Nanofluid slip flow over a stretching cylinder with schmidt and péclet number effects NANOFLUID SLIP FLOW OVER A STRETCHING CYLINDER WITH SCHMIDT AND PÉCLET NUMBER EFFECTS  constant of the expansion/contraction convection

A mathematical model is presented for three-dimensional unsteady boundary layer slip flow of Newtonian nanofluids containing gyrotactic microorganisms over a stretching cylinder. Both hydrodynamic and thermal slips are included. By applying suitable similarity transformations, the governing equations are transformed into a set of nonlinear ordinary differential equations with appropriate boundary conditions. The transformed nonlinear ordinary differential boundary value problem is then solved using the Runge-Kutta-Fehlberg fourth-fifth order numerical method in Maple 18 symbolic software. The effects of the controlling parameters on the dimensionless velocity, temperature, nanoparticle volume fractions and microorganism motile density functions have been illustrated graphically. Comparisons of the present paper with the existing published results indicate good agreement and supports the validity and the accuracy of our numerical computations. Increasing bioconvection Schmidt number is observed to depress motile micro-organism density function. Increasing thermal slip parameter leads to a decrease in temperature. Thermal slip also exerts a strong influence on nano-particle concentration. The flow is accelerated with positive unsteadiness parameter (accelerating cylinder) and temperature and micro-organism density function are also increased. However nano-particle concentration is reduced with positive unsteadiness parameter. Increasing hydrodynamic slip is observed to boost temperatures and micro-organism density whereas it decelerates the flow and reduces nanoparticle concentrations. The study is relevant to nano-biopolymer manufacturing processes.

