Numerical investigation of heat and mass transfer of variable viscosity Casson nanofluid flow through a microchannel filled with a porous medium

: Thermal behaviours and hydrodynamics of non-Newtonian nanofluids flow through permeable microchannels have large scale utilizations in industries, engineering and bio-medicals. Therefore, this paper presents the numerical investigation of heat and mass transfer of variable viscosity Casson nanofluid flow through a porous medium microchannel with the Cattaneo-Christov heat flux theory. The highly nonlinear PDEs corresponding to the continuity, momentum, energy and concentration equations are formulated and solved numerically via the second order implicit finite difference scheme known as the Keller-Box method. Accordingly, the numerical simulations reveal that variable viscosity parameter, thermal Grashof number, solutal Grashof number, thermophoresis parameter, Schmidt number and Casson fluid parameter show increasing effects on both velocity and temperature of the nanofluid. Furthermore, the temperature profile escalates with increasing values of the Eckert number and the thermal relaxation time parameter. Thus, the Cattaneo-Christov heat flux model is beneficial in warming the transport system of microfluidics when compared to that of the classical Fourier heat conduction law. The temperature profile however, indicates a retarding behavior with increasing values of the Brownian motion parameter, Prandtl number and porous medium parameters namely Forchheimer number and porous medium shape parameter and hence, the porous medium quite effectively controls the nanofluid temperature distribution which plays substantial roles in cooling the transport system of microfluidics. Moreover, the concentration profile shows an increasing pattern with escalating values of the Prandtl number, Schmidt number and thermophoresis parameter but it demonstrates a decreasing trend with the Casson


