Hydromagnetic unsteady slip stagnation flow of nanofluid with suspension of mixed bio-convection

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Propulsion and Power Research 2019;xxx(xxx):1e11 http://ppr.buaa.edu.cn/

Propulsion and Power Research w w w. s c i e n c e d i r e c t . c om

ORGINAL ARTICLE Q3

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Hydromagnetic unsteady slip stagnation flow of nanofluid with suspension of mixed bioconvection R. Kumara, S. Sooda, C.S.K. Rajub, S.A. Shehzadc,* a

Department of Mathematics, Central University of Himachal Pardesh, Dharamshala, 176215, India Department of Mathematics, GITAM University, Bangalore, 562163, Karnataka, India c Department of Mathematics, COMSATS Institute of Information Technology, Sahiwal, 57000, Pakistan b

Received 3 January 2018; accepted 16 October 2018 Available online XXXX

KEYWORDS Unsteady flow; Stagnation point region; Mixed convection; Microorganisms; Nanoparticles; Slip regime; Keller-box scheme

Abstract In this paper, stagnation point region (suspended with nanofluid and microorganisms) is examined subjected to velocity and thermal slips. The flow is assumed to be flowing across an exponentially stretching flat surface under magnetic field environment. Governing transport equations (framed under Buongiorno’s model) are converted into self-similar form through suitable unsteady exponential similarity transformations which are then solved by employing implicit finite difference scheme (IFDS) known as Keller-box method (KBM). A parametric analysis is performed along with limited details of KBM. Influence of pertinent constraints on local skin friction, local Nusselt number, local Sherwood number and local density number of micro-organisms are also uncovered through graphs. A significant decrease in skin-friction coefficients has been detected with respect to thermal slip in comparison to velocity slip. ª 2019 Beihang University. Production and hosting by Elsevier B.V. on behalf of KeAi. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.

1. Introduction

E-mail address: [email protected] (S.A. Shehzad). Peer review under responsibility of Beihang University.

Production and Hosting by Elsevier on behalf of KeAi

Hydromagnetic mixed convection boundary layer flow of an electrically conducting nanofluid with heat/mass transportation effects past moving surfaces is of paramount importance in various scientific and technological fields.

https://doi.org/10.1016/j.jppr.2018.10.001 2212-540X/ª 2019 Beihang University. Production and hosting by Elsevier B.V. on behalf of KeAi. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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Nomenclature A unsteady parameter a, b, c constants C0, n0, T0 constants b1 chemotaxtis constant cf skin friction coefficient cfx local skin friction coefficient g gravity vector B0 magnetic field strength D1 thermal slip factor DB Brownian diffusion coefficient Dm diffusivity of micoorganisms DT thermophoretic diffudion coefficient D v Dt Z vt þ V$V material deriviative T dimensionless stream function k thermal conductivity bioconvection Schmidt number Scb M magnetic field strength N1 velocity slip factor Nb Brownian motion parameter Nn Density number of motile microorganisms Local density number of motile microorganisms Nnx Nr buoyancy ratio parameter Nt thermophoresis parameter Nu Nusselt number Nux local Nusselt number P pressure Peb bioconvection Peclet number Pr Prandtl number qm surface microorganisms flux surface mass flux qp qw surface heat flux Rex local Reynolds number Rhb bioconvection Rayleigh number

These include liquid metal fluids, micro-MHD pumps, high temperature plasmas, aerodynamic sensors, magnetogravimetric separations, nanostructured magnetorheological liquids, coating of metals and reactor cooling [1]. The external magnetic field via Lorentz force provides MHD parameter which helps in controlling the cooling rate and hence in the production of a desired quality product ([2e4]). The pioneering work of Choi [5] on nanofluids has revolutionized the treatment of classical fluid dynamics problems. In the past, several investigators have shown great interest in the analysis of nanofluid flows through various geometries ([6e17]). In fluid dynamics, no-slip boundary conditions have not been found appropriate for flows which contain the suspension of particulates such as polymer solutions, emulsions, and fluids which preserve these boundary slips (velocity-thermal) explore applications in several technological problems like polishing of artificial heart valves and internal cavities [18]. Further, stagnation point region in the flow field is a region which experiences zero velocity due to adiabatic and reversible process, and has maximum pressure along with largest heat, and mass transfer rates. These

Sc Sh Shx T Ts TN u,v UN V Vs Wc

Schmidt number Sherwood number local Sherwood number nanofluid temperature dimensionless thermal slip parameter ambient fluid temperature velocity components external flow velocity mass average velocity in given volume element dimensionless velocity slip parameter maximum swimming speed of microorganisms in nanofluid