A mathematical model is presented for three-dimensional unsteady boundary layer slip flow of Newtonian nanofluids containing gyrotactic microorganisms over a stretching cylinder. Both hydrodynamic and thermal slips are included. By applying suitable similarity transformations, the governing equations are transformed into a set of nonlinear ordinary differential equations with appropriate boundary conditions. The transformed nonlinear ordinary differential boundary value problem is then solved using the Runge-Kutta-Fehlberg fourth-fifth order numerical method in Maple 18 symbolic software. The effects of the controlling parameters on the dimensionless velocity, temperature, nanoparticle volume fractions and microorganism motile density functions have been illustrated graphically. Comparisons of the present paper with the existing published results indicate good agreement and supports the validity and the accuracy of our numerical computations. Increasing bioconvection Schmidt number is observed to depress motile micro-organism density function. Increasing thermal slip parameter leads to a decrease in temperature. Thermal slip also exerts a strong influence on nano-particle concentration. The flow is accelerated with positive unsteadiness parameter (accelerating cylinder) and temperature and micro-organism density function are also increased. However nano-particle concentration is reduced with positive unsteadiness parameter. Increasing hydrodynamic slip is observed to boost temperatures and micro-organism density whereas it decelerates the flow and reduces nanoparticle concentrations. The study is relevant to nano-biopolymer manufacturing processes.   5 and Ishak et al., 6 Bég et al. 7 (with magnetohydrodynamic and cross-diffusion effects) and Daskalakis 8 . Akl 9 quite recently investigated unsteady boundary layer flow due to a stretching cylinder with prescribed temperature and obtained the solution analytically.
Bég et al. 10  generation, micro-manufacturing, thermal therapy for cancer treatment, chemical and metallurgical sectors, microelectronics, aerospace and manufacturing. 12 Representative works on convective boundary layer flow and application of nanofluids were conducted by Buongiorno, 13 Das et al., 14 Kakaç and Pramuanjaroenkij, 15 Saidur et al. 16 and Wen et al. 17 Further studies have been communicated by Mahian et al., 18 Nield and Bejan, 19 Haddad et al., 20 Sheremet and Pop 21 and many others. There are two types of model for nanofluids which have been commonly used by the researchers, namely Buongiorno's model 13 23 Li et al., 24 Vanaki et al., 25 Zhao et al., 26 Serna, 27 , Mohyud-Din 28 and others. Ferdows et al. 29 investigated radiative magnetohydrodynamic nano-polymer stretching flows. Uddin et al. 30 studied numerically the stretching fluid dynamics of magnetic nano-bio-polymers.
Bioconvection refers to a macroscopic convection motion of fluid affected by density gradients induced by hydrodynamic propulsion i.e. swimming, of motile microorganisms (see Kuznetsov 31 ). Adding microorganisms (such as algae and bacteria) to base fluids (e.g. water) creates the process of bioconvection which is directionally-orientated swimming typically towards an imposed or naturally present stimulus e.g. light, gravity, magnetic field and chemical concentration (oxygen). The density of the microorganism is inclined to be greater than that of the free stream fluid and this can cause an unstable density profile with subsequent upending of the fluid against gravity (see Raees et al. 32 ). The base fluid has to be water for the majority of microorganisms to survive and be active and it is assumed nanoparticle suspension remains stable and do not agglomerate for a couple of weeks (see Anoop et al. 33 ). For bioconvection to take place, the suspension must be dilute since nanoparticles would increase the suspension's viscosity and viscosity tends to dominates bioconvection instability (see Pedley 34 ).
A recent innovation for microfluidic devices is to combine nanofluids with bioconvection phenomena (see Xu and Pop 35 ). Aziz et al. 36 have studied theoretically the natural bioconvection boundary layer flow of nanofluids and verified that the bioconvection parameters influence mass, heat, and motile microorganism transport rates. Latiff et al. 37 studied unsteady forced bioconvection slip flow of a micropolar nanofluid from a stretching/shrinking sheet. Bioconvection may have also have a role to play in biomicrosystems for mass transport augmentation and microfluidic devices such as bacteriapowered micromixers (see Tham et al. 38 ). Other significant applications of nanofluid bioconvection arise in the synthesis of novel pharmacological agents (drugs) as elaborated by Saranya and Radha 39 and earlier for nano-bio-gels as discussed by Oh et al. 40 Microorganisms can be deployed strategically to enhance biodegradable polymeric nanomaterials and improve various desirable medical characteristics such as bioavailability, biocompatibility, encapsulation, DNA embedding in gene therapy, protein deliverability etc.
The intelligent manufacture of bio-nano-polymers allows drugs to be developed which achieve a "controlled release" and this has been shown to increase therapeutic influence in patients. Examples of such bio-nano-polymers are poly (lactic-co-glycolic acid), polylactic acid, chitosan, gelatin, poly hydroxy alkaonates, poly caprolactone and poly alkyl cyanoacrylate.
To optimize the fabrication of bio-nano-materials, numerical and physico-mathematical simulation has an important role to play. This is a strong motivation for the present study in  The nanoparticles fraction on the ambient is assumed to obey the passively controlled model proposed by Kuznetsov and Nield 42 , while the nanoparticles and temperature distribution on the ambient is assumed to be a constant , CT  respectively. It is worth mentioning that the micro-organisms can only survive in water. This indicates that the base fluid has to be water.
Under these assumptions and the nanofluid model of Kuznetsov and Nield,42 the relevant transport equations are the conservation of total mass, momentum, thermal energy, nanoparticle volume fraction and microorganisms concentration (density) which may be stated in vector form as follows: (see Xu and Pop 36 ) is the velocity vector of the nanofluid flow in the x  direction, y  direction and the z  direction respectively, p is the pressure, T is the temperature, C is the nanoparticle volumetric fraction, n is the density of the motile microorganism,  is the nanofluid density,  is the kinematic viscosity of the suspension of nanofluid and microorganisms,  is the thermal diffusivity of the nanofluid, () 22 22

11
, The relevant boundary conditions corresponding to the physical problem may be stipulated following Zaimi et al. 43 as: where w T is the constant surface temperature, 1 N is the velocity slip factor, 1 D is the variable thermal slip factor, w n is the constant surface density of the motile microorganism, and and TC  denote constant temperature and nanoparticle volume fraction far from the surface of the cylinder, respectively.

SIMILARITY TRANSFORMATION OF MATHEMATICAL MODEL
To proceed, we introduce the following transformations: (Zaimi et al. 43 , Abbas et al. 44 ) ,, Eqn. (6) is satisfies automatically and since there is no longitudinal pressure gradient, Using (13), we have transformed Eqs. (8)-(11) into a system of ordinary differential equations: The boundary conditions (13) are transformed into: Here, the controlling parameters involved in the above dimensionless Eqs. (14)

PHYSICAL QUANTITIES
The quantities of engineering interest in bio-nano-materials processing i.e. Sakiadis-type flows are the wall parameters. These are respectively local skin friction coefficient , Nusselt number x Nu , local nano-particle mass transfer rate i.e. local Sherwood number, and finally the local density number of motile micro-organisms, x Nn defined as: where , w  w q , m q and n q represent the shear stress, surface heat flux, surface mass flux and the surface motile microorganism flux and are defined by: Substitute Eqns. (21) and (13) into (20) we obtain:

