A generalized perspective of Fourier and Fick’s laws: Magnetized effects of Cattaneo-Christov models on transient nanofluid flow between two parallel plates with Brownian motion and thermophoresis

Abstract Present research article reports the magnetized impacts of Cattaneo-Christov double diffusion models on heat and mass transfer behaviour of viscous incompressible, time-dependent, two-dimensional Casson nanofluid flow through the channel with Joule heating and viscous dissipation effects numerically. The classical transport models such as Fourier and Fick’s laws of heat and mass diffusions are generalized in terms of Cattaneo-Christov double diffusion models by accounting the thermal and concentration relaxation times. The present physical problem is examined in the presence of Lorentz forces to investigate the effects of magnetic field on double diffusion process along with Joule heating. The non-Newtonian Casson nanofluid flow between two parallel plates gives the system of time-dependent, highly nonlinear, coupled partial differential equations and is solved by utilizing RK-SM and bvp4c schemes. Present results show that, the temperature and concentration distributions are fewer in case of Cattaneo-Christov heat and mass flux models when compared to the Fourier’s and Fick’s laws of heat and mass diffusions. The concentration field is a diminishing function of thermophoresis parameter and it is an increasing function of Brownian motion parameter. Finally, an excellent comparison between the present solutions and previously published results show the accuracy of the results and methods used to achieve the objective of the present work.