Greek letters a b ε g l h m y u r rf rm rp s t tw q j

thermal diffusivity thermal expansion coefficient velocity ratio parameter average volume of a microorganism mixed convection parameter similarity variable viscosity of the fluid kinematic viscosity bioconvection constant nanofluid density base fluid density microorganisms density microorganisms density electrical conductivity of the fluid ratio of effective heat capacitance of nanoparticles to fluid Wall shear stress dimensionless temperature stream function

stagnation point regions can be realized in the numerous flow fields such as flows along the tips of rockets, aircrafts, submarines and airships. Moreover, scientific and industrial applications can be seen in forced cooling of nuclear reactors (caused by emergency shutdown) and electronic devices by fan, wind current blow into solar central receiver and many other hydrodynamic phenomena [19]. Several investigators have reported the analysis of fluid flow and heat transfer near the stagnation region such as Ref. [20e22]. On the other hand, bioconvection is a novel branch of biological and industrial fluid mechanics in which bioconvection is induced by the macroscopic convective motions in fluid due to the suspension of upwardly swimming mobile microorganisms (denser than the medium). Therefore, the suspended self-driven microorganisms agglomerate at the upper surface of the fluid, and hence results in unstable density stratification which causes the occurrence of bioconvection plumes. The base fluid has to be water so that these suspended self-propelled microorganisms become capable of surviving in the base fluid. A Continuum model for the bioconvection (induced by suspended gyrotactic

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microorganisms) was developed by Pedley et al. [23]. In bottom heavy microorganisms, the center of mass of an organism is displaced from the centre of buoyancy in a reverse direction of their swimming. These bottom heavy microorganisms have been confidently used in bioconvection experiments by Hill and Pedley [24]. Recently, astounding physiognomy and applications of the nanoparticles and microorganisms mixing in base fluid towards the stagnation point region have galvanized and magnetized various researchers to focus only on this particular field. The credit of introducing this novel concept of biologically induced convection in nanofluids goes to Kuznetsov ([25,26]). Kuznetsov [27] in his large scale motion analysis of fluid (nanoparticles and gyrotactic microorganisms) found that self-propelled motile microbes intensify the mixing of nanoparticles in base fluid and prevent the nanoparticles from accumulation. Also, the physical phenomenon created by the simultaneous interactions among nanoparticles, microorganisms and buoyancy forces requires understanding at the fundamental level due to the formation of spontaneous patterns and density stratification. Some research works on this idea have been addressed in Aziz et al. [28], Shaw et al. [29], Kuznetsov [30] and Akbar et al. [31]. Motivated by the above mentioned investigations on nano-bioconvection flows, the purpose of the present study is to examine the behaviour of unsteady stagnation point boundary layer flow of a water based fluid over an exponentially stretching sheet which comprise of nanoparticles and mobile microbes under magnetic field impact, velocity slip and temperature jump. Here, the unidirectional exponentially stretching forces on the surface and free stream have been considered respectively as Uw ðx; tÞZbexpðx=LÞ 1ct and Ue ðx; tÞ Z aexpðx=LÞ 1ct , where constant a > 0 depicts

3

stagnation flow strength ([32,33]). Nanofluid temperature, concentrations of nanoparticles and microorganisms (at the surface of the sheet) are taken as C0 expðx=LÞ and nw Z nN þ Tw ZTN þ T0 expðx=LÞ 1ct ; Cw ZCN þ 1ct n0 expðx=LÞ . 1ct

2. Geometry and mathematical formulation An unsteady two-dimensional stagnation flow of a viscous electrically conducting and incompressible nanofluid subjected to slip regime is considered as shown in Fig. 1. Nanoparticles and mobile microorganisms are assumed to be suspended in the base fluid. Geometrically, x-axis is aligned along the sheet and y-axis is normal to it. The origin of this coordinate system and stagnation point are chosen as a common fixed point that remains unaltered due to equal strength of stretching forces. The sheet is assumed to be lying transversely under a strong magnetic field of uniform strength B0. Following Aziz et al. [28] under Boussinesq’s approximation, the unsteady conservation equations for a bioconvective nanofluid stagnation flow across an exponentially stretching sheet can be put in vector form as below. Mass conservation V $ V Z0;

ð1Þ

Momentum conservation


DV 2 rf Z  VP þ mV V þ ð1  CN Þrf bðT  TN Þ Dt       rp  rf ðC  CN Þ  rm  rf gðn  nN Þ g

ð2Þ

sB2o V ; Energy conservation   DT rcp f Z kV2 T  cp $jp VT ; Dt

ð3Þ

Nanoparticle conservation DC 1 Z  V$jp; Dt rp

ð4Þ

Cell conservation vn Z  V$jm ; vt jp Z jp; B þ jp; T Z  rp DB VC  rp DT jm Z nV þ nU  Dm Vn $

Fig. 1

Geometry of the problem.