Introduction
Nowadays, with increasing energy prices and a demand for energy efficiency, many efforts are made for energy saving and reduction of production costs and hence augmentation of convective heat as well as energy storage are the charming issues in engineering associated with the energy conservation [1] .Consequently, during the past decades numerous techniques were devised to advance the performance of industrial equipment particularly in various heat energy transforming devices or heat-exchangers.This scientific revolution ensured the strong industrial productivity growth which in turn has brought an improved societal quality of life worldwide.In general, the goal is improving the thermal efficiency of heat conversion devices which is referred to as heat transfer rate augmentation which meliorate the overall performance of the industrial system including reducing the initial and capital costs of the heat transfer devices or heat ex-changers.
For internal flows such as fluids flow in tubes or channels, the convectional rate of heat transfer can be augmented through the techniques that do not require additional external power such as refinement of flow channel geometry and fluid additives [2] .As far as channel geometry refinement is concerned, microchannels have been identified as the most essential one to transport fluids in a miniaturization system.To this end, in 1981 the concept of microchannels was predominantly demonstrated by Tuckerman and Pease [3] who achieved high heat flux removal capacity of about 800   2 ⁄ within heat ex-changers by utilizing a channel with hydraulic diameter of 100  .Microchannels are increasingly used in several industrial and engineering applications that span from cooling of microelectronics to bio-technological applications [4] .Therefore, there are numerous research studies that are reporting the investigations of various fluids flow through microchannels.For example, the combined effects of viscous-Joule dissipation and slip wall on the electro-osmotic peristaltic flow of the Casson fluid in a rotating microchannel is investigated by Reddy et al. [5] .Later on, Kmiotek and Kucab-Pietal [6] presented the study of heat transfer phenomena in a microchannel with the presence of slender porous material.
Although they are well known by their great heat removal capacities, fluid flows through microchannels encounter excessive pressure drop and thus it involves a great pumping power [7] .In addition, conventional base fluids like water, ethylene-glycol and oils are poor in heat transfer capacities because of their low thermal conductivity [8] .Due to these facts, new technological fluids with enhanced thermophysical properties such as thermal conductivity and dynamic viscosity are of prodigious attention for microchannel flows.In this regard, insertion of nanometer-sized (1 = 1 × 10 −9 ) solid particles into the conventional fluids is one of the most fruitful convective heat transfer enhancement method.Thus, with persistent diminishment as well as growing heat eliminations in novel brands of devices, there is a need for best effective heat transfer fluids in microchannels.In 1995, Choi and Eastman [9] was the first scientist who introduced the idea of nanofluids meaning nanofluids are engineered suspensions/dispersions of nanoparticles into the common base fluids.Since then, nanofluid flows through microchannels have been utilized in the cooling of various technological and industrial processes [10] .Therefore, now days, numerous researchers including [11][12][13][14][15][16][17][18] are working on the analysis of nanofluids flow as well as heat transfer characteristics through microchannels.
The rates of heat transfer through microchannels become more enhanced through the amalgamation of nanofluids and porous media.Actually, the convective flows in porous media have a significant applications including extraction of crude oil, extraction of geothermal energy, pollution of groundwater, radioactive nuclear waste storage, cooling in transpiration, filtration in chemical industries, purification, transportation processes in aquifers, and fiber insulation [19,20] .Consequently, nowadays the analysis of thermal behaviors and flow of nanofluids trough porous media received a great number of attentions.For example, Algehyne et al. [21] studied the hydrodynamics of chemically reacting water based alumina nanoparticles past over a curved porous geometry under multiple convective constraints by adopting the Buongiorno's and Koo-Kleinstreuer-Li's nanofluid models.The nonlinear governing equations were numerically tackled by employing the irregular generalized differential quadrature scheme together with the Newton-Raphson method.Their outcomes indicated that the nanofluid velocity decreased as the slip-velocity and drag forces escalated.Also, Rashad et al. [22] investigated the partial slip and MHD combined convective flow of Cu-water nanofluid and heat transfer characteristics inside a lid-driven porous enclosure.
Maneengam et al. [23] numerically investigated the influences of Lorentz and Buoyancy forces on the hybrid fluid comprising Al 2 O 3 − Cu nanoparticles through a lid-driven container having obstacles of various shapes.The numerical simulation was given via the Galerkin finite element method and it was observed that the triangular shape of the obstacle enhanced the thermal performance.That is, the Nusselt number increased by 15.54% when the baffle altered its shape from the elliptic to the triangular.Moreover, the electro-magneto-hydrodynamic investigation of nanofluid motion over a Riga plate filled with a Darcy-Forchheimer porous medium was presented by Rasool et al. [24] .The governing partial differential equations were transformed into the ordinary differential equations by using appropriate similarity variables and thereafter tackled numerically.Thus, nanofluid velocity is remarkably influenced by the Darcy-Forchheimer porous medium.In addition, the Darcy-Forchheimer and the Lorentz forces enhanced the skin friction coefficient.Furthermore, Makinde et al. [25] presented the numerical investigation of steady hydromagnetic nanofluid convection inside the micro-porous-channel with injection/suction, radiative heat and heat absorption/generation.Besides, to read more on heat transfer phenomena as well as motion of fluids through porous media the references [26][27][28][29][30] are preferable.
Casson fluid model can be regarded as the most appropriate to industrial applications for instance, exploring the mechanism of pseudo plastic yield stress liquids, in food processing, metallurgy and drilling and bio-engineering operations [31] .Therefore, nowadays a reasonable number of communications can be quoted highlighting Casson fluid model in the existing literature.For instance, Thammanna et al. [32] presented the transient analysis of magnetohydrodynamic stretched flow of couple stress Casson fluid with chemical reaction.Similarly, Mahanthesh et al. [33] addressed the boundary layer flow and heat transfer in Casson fluid submerged with dust particles over three different geometries (vertical cone, wedge and plate).The governing equations were solved by shooting method coupled with the Runge-Kutta-Fehlberg-45 integration scheme.According to their results, a rise in Casson fluid parameter enhances the fluid temperature and the magnetic field improves heat transfer rate.Besides, very recent papers like references [34,35] comprise similar Casson fluid analysis.
The above literature review can establish that the analysis of various nanofluids flow as well as heat and mass transfer phenomena through microchannels because of free or forced convection was presented in detail.However, limited studies on mixed convection as well as heat and mass transfer characteristics of Casson fluid through a vertical microchannel embedded with saturated porous medium have been carried out.Even those investigations are rare in considering temperature dependent dynamic viscosity and the Cattaneo-Christov heat-mass flux theory.Therefore, this paper mainly emphases on the analysis of mixed convection of Casson nanofluid flow as well as heat and mass transfer characteristics in a vertical microchannel filled with a saturated porous medium.The novelty of the present study is to consider temperature dependent dynamic viscosity, Darcy-Forchheimer porous medium, non-uniform temperature at the permeable walls and nanofluid injection/suction mechanism.Moreover, frame-in-different generalization of the classical Fourier heat conduction law and Fick molecular mass diffusion law which is also known as the Cattaneo-Chirstov heat-mass flux theory is employed in formulating the governing equations for energy and concentration.

