Application of Fractional Derivative Without Singular and Local Kernel to Enhanced Heat Transfer in CNTs Nanofluid Over an Inclined Plate

. Nanofluids are a novel class of heat transfer fluid that plays a vital role in industries. In mathematical investigations, these fluids are modeled in terms of traditional integer-order partial differential equations (PDEs). It is recognized that traditional PDEs cannot decode the complex behavior of physical flow parameters and memory effects. Therefore, this article intends to study the mixed convection heat transfer in nanofluid over an inclined vertical plate via fractional derivatives approach. The problem in hand is modeled in connection with Atangana-Baleanu fractional derivatives without singular and local kernel having strong memory. The human blood is considered as base fluid dispersing carbon nanotube (CNTs) (single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes(MWCNTs )) into it to form blood-CNTs nanofluid. The nanofluids are considered to flow in a saturated porous medium under the influence of an applied magnetic field. The exact analytical expressions for velocity and temperature profiles are acquired using the Laplace transform technique and plotted in various graphs. The empirical results indicate that the memory effect decreases with increasing fractional parameters in the case of both temperature and velocity profiles. Moreover, the temperature profile is higher for blood-SWCNTs by reason of higher thermal conductivity whereas, this trend is opposite in case of velocity profile due densities difference. to non-locality, non-singularity and strong heredity and memory effect. A fractional Casson fluid model is developed for human blood-CNTs nanofluid associated with physical initial and boundary conditions. The model is solved for exact solutions via the Laplace transform technique. The analytical results are displayed in graphs with physical arguments.


Introduction
In mixed convection regimes, enhanced heat transfer is significant for energy savings operations in industries. The primary constraint of traditional heat transfer fluid is poor thermal 3 carried out entropy generation and the second law of, thermodynamics application for kerosene oil-SWCNTs and kerosene oil-MWCNTs flow in a rotating microchannel. They considered source/sink, radiation and magnetic field effect. Their results shown that the velocity function reduced with Reynold number and entropy generation increases with Reynold, and Brinkman numbers. The interesting applications of CNTs nanofluid can be found in the review papers [13][14][15] and the reference therein.
The CNTs feature considerable mechanical and electrical thermal conduct forming a hexagonal cylinder network of carbon atoms 100 nm in length and 1nm in the bore. The major application of CNTs listed additives in polymers, nanolithography, hydrogen storage, supercapacitor, lithium-battery anodes and drug delivery [16]. Murshed [17] mention in the review paper that CNTs nanofluids have six-time higher thermal conductivity compared to other materials at ambient temperature. The CNTs nanofluids sufficiently investigated in the literature (see for example Xie et al. [18], Sarafraz et al. [19], Selimefendigil, and Öztop [20], Ghazali et al. [21] and Abdeen et al. [22]) but without memory and heredity effect. This is since, in mathematical studies, the traditional models with integer-order PDEs are utilized. These models can improve by using the applications of fractional derivatives. It is approved in the previous literature that fractional derivatives model can explain efficiently the real-world problem comprising electrical networks, diffusive transport, probability, electromagnetic theory, rheology, viscoelastic materials and fluid flow [23][24][25][26][27][28][29]. In the literature, several approaches for fractional derivatives are presented but the most common are the Riemann-Liouville [30], the Caputo [31,32], the Caputo-Fabrizio [33] and Atangana-Baleanu [34] fractional derivatives approaches. Among them, the most recent is Atangana-Baleanu fractional derivative without local and singular kernel having strong heredity and memory effect.
For the problem in hand, the Atangana-Baleanu fractional derivative approach is chosen due to non-locality, non-singularity and strong heredity and memory effect. A fractional Casson fluid model is developed for human blood-CNTs nanofluid associated with physical initial and boundary conditions. The model is solved for exact solutions via the Laplace transform technique. The analytical results are displayed in graphs with physical arguments. 4