ð5Þ VT ; T

ð6Þ ð7Þ

In Eq. (6), mass flux diffusion vector of nanoparticles ðjp Þ is the sum of mass flux due to Brownian diffusion ðjp; B Þ and thermophoretic diffusion ðjp; T Þ ([30]), and in Eq. (7), mass flux for the mobile microorganisms ðjm Þ is the sum of microorganisms fluxes due to fluid convection (macroscopic

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fluid motion) and directional swimming of microbes (self tonanofluid. propelled) relative Here, U Z bD1 WCc VC, in which D C ZCw  CN and b1 Wc is assumed to be constant. The relevant boundary conditions to this problem are: t<0

b f ð0ÞZ0; f 0 ð0ÞZ þ Vs f 00 ð0Þ; a qð0ÞZ1 þ Ts q0 ð0Þ; 4ð0ÞZ1; xð0ÞZ1; f 0 ðNÞ Z 1; qðNÞZ0; 4ðNÞZ0; xðNÞZ0;

:

ð16Þ

ð17Þ

where prime signifies differentiation with respect toh.

T ZTN ;

u Z vZ0;

CZCN ;

nZnN ;

ð8Þ

t0: vZ0; CZCw ;

vu uZUw þ nN1 ; vy

vT T ZTw þ D1 ; vy

ð9Þ

ð10Þ

Let us consider the coming similarity transformations in terms of dimensionless variables and functions in order to reduce the appearing partial differential equations into selfsimilar form: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aexpðx=LÞ T  TN y; qðhÞZ ; hZ 2nLð1  ctÞ Tw  TN C  CN n  nN ; xðhÞZ ; Cw  CN nw  nN sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2anLexpðx=LÞ f ðhÞ: jZ ð1  ctÞ

4ðhÞZ

ð11Þ

000

f þ f f 00  2f 02 þ 2 þ M ð1  f 0 Þ  Aðhf 00 þ 2f 0  2Þ þlðq  Nr4  Rhb xÞZ0;

ð12Þ

q00  APrðhq0 þ 2qÞ þ Prðq0 f  f 0 qÞ þNbf0 q0 þ Ntq02 Z0;

ð13Þ

400  AScðh40 þ 24Þ þ Scð40 f  f 0 4Þ Nt 00 q Z0; Nb

x00  AScb ðhx0 þ 2xÞ þ Scb ðx0 f  f 0 xÞ Peb ½400 ðx þ uÞ þ x0 40 Z0:

n nN Ue L PrZ ; uZ ; ; Rex Z a n nw  nN n n ScZ ; Scb Z ; Ts ZD1 DB Dm

The ensuing system (of ordinary differential equations) is obtained after incorporating the above transformations into governing equations:

þ

Lc 2sB20 L tDB ðCw  CN Þ ; ;MZ ; NbZ Ue rf Ue a rffiffiffiffiffiffiffiffi 2Lð1  CN ÞbgðTw  TN Þ nUe lZ ; Vs ZN1 ; 2 2L Ue   rp  rf ðCw  CN Þ tDT ðTw  TN Þ ; NrZ NtZ aTN rf bð1  CN ÞðTw  TN Þ   gðnw  nN Þ rm  rf bWc b ; εZ ; Peb Z ; Rhb Z a Dm brf ð1  CN ÞðTw  TN Þ AZ

nZnw at yZ0;

uZUe ; T /TN ; C/CN ; n/nN ; as y/N;

The physical and dynamical parameters (non-dimensional) which appeared in above reduced equations are given below:

ð14Þ

ð15Þ

The set of boundary conditions supporting the conditions at surface and free stream have the following dimensionless form:

rffiffiffiffiffiffiffiffi Ue : 2nL

The quantities of physical and practical interest which have a line of attraction for several scientists and engineers for present problem are coefficients of skin friction, rates of heat and mass transfer, and rate of microorganisms transfer. The respective mathematical picture of these transfer rates already exists in literature and has the following well known forms: cf Z

tw Lqw ; ; NuZ kðTw  TN Þ rf U 2e

Lqp Lqm ShZ ; NnZ : ðCw  CN Þ Dn ðnw  nN Þ

ð18Þ

From previous investigations on nanofluid flow, it can be concluded that nanofluid heat flux at the wall is not only associated with conduction heat transfer but also with heat transfer due to nanoparticle diffusion. Likewise, nanoparticle flux at the wall is due to Brownian motion and thermophoretic force [34]. Similarly, motile microorganisms flux at the sheet surface is due to their self-propagation and diffusion. Recently, some research works have been appeared in literature on the dynamics of boundary layer flows of bioconvective nanofluids such as Mutuku and Makinde [35], Ahmed et al. [36], Uddin et al. [37] and Beg et al. [38]. In these works, the influence of diffusion processes on heat and mass transfer rates and effect of microorganisms self-propagation on microorganisms transfer rate

