is discussed. The generalized Fourier law is substituted in temperature profile. A portion of the dimensionless factors are utilized to make the governing equations into dimensionless structures. The solutions for temperature and velocity profiles of Caputo–Fabrizio model are acquired through the Laplace transform method. These solutions are greatly affected through the variation of different dimensionless variables like Prandtl number, Grashof number, and second-grade fluid parameter. Finally, the influence of embedded parameters is shown by plotting graphs through Mathcad. From the graphical results it is concluded that, the temperature of the fluid decreases with the increasing values of the Prandtl number and Second grade fluid parameter and increases with the passage of time. The velocity of the fluid increases with increasing values of the Grashof number, second grade parameter and time while decreases with increasing values of fractional parameter and Prandtl number.


Problem's formation
Consider the second grade fluid of unsteady flow closer to an infinite upright plate with exponential heating the straight upward direction of the plate.Consider as x-axis direction and y-axis is put up with perpendicular to the flat established by the plate when t = 0 both the fluid and the plate are remainder to the constant temperature T infinity.After some time, the plate begins to move at the specified speed.The governing equation is provided by, according to the standard Boussinesq's approximation 17 .Fluid flow problem is represented geometrically by Fig. 1.
Following are the IC's and BC's; Substituting the bellow dimensionless factors, Introducing the above unit less factor in (1)-( 3) and drop the star, we acquired The fractional generalization of Fourier law is given by Henry et al. 18 and Hristov 19,20 , which addresses the thermal heat flux.
(1) www.nature.com/scientificreports/Caputo-Fabrizio fractional derivative and its Laplace transform is defined as 21 Using the above CF definition in Eq. ( 6) in D γ t we obtain the following form,γ is fractional parameter, The dimensionless initial and boundary condition are follow;

Temperature profile
Put Eq. ( 7) in Eq. ( 8) and taking Laplace transform to Eq. ( 8) and also to associated IC and BC, we acquired where a 1 = 1 1−γ .The transform IC and BC are following, After solving Eq. ( 10) and using Eq. ( 11), we acquired Equation (12) inverse Laplace transformation is as under: where and

Velocity profile
Taking LT on Eq. ( 5) and their associated IC and BC, and substitute Eq. ( 13) we acquired the following ODE, After solving Eq. ( 16) and introduce the related transform IC and BC, we acquired where Pr(q+γ a 1 ) a 1 .

Discussion and results
In this section, we examine physical comprehension.The physical sketch of the problem is given in The effects of time on the temperature profile are depicted in Fig. 4 plots.It has been noted that as the value of time increases, so does the temperature profile.We have discovered that as the values of time are increased, the boundary layer becomes thicker.Close to the plate, when it asymptotically approaches zero, is where the temperature is the highest.The behavior of several embedded elements on velocity profiles is depicted in Figs. 5, 6, 7, 8 and 9.The impact of a fractional parameter on the velocity field is depicted in Fig. 5.When the values of the fractional parameter increases the momentum boundary layer thickness increases.So, with increasing values of the fractional factor, it can be seen that the fluid's velocity decreases.After a certain threshold for y, they exhibit the opposite effects, i.e., the velocity increases as the value of the fractional parameter is increased.
T(y, t) = T 1 (y,t)* T 2 (y,t) 0 < α< 1, where Figure 6 depicts the properties of the Grashof number.For larger values of the Grashof number, the fluid's velocity increases.Since the Grashof number is the ratio of the buoyancy forces to the viscous forces so the increasing values of the Grashof number, decreases the viscosity and as consequence of decrease in viscosity the velocity of the fluid increases.The Grashof number is used to describe the association of the mass and heat transfer thermally induced by natural convection.Physically, this is due to the forces of viscosity and thermal upgrading.The existence of convective current electrified the fluid more as the Grashof number increased.Convective current boosts the fluid's velocity via increasing buoyancy force.
The effect of the second-grade fluid component on velocity is seen in Fig. 7.With a higher value of secondgrade fluid, the fluid's motion decreases.This demonstrates the fall in momentum limit layer thickness.The effect of the Prandtl number on the momentum limit layer is seen in Fig. 8.It shows that the fluid velocity decreases as the Prandtl number rises.The Prandtl number is the ratio of the momentum diffusivity to the thermal conductivity so when the Prandtl number increases it causes an increase in the momentum diffusivity.The Prandtl  number and heat conductivity have an inverse relationship.The fluid will have lower heat conductivity for larger Prandtl numbers, which results in a higher viscosity.The velocity decreases as a result.Figure 9 emphasizes how time affects the velocity field.It has been noted that when time goes faster, fluid velocity also accelerates.Fluid velocity increases closer to the plate and as we move away from it, it asymptotically decreases to zero.

Conclusions
We looked into the generalized Fourier law, thermal heat flux, and natural convection flow of a second-grade fluid in this paper's for closed-form solution.Using the Caputo-Fabrizio fractional derivative, the well-known model is converted to a fractional model.The Laplace transform is used to obtain the solutions to the momentum Fig.1.To determine the effects of various embedded components, such as the Grashof number, parameter for secondgrade fluid, the Prandtl number, the fractional parameter, and the time, certain mathematical calculations have been made.Every graph is plotted against y.The effects of different factors on the temperature profile are shown in Figs.2, 3 and 4. The behavior of the Prandtl number on the temperature field is shown in Fig.2.It shows that when the Prandtl number increases, the temperature profile slows down.Physically, this phenomenon results from a drop in thermal conductivity as a result of a drop in fluid temperature.The dimensionless Prandtl number is an intrinsic characteristic of the fluid.The free flowing fluids have small Prandtl number with high heat conductivity.The Prandtl number controls the relative thickness of the thermal and momentum boundary layers.The Prandtl number is the ratio of the momentum diffusivity to the thermal conductivity so when the Prandtl number increases it causes a decrease in the thermal conductivity.Hence the temperature of the fluid decreases with the increase in the Prandtl number.The effects of a fractional parameter on temperature are plotted in Fig.3.When the values of the fractional parameter increases the thermal boundary layer increases.Therefore when the values of fractional increases, the fluid's temperature decreases.

Figure 2 .
Figure 2. Analysis of Prandtl numer Pr on temperature profile.

Figure 3 .
Figure 3. Analysis of fractional parameter γ on temperature profile.

Figure 4 .
Figure 4. Analysis of time t on temperature profile.

Figure 5 .
Figure 5. Analysis of fractional parameter γ on velocity profile.

Figure 6 .
Figure 6.Analysis of Grashof number Gr on velocity profile.

Figure 7 .
Figure 7. Analysis of second grade parameter α 1 on velocity profile.

Figure 8 .
Figure 8. Analysis of Prandtl numer Pr on velocity profile.

Figure 9 .
Figure 9. Analysis of time t on velocity profile.