Introduction
In most of the engineering systems, the process of heat transfer is suppressed due to the use of low thermal conductivity uids and these uids behaves like obstacles to magnify the thermal energy ow in the engineering systems. Therefore, to overcome these troubles in heat transfer process, it is very momentous to use the uids which have larger thermal conductivity than the conventional uids. Thus, scientists and researchers named that special class of uids as "nano uids". It is obvious that, nano uids have higher thermal conductivity when compared to the common pure liquids. However, with the tremendous improvement and modi cation in the modern nano uid technology, the nano-materials are adequately used in most of the scienti c and bio-engineering applications such as probing of DNA structure, tumour obliteration via heating, MRI contrast enrichment, bio detection of pathogens, proteins detection, tissue engineering and so on [1][2][3]. Further, the nano uid technology is used in lubrication systems, transportation, industries, drug delivery, defense, medicine, cosmetics, catalysis, food packing, automobiles and many other [4][5][6].
Nano uids are the compounds of base uids and nanoparticles with the mean sizes of the range 1-100 nm. To intensify the thermal conductivity of the uids, the working uids are transformed to nano uids, owing to their strong thermal conductivity than prevalent uids in the given medium. Usually, the suspension of nano-materials includes the nanoparticles volume fraction (commonly < 5%) of metal and which causes the magni cation of ther-mal conductivity of nanomaterials. This fact was proved by Choi and Eastman [7]. With this idea of nano uids, many of the researchers studied the heat and mass transfer process in various geometries by considering water as base uid of Newtonian or viscous type in large scale. However, when the uids are deformed between two boundaries approaching to each other or moving away from one another, such behaviour of uid ows is named as squeezing ow. Generally, squeezing ows are produced by the application of normal stress to the moving plates/surfaces. Further, squeezing ows have number of applications in the eld of engineering, industry and biomedicine. Particularly, in lubrication systems, polymer processing, ow of blood in vessels owing to the blood pressure, foam formation, food processing, hydro-dynamical machines, cooling towers, chemical processing, and bi-axial expansion of bubble boundaries, heating/cooling processes, injection moulding, compression, moisture migration, dampers and so on. Also, a few of other squeezing ows examples comprise the diarthrodial joints and valves related to the eld of mathematical biology and bioengineering [8]. In many of the extreme operating systems the occurrence of unpredictable deviation of lubrication viscosity with respect to temperature in squeezing ow can be controlled by applying the external uniform magnetic eld.
Following the pioneering work of Stefan [9], the thermal ow behaviour of viscous uids in squeezing ow can be easily analyzed. However, for the rst time Stefan [9] made an attempt to demonstrate the thermodynamic characteristic properties of squeezed ows in lubrication systems. Further, Stefan developed squeezed ow model by utilizing lubrication assumptions. Additionally, Stefan's work was continued by Reynolds [10] to study the behaviour of uid ow between two elliptic plates. Further, a similar type problem of uid ow through the rectangular channel was investigated by Archibald [11]. Leider and Bird [12] theoretically discussed the squeezed phenomena by considering the power-law liquid ow between two parallel disks. Their study reports that, the stress overshoot situation should be adequately explained via suitable rheological model to understand the fast squeezing phenomena. Domairry and Aziz [13] investigated the similar problem of viscous incompressible squeezing ow between two parallel disks in which they presented the inuence of inertia, magnetic eld on injection/suction process. Also, their results show that, the component of axial velocity ampli ed for the enhancing similarity variable in suction & injection cases. Further, Siddiqui et al. [14] used the homotopy perturbation method to investigate the ow behaviour of magneto-hydrodynamic, two-dimensional squeezing ow of viscous uid between two parallel plates. Their investigation disclosed that, the longitudinal velocity diminished for the increasing values of magnetic parameter.
The time-dependent, two-dimensional axisymmetric squeezing ow of viscous uids between two parallel plates was analytically discussed by Rashidi et al. [15] by using analytical technique. Their discussion portrayed that, the squeezing parameter behaves like a function of ow velocity. Further, the heat transfer characteristics of viscoelastic uid ow between two plates were reported by Kaushik et al. [16]. Their investigation reports that, the velocity of viscoelastic uid diminishes in the transient state due to the presence of strong elastic behaviour. The unsteady magneto-hydrodynamic electrically conducting Je rey uid ow between two parallel plates was analyzed by Muhammad et al. [17]. Their study states that, the magnifying Deborah number enhances the ow eld in the channel. The analytical solutions for the magnetohydrodynamic incompressible axisymmetric ow between two in nite parallel disks via homotopy analysis method were discussed by Joneidi et al. [18]. Clearly their investigation remarked that, ow velocity shows the cross ow behaviour for enhancing magnetic eld parameter at central portion of the channel. Hayat et al. [19] discussed the impact of magnetic eld on the squeezed ow of second order uid between two parallel disks via homotopy analysis method. It is noticed from their investigation that, the velocity eld has opposite behaviour in suction and blowing cases. Duwairi et al. [20] analyzed the in uence of squeezing parameter on heat transfer behaviour of viscous incompressible uid ow through channel with viscous dissipation impact. However, their demonstration predicts that magnifying squeezing parameter diminish the heat transfer rate and increases the skin-friction coe cient and rising extrusion parameter diminishes the Nusselt number and increases the momentum transport coe cient in the ow region.
