MHD Flow of Sodium Alginate-Based Casson Type Nanofluid Passing Through A Porous Medium With Newtonian Heating

Casson nanofluid, unsteady flow over an isothermal vertical plate with Newtonian heating (NH) is investigated. Sodium alginate (base fluid)is taken as counter example of Casson fluid. MHD and porosity effects are considered. Effects of thermal radiation along with heat generation are examined. Sodium alginate with Silver, Titanium oxide, Copper and Aluminum oxide are added as nano particles. Initial value problem with physical boundary condition is solved by using Laplace transform method. Exact results are obtained for temperature and velocity fields. Skin-friction and Nusselt number are calculated. The obtained results are analyzed graphically for emerging flow parameters and discussed. It is bring into being that temperature and velocity profile are decreasing with increasing nano particles volume fraction.

nanofluid was first investigated by Choi 33 . He defined that the fluids occupying the sizes of particles less than 100 nm is called nanofluid. The categorieswith different attitude of nano particles are particle material, Base fluid, size and concentration, of the nanofluid. Suspend these nano particles into any type of conventional fluid like oil, water, ethylene glycol to make nanofluids. The reason why nano size particles are preferred over micro size particles has been explained by 34 . Nano particles over micro particles, good improvement have seen in thermo physical properties. Nanofluids have various applications such as in air conditioning cooling, automotive, power plant cooling, improving diesel generator efficiency etc. 35 . Usually water, ethylene glycol are utilized as heat transfer base fluids. Different substances are used for the production of nanoparticles, which are generally divided into metallic i.e. copper 36 , metal-oxide i.e. CuO 37 , chalcogenides sulphides, selenides and telluride's, mentioned 38 and different particles, such like carbon nanotubes 39 . In literature the size of one particle is in between 20 nm 40 and 100 nm 41 .
Casson fluid model was first presented by Casson in 1959. Casson fluids in tubes was first studied by Oka 42 . Examples of Casson fluids are honey, blood, soup, jelly, stuffs, slurries, artificial fibers etc. Cassonnanofluid flow with Newtonian heatingpresented by 43 . Sarojamma et al. 44 investigated Casson nanofluid past over perpendicular cylinder in the occurrence of a transverse magnetic field with internal heat generation or absorption.
Khalid et al. 45 examined unsteady MHD Casson fluid withfree convection flow in a porous medium. Bhattacharyya et al. 46 studied systematically magnetohydrodynamic Casson fluid flow over a stretching shrinking sheet with wall mass transfer. Arthur et al. 47 studied Casson fluid flow in excess of a perpendicular porous surface, chemical reaction in the existence of magnetic field. Recently, Fetecau et al. 48 has investigated fractional nanofluids for natural convection flow over an isothermal perpendicular plate with thermal radiation. Hussanan et al. 49 investigates the unsteady heat transfer flow of a non-Newtonian Casson fluid over an oscillating perpendicular plate with Newtonian heating. Recently, Imran et al. 50 analyzed the effect of Newtonian heating with slip condition on MHD flow of Casson fluid. MHD flow of Casson fluid with heat transfer and Newtonian heating is analyzed by Hussanan et al. 51 . The effect of Newtonian heating for nanofluid is recently investigated by 43,52 . But no work is done until now on heat transfer enhancement in Sodium alginate fluid with additional effects of NH, MHD, porosity, heat generation, and thermal radiation. Silver (Ag), Titanium oxide (TiO 2 ), Copper (Cu) and Aluminum oxide (Al 2 O 3 ) are nano particles suspended in base fluid. Problem is solved and interpreted graphically with some conclusions.