Mathematical analysis and problem formulation
Let us consider steady mixed convection of Casson nanofluid through a permeable vertical microchannel.Assume that the permeable walls of the microchannel are positioned at  = 0 and  =  as demonstrated in Figure 1, where  is the width of the microchannel.Also consider the fluid motion is induced by the pressure gradient and the thermal and solutal buoyancy forces.The nanofluid injection ( = 0) and suction ( = ) at the walls of the microchannel is also taken into account.Moreover, it is assumed that there is no slip condition at the walls whereas non uniform temperatures at the walls are considered in such a way that  0 is temperature at  = 0 and   is temperature at  =  with  0 <   .The dynamic viscosity of the nanofluid is considered to be temperature dependent and written as () =  0  −(− 0 ) .Here  represents viscosity variation coefficient whereas  0 denotes the left wall dynamic viscosity.
The non-Newtonian fluid known as Casson was pioneered by Casson [36] , in 1959 when he was investigating the flow equations for a pigment oil suspension of printing ink.According to Raza et al. [27] the shear stress tensor of the Casson fluid model is given as follows.
Here   =   .  where   designates rate of deformation at the (, ) ℎ component while   represents fluid stress yield.Besides,  denotes product of the component of deformation rate with itself and   is a critical value of this product.Similarly,   denotes plastic dynamic viscosity of the non-Newtonian fluid.Considering the case  <   , Equation (1) takes the form: where  = ).
For two dimensional flows, ).Therefore, Equation (2) becomes: Therefore, by using all the above assumptions and considering the Cattaneo-Christov heat-mass flux theory, under the usual Oberbeck-Boussinesq approximations, the governing PDEs for continuity, energy and concentration equations are given as follows.
= 0 (4) With the boundary conditions: where  is velocity in the axial direction,  0 is uniform injection/suction velocity,  is width of the microchannel,  is density of nanofluid,  is pressure,  is Temperature of the nanofluid,  is concentration of nanoparticles,   is specific heat at constant pressure,   =    ⁄ is nanofluid thermal diffusivity, with  signifies thermal conductivity, г is the ratio of nanoparticles heat capacity and base fluid heat capacity,  is porous medium permeability,  is acceleration due to gravity,   is the Brownian diffusion coefficient,   is the thermal diffusion coefficient,  1 is thermal expansion coefficient,  2 is solutal expansion coefficient,   is thermal relaxation time parameter,   is solutal relaxation time parameter and  is porous medium inertia resistance coefficient however,  = 0 yields the usual Darcy law.

Non-dimensionalization
We define the following dimensionless variables for the sake of non-dimensionalization.
= 0 (10) With the dimensionless boundary conditions: where,  is the suction/injection Reynolds number,  is thermal Grashof number,  is solutal Grashof number,  is the Eckert number,  is the Prandtl number,  is dimensionless pressure gradient parameter,  is dimensionless viscosity variation parameter, λ  is dimensionless thermal relaxation time parameter, λ  is dimensionless solutal relaxation time parameter,  is porous medium shape factor parameter,  is the Forchheimer number,  is the Schmidt number,  is the Brownian motion parameter and  is the thermophoresis parameter.
Actually, the continuity Equation (10),   = 0 suggests that  is as a function of  only.Therefore, the dimensionless governing Equations ( 11)-( 14) are ODEs with respect to  only and written as follows.
With the dimensionless boundary conditions:  = 0,  = 0, ′ + ′ = 0 at = 0,  = 0,  = 1,  = 1 at  = 0 There are also physical quantities of engineering interests including coefficient of the skin friction   , the Nusselt number  (wall heat transfer rate) and the Sherwood number ℎ (wall mass transfer rate).Therefore, the non-dimensional forms are given below.