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(at the surface of sheet) have been respectively ignored. Therefore, these transfer rates are revisited, and hence following Buongiorno [39]; heat flux, nanoparticle flux and microorganisms flux along with shearing stress have the following revised form:     vu vC DT vT tw Zm ; qp Z  DB  ; vy yZ0 vy TN vy yZ0    vT vC rp DT vT þ h  rp DB  qw Z  k : ð19Þ vy vy TN vy yZ0   b1 W c vC vn  Dm qm Z n : vy yZ0 Cw  CN vy Here h Z cp T. After using Eqs. (11) and (19) in Eq. (18), we obtain the dimensionless transfer rates as:

pffiffiffiffiffiffiffiffiffiffi 2Rex cf Zf 00 ð0Þ; pffiffiffi Nu N ux Z 2 pffiffiffiffiffiffiffiZ  q0 ð0Þ  Nbf0 ð0Þ  Ntq0 ð0Þ; Rex pffiffiffi Sh Nt Shx Z 2 pffiffiffiffiffiffiffiZ  f0 ð0Þ  q0 ð0Þ; Nb Rex pffiffiffi Nn N nx Z 2 pffiffiffiffiffiffiffiZ  x0 ð0Þ þ Peb ðxð0Þ þ uÞf0 ð0Þ: Rex

cfx Z

3. Keller box method and discussion of results Since, the boundary value problem depicted by Eqs. 12e17 are coupled and have strong nonlinearity so closed form solution is not achievable. Hence, we relied on a numerical method to solve this highly nonlinear set of equations with associated boundary conditions. Several numerical schemes are available in literature to investigate the boundary layer flow problems. But, we have given preference to an implicit finite difference scheme devised by Keller [40] (also termed as Keller box method (KBM)). This method has second order accuracy in all variables with arbitrary mesh spacing, and is unconditionally stable. Another important trait of this scheme is that one has to do lesser amount of computational work in comparison to other methods in order to attain the prescribed accuracy (Vajravelu and Prasad [41]). In this method, an iteration process is required in order to achieve the convergence of order 106 and a grid of size D hZ 0:005 was found to be adequate. The detailed procedure of this scheme has been given in Kumar et al. [42], and therefore, has been ignored here to protect the space. Due to the absence of related works, a grid independence for the results has been presented for different grid sizes in Table 1 to establish the accuracy of results. The numerical computation is accomplished for various values of parameters in order to have a clear picture about the behaviour of nanofluid velocity, temperature, nanoparticle concentration and micro-organisms distributions. During the entire computational process, the values assigned to basic set of

5 Table 1

A grid independence study.

Step size

f 00ð0Þ

Dh

hN Z 3

hN Z 4

hN Z 5

hN Z 6

0.05 0.01 0.005 0.001

0.081123 0.081091 0.081086 0.081086

0.081122 0.081091 0.081085 0.081086

0.081122 0.081091 0.081085 0.081086

0.081129 0.081091 0.081085 0.081086

parameters are: AZ0:5; PrZ7; M ZlZScZPeb Z Scb Z 2; εZ1; Ts ZNrZNbZNtZRhb ZuZVs Z0:2; unless and otherwise stated. Distributions of dimensionless nanofluid velocity (f’(h)) for different values of velocity ratio parameter (ε), velocity slip parameter (Vs), unsteady parameter (A), magnetic field parameter (M ), bioconvection Peclet number (Peb), and bioconvection Rayleigh number (Rhb) have been shown through Figs. 2e7 respectively for both buoyancy assisting (l > 0) and opposing (l < 0) flows. Since, in all these figures, the profiles approach free-stream boundary conditions asymptotically, therefore the numerical results are accurate. In buoyancy assisting flows (l > 0), fluid gets accelerated because pressure gradient is favourable and buoyancy forces act in the direction of mainstream. On the other hand, for opposing flows the pressure gradient is adverse which retards fluid motion in the boundary layer. In the present study, Figs. 2 and 3 are very interesting and important as these provide a critical analysis for the physical phenomenon. Fig. 2 illustrates the influence of velocity ratio parameter (ε) on the velocity boundary layer profiles f’(h). It is very clear from this figure that f’(h) increases with the increasing ε. In order to understand the hidden physics behind this figure, we have to explore the interactions between velocity ratio parameter (ε) and velocity slip (Vs). Since ε is the ratio of stretching velocity (b) to free stream velocity (a) with which fluid strikes the surface of the sheet, therefore, (ε Z 0) depicts that these forces are of equal magnitude. In this case, the effect of velocity slip is minimum and we have a different type of boundary layer

