Radiative flow of non Newtonian nanofluids within inclined porous enclosures with time fractional derivative

An unsteady convection-radiation interaction flow of power-law non-Newtonian nanofluids using the time-fractional derivative is examined. The flow domain is an enclosure that has a free surface located at the top boundaries. Also, the geometry is filled by aluminum foam as a porous medium and the overall thermal conductivity as well as the heat capacity are approximated using a linear combination of the properties of the fluid and porous phases. Additionally, the dynamic viscosity and thermal conductivity of the mixture are expressed as a function of velocity gradients with a fractional power. Marangoni influences are imposed to the top free surface while the bottom boundaries are partially heated. Steps of the solution methodology are consisting of approximation of the time fractional derivatives using the conformable definition, using the finite differences method to discretize the governing system and implementation the resulting algebraic system. The main outcomes reveled that as the fractional order approaches to one, the maximum values of the stream function, the bulk-averaged temperature and cup-mixing temperature are reduces, regardless values of the time.

This paper aims to use the fractional derivative approaches to examine the radiation and Marangoni influences on the power-law non-Newtonian nanofluid flow within an inclined domains. The geometry has a free surface where the surface tension is a function of the temperature gradients and is filled by a porous medium. The worked liquid is consisting of carboxymethyl cellulose (CMC) as a non-Newtonian base fluid while CuO elements are considered as nanoparticles. Also, the aluminum foam is considered as porous elements while the radiation flux is considered in the normal direction. The conformable operator is used for estimating the time fractional derivatives while the dimensionless governing system is solved numerically using an implicit FDM. The novelty of this work appears in simulating important impacts such as Marangoni influences on the flow of unused nanofluids frequently using the fractional partial differential equations that is did not presented before, and is more attractive for the researchers. Also, the results of the current simulations can be effective in various industrial practices such as oil recovery and materials processing. Further, a good survey on applications of the fractional calculus to oil industry is presented in Martínez-Salgado et al. 65 .

Description and formulation of the problem
The flow domain is illustrated in Fig. 1. This situation is consisting of an enclosure that has a free-surface and partially heated from below. The following hypotheses are considered to formulate the mathematical model of this physical case: • Height of the enclosure is H and the inclination angle is γ.
• Length of the heated section is b and its location is denoted by d.
• A low temperature condition ( T = T c ) is decreed to the side walls and the bottom wall is partially heated ( T = T h ) and thermally insulated. • The free surface (top wall) has a heat transfer based on the Newton's low cooling.
• The surface tension σ at the free surface is a function in the nanofluid temperature and it is expressed as:  www.nature.com/scientificreports/ • The nanofluid flow is unsteady, laminar and two dimensional.
• The non-Newtonian power law nanofluids are represented by the single-phase model.
• The base fluid is carboxymethyl cellulose (CMC) while CuO is assumed as nanoparticles.
• The thermophysical properties of the components of the nanofluid are given in Table 1 while the dynamical properties of the CMC-water are included in Table 2. • The domain is filled by homogeneous aluminum foam and the Darcy model is applied.
• The thermal conductivity of the porous medium is considered variable and the thermal radiation is taken in Y-direction. • The thermal equilibrium state is satisfied between the porous and nanofluid phase.
The mathematical formulations of the present case are modeled using the continuity, momentum and energy equations based on the previous assumptions; those are written as, see 10,11,66,67 : In Eq. (6), I 2 = 1 2 tr D 2 is the second invariant of the deformation tensor where D = 1 2 ∇V + (∇V ) T and tr denotes the trace of a second-order tensor. Here, it should be mentioned that the form of the dynamic viscosity (Eq. 6) is given in Zhuang and Zhu 10 .
Also, n is the power-law index where n < 1 and n > 1 correspond to the case of shear thinning fluids and shear thickening fluids, respectively. More specific: The overall thermal conductivity is depending on the features of the power-law (see Ming et al. 42 ) as:  www.nature.com/scientificreports/ Introducing the following boundary conditions: Introducing the next dimensionless quantities: The next system is obtained by using Eq. (11) The dimensionless boundary conditions are: www.nature.com/scientificreports/ Here it should be mentioned that the Roseland approximation is applied for the radiation flux, as follows: In Eq. (22), σ * is the Stephan-Boltzman constant and k * is the mean absorption coefficient. Also, the following correlations are applied for the nanofluid properties: Heat transfer coefficients. In the current case, the definition of the local Nusselt number is depending on two sources of the heat flux, namely, the heat flux due to the heated section and the heat flux due to the thermal radiation. The overall heat flux is expressed as: Consequently, the local Nusselt number at the heated section is denoted as: The average Nusselt number for the CMC-nanofluid is defined as follow: Thermal mixing. In this part, the cup-mixing and bulk-averaged temperatures are defined as: If the non-dimensional temperature changes between 0 and 1, then the value of θ CUP and θ avr cannot run over 1.
Entropy generation. The entropy equations can be writing in the following form: Using the Fourier law of the heat conduction q = −k eff ∇T and substituting Eq. (24) for the heat flux in Y-direction as well as using the dimensionless variables and the characteristics entropy , the entropy generation is given by: www.nature.com/scientificreports/ is ratio of the irreversibility distribution. In addition the local and average Bejan number are expressed as:

Numerical method and validation
An implicit scheme based on the finite differences technique is presented for the governing system of the fractional PDE's. Firstly, the time-fractional derivatives are approximated using the conformable definition then the first upwind and the second differences approaches are used for the both the first and second derivatives. The FDM for the time fractional derivatives is expressed as: In addition the FDM for the diffusion terms in the RHS of Eqs. (12)-(20) are given as: Finally, the following algebraic system is obtained: Here, the following algorithm is used to implement the obtained discretized equations: (a) Select a suitable grid. It is recommended to start with 31 × 31. The alternating direction implicit (ADI) is applied to solve the resulting system while the time step is selected to be 10 −4 . A grid independency investigation is performed and presented in Table 3. It is noted that the grid size of ( 121 × 121) is suitable for all the computations. Additionally, there are many validation tests are carried out for the obtained results. Table 4 shows comparisons of the average Nusselt number (at β = 1 ) with those obtained by Biswas and Manna 67 . Also, Fig. 2 shows graphical comparisons with Biswas and Manna 67 . All these validation tests show that there are excellent agreements between the outcomes. www.nature.com/scientificreports/

Results and discussion
A comprehensive discussion of the obtained outcomes is presented in this section. Here, it is interested with the influences of the time parameter ( 0.1 ≤ τ ≤ 0.5) (unsteady state), variations of the fractional order ( 0.8 ≤ β ≤ 1) , the power-law index ( 0.76 ≤ N ≤ 1) , the radiation parameter ( 0 ≤ Rd ≤ 3) and the inclination angle ( 0 ≤ γ ≤ π/2) . Also, the corresponding value of the Prandtl number is set as Pr = 204 . The outcomes presentation tools are the contours of the streamlines, isotherms, entropy due to fluid friction and local Bejan number. Also, graphical illustrations for the average Nusselt number, cup-mixing temperature, bulk-averaged temperature, total entropy and average Bejan number are taken into account. Impacts of the fraction derivatives order β and dimensionless time parameter  = 0, 30 ). However, as γ is varied ( γ = 60) , a minor clockwise vortex is formulated near the right wall. This cell is enlarged as γ is increased until a symmetrically flow is obtained at γ = 90 . The temperature distributions show an enhancement in the temperature gradients as γ is increased indicating a good rate of the heat transfer at γ = 90 . The fluid friction entropy indicates that the fluid friction irreversibility is occurred near the left and bottom walls while as γ is increased, values of the fluid friction entropy is enhanced due to the enhancement of the velocity gradients. Figure 6 displays the profiles of the average Bejan number for the variations of the inclination angle γ and the power-law index N . It is noted that Be av > 0.5 for all values of γ and N which indicating to the dominance of the heat transfer entropy comparing with the fluid friction entropy. The results, also, disclosed that the increase in the power-law index enhances the temperature gradients and hence the average Bejan number is augmented. Figure 7 exhibits that the total entropy confined the flow domain is a decreasing function in the power-law index N due to the increase in the dynamic viscosity while as the inclination angle γ is growing, an enhancement in the temperature differences are obtained and hence S total is supported. Impacts of γ and N on values of the average Nusselt number Nu av are examined with the help of Fig. 8. The figure revealed that the growing in the power-law index N causes a reduction in the rate of the heat transfer while the thermal boundary layer near the heated section is enhanced as γ approaches to 90. The cup-mixing temperature shows the inverse behavior of the average Nusselt number when the impacts of γ is examined. These observations are presented in Fig. 9. It is, also, noted that the power-law index N has a negative effects on the cup-mixing temperature. In the same context, Fig. 10   www.nature.com/scientificreports/ www.nature.com/scientificreports/ while the opposite observations are found when the power-law index N is growing. All these behaviors are due the increase in the overall dynamic viscosity that reduces the convective-radiation mode.

Conclusions
Using the fractional derivatives basics, the unsteady convective-radiation flow confined an enclosure filled with CMC-water power-law non-Newtonian nanofluids was investigated. The fractional derivatives were taken on the time while the conformable definitions were used to approximate the calculations. The Marangoni effects are imposed to the top-free surface of the domain while the bottom boundaries are partially heated. The onephase model in which the overall dynamic viscosity and thermal conductivity are functions of the power-law index is presented while the Rosseland approximation is used for the thermal radiation. Beside the cup-mixing temperature and the bulk-averaged temperature, the entropy of the system is examined for the variations of the controlling parameter. The main outcomes of this study revealed that the increase in the fractional order enhances the average Nusselt number while the maximum values of the stream function, the cup-mixing temperature and the bulk-averaged temperature are reduced as β approaches to one, regardless values of the time. Also, presence of the radiation parameter within the domain accelerates the mixture flow and enhances the thermal boundary layer. Additionally, the increase in the power-law index reduces the convective mode, the total entropy, the cup-mixing temperature and the bulk-averaged temperature while the average Nusselt number is enhanced. www.nature.com/scientificreports/  www.nature.com/scientificreports/ www.nature.com/scientificreports/