MAPLE 18 NUMERICAL SOLUTION, SPECIAL CASES AND VALIDATION
Numerical solutions to the ordinary differential Eq. and also setting Re 1  we retrieve the model of Fang et al. 41 Furthermore for 0 S  (steady case), 0 Nt Nb  (nano-particle absence) and disregarding Eqns. (16) and (17)

NUMERICAL RESULTS AND DISCUSSION
In this section, we present the effects of parameters ,, These results are consistent with published work by Mukhopadhyay 51 and also Wang. 52 Furthermore, the magnitude of the wall shear stress decreases with an increase in the hydrodynamic slip factor.    The case of 0 b  applies to the classical non-slip scenario. The momentum boundary-layer thickness decreases with an increase in the velocity slip parameter. In Fig. 2(b), for both S>0 (accelerating cylinder) or S<0 (decelerating cylinder), with an increasing the velocity slip, b, the dimensionless temperature, (), is markedly enhanced. However temperatures are somewhat greater for the accelerating cylinder case as compared with the decelerating case.
The thermal boundary-layer thickness therefore increases with an increase in the velocity slip parameter. The deceleration in the velocity field, f / (), implies that momentum diffusion is reduced. This benefits the transport of heat via thermal diffusion which manifests in a heating in the bio-nano-boundary layer regime and an associated elevation in temperature. Thermal diffusivity is dominant in this case i.e. heat conduction is stronger than heat convection. Both velocity and temperature profiles are found to decay smoothly from maxima at the cylinder surface to the free stream, indicating that a sufficiently large infinity boundary condition has been imposed in the MAPLE18 computational domain. The dimensionless nanoparticle volume fraction (nano-particle concentration),  (), as depicted in Fig. 2(c) is observed to be reduced with increasing velocity slip. The nano-particle concentration boundary-layer thickness will therefore be increased with a rise in the velocity slip parameter. The dimensionless concentration also decreases for accelerated flow (S>0) whereas it is elevated for decelerated (S<0) flow. The prescribed Schmidt number in Fig. 2 is 20.0. Schmidt number (Sc) expresses the ratio of momentum diffusivity to species diffusivity i.e. viscous diffusion rate to molecular (nano-particle) diffusion rate. Sc is also the ratio of the shear component for diffusivity viscosity/density to the diffusivity for mass transfer D. It physically relates the relative thickness of the hydrodynamic layer and mass-transfer boundary layer.
We further note that Pr is prescribed as 6.8 as this quite accurately represents water-based nano-bio-polymers. The deceleration in the flow with increasing hydrodynamic slip also acts to decrease molecular diffusion rate (via the Schmidt number) and this will result in decreasing nano-particle boundary layer thickness. The depletion in nanoparticle concentration will cause a corresponding elevation in nano-particle mass transfer rate at the cylinder surface (wall). From Fig. 2(d), it is evident that the dimensionless microorganism number density function, () increases as velocity slip increases i.e. with flow deceleration.
The behavior is different from the nano-particle concentration field. Unlike the diffusion of nano-particles (which is molecular in nature), the micro-organisms move by flagellar propulsion which is encouraged in slower flows. They are therefore able to propel more evenly through the boundary layer for slower flow. The microorganism boundary layer thickness also increases with increasing velocity slip. The implication is that more homogenous distributions of micro-organisms through the boundary layer regime are achieved with deceleration in the flow. This is desirable in the manufacture of biodegradable nano-polymers as further elaborated by Thomas and Yang. 3 It is also observed that () values are greater for the accelerating cylinder case (S >0) as compared with the decelerating cylinder case (S<0). Therefore contrary responses in the micro-organism number density magnitudes are induced depending on whether the boundary layer flow is accelerating (which it does for no-slip) or the cylinder is accelerating. The former is associated with slip absence (or presence which causes deceleration in the flow) whereas the latter is connected to the unsteadiness in the cylinder stretching motion. respectively. It is apparent that the thermal slip parameter leads to a decline in dimensionless temperature ( Fig. 3(a)). The greatest effect is as expected at the cylinder surface. Physically, as the thermal slip parameter rises, the fluid flow within the boundary layer will be less sensitive to the heating effects of the cylinder surface and a reduced quantity of thermal energy (heat) will be transmitted from the hot cylinder to the fluid, resulting in a fall in temperatures i.e. cooling and thinning of the thermal boundary layer (decrease in thermal boundary layer thickness). For an accelerating stretching cylinder (S>0), the temperatures are substantially higher than for a decelerating stretching cylinder (S<0). The dimensionless nano-particle concentration is found to be strongly increased with greater thermal slip. Nanoparticle concentration however is enhanced for the decelerating stretching cylinder case whereas it is depressed for the accelerating cylinder case. Micro-organism number density function (Fig 3(c)) is however significantly decreased with increasing thermal slip effect. For an accelerating stretching cylinder (S>0), the micro-organism density is (as with temperature) unlike nano-particle concentration, substantially higher than for a decelerating stretching cylinder (S<0). Micro-organism number density and temperature profiles are very similar indicating that fields respond in a similar fashion in the external boundary layer regime on the stretching cylinder. Thermal diffusion and micro-organism propulsion obey similar physics in the flow as opposed to nano-particle diffusion which has a distinctly different response. An increase in thermal slip essentially thickens both the thermal and micro-organism number density boundary layers whereas it thins the nano-particle concentration boundary layer thickness. Therefore biotechnological engineers can achieve very different thermo-fluid characteristics in nano-bio-polymers by judiciously utilizing thermal slip at the cylinder wall and also via the rate of cylinder stretching (unsteadiness). arises only in the nano-particle species conservation eqn. (16). It is defined as , other words the ratio of momentum diffusivity to diffusivity of microorganisms. For Sb >1 as studied in Fig. 4, momentum diffusivity exceeds micro-organism diffusivity. As this parameter increases the difference in diffusivity is amplified and momentum diffusion rate increasingly dominates the micro-organism diffusion rate leading to a reduction in microorganism density number magnitudes, (). There is a corresponding diminishing in the thickness of the micro-organism number density boundary layer. For an accelerating stretching cylinder (S>0), the micro-organism density is significantly higher than for a decelerating stretching cylinder (S<0). Therefore for higher Pe values the micro-organism speed will be reduced and/or the diffusivity of micro-organisms will be decreased. This will result in reduced concentrations of micro-organisms in the boundary layer and an elevation in motile micro-organism mass transfer rate,   1   , to the cylinder surface, as observed in Figure 6. with bioconvection Schmidt or biconvection Péclet number, these distributions have been omitted.

