Analysis of Magnetic and non Magnetic Nanoparticles with Newtonian/non-Newtonian Base Fluids Over a Nonlinear Stretching Sheet

Unsteady mixed convectional boundary layer ﬂow of Casson nanoﬂuid having magnetic( F e 3 O 4 ) and non-magnetic( Al 2 O 3 ) nanoparticles suspension within two diﬀerent types of base ﬂuids, water(Newtonian) and Sodium Alginate(non-Newtonian), which incorporates viscous and ohmic dissipation eﬀects over a permeable nonlinear stretching sheet with magnetic ﬁeld eﬀects. Some suitable non-dimensional similarity transforma-tions are applied to convert the governing PDEs into a system of nonlinear coupled ODEs and then solved by diﬀerential transformation method(DTM) association with Pade-approximation. To validate the present results for limited cases, a comparison was made with previously existing literature and found to be in well agreement. The impact of radiation, internal heat sink/ source, viscous and ohmic dissipation was discussed for magnetic and non-magnetic nanoﬂuid categories by tabular and graphical demonstrations. We have presented the tabular results of various emerging parameters to discuss the nature of skin friction, Nusselt and Sherwood numbers. It was also observed that performance of non-Newtonian(Sodium Alginate) ﬂuid in heat and mass transfer is better than Newtonian(water) based ﬂuid but no major diﬀerence was seen in heat and mass transfer when comparison was made with Magnetic and Non-magnetic nanoparticles.


Introduction
A fluid is a special kind of matter having no fixed shape that can deform easily because of external pressure. Mainly fluids are categorized as Newtonian and non-Newtonian fluids which have a lot of industrial examples such as food industry, polymer industry, paper production and many more extensive applications. On adverse condition there is no existing model that can fulfill all the needs of physical characteristics of non-Newtonian fluids. Several models are proposed to overcome this deficiency such as Maxwell, Williamson, Viscoelastic ,Walter B-Model etc. For better and comprehensive interpretation of non-Newtonian fluids, Casson Model was presented in 1959 [1]. Casson model has shear thinning properties which is supposed to have infinite viscosity at zero shear rate, a yield stress, below which no flow occurs and a zero viscosity at an infinite shear rate. If the yield stress is greater than the shear stress then it would be referred as a solid but if the yield stress is smaller than the shear stress, the fluid will start to flow. Casson Model was fascinated by many researchers and scientists due to this unique property. Blood, Ketchup, concentrated juices, jellies etc are common examples of Casson fluid. This fluid has ability to capture complex rheological properties of a fluid as compared with other simplified fluid Models like power law 2nd & 3rd third grade fluid models etc.
With an expanding universal race, industrial and engineering developments require efficient energy utilization and its implementation. For this purpose industries needs to evolve progressive heat transfer fluids with consequently higher thermal conductivities. Water, oil, glycol etc typically possess lower thermal conductivity which is a great barrier in the thermal reforms and applications. To overcome this deficiency, Maxwell initially initiated the idea of using small size solid particles in the regular base fluid to enlarge the thermal conductivity. Although it was observed that suspension of large and micro or milli-sized particles creates disruption technically. For example (i): extensive drop in pressure (ii): pipelines erosion (iii): clogging micro channels of devices (iv): faster settling time (v): absorption of surfaces. Keeping a constant view on the hindrances Choi and Eastman [2 − 4] not only introduced the concept of enhancing thermal conductivity using nano/microsized particles but also explicitly verified that the inclusion of nano-sized particles in the regular fluid increases the thermal conductivity. This mixture composed of nano-particles with traditional fluid which was named as Nanofluids. Nanofluids was preferred due to many scientific reasons like nanofluids are more stable as they have longer suspension time, reduce the demands of pumping power, having the capacity of saving energy etc. Although nano-particles have numerous applications such as improving the efficiency of diesel generators, automotive, air conditioning, power plant cooling, solar water heating, drilling thermal storage etc.