Fig. 2

Effect of velocity ratio parameter on nanofluid velocity.

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Fig. 3

Effect of velocity slip on nanofluid velocity (ε Z 1.2).

Fig. 4

Effect of unsteadiness on nanofluid velocity.

structures for l > 0 and l < 0. But for ε > 0 or ε < 0, the impact of velocity slip is comparable with that of velocity ratio parameter and we have inverted boundary layer profiles. Fig. 3 communicates the effect of velocity slip Vs on the velocity f’(h) for ε Z 1.2. Here, it is interesting to note that f’(h) is reduced for l > 0 and l < 0 with increasing values

Fig. 5

Effect of magnetic field strength on nanofluid velocity.

Fig. 6

Effect of bioconvection Peclet number on nanofluid velocity.

Fig. 7 Effect of bioconvection Rayleigh number on nanofluid velocity.

of Vs. This has been due to the fact that stretching forces (b) are of higher weight than that of free stream forces (a) which in turn allow the fluid to slip at the surface. Figs. 4e6 demonstrate that nanofluid velocity is raised with increasing values of A, M and Peb for buoyancy assisting (l > 0) flows whereas f’(h) decreases for opposing flow (l < 0). The velocity boundary layer thickness

Fig. 8

Effect of magnetic field strength on nanofluid temperature.

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Fig. 9

7

Effect of velocity slip on nanofluid temperature. Fig. 11

becomes thinner with M and Peb. This has been due to the physical reason that in assisting flows the Lorentz force is positive hence an increase in M causes acceleration in velocity profiles whereas in buoyancy opposing flows the velocity profiles are depreciated. The considered work is reduced for thermodynamic fluid flow case when M Z 0. Opposite trends have been noticed in Fig. 7 (for f’(h) profiles) with respect to bioconvection Rayleigh number (Rhb). A prominent depreciation in f’(h) profiles has been due to the natural evidence that bioconvection plumes hamper the motion of nanofluid away from the sheet. Unique interception points have also been detected in f’(h) for each buoyancy assisting and opposing flows for A and Rhb after which velocity boundary layer reverses its nature. Figs. 8e10 are plotted to explore the dominance of magnetic field (M ), velocity slip Vs and temperature jump (Ts) on the thermal boundary layer profiles (q(h)) for altered values of unsteadiness parameter (A). It is observed from these figures that (q(h)) is reduced with M, Vs and Ts. Temperature profiles are prominently curtailed with the increasing values of unsteadiness parameter (A). The role of velocity slip (Vs) in diminishing the profiles of q(h) is obvious because with higher magnitude of velocity slip,

Fig. 10

Effect of thermal slip on nanofluid temperature.

Effect of unsteadiness on nanofluid concentration.

lesser amount of thermal energy is transmitted from the hot surface. Profiles of nanoparticle concentration (f(h)) for different scales of unsteadiness (A) and thermophoresis parameter (Nt) are plotted in Fig. 11. As seen in Fig. 11, the nanoparticle concentration is dwindled with the rising unsteadiness (A), however profiles are enlarged with an increment in Nt. Since, Nt is defined as fraction of nanoparticle diffusion (thermophoresis effect) to the thermal diffusion in nanoliquid, therefore, an increase in Nt points to an elevation in thermophoresis forces which tends to move the solid particles from hot region to cold one, which in turn amplify f(h). Fig. 12 elaborates the influence of bioconvection Peclet number (Peb) and unsteadiness (A) on the density of mobile microbes (x(h)). From this figure, it is viewed that dimensionless density of microbes are reduced with an increase in Peb and A. Physically, Peb is the measure of relative strength of directional and random swimming of motile microbes. So, higher value of Peb exemplify greater directional movement of microbes, and this results is diminished x(h) profiles.

Fig. 12 Effect of bioconvection Peclet number on microorganisms concentration.

