Model and Comparative Study for Flow of Viscoelastic Nanofluids with Cattaneo-Christov Double Diffusion

Here two classes of viscoelastic fluids have been analyzed in the presence of Cattaneo-Christov double diffusion expressions of heat and mass transfer. A linearly stretched sheet has been used to create the flow. Thermal and concentration diffusions are characterized firstly by introducing Cattaneo-Christov fluxes. Novel features regarding Brownian motion and thermophoresis are retained. The conversion of nonlinear partial differential system to nonlinear ordinary differential system has been taken into place by using suitable transformations. The resulting nonlinear systems have been solved via convergent approach. Graphs have been sketched in order to investigate how the velocity, temperature and concentration profiles are affected by distinct physical flow parameters. Numerical values of skin friction coefficient and heat and mass transfer rates at the wall are also computed and discussed. Our observations demonstrate that the temperature and concentration fields are decreasing functions of thermal and concentration relaxation parameters.


Introduction
There is a significant advancement in the nanotechnology due to its rich applications in the industrial and physiological processes. The modern researchers are engaged to explore the mechanisms through the nanomaterials. A solid-liquid mixture of tiny size nanoparticles and base liquid is known as nanofluid. The colloids of base liquid and nanoparticles have important physical characteristics which enhance their potential role in the applications of ceramics, drug delivery, paintings, coatings etc. Nanofluids are declared as super coolants because their heat absorption capacity is much higher than traditional liquids. The reduction of the system and enhancement in its performance can be achieved with the implications of nanoliquids. The term nanofluid was first introduced by Choi and Eastman [1] and they illustrated that the thermal properties of base liquids are enhanced when we add up the nanoparticles in it. Buongiorno [2] developed the model of nanoparticles by considering the thermophoretic and Brownian motion aspects. Further the recent developments on nanoliquids can be seen in the investigations [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 and double diffusion of heat and mass transfer by Cattaneo-Christov theory. Thus to the best of the author's knowledge, no such attempt has been discussed in the literature yet. Transformation procedure is utilized to convert the partial differential system into the set of nonlinear ordinary differential system. The governing nonlinear system has been solved through the homotopy analysis method (HAM) [42][43][44][45][46][47][48][49][50]. Convergence of computed solutions is checked by plots and numerical data. The contributions of various pertinent parameters are studied and discussed. Heat and mass transfer rates at the surface are also analyzed through numerical values.

Formulation
Let us consider the steady two-dimensional (2D) flow of viscoelastic nanofluids over a linear stretching sheet with constant surface temperature and concentration. The flow models for elastico-viscous and second grade materials are considered. The Brownian motion and thermophoresis are taken into consideration. Here x-axis is along the stretching surface while yaxis is normal to the x-axis. The stretching velocity is u w (x) = ax with a > 0 as the constant. The heat and mass transfer mechanisms are examined through Cattaneo-Christov double diffusion expressions. Governing equations of mass, momentum, energy and nanoparticles concentration for boundary layer considerations are @u @x þ @v @y Note that u and v represent the flow velocities in the horizontal and vertical directions respectively while ν(= μ / ρ f ), μ, ρ f and k 0 = −α 1 / ρ f denote kinematic viscosity, dynamic viscosity, density of base liquid and elastic parameter respectively. Here (k 0 > 0) depicts elastico-viscous fluid, (k 0 < 0) demonstrates second grade fluid and (k 0 = 0) corresponds to Newtonian fluid. The Cattaneo-Christov double diffusion theory has been introduced in characterizing thermal and concentration diffusions with heat and mass fluxes relaxations respectively. Then the frame indifferent generalization regarding Fourier's law and Fick's law (which is named as Cattaneo-Christov anomalous diffusion expressions) are derived as follows: q þ l E @q @t þ V:rq À q:rV þ ðr:VÞq J þ l C @J @t þ V:rJ À J:rV þ ðr: where q and J stand for heat and mass fluxes respectively, k for thermal conductivity, D B for Brownian diffusivity, λ E and λ C for relaxation time of heat and mass fluxes respectively. Classical Fourier's and Fick's laws are deduced by inserting λ E = λ C = 0 in Eqs (3) and (4). By considering the incompressibility condition (r.V = 0) and steady flow with ð @q @t ¼ 0Þ and ð @J @t ¼ 0Þ, Eqs (3) and (4 can be rewritten as q þ l E ðV:rq À q:rVÞ ¼ À krT; J þ l C ðV:rJ À J: Now by taking the Brownian motion and thermophoresis effects in Eqs (5) and (6), then the two dimensional energy and concentration equations take the following forms: Here one has the following prescribed conditions: where and in which α = k/(ρc) f , (ρc) f and (ρc) p stand for thermal diffusivity, heat capacity of liquid and effective heat capacity of nanoparticles respectively, D B for Brownian diffusivity, C for concentration, D T for thermophoretic diffusion coefficient, T w and C w for constant surface temperature and concentration respectively and T 1 and C 1 represent the ambient fluid temperature and concentration respectively. Selecting u ¼ axf 0 ðzÞ; v ¼ À ðanÞ 1=2 f ðzÞ; z ¼ a n À Á 1=2 y; Eq (1) is identically verified and Eqs (2) and (7)- (12) have been reduced to where (k Ã ) stands for viscoelastic parameter, (Pr) for Prandtl number, (N b ) for Brownian motion parameter, (N t ) for thermophoresis parameter, (δ e ) for thermal relaxation parameter, (Sc) for Schmidt number and (δ c ) for concentration relaxation parameter. It is examined that (k parameters can be specified by using the definitions given below: Skin friction coefficient is given by The dimensionless form of skin friction coefficient is stated below: Re 1=2 where local Reynolds number is denoted by Re x = u w x/ν.