Numerical solutions
In this study the numerical simulation is done via the Keller-Box method.The Keller-Box is second order accurate implicit finite difference scheme which was named after the pioneer work of Cebeci and Bradshaw [37] .Indeed, the Keller-Box is stable unconditionally and comprises attractive extrapolation features with arbitrary spacing.The scheme consists the following four crucial steps.

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Reducing the second order ODEs into a system of first order equations.

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Finite difference discretization of a system of first order equations.

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Linearizing the resulting algebraic equations by using the Newton method and writing in matrixvector form.

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Solving the linearized system of equations using the block-tridiagonal elimination technique.
Therefore, the Keller-Box method is employed to solve the non-linear ODEs (15)-( 17) along the boundary conditions (18).

The velocity, temperature and concentration profiles
The influence of the Casson fluid parameter on velocity and temperature of the nanofluid is portrayed in Figure 2a, b respectively.Accordingly, within the microchannel core region both velocity and temperature of the nanofluid increase with .This is the case because as  increases, the yield stress dominates the dynamic viscosity of thenanofluid.Remarkably, as  → ∞ the Casson fluid have a tendency of performing like Newtonian fluid.This result is similar to the findings of Reddy et al. [5] and Roja et al. [38] .As values of the variable viscosity parameter  rise, both velocity and temperature of the nanofluid increase significantly as displayed in Figure 3a, b respectively.Indeed, this result is expected because () =  0  − which implies that the dynamic viscosity decreases as  increases and hence it is favourable for fluid motion that also in turn leads to an increase in the nanofluid temperature.Similar result was reported by Mahmoudi et al. [39] .3b).The secret behind this result is the fact called effect of the cross-diffusion meaning small increase in temperature of the nanofluid may result in small decrease in the concentration of the nanoparticles and vice-versa.By the same argument, the nanoparticles concentration profile decreases when the magnitude of  increases as demonstrated in Figure 4b.The impacts of thermal buoyancy parameter  (thermal Grashof number) and solutal buoyance parameter  (solutal Grashof number) respectively, on the nanofluid velocity and temperature are presented in Figures 5a, b and 6a, b.Consequently, these figures show that the nanofluid velocity and temperature escalate with rising magnitudes of  and  .Physically, as  and  increase, the buoyance forces due to the differences in temperature and concentration respectively also increase that obviously increases the nanofluid velocity which in turn rises the viscous heating within fluid layers.So, the temperature profile also enhances inside the microchannel core region.However, Figure 7a, b demonstrates the reverse situations in the case of concentration of the nanoparticles.The influences of thermophoresis parameter  on the nanofluid velocity and temperature are portrayed in Figure 8a, b respectively.Hence, the nanofluid velocity and temperature escalates as the amounts of  increase.Physically, the thermophoretic force gets stronger when the amounts of  rises that will lead to the migration of nanoparticles from hot microchannel walls to the cold fluid throughout the core flow region.Therefore, the temperature profile enhances which also in turn enhances the velocity profile with increasing amounts of .  Figure 9a,b are graphs that show the nanofluid velocity and temperature fall down as the magnitudes of the parameter of Brownian motion  upsurge.Physically, when the values of  escalates, the random and non-uniform movements of the nanoparticles inside the microchannel core region also enhance that will lead to the increment of collisions between the moving base fluid molecules and the nanoparticles.Therefore, these increments of collisions may cause the retardation of fluid motion and hence its velocity.Moreover, when nanofluid moves slowly its temperature also reduces due to the lessened fluid kinetic energy.Figure 10a indicates that as the amounts of  upsurges, the nanoparticles concentration also enhances.An argument for this result may be the fact that when the amounts of  rises, the thermophoretic force gets stronger that will lead to the migration of nanoparticles from hot microchannel walls to the cold fluid and thus concentration of the nanoparticles enhances throughout the core flow region.But Figure 10b reveals that the nanoparticles concentration diminishes as the magnitudes of the parameter of Brownian motion  rise.Physically, when the values of  escalates, the random and non-uniform movements of the nanoparticles inside the microchannel core flow region also enhance that will lead to the increment of collisions between the moving base fluid molecules and the nanoparticles.Therefore, these increments of collisions and random movement of nanoparticles may cause the diminishing of nanoparticles concentration throughout the core flow region.12a).Indeed, the justification is the fact that as the amount of  enhances the amount of molecular mass diffusion within the fluid reduces as a result of which the nanoparticles concentration stays higher throughout the microchannel core flow region.Figure 12b portrays an effect of the solutal relaxation time parameter  on the concentration profile.As it can be seen from the graph, when magnitude of  increases the nanoparticles concentration decreases.
Figure 13a depicts that the nanofluid velocity reduces considerably with increasing amount of  (porous medium shape factor parameter). Mahmoudi et al. [39] and Kasaejan et al. [42 ] presented alike findings.This is the case because permeability of the porous medium and  are inversely related so that the nanofluid velocity decreases with increasing amount of .Likewise, the nanofluid velocity declines noticeably with increasing amount of  (the Forchheimer number), since F represents the inertial resistivity force which obviously opposes the nanofluid motion and therefore, declines the velocity profile.9a.This is the case since  describes viscous heating between the fluid layers and thus the nanofluid temperature enhances as the value of  rises.From the references [39][40][41] , a comparable outcome was reported.Figure 14b displays a reverse situation in the case of the nanoparticles concentration.
The influences of the Prandtl number  on the nanofluid temperature and the nanoparticles concentration are displayed in Figure 15a, b respectively.Therefore, the nanofluid temperature decreases with escalating amounts of  because high amount of  implies less amount of fluid thermal diffusivity within the microchannel core flow region as a result of which the nanofluid temperature remains lower throughout the core flow region (see Figures 15a).Analogous result was given in Kasaejan et al. [42] and Menni et al. [43] while Mahmoudi et al. [39] reported a conflicting result.Contrastingly, Figure 15b presents that the nanoparticles concentration escalates when the size of  upsurges.
The influences of thermal relaxation time parameter  on the nanofluid temperature and the nanoparticles concentration are indicated in Figure 16a, b respectively.Accordingly, Figure 16a indicates that larger value of  yields larger amount of nanofluid temperature.In fact, thermal relaxation time is amount of time that the fluid requires to transfer heat into its surroundings and therefore, bigger value of  indicates the fluid needs more extra time to transfer heat so that its temperature remains higher.Here, for  = 0, heat transfers slowly throughout the microchannel walls and hence fluid temperature distribution is lower for the classical Fourier heat conduction law.That is, the Cattaneo-Christov heat flux model is beneficial for the microfluidics systems with high heat.An equivalent finding was reported by Mahanthesh et al. [33] and Nayak et al. [44] .The opposite scenario was observed for the nanoparticles concentration with  (see Figure 16b).