CONCLUSIONS
The unsteady bioconvective slip flow of a nanofluid (containing both nanoparticles and gyrotactic microorganisms) in the external boundary layer from a stretching cylinder, is studied as a simulation of bio-nano-polymer fabrication. The Buongiorno nanofluid model is employed with physically more realistic passively controlled boundary conditions. Both thermal and hydrodynamic slip effects at the cylinder surface are considered. The governing transport equations are transformed into a set of ordinary differential equations using similarity variables. The transformed well-posed ninth order boundary value problem is solved using the Runge-Kutta-Felhberg fourth-fifth order numerical method in MAPLE18 symbolic software. Validation with previous computations is included. The computations have shown that increasing bioconvection Schmidt number reduces motile micro-organism density function. Increasing hydrodynamic slip enhances temperatures and motile microorganism density function, but decreases nanoparticle volume fraction (nano-particle concentration) values. Increasing thermal slip reduces temperatures and furthermore for an accelerating stretching cylinder (S>0), the temperatures are greater than for a decelerating stretching cylinder (S<0). Nano-particle concentration is conversely elevated with greater thermal slip whereas micro-organism number density function is greatly depressed with increasing thermal slip effect. At any bioconvection Schmidt number, for an accelerating stretching cylinder (S>0), the micro-organism density is much higher than for a decelerating stretching cylinder (S<0). Local Nusselt number is reduced with increasing hydrodynamic and thermal slip and also for an accelerating cylinder. The local microorganism transfer rate is increased with greater values of bioconvection Péclet number whereas it is suppressed with greater bioconvection Schmidt number and for an accelerating cylinder (positive values of unsteadiness parameter). The present work has been confined to constant fluid properties and ignored electromagnetic effects. For future work, the present model may therefore be extended to consider variable fluid properties and also multi-physical effects e.g. chemical reaction, magnetohydrodynamics, second order slip and melting effects. These are also relevant to bio-nano-polymer processing applications and efforts in this regard are under way.