The process of simultaneous heat and mass transfer of viscous uid between two parallel plates with squeezing e ect was investigated by Mustafa et al. [21]. Their analytical investigation reports that, the magnitude of Nusselt number ampli es with rising Eckert and Prandtl numbers. Further, their study shows that, the concentration distribution is suppressed for destructive chemical reaction and magni ed for the constructive chemical reaction parameters. Khan et al. [22] discussed the two-dimensional incompressible magnetized squeezed ow of viscous Casson uid through the channel with porous medium impacts. However, their analysis shows that, the component of normal velocity diminished for the magnifying squeez-ing parameter values. Similar study was continued by Khan et al. [23] by considering the long in nite parallel plates via variation of parameters method. The characteristics properties and heat transfer behaviour of incompressible nano uid ow between two non-parallel stretching walls was studied by Dogonchi and Ganji [24] via Duan-Rach method. Their investigation shows that Nusselt number, thermal eld and viscosity of the uid intensi ed with increment in stretching parameter. Also, their results shows that, for the cases of convergent and divergent channels, the temperature eld rises with enhancement in heat generation parameter and further temperature eld diminished for the magnifying radiation parameter values in the ow region. Dogonchi and Ganji [25] analytically investigated the double di usion ow characteristics of nano uid ow between two non-parallel walls with Joule heating impact by using Duan-Rach approach. Further, the behaviour of nano uid was studied in terms of Brownian motion and thermophoresis e ects with important mechanism. It is observed from their investigation that, the rate of local thermal energy transfer rises with upsurge in Schmidt number and concentration distribution upsurges with magnifying Brownian motion parameter. Further, Dogonchi and Ganji [25] demonstrated the inuences of MHD on the time-dependent squeezing ow of nano uid between two parallel plates by considering the impact of thermal radiation. Also, their study includes the dissipation e ects to account the heat dissipation in the ow region. It is remarked from their investigation that, the thermal eld and local heat transfer rate upsurges with increment in thermal radiation parameter. However, some of the pioneering e orts made in this direction to analyze the di usion characteristics of nano uids are [26][27][28][29][30][31][32][33].
The phenomena of combined heat and mass transfer has plenty of applications in many of the scienti c and industrial processes, like refrigeration, heat exchangers, power collector equipment, air conditioners, crops damage, food processing and many others. In many of the physical problems, the process of heat transfer from one object to another object or in the same body occurs because of the temperature variation. Having this physical situation in account, Fourier's introduced his renowned law, referred as Fourier's law of heat conduction [34]. It is clear from the literature that, the Fourier's law is insu cient to predict the heat transfer behaviour completely. This is due to the occurrence initial disturbance instantly in all parts of the whole medium or object. In practical, there is no such material or object which obeys the Fourier's law. Therefore, to solve this problem Cattaneo [35] introduced the time relaxation factor so-called thermal inertia in conventional Fourier's law.
This relaxation factor indicates the resistive behaviour of the medium to change produced by the thermal gradient which is imposed externally to the medium, thus, this situation causes the thermal time delay in the medium. However, this amendment states that, the process of heat transfer is analogous to the propagation of thermal wave with con ned speed in the medium. Further, Fourier's heat equation is parabolic in nature. Whereas, Cattaneo's heat transfer equation is governed by the hyperbolic equation. Christov [36] further improved the Cattaneo theory by introducing the time derivatives in Cattaneo's theory with Oldroyd upper convective derivatives that conserved the physical-invariant modeling. Further, the modi ed Cattaneo-Christov theory is subjected to stability along with uniqueness analysis by Ciarletta and Straughan [37]. Following are the some of the pioneering e orts made by the researchers to investigate the e ects of Cattaneo-Christov double di usion models on heat and mass transfer characteristics of viscous and non-viscous uids.
The impacts of Cattaneo-Christov heat ux model on thermal convection in horizontal layer of incompressible Newtonian uid was studied by Straughan [38] using D2 Chebyshev tau numerical method. However, literature [38] reports that, the easier occurrence of thermal convection takes place for the amplifying Cattaneo parameter. Hayat et al. [39] studied the impacts of Cattaneo-Christov heat ux model on ow of upperconvected Maxwell uid (UCM) over a sheet with analytical technique. Their study shows that, the thermal eld is stronger in Fourier's model when compared to Cattaneo-Christov double di usion models. The characteristics of homogeneous-heterogeneous chemical reactions in case of Oldroyd-B uid ow over a stretching surface with Cattaneo-Christov double di usion models under magnetic eld was discussed by Hayat et al. [40]. Further, the detailed study on Cattaneo-Christov double di usion models can be found in [41][42][43][44].
Dogonchi and Ganji [45] utilized the Duan-Rach approach to investigate the in uence of generalized Fourier's heat di usion model on nano uid ow between two parallel plates with magnetic e ects. Farooq et al. [46] discussed the impacts of Cattaneo-Christov double di usion models in the squeezing ow of Newtonian uid past a stretched sheet in the presence of Darcy porous medium. Muhammad et al. [47] reported the phenomena of solutal and thermal strati cation in the squeezing ow of twodimensional viscous uid about a stretching surface with generalized thermal and mass ux models via optimal homotopy method. Recently, Farooq et al. [48] described the e ects of Cattaneo-Christov double di usion models on the squeezing ow of unsteady, viscous Newtonian uid between two parallel plates by considering Darcy porous material and time-dependent chemical reaction. Akmal et al. [49] discussed the in uence of Cattaneo-Christov heat and mass ux model on Newtonian nano uid ow between two parallel plates with entropy generation analysis and magnetic e ects. Their discussion reports that, the entropy generation and temperature eld decayed for the magnifying thermal relaxation time parameter.
However, the technological applications of squeezing ows as mentioned before motivated us to consider this particular ow problem. Thus, the main objective of this article is to address the combined e ects of heat and mass transfer on magnetized squeezed ow of viscous Casson nano uid ow between two parallel plates via generalized Cattaneo-Christov double di usion models with timedependent chemical reaction e ects. The novelty of the present numerical work is to explore the e ects Cattaneo-Christov double di usion models in energy and concentration equations in place of classical Fourier's and Fick's laws owing to their drawbacks as mentioned before. After the comprehensive literature survey, it is observed that, the physical problem considered in this paper is not yet reported in the literature. Thus, authors made an attempt to solve this physical problem governed by time-dependent coupled nonlinear equations using Runge-Kutta fourth order method with shooting scheme (RK-SM) and bvp4c techniques. Further, the impacts of di erent ow parameters involved in the system are discussed physically and shown graphically.