Skin friction coefficient (wall shear stress)
Figures 17a-19b illustrate the impacts of the embedded governing thermophysical parameters on the skin friction coefficient at the left wall ( = 0) as well as right wall ( = 1) of the microchannel.As a consequence, the figures present that the skin friction coefficient   at  = 0 and  = 1 increase as the magnitudes of the pressure gradient parameter  , Eckert number  , Forchheimer number  and injection/suction Reynolds number  increase.Mahmoudi et al. [39] found a similar research result.Thermal Grashof number  shows a decreasing effect on   at  = 0 and  = 1.Moreover,   rises at  = 0 (see Figure 19a) but   falls down at  = 1 (refer Figure 19b) with increasing values of .

The nusselt number (wall heat transfer rate)
Figures 20a-23b illustrate the impacts of the embedded governing thermophysical parameters on the Nusselt number  at the left wall ( = 0) as well as right wall ( = 1) of the microchannel.As a consequence, the figures reveal that  at  = 0 and  = 1 increase as the values of  ,  and  increase.Mahmoudi et al. [39] found a similar research result.Moreover, as the magnitude of , , , and  increase,  at  = 0 shows a rising tendency but it shows a decreasing trend at  = 1.
Besides,  and e indicate the opposite scenarios on  at  = 0 and at  = 1.

