Effects of Varying Viscosity and Mixed Convection on Nanotubes-water Flow With Reactions by a Stretching Cylinder: a Comparative Study

: The addressed work explains SWCNT (Single walled carbon nanotubnes) and MWCNT (Multi walled carbon nanotubnes) nanoﬂuid ﬂow under the inﬂuences of temperature dependent viscosity and mixed convection. Comparative study of SWCNT and MWCNT suspended in base liquid is presented. Further heat and mass transfer are addressed for nanoﬂuid effected by radiation, heat generation/absorption and diffusion species. Mathematical development of problem is taken in cylindrical coordinates. System of highly nonlinear differential equations are constructed via appropriate trans-formations. The system of equations are tackled numerically by bvp4c MATLAB solver. The ﬁndings of the study show that volume fraction contributes to decline the ﬂuid ﬂow by cylindrical shaped nanoparticles. In addition, ﬂuid ﬂow decelerates via curvature and magnetic parameters while it boots by Grashof number and volume fraction. Further more, temperature dependent viscosity variable corresponds to decrease the temperature close to the surface and it develops away from the surface. The temperature advances in MWCNT-liquid than SWCNT-liquid. Volume fraction and magnetic parameters correspond to skin friction coefﬁcient enhancement. Heat transfer rate increases for larger curvature and heat generation parameters and reverse trend holds against radiation param-1


Introduction
The challenges and demands of advanced industries have been attracted the attention of researchers. The solar collectors used in devices are mostly affected by poor conductivity and low heat up capability of ordinary liquids. Choi [1] was the first one to take lead to enhance thermal conductivity in this way by mixing nano-sized particles in ordinary liquids. Nanoliquids are composed by suspension of chemical stable nano-scaled materials namely oxides, metals and carbides etc in base liquids. These novel fluids are termed as nanoliquids which enhance thermal characteristics of base liquids (engine oil, water, glycol etc). Ordinary liquids namely engine oil, water, glycol etc having low conductivity. Suspension of nanoparticles in low thermal conductivity liquids advance the thermal conductivity and hence accelerates the performance of industrial liquids. Further studies of nanofluids are cited therein [2][3][4][5][6][7][8][9][10][11][12]. Khan et al. [13] addressed MHD nanoliquid with entropy generation in rotating frame. Sohail et al. [14] investigated Darcy Forchheimer hybrid nanoliquid in porous medium under the impact of entropy analysis. Khan et al. [15] analyzed non-axisymmetric Homann stagnation point nanofluid flow through multiple solutions. Huda et al. [16] elaborated Cattaneo-Christov model for nanofluid with moving needle. Reactive stretched flow of Al 2 O 3 − water in porous space is examined by Lia et al. [17].
Magnetic field application in fluid flow analysis has gained attention of scientists due to its wide range of utilization in many fields namely industries, drug delivery, MHD (Magnetohydrodynamics) generator, mechanical and physiological phenomena and many others. Nadeem et al. [18] studied MHD nanoliquid flow numerically. Malvandi et al. [19] studied mixed convection of MHD nanofluid saturate in vertical annulus. Activation energy in MHD squeezed flow with binary chemical reactions was studied by Ahmad et al. [20]. Hayat et al. [21] investigated magnetohydrodynamics third grade nanoliquid convective flow by nonlinear stretched plate. Partial slip in MHD nanoliquid flow with viscous dissipation near stagnation point was investigated by Emad et al. [22].
Radiation does not need any medium to transmit. It depends on shape, temperature and propagates by electromagnetic waves. It is practiced that system in industries having little temperature difference in fluid caused problems. To overcome this difficulty the researchers incorporated a term named as radiation parameter. The variation in temperature of fluid and wall can be novel by this parameter. Cortell [23] summarized influence of heat generation and radiation in convective flow. A summary about this title is cited in the studies [24][25][26][27].
Chemical reactions are categorized mainly in two types namely homogeneous and heterogeneous reactions. Reactions which encounter catalyst in same phase (namely gases, liquids, solids) correspond to homogeneous and reactions which happen in two or several different phases (like solid and gas, solid and liquid) as heterogeneous reactions. Some utilization of chemical reactions are found in iron oxidation, polymer and metallurgical industries. Reactions species have composite link for formation and usage of reactant species. Generally reactions rate depend on the magnitude of mass itself. A simple isothermal model proposed by Merkin et al. [28] investigates homogeneous-heterogeneous reactions in flow. Influence of chemical reaction in liquid flow was studied by Bhattacharyya [29]. Chemical reactive fluid flow was reported by Rashidi et al. [30] to explore mixed convection for heat and mass transfer. Convective flow with homogeneousheterogeneous reactions saturated a porous medium was analyzed by Hayat et al. [31]. Zakir et al. [32]   Further more, CNTs particles and base liquid are in thermal equilibrium. Linearly stretching cylinder (i.e. U w = U 0 z l ) is along axial direction (z − axis) while liquid is assumed to deform in radial direction (r − axis).
In above a ⋆ and T r indicate reference condition of the liquid. Generally, positive values of a ⋆ stand for liquid and negative values of a ⋆ show gases.
For heterogeneous-homogeneous reactions, the model of isothermal is defined by At surface of catalyst the first-order reaction is The governing problems via boundary layer approximation in cylindrical coordinates are as follows: with conditions where ponents, the density, the specific heat, the dynamic and kinematics viscosity, the thermal conductivity, the stretching velocity, the reference velocity, the characteristics length, the radius of cylinder, the strength of magnetic field, the electric transport, the gravitational acceleration, the thermal coefficient, the heat generate per unit volume, the wall and ambient temperatures, the Stefan Boltzmann constant, the mean absorption coefficient, the reference temperature and the diffusion coefficients respectively.
Following model by Xue [33] one has , in which φ denotes the volume fraction of nanosized particle, ρ f the base liquid density, k f the base liquid conductivity, k CN T represents nanotubes thermal conductivity, σ CN T and σ f the electric conductivity of base liquid and nanoparticle respectively. Letting Eq. (4) is trivially confirmed and Eqs.(5 − 8) reduce to with where γ = 1 One has, for equal diffusion coefficients D A⋆ and D B ⋆ as [28] Φ ⋆ (η) + h ⋆ (η) = 1.