Stress tensor of Casson fluid
The thermodynamic characteristic properties of Casson uid was rst examined by Casson in the year 1959 [50]. From the literature [50] it is observed that, Casson uid is a shear thinning uid with in nite viscosity at zero shear rates and a zero viscosity at an in nite rate of shear. However, the following rheological equation of Casson uid ow [22,50] is used to construct the ow equations for the present problem.
In this model, the symbol π = e ij . e ij denotes the product of deformation components, e ij denotes (i, j) th deformation rate component, critical value is indicated by πc, Py and µ B indicate the yield stress and viscosity. Here, yield stress is the minimum stress required to produce the ow and below which no ow occurs.

. Mathematical formulation of the problem
The impacts of Cattaneo-Christov heat and mass ux models on incompressible squeezed ow of magnetized twodimensional time-dependent viscous Casson nano uid between two parallel plates separated by a distance ±h (t) = ±l( − αt) / with Brownian motion and thermophoresis e ects have been examined numerically. Owing to the motion of the upper plate with velocity −αl √ −αt at a distance ±l( − αt) / towards or away from the lower stationary plate situated at y = , uid ow occurs. In this case the symbol "l" signi es the initial position (when time t = ). Squeezing phenomena occurs when α > till plates reach t = /α and plates are separated when α < . The Cartesian coordinate system is used to study the mathematical behaviour of this physical situation and in which x-coordinate is aligned along the axial ow direction of the channel, while y-axis is taken normal to axial coordinate. Fluid ow is regulated by applying the Bo( − αt) − / perpendicular to plates. Further, the squeezing rate parameter α has the dimension of (time) − and αt < . However, Figure 1(a) clearly illustrates the ow con guration with all necessary conditions and associated coordinate system of the present problem. Further, Figure 1(b) identically demonstrates the impact of applied magnetic eld on the ow behaviour before and after its application. Further, the Cattaneo-Christov double di usion models are used to study the heat and mass ow behaviour of Casson nano uid in a channel instead of conventional transport models. However, the thermal ux q and mass ux J accomplish the succeeding equations in terms of Cattaneo-Christov models [47,48].
In the above Eqs. (2) and (3), heat ux is denoted by q and J represents the mass ux. Also, the symbols ϵ T and ϵ C describes the relaxation times with respect to temperature and concentration elds. V is the velocity vector, the thermal conductivity of the uid is denoted by κ and D B be the coe cient of Brownian motion.
However, the classical Fourier's and Fick's laws can be sought back by imposing the condition ϵ T = ϵ C = on the Eqs. (2) and (3), respectively. Further, by applying the law of conservation mass condition, ∇.V = , the Eqs. (2) and (3) transformed to the following form [47,48].
Additionally, the heat dissipation owing to frictional forces generated through shear stress is accounted. Also, magnetic eld and Joule heating e ects are accounted respectively, in the momentum and energy equations. Further, a homogeneous chemical reaction of rst order is assumed to be occurring. With these assumptions and by making the use of Casson uid ow model described in section 2, the governing equations of transient squeezing ow of Casson nano uid between two parallel plates with necessary conditions are obtained as follows.

Energy di usion equation:
∂T

Mass di usion equation:
In the above Eqs. (6)-=(10), u and v are the axial and radial velocities along x and y axis, ρ nf is nano uid density, ν nf depicts kinematic viscosity of nano uid, µ nf is the dynamic viscosity nano uid, p is the pressure, T and C are the nano uid temperature and concentration, β denotes the Casson uid parameter, D B is the coe cient of Brownian motion, D T is the thermophoresis di usion coe cient, σ describes the electrical conductivity, Bo magnetic eld, κ denotes the thermal conductivity, α is squeezing rate, T H , C H describes temperature and concentration of the upper plate and k is the time-dependent chemical reaction coe cient Further, the values of Σ T and Σ C in the Eqs. (9) and (10) are examined as follows: Thus, the required dimensional temperature and concentration equations are obtained by replacing the examined values of Σ T and Σ C in the corresponding energy and concentration equations. In view of this reason, the Eqs. (11) and (12) are replaced in the Eqs. (9) and (10) to the obtained modi ed heat and mass transport equations, thus, we have [47,48].
The associated boundary conditions for the timedependent squeezing ow are as follows: In the Eq. 15(a), vw = dh dt indicate the velocity with which upper plate is moving towards or away from the lower plate which is kept at a distance y = from the upper plate at y = h (t). Further, the value of the vw is examined as

. Similarity transformation approach
The physical problem discussed in this paper is governed by the time-dependent, coupled nonlinear partial di erential equations (PDEs) of higher order. Owing to the complexity of governing equations, the analytical or direct methods are insu cient to solve these PDEs completely. However, to tackle this problem, the complex PDEs are rst converted to the group of nonlinear ordinary di erential equations (ODEs) by utilizing the applicable transformations. Further, the reduced nonlinear ODEs are solved by employing RK-SM [51] and bvp4c [52] methods. Thus, following suitable similarity transformations [21] are used to this end.
The pressure gradient term in the Eqs. (7) and (8) is eliminated by using the Eqs. (6) and (16) and the resulting momentum equation is of the following form: Further, the thermal and concentration di usion equations are articulated in terms of similarity variable. To this end, using Eq. (16) into the Eqs. (13) and (14), the transformed temperature and concentration equations are as follows: Similarly, the relevant boundary conditions are reduced to dimensionless form by replacing the Eq.  (Schmidt number). Thus, the considered physical problem under the application of magnetic eld is completely portrayed in terms of squeezing parameter S and which in turn governed by the squeezing rate α. The condition S > related to motion of plates away from each other and S < corresponds to the plates approaching towards each other, so-called squeezing process. Also, the e ects of Cattaneo-Christov double di usion models on ow behaviour are examined in terms of thermal and concentration relaxation time parameters. Further, the characteristics of nano uids are depicted in terms of Brownian motion and thermophoresis parameters. Prandtl, Hartmann and Eckert numbers are utilized to control the ow and heat transfer process. Also, Schmidt number is used to describe the ows when there exist simultaneous heat and mass transfer phenomenon. Further, the homogenous chemical reaction of rst order is illustrated in terms of time-dependent chemical reaction parameter.