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Fig. 13

Variation in local skin friction coefficient with Vs and Ts. Fig. 15

Fig. 13 depicts the variations of local skin friction coefficient (cfx) with velocity slip (Vs) and thermal jump (Ts). It is observed that when Ts Z 0 no variation can be seen in cfx with increasing values of Vs, but for a fixed Ts > 0, a nominal decrease is observed in the values of cfx with Vs. On the other hand, a significant decrease in cfx has been found for higher values of Ts which implies that Ts can be utilized for reducing the skin friction coefficient. The effects of mixed convection parameter (l) and unsteadiness (A) on local skin friction coefficient (cfx) are elucidated through Fig. 14. It is inferred that skin friction increases with A and l, but the increase is more significant for l. Since buoyancy assisting flows have flow supportive pressure gradient, therefore, flow gets accelerated. So it is self-evident that skin friction is reduced. That is why the magnitude of skin friction is lower for l > 0 in comparison to l < 0. Fig. 15 explains the influence of magnetic field strength (M ) and bioconvection Rayleigh number (Rhb) on cfx. It is recognized that increasing values of M has an increasing effect on cfx, but cfx reduces with Rhb. The influences of bioconvection Peclet number (Peb) and bioconvection Schmidt number (Scb) on cfx are confirmed through Fig. 16. From this figure, it is noticed that local skin friction is

Fig. 14

Variation in local skin friction coefficient with l and A.

Variation in local skin friction coefficient with M and Rhb.

intensified with an augmentation in Peb and Scb. It is also detected that cfx has lesser magnitude in the absence of bioconvection parameters. Supremacy of unsteadiness (A), thermal jump (Ts), mixed convection parameter (l) and velocity slip (Vs) on local Nusselt number Nux can be perceived through Figs. 17 and 18. Fig. 17 indicates that the heat transfer rates are notably reduced with increasing thermal slip parameter (Ts) and effect of unsteadiness (A) on heat transfer rates is very small at higher values of Ts, however at lower values of Ts, an apparent augmentation in Nux is observed with increasing A. Further, it is witnessed from Fig. 18 that Nux reduced with an increase in Vs and l. For buoyancy opposing flow (l < 0), the local Nusselt number is maximum for no-slip conditions (Vs Z 0). But as velocity slip is introduced, Nux starts to reduce with the rising values of Vs. It is very striking to mention that Nux is reduced predominantly in the neighbourhood of l Z 0 with the raised values of Vs and forced convection (l Z 0) control heat transfer rates at the surface of the sheet. The variability in local Sherwood number (Shx) with controlled parameters can be identified through Figs. 19e21. Local Sherwood number is declined with augmenting values of velocity slip (Vs), mixed convection

Fig. 16

Variation in local skin friction coefficient with Peb and Scb.

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Fig. 17

Fig. 18

Variation in local Nusselt number with A and Ts. Fig. 20

Variation in local Sherwood number with Scb and Nt.

Fig. 21

Variation in local Sherwood number with Peb and A.

Variation in local Nusselt number with l and Vs.

parameter (l), (Scb) bioconvection Schmidt number and (Peb) bioconvection Peclet number, whereas Shx increases with increasing thermophoresis parameter (Nt) and unsteadiness (A). A sudden increase is observed in Shx in Fig. 19 for forced convection flow i.e. for l Z 0.

Fig. 19

9

Variation in local Sherwood number with l and Vs.

The response of nanofluid and bioconvection parameters on local density number of motile microbes (Nnx) are displayed through Figs. 22e24. The local density number of microbes are reduced with mixed convection parameter (l), velocity slip (Vs), bioconvection Peclet number (Peb) and bioconvection Rayleigh number (Rhb), whereas enhancing

Fig. 22 Variation in local density number of motile microorganisms with Peb and Scb.

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2) An increment in the values of M, Vs, Ts leads to a decrement in dimensionless temperature. 3) An augmentation in bioconvection Peclet number (Peb) suppresses motile micro-organisms concentration. 4) Local skin friction, heat/mass transfer and motile microorganisms transfer rates are enhanced with an increase in A and reduced with an increase in l. 5) Thermal slip Ts lower cfx significantly in comparison to velocity slip. 6) Local Nusselt number Nux is diminished very much for lower values of A if magnitude of Ts is elevated. 7) Local density number due to motile microorganisms can be curtailed largely if Peb is raised and Scb is lowered. Fig. 23 Variation in local density number of motile microorganisms with l and Vs.

Acknowledgements Second author is thankful to Central University of Himachal Q2 Pradesh, Dharamshala, India for providing fellowship to conduct this research.