Eqs. (13)-(14) imply that
Skin friction coefficient and Nusselt number are where τ w and q w are defined as; where Re 1 2 z = U 0 ν f l z represents the local Reynolds number.

Discussion
This section elaborates the graphical discussion for SWCNT and MWCNT nanoflu-  Table 1.
The outcomes of curvature, Hartman number, Grashof number, volume fraction and others involved dimensionless variables are elaborated for the distributions Velocity: Curvature variable γ declines the velocity distribution. (See Fig. 2).
Clearly γ and R inversely relate to each other and therefore the resistive force enhances for the liquid flow. Therefore the velocity field for SWCNT and MWC-NT behaves similarly. Temperature: The lines for temperature via heat variable is addressed in Fig. 6 with SWCNTs and MWCNTs as nanoparticles. Thermal layers and temperature gradient enhance for larger Q r . Heat in liquid increases as the values Q r higher. Temperature variations is noted similar for both type CNTs. The temperature curves follow free stream condition for larger η. Fig. 7 addresses the influence of viscous dissipation (i.e. Eckert number Ec) on temperature. The temperature rises when Ec is increased. Larger Ec represent high kinetic energy which corresponds to enhance the temperature of liquid. Moreover, same temperature is noted for cylinder shaped SWCNTs and MWCNTs nanofluids. Fig. 8 shows the effect of curvature variable on temperature. The fluid heats up by increasing γ. Heat in MWCNTs-water is noted higher in comparison to SWCNTs-water. Temperature enhances against more heat generation parameter Q r (see Fig. 9). Temperature in case of SWCNTs and MWCNTs are noted same for the base fluid. The curves of radiation variable R d for θ(η) is shown in Fig. 10. Outcomes of radiation variable results to enhancement in θ(η). Higher values of radiation variable results decrease in absorption coefficient. Hence the temperature field increases. Fig. 11 is sketched for the temperature dependent viscosity variable on thermal field. Figure   shows that increment in θ r , contributes to enhancement of temperature. Physically it reflects that larger viscosity mean higher resistance for flow and consequently the kinetic energy enhances. Temperature close to the surface first declines and then advances far away from cylinder. Same temperature is noted for both CNTs.
Concentration: Fig. 12 addresses the concentration via curvature variable γ.
Concentration enhances for larger values of γ. Fig. 13 shows the curves for concentration gradient via homogeneous variable K. The concentration decreases for K. In fact there is direct relation between chemical reaction and K. Concentration for larger heterogeneous variable Ks can be seen in Fig. 14. Same behavior is noted for Ks. The concentration function boosts for Sc (see Fig. 15). There is inverse relationship between Schmidt number and small mass diffusivity. Further same mass transfer rate is noted for both MWCNTs and SWCNT. Fig. 16 shows the behavior of skin friction coefficients via M f and φ. Skin friction for the fluid flow enhances for larges values of M f and φ. Fig. 17 addresses the Nusselt number via variables R d and Q r . Nusselt number increases for larger Q r and it decreases for R d . Nusselt number for γ and Q r is opposite (see Fig. 18).

Main findings
This article addressed a comprehensive study of SWCNT and MWCNT suspended in base fluid water transport towards a stretching cylinder effected by mixed convection, MHD, viscous dissipation, thermal radiation and homogeneous-heterogeneous reactions. Further more the base fluid is documented variable viscosity. Cylindrical coordinates are adopted for mathematical development. Nonlinear coupled differential equations are obtained through similarity transformations. These developed equations are tackled via bvp4c implicit finite difference MATLAB soft-