Generally nanoparticles are formed by T iO 2 , AIN, Al 2 O 3 , Cu, SiO, and graphite. On comparison, we can find these nanoparticles possess elevated thermal conductivity than conventional base fluid. Eastman et al. [5] further observed that 40% thermal conductivity can be enhanced by adding Cu(10nm) particles into an ethylene glycol. With the passage of time many researchers reported that by adding (1 − 5)% nanoparticles, more than 20% thermal conductivity can be increased in ordinary fluids. Boungiorno's model [6] is the most popular model who studied thermal conductivity of nanofluids experimentally. Further the same author investigated the Brownian diffusion and thermophorsis effects for convective transports in nanofluids and proposed an analytical model [7] and concluded that Brownian motion and thermophorsis would be used to expand thermal conductivity in base fluid. Much related work can be seen in [8 − 12]. When magnetic nanoparticle, in the presence of magnetic field are added to base fluid, form a nanofluid subclass. Since, they frequently respond to applied magnetic field. By adding various types of magnetic nanoparticles, ferrofluids can be synthesised. Ferrofluid has many practical applications within a scientific field [13]. Some common examples are sensors, magnetic based devices, biomedical engineering and medicines. Casson fluid flow towards MHD nonlinear stretching sheet with slip effects was studied by [14]. An analytical steady state solution of nanofluid flow of sisko model was presented by [15]. There are basically three subclasses that state how material is influenced with magnetic field: • Diamagnetism: water, copper, quartz, lead, etc are diamagnetic. They are very less influenced by the Magnetic flux.
• Paramagnetism: sodium, platinum, oxygen, iron oxide(F eO), are paramagnetic. They are much more affected by field of force than diamagnetism • Ferromagnetism: iron gadolinium, magnetite (F e 3 O 4 ), CoF e 3 O 4 , nickel, and M nBi are strenuously affected by the magnetic field and highly polarized in the magnetic field direction and interestingly they retain their state of polarization even when field is disconnected.
All above type of fluids are together known as ferromagnetic fluids [21]. Ferrofluids [13] are normally referred as magnetic fluids. Ferrofluids have nicely proven their effectiveness in magnetically targeting the drug delivery to specific part of human body, targeting cancer cells or tumors, monitoring brains activities, removal of toxins for the treatment of cancer etc. Ferrofluid as magnetic nanoparticle has tremendous medical applications as they have ability to reach even in small capillaries of human body. Another useful application of ferrofluid is magnetic ink which is less expensive and high speed for silent printers. Ferrofluids can also be used to control the underground flow via externally applied magnetic field. In [16] field dependent swirling viscous ferrofluid along a porous rotating disk was studied. The results were obtained for electrically non-conducting fluids. Some more interesting work relating effectiveness of ferrofluids can be found in [17 − 22].
A study of magnetic& non-magnetic nanoparticles with Newtonian & non-Newtonian based fluids was firstly discussed by Hakeem et. al in [24]. By numerical investigation they concluded that magnetic nanoparticles are more influenced by the magnetic field as compared by non-magnetic nanoparticles. They also claimed that non-Newtonian based nanofluids have greater skin friction and Nusselt number than Newtonian based nanofluids. Same author [25] investigates the 2 − D steady convective flow and heat transfer with two sorts of Newtonian(Water) and non-Newtonian(Sodium Alginate) base fluids along a flat plate. Non magnetic (Al 2 O 3 ) and magnetite (F e 3 O 4 ) nanoparticles were added in base fluids and discussed the consequences of thermal radiation and slip conditions. Despite of already existing research, heat transfer fluids are still needed to improve thermal enhancement. Due to thermophysical properties of Al 2 O 3 (non-magnetic) and F e 3 O 4 (magnetite) are considered to be most suitable option and hence selected for present research. The non-dimensional form of conservation equations have attained worth attention due to various transport mechanisms in the flow of nanofluids. Although non-dimensionalization can have many advantages, some of which are listed as below: (i) Analysis of any system without looking at their material properties. (ii) Flow parameters are easy to understand. (iii) Shape and size of the geometry can be more generalized in nondimensionalization. (iv) Physical problem can be guessed by experiment.

Formulation of Flow Model
Mixed convectional, unsteady, laminar and incompressible flow of water and sodium alginate based Casson fluid having magnetic and non-magnetic nanoparticles is considered along a nonlinear stretched sheet fixed in a penetrable medium. The base fluids as well as magnetic & non-magnetic nanofluids are presumed in thermal equilibrium. The x-axis is in the direction of stretched sheet and y-axis is normal to the sheet (see fig.1). The velocity of the sheet is U w = c1x n 1−γt where c 1 , γ ≥ 0 are constants , t is the time and n > 0 is representing the nonlinear parameter of the sheet. A magnetic flux of intensity B 0 is enforced perpendicular to a sheet and defined as Conversion laws under the boundary layer estimations in a dimensional mode are [28]: where velocity components are u and v, effective thermal conductivity is k nf , dynamic viscosity is µ nf , effective density is ρ nf , effective heat capacity is (ρC p ) nf , and α nf = knf (ρCp)nf is the thermal diffusivity of the nano-fluids as mentioned below [24, 25] where ϕ is volume fraction of the nano-particles. Strength of magnetic field induction is B, β is Casson parameter , B T , B c are thermal and concentration expansion coefficients, q r is radiation heat flux , mass diffusion co-efficients D m , and mean temperature and D 1 is thermal diffusion co-efficients and k 0 is the chemical are variable magnetic field and variable permeability parameters respectively. The above governing equations (1 − 4) are associated with the Bc's as follows: are convective heat and mass transfer with N 0 , h 0 and h 1 are constants where c 2 and c 3 are temperature and concentration references. The radioactive heating flux in the temperature equation is calculated and simplified by the Rosseland approximation i.e The heat variation in the flow is supposed to be less enough so T 4 is taken a linear relation of temperature ignoring terms having higher power index while expanding T 4 at T ∞ is obtained as, [28, 32, 33] we get, Also q ′′′ is modeled as [32, 33] Introducing similarity transformation, [28] The system of transformed non-linear ODEs are: (1 + 4 Also the transformed BC's are: where dimensionless parameters in (7 − 9) equations are is the slip parameter, Schmidth and Soret numbers and chemical reaction parameter respectively.