Description of the Proposed Model
Consider the unsteady mixed convection flow of blood bases CNTs nanofluid over an inclined vertical plate with isothermal temperature T  (room temperature/ ambient temperature).
The half-space of the plate is filled with packed with human blood with SWCNTs and MWCNT with a saturated porous medium. The nanofluid is assumed to be electrically conducting. Hence, a magnetic field ( ) 2 0 sin nf B  of strength 0 B and direction  is applied to the flow direction. The applied magnetic field due to polarization is ignored due to the very small Reynolds number.
At the beginning at 0 t  , the system is in the rest position. But since the short interval t + , the inclined plate oscillates with  and the ambient temperature of the plate T  rises to W T . By the virtue of rising in temperature and oscillation of the plate, the mixed convection uncoils and the nanofluid starts motion in the upper direction as exhibited in Figure 1.    [37], Darcy's law [38], Fourier law of heat conduction [39] and Boussinesq approximation [40] the governing equations of the proposed problem as given by [36] subject to the following initial and boundary conditions ( ) ( ) where nf  is the density, ( ) nf is used for nanofluid where the subscripts f and s will be used for base fluid and solid nanoparticles respectively. The mathematical models for the thermophysical properties of nanofluid are given in Table 1 whereas its numerical values are given in Table 2. 6 Table 1 Mathematical model for thermophysical properties of nanofluid [41].

Physical Quantity
Mathematical model Density  , , , where ( ) ( ) is the dimensionless Casson fluid parameter, magnetic number, permeability parameter, thermal Grashof number, and Prandtl number respectively and 0 4  , and 5  are constant terms produced during Calculi. The time-fractional form of Eqs. (6) and (7) is given by [42,43] ( ) ( ) ( ) ( ) where ( ) is the non-local and non-singular Mittag-Leffler function used as the kernel in the construction of Eq. (12). The Laplace transform of Eq. (12) is given by [44] ( )  . It worth mentioning here that for 1  = , the model presented in Eqs (10) and (11) is reduced back to the classical form exhibited in Eqs. (2) and (3). This validated the time-fractional model proposed for CNT's-blood nanofluid.

Solutions of the Problem
The Laplace transform technique is adopted to find the exact solutions for the proposed problem.

Solution of Energy Equation
Applying the Laplace transform to Eq. (11) in the light of Eqs. (12) and (14) and using the corresponding initial and boundary conditions from Eqs. (8) and (9) Equations (16) Equation (20) and It is mentionable here that Eq. (20) satisfy all the imposed physical conditions which validate our solutions.

Solutions of Momentum Equation
In this section, the same procedure of solutions as of energy equation is acquired.

Discussion of Results
In this section, the effects of various flow parameters (for instance, fractional parameter  The M is a dimensionless number which is accorded with the Lorentz force that counters the nanofluid velocity. Higher the M higher will be the Lorentz force which will resist the motion. This is why the velocity retarded in both the cases of CNTs with increasing M [42]. Likewise, the inclination of the magnetic field  weakens the impact of M that carries off the Lorentz force.

 =
(normal magnetic field) the influence of the Lorentz force is the strongest as depicted in Figure (9) [43].
Figure (10) presents the effect of K on the velocity profile for both the cases of CNTs. It is witnessed that greater values of K magnifying the velocity field. This is on account of a reduction of resistance of porous medium and causes improvement in the momentum boundary layer. Physically, in this view, the velocity field enhanced [38]. Finally, Figure (11

) depicts the consequences of
Gr on the velocity profile. it is the ratio of buoyancy and viscous forces. The higher Gr leads to enhancement in buoyancy forces that give grow the induced flows [35].

Concluding Remarks
In the study, a fractional initial and boundary values problem is modeled for the flow of human • The temperature profile increase with an increasing volume fraction of CNTs although decreases with increasing fractional parameter (for both cases of CNTs) because of variation in the thermal boundary layer.
• The velocity profile increases with increased permeability of the porous medium and thermal Grashof number due to the improvement in the velocity boundary layer.            Consequences of  on ( )