References

Fig. 24 Variation in local density number of motile microorganisms with Rhb and A.

influence of unsteadiness and (Scb) bioconvection Schmidt number on Nnx can be seen. Here, Nnx is maximum for greater values of Scb but for lower values of Peb.

4. Conclusions In the present paper, a detailed examination of an unsteady bioconvective boundary layer slip flow of hydromagnetic nanofluid due to a stagnation point region has been carried out. Similarity transformations in exponential form have been utilized to transform the governing partial differential equations into self-similar form. A very effective numerical scheme called Keller box algorithm has been handeled to obtain the grid independent numerical solutions. This work will find applications in extrusion of plastic sheets, drawing plastic films, glass blowing, metal spinning, paper production etc. The summary of this work is extracted as follows: 1) Nanofluid temperature, nanoparticle concentration and microbes concentration fall with A, whereas nanofluid velocity enhances with A for a buoyancy aiding flow.

[1] W.A. Khan, O.D. Makinde, Z.H. Khan, MHD boundary layer flow of a nanofluid containing gyrotactic microorganisms past a vertical plate with Navier slip, Int. J. Heat Mass Transf. 74 (2014) 285e291. [2] K. Bhattacharyya, I. Pop, MHD boundary layer flow due to an exponentially stretching sheet, Magnetohydrodynamics 47 (2011) 337e344. [3] G.K. Ramesh, B.J. Gireesha, S.A. Shehzad, F.M. Abbasi, Analysis of heat transfer phenomenon in magnetohydrodynamic casson fluid flow through Cattaneo-Christov heat diffusion theory, Commun. Theor. Phys. 68 (2017) 91e95. [4] G.K. Ramesh, K.G. Kumar, S.A. Shehzad, B.J. Gireesha, Enhancement of radiation on hydromagnetic Casson fluid flow towards a stretched cylinder with suspension of liquid-particles, Can. J. Phys. 96 (2018) 18e24. [5] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, development and applications of non-Newtonian flow, ASME Fluid Eng. Division USA 66 (1999) 99e105. [6] R.U. Haq, N.F.M. Noor, Z.H. Khan, Numerical simulation of water based magnetite nanoparticles between two parallel disks, Adv. Powder Technol. 27 (2016) 1568e1575. [7] F.A. Soomro, R.U. Haq, Z.A. Khan, Q. Zhang, Passive control of nanoparticle due to convective heat transfer of Prandtl fluid model at the stretching surface, Chin. J. Phys. 55 (2017) 1561e1568. [8] M. Usman, R.U. Haq, M. Hamid, W. Wang, Least square study of heat transfer of water based Cu and Ag nanoparticles along a converging/diverging channel, J. Mol. Liquids 249 (2018) 856e867. [9] R.U. Haq, S.N. Kazmi, T. Mekkaoui, Thermal management of water based SWCNTs enclosed in a partially heated trapezoidal cavity via FEM, Int. J. Heat Mass Transf. 112 (2017) 972e982. [10] R.U. Haq, F.A. Soomro, Z. Hammouch, Heat transfer analysis of CuO-water enclosed in a partially heated rhombus with heated square obstacle, Int. J. Heat Mass Transf. 118 (2018) 773e784. [11] K. Bhattacharyya, G.C. Layek, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet, Phys. Res. Int. 2014 (2014), 592536. [12] G.K. Ramesh, G.S. Roopa, S.A. Shehzad, F.M. Abbasi, An electromagneto-hydrodynamic flow Maxwell nanoliquid past a Riga plate: a numerical study, J. Braz. Soc. Mech. Sci. Eng. 39 (2017) 4547e4554.