c1(n+1) are the Biot numbers. Also,A 1 to A 7 defined as below are the nanofluid parameters.
The significant physical parameters of engineering concerns are dimensionless skin friction coefficient C f x , Nusselt numbers N ux and sherwood numberSh x that physically indicates shear stress, rate of heat & mass transfer respectively and are defined as under: (Tw−T∞) and where τ wx is the shear stress at the sheet along x-axis : τ wx = µ nf ( ∂u ∂y ) y=0 , q w and q s are surface heat & mass fluxes defined as: Upon fixing the local Reynold number Re x = x Uw νf and using the above defined relations we have  Table-

Results and Discussion
This work specifically scrutinized the nature of Newtonian(water) and non-Newtonian(sodium alginate) suspension containing magnetic F e 3 O 4 (making as ferrofluid) and non magnetic Al 2 O 3 nanoparticles along-with mass & heat transfer analysis. Time dependent, mixed convectional boundary layer flow of a casson nanofluid through non-linear stretching sheet in a porous media with effects of field of magnetic force, chemical reaction, viscous and ohmic dissipation was explored. For plausible understanding of physical problem, numerical computations are obtained to deliberate the impacts of various 2 − D governing parameters on boundary layer profiles and results are presented in terms of tables and graphs. To validate the method the current outcomes are correlated with already published literature and displayed in the table (2 − 4). Table 2 and 3 shows the analogy of skin friction for variate values of Casson, magnetic field and unsteady parameter. These tables are in good agreement with (9, 14, 17, 22, 27, 28) . Table 4 highlights the comparison of N ux for different Prandtl number that revealed good agreement with (3, 12, 27, 28). Table [ and other is sodium Alginate(N aAlg) having Newtonian and non-Newtonian features respectively. In this figure we noticed the velocity profile for fluids having magnetic and non-magnetic nanoparticles decelerated but reverse trend was followed by the thermal and concentration profiles. This was due to the applied transverse magnetic field. As electrically transmitted nanoparticles generates Lorentz force which slower down the motion of nanoparticles in the boundary layer but it starts acceleration in the other two profiles. Fig(3) demonstrate the behavior of unsteady parameter S. As S enhances, all the three profiles started to decrease. It means less mass exchanged from the fluid to the boundary layer. It is quite obvious from physical situation of unsteadiness, when S rises, more heat is released by the sheet and hence the temperature starts diminishing. On the other hand one can say that for increasing values of S cooling rate is more agile. So, we can conclude that with the gradual increment of unsteady parameter S, the thermal profile decreases due to less heat transferring from sheet to fluid. Almost similar behavior was seen for both magnetic and non-magnetic nano-fluids in newtonian and non-newtonian based liquids. Fig(4) precisely delineate the behavior observed by the Casson parameter β. From figure it is obvious that by increasing the values of β the velocity of fluid started to decrease. This happens due to the inverse relationship of β with yield stress. When the values of β increases the yield stress decreases and hence decline the momentum profile for both geometries(magnetic & non-magnetic nano-particles). Also, fluid plasticity is reduced by increasing the values of β. In Fig(5) we had seen that velocity profile depreciate for increasing values of δ but reverse effects was observed for thermal and concentration profiles. From a physical situation we have seen that as slip (δ = 0) occurs , the velocity of fluid that is adjoining to the sheet becomes smaller than the fluid's velocity ⊥ to the sheet. Simply, increasing values of δ allows more the fluid to flow around the sheet, this causes declination in momentum profile. In temperature profile Bi 1 > 0.1 was considered (since Bi 1 < 0.1 the internal resistance to transferring heat is not effective or effects are considered negligible). Strong or large value of Bi 1 indicates effective internal diffusion resistance. From figure we can notice that both temperature profiles for magnetic/non-magnetic nanofluids increases for increasing values of Bi 1 . Similar physical trend was observed in concentration boundary layer which indicates that momentum diffusion is larger than thermal diffusion and hence concentration profile increases with rising values of Bi 2 . Fig(6) depicts the influence of porosity parameter (Kp) for velocity profile. It is obvious from figure the velocity distribution decelerates as we increase the porosity parameter