Convergence Analysis
Here the expressions (38)-(40) contain ℏ f , ℏ θ and ℏ ϕ . Moreover the convergence is accelerated by the auxiliary parameters ℏ f , ℏ θ and ℏ ϕ in series solutions. For the purpose of determining appropriate values of ℏ f , ℏ θ and ℏ ϕ , the ℏ-curves at 20th order of deformations are sketched to see the appropriate ranges of ℏ f , ℏ θ and ℏ ϕ . It is apparent from Figs 1 and 2 that the admissible  Table 1 shows that the convergent series solutions of velocity, temperature and concentration fields require the 19th order of approximations for elastico-viscous fluid situation whereas the 29th order of deformations are enough for the convergent homotopy solutions of second grade material situation (see Table 2).        parameter (N t ) on temperature field θ(z) for both fluids. Larger values of thermophoresis parameter (N t ) constitute a higher temperature field and more thermal layer thickness. The reason behind this phenomenon is that an enhancement in thermophoresis parameter (N t ) yields a stronger thermophoretic force which allows deeper migration of nanoparticles in the fluid which is far away from the surface forms a higher temperature field and more thickness of thermal layer for both fluids. Moreover thermal layer thickness is lower for second grade parameter (k It is also noticed that concentration layer thickness is lower for second grade parameter (k Ã < 0) when compared with the elastico-viscous parameter (k Ã > 0). From Fig 11 we observed that the larger Schmidt number forms a decay in the concentration field ϕ(z) and its related thickness of concentration layer for both fluids. Physically Schmidt number is based on Brownian diffusivity. An increase in Schmidt number (Sc) yields a weaker Brownian diffusivity. Such weaker Brownian diffusivity corresponds to lower concentration field ϕ(z) for both fluids. It is also observed that concentration field ϕ(z) is lower for negative values of (k Ã ) when compared with the positive values of (k Ã ). From Fig 12 it is clearly examined that a weaker concentration field ϕ(z) is generated by using larger Brownian motion parameter (N b ) for both fluids. In nanofluid flow, due to the existence of nanoparticles, the Brownian motion takes place and with the increase in Brownian motion parameter (N b ) the Brownian motion is affected and hence the concentration layer thickness reduces. It is also examined that concentration field is less for (k Ã < 0) in comparison to (k Ã > 0). Fig 13 shows that the higher thermophoresis parameter (N t ) yields a stronger concentration field ϕ(z) for both fluid cases. Moreover the concentration field ϕ(z) is weaker for (k Ã < 0) when compared with (k Ã > 0). Table 3 shows  Table 3 presents a good agreement of HAM solution with the existing optimal homotopy analysis method (OHAM) solution in a limiting sense. Table 4 is calculated in order to investigate the numerical computations of skin friction coefficient À Re 1=2

Discussion
x C f for several values of (k Ã ). Here we noticed that the skin friction coefficient is higher in case of  Tables 5 and 6 show the numerical computations of heat transfer rate -θ 0 (0) for various values of thermal relaxation parameter (δ e ) in case of elastico-viscous (k Ã > 0) and second grade (k Ã < 0) materials respectively. Here we noticed that the heat transfer rate -θ 0 (0) has higher values for larger (δ e ) in both materials. It is also observed that the values of heat transfer rate -θ 0 (0) for negative values of (k Ã ) are higher when compared with the positive values of (k Ã ).

Conclusions
Boundary-layer flow of viscoelastic nanofluids bounded by a linear stretchable surface with Cattaneo-Christov double diffusion has been discussed. The key points of the presented study are given below: • An enhancement in the positive values of viscoelastic parameter (k Ã ) demonstrate a decreasing behavior for the velocity field f 0 (z) while opposite behavior is noted for the negative values of viscoelastic parameter (k Ã ).
• Larger values of Prandtl number (Pr) show decreasing trend for temperature profile θ(z) and its related thickness of thermal layer.
• Both temperature field θ(z) and its associated thermal layer thickness are reduced for larger thermal relaxation parameter (δ e ).
• Both temperature θ(z) and concentration ϕ(z) fields show opposite behavior for increasing values of Brownian motion parameter (N b ).
• Higher concentration relaxation parameter (δ c ) causes a decay in the concentration field ϕ (z).