. Physical quantities of engineering interest
Physical quantities of engineering interest such as, momentum transport coe cient, heat and mass transfer rates for the present problem are de ned as follows [21].
Thus, in terms of Eq. (16), the Eq. (21) can be expressed as follows:

Numerical solution procedure
The channel ow of unsteady two-dimensional Casson nano uid with Cattaneo-Christov heat and mass ux models with time-dependent chemical reaction and Joule dissipation is associated with highly nonlinear coupled ow equations. Due to the inadequacy of the analytical methods, the present problem is solved via numerical approaches namely, RK-SM and bvp4c method. Thus, the Eqs. (17)- (19) with Eq. (20) are solved using RK-SM and bvp4c techniques. However, the RK-SM method begins by discretising the coupled nonlinear higher order ODEs into the set of rst order ODEs. Thus, following are the discretized rst order ODEs used for numerical calculations.
Further, the current shooting technique initiates by converting boundary value problem (BVP) into an initial value problem (IVP) by predicting appropriate guesses of F ( ), F ( ) , θ ( ) and ϕ ( ). Consequently, the converted IVP is solved by employing RK-SM. Further, in RK-SM scheme, the convergence condition is governed by the suitable initial guesses of the unknown quantities. However, for the accurate similarity solutions over the nondimensional time t, the convergence criteria is xed as − and h = . be the step size. Further, once the convergence criteria is reached, the set of ODEs are integrated by using the RK-SM to obtain the required results.

. Comparative results between Runge-Kutta scheme and bvp4c matlab function
The comparative results obtained based on Runge-Kutta scheme and bvp4c matlab function are illustrated in the Tables 2 and 3 Table 2. Similarly, F(η), θ(η) and ϕ(η) values evaluated based on bvp4c technique is tabulated in the Table 3. The analysis of Tables 2 and 3 portrays that, there exists an excellent agreement between RK-SM and bvp4c methods. Thus, with this comparison it is remarked that, the numerical values obtained based on RK-SM and bvp4c are highly accurate and found to be reasonable. However, this comparison shows the accuracy of RK-SM and bvp4c methods for solving ow equations.