Mathematical Modeling and solution of the Problem
Sodium alginate with Silver (Ag), Titanium oxide (TiO 2 ), Copper (Cu) and Aluminum oxide (Al 2 O 3 ) nano particles is considered. Heat transfer, thermal radiation and heat generation are taken. Unsteady flow is over an infinite vertical plate (ξ > 0) embedded in a saturated porous medium. MHD effect with uniform magnetic field B of strength B 0 and small magnetic Reynolds number. Initially both the plate and fluid are at rest with constant temperature Θ ∞ . At time t = 0 + the plate originates oscillation in its plane ξ = 0 according to condition After some time, plate temperature is raised to Θ w . The fluid is electrically conducting. Therefore, by Maxwell equations e By using Ohm's law nf The quantities ρ nf , μ e and σ are assumed constants. Magnetic field B is normal to V. The Reynolds number is so small that flow is laminar. Hence,   where π = e ab e ab and e ab is the (a, b) ah factor of the deformation rate, π is represent the product of the factor of deformation rate with itself, π c is represent the critical value of this product based on the non-Newtonian model, μ η is represent the plastic dynamic viscosity of the non-Newtonian fluid and P λ is yield stress of fluid. Under these conditions alongside with the assumption that the viscous dissipation term in the energy equation is neglected, we get the following system 56 : nf t nf nf s where k * is absorption coefficient and σ * is Stefan-Boltzmann constant. Where Q 0 is the heat generation term, ρ nf is the density of nanofluids, μ nf is the dynamic viscosity, u is the fluid velocity in the x-axis perpendicular direction, γ is the Casson fluid parameter, ψ(0 < ψ < 1), K > 0, ψ is the porous medium and K is the permeability of porous medium, h s is a constant heat transfer coefficient, Θ w is the constant plate temperature (Θ w < Θ ∞ , Θ w > Θ ∞ due to the cooled or heated plate, respectively), g is the acceleration due to gravity, and β nf is the thermal expansion coefficient of the nanofluid. Expressions for (ρc p ) nf , (ρβ) nf , μ nf , ρ nf , σ nf , k nf are given by 24 : where φ the volume fraction of nano particles, ρ f and ρ s is represent the density of base fluid and particle respectively, and c p is specific heat on constant pressure. k nf , k f , and k s are the thermal conductivities of the nanofluid, the base-fluid, and the solid particles, respectively. The expressions of Eq. (10) are classified to nano particles 57 . For supplementary nano particles with unlike thermal conductivity, dynamic viscosity, see to Table 1 [58][59][60] . the dimensionless variables are 56 , Into Eqs (7-9), we get ; 0, 0 0, 0 as (14) where

Particular Cases
In order to link our found solutions with published literature, the following particular cases are examined by taking some parameters absent. Making Gr = γ = 0 and Re = 1 in Eq. (22), reduces to:   , in Eq. (22), it moderates to: Identical to 58 ,Eq. (35).

Discussion
In this section different parameters including γ, φ, Gr, M, K, Pr, Nr Figs 2-11 are plotted. Geometry of problem is shown in Fig. 1. The influence of γ on u(y, t) which shows oscillatory behavior increasing first then decreasing is highlighted in Fig. 2. Figures 3 and 4 show effects of φ on u(ξ, t) and θ(ξ, t).ϕ is take in between 0 ≤ φ ≤ 0.04 due to sedimentation when the range goes above from 0.08. It is observed in both cases if the nano particles volume fraction φ is increased it leads to the decreasing of temperature and velocity profile. Figure 5 highlights the effect of Gr for Sodium alginate -based, Casson nanofluids on velocity profile. It is found that with increasing Gr, velocity increases. Because increasing effect in Gr, due to increase of buoyancy forces and decrease of viscous forces. Figure 6 the effect of M = 0, 1, 2 on the velocity profile. u(ξ, t) decreases due to increasing dragging force. M = 0, shows absence of MHD. Figure 7 shows K effect of on u(ξ, t). Velocity decrease due to decreasing friction. Figure 8 highlights that profile of velocity is increased with increasing radiation parameter Nr. The effect is studied for TiO 2 nano particle.           The impact of two different types of nano particles (Al 2 O 3 Sodium alginate -based Casson nanofluid and Ag-Sodium alginate -based nanofluid) on profile of velocity is studied in Fig. 9. The profile of velocity is greater for Al 2 O 3 Sodium alginate -based Casson nanofluid and lower profile velocity for Ag-Sodium alginate -based nanofluid is observed. Figure 10 highlights the comparison of both (Cu Sodium alginate -based Casson nanofluid and Ag-Sodium alginate -based nanofluid) on u(ξ, t). Velocity of Ag-Sodium alginate -based nanofluid is lower than copper Sodium alginate -based nanofluid. This shows that Cu nano particles have more thermal diffusivity compare to Ag which is physically true. Furthermore, the same comparison is study for Al 2 O 3 and TiO 2 models in Fig. 11, which shows that Aluminum oxide Al 2 O 3 nano particles have high thermal diffusivity as compare to Titanium oxide TiO 2 .

Conclusion
The following remarks are concluded from this work: • u(ξ, t) decreases as γ increases • Temperature and velocity profile are decreasing with increasing nano particles volume Fraction φ. • Al 2 O 3 nanofluid has higher velocity from TiO 2 nanofluid and Cu nanofluid has higher velocity from Ag nanofluid. • The pours medium K and MHD Μ show opposite behavior.