The sherwood number (wall mass transfer rate)
Figures 24a-27b portray the influences of the embedded governing thermophysical parameters on the Sherwood number ℎ at the left wall ( = 0) as well as right wall ( = 1) of the microchannel.As a result, these graphs depict that ℎ at  = 0 and  = 1 increase as the values of , ,  and  increase.Nonetheless,  and  show a diminishing consequence on ℎ at  = 0 and  = 1.Moreover, as the magnitude of , , , and  increase,  at  = 0 shows a rising tendency but it shows a decreasing trend at  = 1.Besides,  and e indicate the opposite scenarios on  at  = 0 and at  = 1.

Conclusions
Thermal behaviours and hydrodynamics of non-Newtonian nanofluids flow through permeable microchannels have large scale utilizations in industries, engineering and bio-medicals.Hence, this paper mainly emphases on the analysis of mixed convection of Casson nanofluid flow as well as heat and mass transfer characteristics in a vertical microchannel filled with a saturated porous medium.The highly nonlinear PDEs corresponding to the continuity, momentum, energy and concentration equations are formulated and solved numerically via the second order implicit finite difference scheme known as the Keller-Box method.Therefore, depending on the results obtained from the present analysis the key conclusions are given as follows.

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Both nanofluid velocity and temperature indicate a rising trend as the values of , , , , ,  and  increase.

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The porous medium dampens the nanofluid motion as well as the nanofluid temperature distributions.

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The temperature profile escalates with increasing values of  and  however it falls with .

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The effect of the Brownian motion is opposite on the nanofluid velocity and temperature profiles.

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The nanoparticles concentration increases with ,  and .

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The skin friction coefficient   shows an increasing behavior as the values of , ,  and  increase.

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The Nusselt number  demonstrates an enhancing pattern when the magnitudes of ,  and  rise.

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and  reveal opposite scenarios on the Nusselt number  at the left and right walls of the microchannel.

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The mass transfer rate ℎ at both walls of the microchannel shows an increasing pattern with increasing values of , ,  and .

Figure 1 .
Figure 1.Coordinate system and physical flow model.

Figure
Figure4adepicts that the nanofluid concentration profile decreases as the values of the Casson fluid parameter  increases.Actually, this is the opposite scenario to the effect of  on the temperature profile (see Figure3b).The secret behind this result is the fact called effect of the cross-diffusion meaning small increase in temperature of the nanofluid may result in small decrease in the concentration of the nanoparticles and vice-versa.By the same argument, the nanoparticles concentration profile decreases when the magnitude of  increases as demonstrated in Figure4b.

Figure 3 .
Figure 3. (a) effects of (a)  and (b)  on concentration profile.

Figure 6 .
Figure 6.(a) effects of (a)  and (b)  on concentration profile.

Figure 8 .
Figure 8.(a) effects of (a)  and (b)  on concentration profile.

Figure 9 .
Figure 9. (a) effects of (a)  and (b)  on velocity profile.

Figure
Figure11a, b displays that the Schmidt number  indicates a rising effect on the nanofluid velocity and temperature respectively.Similarly, concentration profile increases with increasing values of  (see Figure12a).Indeed, the justification is the fact that as the amount of  enhances the amount of molecular mass diffusion within the fluid reduces as a result of which the nanoparticles concentration stays higher throughout the microchannel core flow region.Figure12bportrays an effect of the solutal relaxation time parameter  on the concentration profile.As it can be seen from the graph, when magnitude of  increases the nanoparticles concentration decreases.

Figures
Figures14a, bare graphs that show the impacts of the Eckert number  on temperature and nanoparticles concentration respectively.The temperature profile enhances with increasing values  as provided in Figure9a.This is the case since  describes viscous heating between the fluid layers and thus the nanofluid temperature enhances as the value of  rises.From the references[39][40][41] , a comparable outcome was reported.Figure14bdisplays a reverse situation in the case of the nanoparticles concentration.
However, the Casson fluid parameter  shows the opposite effects on   at  = 0 and  = 1 as presented in Figure 17a, b respectively.