, respectively. The numerical values of F(η), θ(η)and ϕ(η) calculated based on RK-SM is shown in the
. Results and discussion: The impact of squeezing parameter (S) on the ow behaviour is clearly described in this paragraph. However, Figures 2-5 describe the e ect of S on velocity eld, temperature and concentration pro les, respectively. Figure 2 clearly portrays that, the radial velocity eld diminished for S > and enhanced for S < . Clearly, the quantity S > associated with the moment of the plates away from one another and S < is related to the moment of plates close to one another. However, the variation of velocity in the ow region is mainly owing to the reason that, for S > uid sucked into the channel and hence velocity increases.
On the other hand, for S < liquid released out from the channel and creates the liquid drop inside the channel and hence ow velocity diminished. Figure 3 shows that, for S > , the axial velocity pro le decreased in the region ≤ η ≤ . and increased in the remaining region of the -. -. -. .
-.   channel. Similarly, for S < , the axial velocity eld enhanced in region ≤ η ≤ . and decreased in region . ≤ η ≤ . Further, from Figure 3 it is observed that, due to the variation in velocity at the boundaries cross ow behaviour is noticed at the central portion of channel. Furthermore, from Figure 4 it is remarked that, the temperature pro le diminished for S > and it is enhanced for S < . Practically, increasing squeezing parameter (S > ) decreases the squeezing force and cause to decay the temperature pro le in the ow region. Further, from Figure 4 it is noticed that, an increment in temperature pro le is quite obvious for S < , because, an increment in squeezing number closely associated with the decay of kinematic viscosity of the uid, an increment in length between the plates, and an increment in speed with which the plates move. Further, Figure 5 portrays that, the concentration pro le increased for S > and suppressed for S < . However, the in uence of S on concentration eld near the boundaries is insigni cant when compared to the other portion of the channel and which clearly observed in Figure 5. Figures 6-9 illustrates the e ect of Casson uid parameter (β) on velocity, temperature and concentration pro les. From Figure 6 it is observed that, the radial velocity pro le diminished for the increasing values of β. This variation in velocity eld is mainly due to the fact that, a small upsurge in β results the enhancement in the uid viscosity and hence this increase in viscosity o ers more opposition to the uid ow inside the channel. Therefore, the radial velocity pro le decreased for the increasing β values. Further, Figure 7 portrays the behaviour of axial ow and it is observed that, the axial velocity pro le suppressed in the region ≤ η < . and increased in the remaining region of the channel for the increasing values of β. Further, it is clearly noticed from Figure 7 that, due to the variation of velocity near boundaries of the channel, a , Ω T = .
-. cross ow trend is observed at the central portion. Further, from Figure 8 it is remarked that, the temperature eld diminished for the increasing values of β. Also, from Figure 9 it is observed that, the concentration pro le increased for the increasing values of Casson uid parameter in the ow region. Figures 10-13 illustrates the e ect of Hartmann number (Ha) on velocity, temperature and concentration proles in the ow region, respectively. Figure 10 portrays that, the radial velocity pro le suppressed for the enhancing values of Ha. This fact is mainly due to the reason that, an increment in magnetic eld leads to the increase of Lorentz forces related to the magnetic eld and these increased Lorentz forces o ers the more resistance to the velocity eld. Therefore, it is obvious to expect that, the velocity eld will suppress for the increasing values of Hartmann number in the ow region. Further, from Figure 11 it is noticed that, the axial velocity pro le diminished in the region ≤ η ≤ . and it is raised in the other portion of the channel for the increasing values of Ha. Due to this variation in axial velocity, there occurs cross ow behaviour at the central portion of the channel. Further, from Figure 12 it is noticed that, the temperature pro le increased for the increasing values of Ha. This variation in temperature pro le is mainly due to the reason that, the presence of Joule heating e ect causes the thickening of temperature boundary layer, so that, the thermal eld enhanced with increasing values of Ha. Consequently, higher values of Ha are preferred for the case where the heating is required and vice-versa, the smaller Table 3: Numerical values of F (η), θ (η) and ϕ (η) for di erent set of S, obtained based on bvp4c scheme with xed values of β = . , Ha = .