which happens due to the presence of the porous media. Generally this porosity factor has the tendency to soak the significant amount of fluid from the boundary layer, so the velocity profile depreciates in the entire flow domain. Analytical results which incorporate with parameter of volume fraction ϕ were plotted in Fig(7) for two different sorts of nanoparticles named as F e 3 O 4 and Al 2 O 3 . It was perceived that velocity of nanofluids decelerate with increasing ϕ but increasing values of ϕ enhances the thermal profile. It is physically true due to the fact that when ϕ increases it enhances viscosity as well as thermal conductivity of the nanoparticles this accelerates the thermal profile and finally causes depreciation in velocity. According to Maxwell and Brickman [5] increasing in volume fraction increases the viscosity and thermal conductivity of nanofluids. This causes decrease in velocity. Fig(8) describe the effects of radiation parameter(R), heat sink / source(A * ) and Viscous dissipation(Ec) factors for F e 3 O 4 and Al 2 O 3 . It was concluded that for increasing values of R, A * and Ec the magnetic and non-magnetic nanofluid profiles get enhanced. Scientifically, the heat flux existence causes the temperature higher so the thermal boundary layer for the whole flow region increases for increasing values of Rosseland diffusion application. By larger values of A * which acts as a responsible factor of heat generation, releases more energy to the fluid and then this uplift the temperature profile. Also, the frictional heat will be generated as the fluid moves faster along the surface this causes hike in temperature boundary layer thickness. Fig(9) exhibits the behavior of Scmidth number, Soret number and chemical reaction parameter. The concentration profile depreciates for growing values of Sc and K r for both F e 3 O 4 and Al 2 O 3 but rising values of Sr parameter both the magnetic & non-magnetic nanofluid profiles indicates the increasing trend. All these above behavior of concentration profiles are according to the existing physical situations. On explanatory note, the rate of mass transfer gets smaller when diffusion coefficient increases, this retards both the concentration profiles of magnetic and non-magnetic nanofluids. Similarly, higher molecular motion leads sharpen process of mass transportation which resulted the deceleration in the concentration profile. But increasing values of Sr both the F e 3 O 4 and Al 2 O 3 profiles shows increasing trend. This indicates the physical reality due to the temperature gradient, mass fluxes move from the smaller to larger concentration species. This predicts that increasing the values of Sr, diffusion species causes increment in concentration profiles.

Conclusion
The current investigation specifically discussed the mass and heat transfer analysis of Newtonian(Water) & non-Newtonian(Sodium Alginate) suspension containing magnetic( F e 3 O 4 ) & non-magnetic(Al 2 O 3 ) nanoparticles . The unsteady convective flow of casson nanofluid along a non-linear stretched sheet by permeable media under the effects of magnetic field, chemical reaction, thermo-diffusion, viscous and ohmic dissipation was thoroughly discussed in this work. Some imperative results are listed as under: • The velocity, temperature and concentration profiles decreases for increasing values of unsteady parameter(S). Almost similar results were seen for both magnetic & non-magnetic nanoparticles.
• The concentration profile starts decreasing for increasing Schmidth number(Sc) and chemical reaction parameter(K r ) but increases for increasing values of slip parameter(Bi 2 ) and Soret number(Sr).
• It was also observed that the performance of non-Newtonian(Sodium Alginate) fluid in heat and mass transfer is better than Newtonian(water) based fluid.
• No major difference was seen in heat and mass transfer when compared with Magnetic nanoparticles and Non-magnetic nanoparticles.
• It would be worth mentioning that all physical behavior of this flow problem coincides very well with already published literature either graphical or in tabular representation.

Declaration
• Data Availability Statement: All the data set for supporting the results and conclusion is provided in the manuscript.
• Competing interests: All the authors declared no Competing interests.
• Funding Statement: The research and publication of this article is self-funded and did not receive any specific funding. c). Mr. Rana Muhammad Akram Muntazir (Author) assisted with structure and language of the manuscript. All authors read and approved the fnal manuscript.
• Acknowledgement The authors sincerely thank to "University of Engineering and Technology Lahore Pakistan".