Results obtained based on bvp c technique
-. -.
-. values of Ha are chosen for the case where the cooling is necessary. Also, it is observed from Figure 13 that, the concentration distribution is diminished for the magnifying values of Ha in the ow region. The e ect of thermal relaxation time parameter is described via Cattaneo-Christov heat and mass ux models using the coupled ow equations. Certainly, ow behaviour in uences the heat and mass transfer process. However, at the same time the temperature eld significantly in uences the concentration distribution in the ow region. This is due to the presence of Brownian motion and thermophoresis e ects in the temperature and concentration equations. Further, these e ects are negligible on velocity eld due to small temperature changes does not a ect the viscosity of the uid. But these e ects are considerably noticed on concentration distribution.
Thus, the in uence of Ω T on thermal and concentration pro les is portrayed in the Figures 14 and 15. Figure  14 describes that, the temperature eld suppressed for the magnifying values of Ω T . This variation in temperature pro le is mainly due to the fact that, when the values of Ω T raises, then the nano particles requires extra additional time to transfer the heat energy to their adjacent particles, this situation describe the non-conducting behaviour of the material medium and causes to decay the temperature pro le in ow regime. However, the case Ω T = corresponds to the ow of heat with in nite speed and hence, temperature eld is pre-dominant when Ω T = . Thus, it is observed that, the temperature pro le suppressed in  case of Cattaneo-Christov heat ux model when compared to classical Fourier's law of heat conduction. Further, it is noticed from Figure 15 that, the concentration pro le increased for the increasing values of Ω T in the ow regime.
The in uence of thermophoresis and Brownian motion forces characterizes the nanoparticles volume fraction di usion and that correlates with one another in the neighborhood of boundaries which in turn considerably in uence the concentration relaxation time parameter in the ow region. However, with the coupled energy and concentration equations, it is noticed that, the Ω C signicantly e ects the thermal and concentration pro les. Further, in uence of Ω C on velocity eld is comparatively neg-  ligible. However, this situation well described in the Figures 16 and 17. It is observed from Figure 16 that, the temperature eld diminished with increase of Ω C . Further it is remarked from Figure 17 that, the concentration prole suppressed for the increasing values of Ω C . The physical behaviour noticed in Figure 17 is justi ed as follows, when Ω C increases, the nano uid particles need more time to di use through the material medium and causes to decay the concentration pro le. However, for Ω C = , di usion process occurs at an in nite speed. Further, the concentration eld dominates in case of Ω C = . Thus, it is observed that, the concentration eld suppressed in case of Cattaneo-Christov mass ux model when compared to Fick's law. , Ω T = Ec = . , N b = . , Pr = . and Sc = . , Ω T = Ec = . , N b = . , Pr = . and Sc = .
Further, the presence of Brownian motion and thermophoresis parameters in temperature and concentration equations, signi cantly a ects the thermal and concentration elds. Thus, the in uence of thermophoresis parameter (Nt) on temperature and concentration elds is portrayed in the Figures 18 and 19. However, it is remarked from Figure 18 that, the temperature pro le increases with the increasing values of N t . This increment in thermal prole is mainly due to the strong thermophoresis forces in ow region and hence these forces cause the increase in temperature distribution in the ow regime. Further, from Figure 19 it is observed that, the nanoparticle concentration pro le diminished for the increasing values of thermophoresis parameter in the ow region. , Ω T = Ec = . , N b = . , Pr = . and Sc = . , Ω T = Ec = . , N b = . , Pr = . and Sc = .
Further, Figures 20 and 21 are presented to describe the e ect of Brownian motion parameter (N b ) on temperature and nanoparticle concentration distributions. However, it is noticed from Figure 20 that, the temperature prole suppressed for the enhancing values of N b . Also, it is remarked from Figure 21 that, the nanoparticle concentration distribution enhanced for the increasing values of Brownian motion parameter (N b ). This variation in concentration eld is mainly due to the reason that, an increment in N b increases the Brownian motion di usion coe cient and which makes the concentration eld to enhance in ow region. Hence, increased concentration eld is noticed for the enhanced Brownian motion parameter. Also, from Figure 21 it is noticed that, the e ect of N b Figure 10: E ect of Ha on F (η) for xed S = . , β = . , N t = Kr = . , Ω T = . , Ω C = . , Ec = . , N b = . , Pr = . and Sc = . , Ω T = .
on nanoparticles concentration is negligibly small beyond the value 1.0, but it is signi cant in the region ≤ η ≤ . .
Further, the in uence of Eckert number (Ec) on temperature and concentration pro les is illustrated in the Figures 22 and 23. It is observed from Figure 22 that, the temperature pro le enhanced for the increasing values of Eckert number. However, this upsurge in temperature prole is expected because Echas direct impact on the heat dissipation process and hence increases the temperature eld in the ow region. Also, the thickness of thermal boundary layer diminished upon the increase of Ec. It is , Ω T = . , Ω C = . , Ec = . , N b = . , Pr = . and Sc = . , Ω T = .
quite obvious that, Eckert number is mainly used to determine the dissipated heat in the given medium/system. Further, from Figure 23 it is remarked that, the concentration pro le diminished for the increasing values of Eckert number in the ow regime. This variation in concentration eld is due to the presence of Brownian motion and thermophoresis e ects in the energy and concentration equations.
Similarly, the impact of Prandtl number (Pr) on temperature and concentration pro les is depicted in the Figures 24 and 25. However, it is remarked from the Figure 24 that, the temperature pro le pro le enhanced for the in-  creasing values of Prandtl number in the ow region. This variation in temperature pro le is mainly due to the fact that, the larger values of Pr considerably reduces the thermal di usivity and decreases the thickness of the thermal boundary layer and which in turn responsible for the increment of thermal pro le in the ow regime. Further, it is expected that, the Pr < related to physical objects with small viscosity & larger thermal di usivity. Also, Pr > related to oils with smaller thermal di usivity and high viscosity. Further, it is noticed from Figure 25 that, the concentration pro le decreased for the increasing values of Pr. This e ect is observed due the coupled e ects of Brownian motion and thermophoresis parameters in temperature and concentration equations.  Further, the e ect of Schmidt number (Sc) on temperature and concentration pro les is illustrated in the Figures 26 and 27. Figure 26 describes that, the temperature pro le decrease for the increasing values of Schmidt number in the ow region. However, it is clear from Figure 26 that, the Schmidt number has no considerable effect on temperature pro le. Furthermore, the in uence of Schmidt number on concentration pro le is depicted in the Figure 27. However, it is remarked from Figure 27 that, the concentration eld suppressed for the increasing values of Schmidt number. This reduction in concentration eld is mainly due to the fact that, the larger values of  Sc decreases the mass di usivity and hence concentration pro le decreases. Due to the presence of thermophoresis and Brownian motion e ects in the energy and concentration equations, the time-dependent chemical reaction parameter inuences the both temperature and concentration elds in the ow region signi cantly. Thus, this e ect was clearly shown in the Figures 28 and 29, respectively. It is noticed from Figure 28 that, the temperature pro le diminished for the destructive chemical reaction (Kr > ) and it is enhanced for the constructive chemical reaction (Kr < ) in the ow regime. Further, from Figure 29 it is observed that, the concentration pro le decreased for the destructive chemical reaction and it is increased for the generative chemical reaction.  However, from the industrial point of view, the quantities of engineering interest such as momentum, heat and mass transport coe cients have large number of advantageous. Also, the heat and mass transfer properties of viscous uids are identi ed mainly in terms of skin-friction, heat and mass rates. Further, the numerical values of skinfriction coe cient F (η) , heat θ (η) and mass transfer rates ϕ (η) at η = for di erent values of control parameters are calculated and tabulated in the Tables 4-6, respectively. From Table 4 it is noticed that, the magnitude of momentum transport coe cient enhanced for the increasing values of squeezing and Hartmann numbers. Further, the magnitude of skin-friction coe cient diminished for  the increasing values of Casson uid parameter in the ow regime. Furthermore, the numerical values of heat transfer rate θ (η) for di erent values of physical parameters are tabulated in the Table 5. It is clearly noticed from Table 5 that, the magnitude of Nusselt number suppressed for the increasing values of Brownian motion and thermal relaxation time parameters. Further, it is noticed that, the magnitude of Nusselt number ampli ed for the increasing values of thermophoresis and concentration relaxation time parameters.
Similarly, the behaviour of Sherwood number for various values of control parameters is illustrated in the Table  6. It is remarked from Table 6 that, the magnitude of mass transfer rate diminished for the increasing values of Brow-  nian motion and concentration relaxation time parameters. Also, it is observed that, the magnitude of Sherwood number enhanced for the magnifying values of Schmidt number and time-dependent chemical reaction parameter in the ow region.

Conclusions
The present numerical investigation address the combined e ects of heat and mass transfer in squeezing ow of magneto-hydrodynamic, unsteady Casson nano uid ow between parallel plates by considering the generalized Cattaneo-Christov heat and mass ux models with tran-  sient chemical reaction process. Due to the moment of upper plate with velocity −αl √ −αt towards or away from the lower plate, the motion of the uid is produced. The timedependent, coupled highly nonlinear PDEs governing the present physical problem are solved by utilizing RK-SM and bvp4c methods. The physical signi cance of the considered problem is described by portraying the behaviour of the di erent control parameters on ow, temperature and concentration distributions. Also, the obtained numerical data for various values of control parameters are expressed in terms of graphs and tables. Thus, the following are the major nding of the present numerical study. • Temperature eld is diminished in Cattaneo-Christov thermal di usion model when compared to classical Fourier's law. • Concentration eld is suppressed in Cattaneo-Christov mass di usion model when compared to conventional Fick's law. • Temperature eld is an increasing function of thermophoresis parameter. • The in uence of S > and S < on axial ow is opposite. • Magnifying Ha diminish the radial and axial velocity eld. Table 5: Numerical values of heat transfer rate for various values of Ω T , Ω C , N t and N b with xed S = . , β = . , Ha = . , Pr = . , Ec = . , Sc = . and Kr = .  Table 6: Numerical values of mass transfer rate for various values of N b , Ω C , Sc, Kr with xed Ω T = . , β = . , S = Ha = . , N t = . , Ec = . , Pr = .   • Temperature eld is a diminishing function of Brownian motion parameter. • E ects of Pr and Ec on temperature and concentration elds are found to be similar.

Future research work
However, it is anticipated that, the present paper gives a motivation for the mathematical modelling of squeezing problems and advantageous in bearings, motors and lubrication, heating/cooling processes, injection modeling, compression, moisture migration, food and polymer processing, viscometers, rheometer and in elongation ows etc. Further, the present investigation may be helpful to analyze the squeezing ow occurring in the bioengineering and biology. For example, the valves, diarthrodial joints and blood ow problems. Also, the present results may be extended to analyze the characteristics of time-dependent retardation or relaxation phenomenon in non-Newtonian uids. Also, the squeezing ow of shear thinning/thickening uids like third-grade, second-grade uids may be demonstrated.

Acknowledgements:
The authors wish to express their gratitude to the reviewers who highlighted important areas for improvement in this earlier draft of the article. Their suggestions have served speci cally to enhance the clarity and depth of the interpretation of results in the revised manuscript. One of the author Usha Shankar wishes to thank Karnataka Power Corporation limited, Raichur Thermal Power Station, Shaktinagar, for their encouragement.