63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

+

MODEL

Suspension of mixed bio-convection [13] M.H.M. Yasin, A. Ishak, I. Pop, MHD heat and mass transfer flow over a permeable stretching/shrinking sheet with radiation effect, J. Magn. Magn. Mater. 407 (2016) 235e240. [14] F. Selimefendigil, H.F. Oztop, A.J. Chamkha, MHD mixed convection and entropy generation of nanofluid filled lid driven cavity under the influence of inclined magnetic fields imposed to its upper and lower diagonal triangular domains, J. Magn. Magn. Mater. 406 (2016) 266e281. [15] F. Mabood, W.A. Khan, A.M. Ismail, MHD boundary layer flow and heat transfer of nanofluids over a nonlinear stretching sheet: a numerical study, J. Magn. Magn. Mater. 374 (2015) 569e576. [16] A.A. Hakeem, N.V. Ganesh, B. Ganga, Magnetic field effect on second order slip flow of nanofluid over a stretching/shrinking sheet with thermal radiation effect, J. Magn. Magn. Mater. 381 (2015) 243e257. [17] R. Ahmad, M. Mustafa, T. Hayat, A. Alsaedi, Numerical study of MHD nanofluid flow and heat transfer past a bidirectional exponentially stretching sheet, J. Magn. Magn. Mater. 407 (2016) 69e74. [18] S. Mukhopadhyay, I.C. Mandal, Magnetohydrodynamic (MHD) mixed convection slip flow and heat transfer over a vertical porous plate, Eng. Sci. Tech. Int. J. 18 (2015) 98e105. [19] N.F.M. Noor, R.U. Haq, S. Nadeem, N. Hashim, Mixed convection stagnation flow of a micropolar nanofluid along a vertically stretching surface with slip effects, Meccanica 50 (2015) 2007e2022. [20] G.K. Ramesh, T. Hayat, A. Alsaedi, Stagnation point flow of Maxwell fluid towards a permeable surface in the presence of nanoparticles, Alexandria Eng. J. 55 (2016) 857e865. [21] R. Kumar, S. Sood, Effect of quadratic density variation on mixed convection stagnation point heat transfer and MHD fluid flow in porous medium towards a permeable shrinking sheet, J. Porous Media 19 (2016) 1083e1097. [22] R. Kumar, S. Sood, Interaction of magnetic field and nonlinear convection in the stagnation point flow over a shrinking sheet, J. Eng. 2016 (2016) 8. Article ID 6752520. [23] T.J. Pedley, N.A. Hill, J.O. Kesseler, The growth of bioconvection patterns in a uniform suspension of gyrotactic micro-organisms, J. Fluid Dynamics 195 (1988) 223e237. [24] N.A. Hill, T.J. Pedley, Bioconvection, Fluid Dynamics Research 37 (2005) 1e20. [25] A.V. Kuznetsov, The onset of nanofluid bioconvection in a suspension containing both nanoparticles and gyrotactic microorganisms, Int. Commun. Heat Mass Transf. 37 (2010) 1421e1425. [26] A.V. Kuznetsov, Nanofluid bioconvection in water-based suspensions containing nanoparticles and oxytactic microorganisms: oscillatory instability, Nanoscale Res. Lett. 6 (2011) 100. [27] A.V. Kuznetsov, Non-oscillatory and oscillatory nanofluid bio-thermal convection in a horizontal layer of finite depth, Eur. J. Mech. B Fluid 30 (2011) 156e165.

11 [28] A. Aziz, W.A. Khan, I. Pop, Free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms, Int. J. Therm. Sci. 56 (2012) 48e57. [29] S. Shaw, P. Sibanda, A. Sutradhar, P.V.S.N. Murthy, Magnetohydrodynamics and Soret effects on bioconvection in a porous medium saturated with a nanofluid containing gyrotactic microorganisms, J. Heat Transf. 136 (2014), 052601. [30] A.V. Kuznetsov, Nanofluid bio-thermal convection: simultaneous effects of gyrotactic and oxytactic micro-organisms, Fluid Dyn. Res. 43 (2011), 055505. [31] N.S. Akbar, Z.H. Khan, Magnetic field analysis in a suspension of gyrotactic microorganisms and nanoparticles over a stretching surface, J. Magn. Magn. Mater. 410 (2016) 72e80. [32] S. Mukhopadhyay, K. Bhattacharyya, G.C. Layek, Mass transfer over an exponentially stretching porous sheet embedded in a stratified medium, Chem. Eng. Commun. 201 (2014) 272e286. [33] K. Bhattacharyya, Boundary layer flow and heat transfer over an exponentially shrinking sheet, Chin. Phys. Lett. 28 (7) (2011), 074701. [34] A. Behseresht, A. Noghrehabadi, M. Ghalambaz, Natural-convection heat and mass transfer from a vertical cone in porous media filled with nanofluids using the practical ranges of nanofluids thermo-physical properties, Chem. Eng. Res. Des. 92 (2014) 447e452. [35] W.N. Mutuku, O.D. Makinde, Hydromagnetic bioconvection of nanofluid over a permeable vertical plate due to gyrotactic microorganisms, Comput. Fluids 95 (2014) 88e97. [36] S.E. Ahmed, A.M. Aly, M.A. Mansour, MHD mixed bioconvection stagnation point flow of nanofluids towards a stretching surface, J. Nanofluids 4 (2015) 528e535. [37] M.J. Uddin, M.N. Kabir, O.A. Beg, Computational investigation of Stefan blowing and multiple-slip effects on buoyancy-driven bioconvection nanofluid flow with microorganisms, Int. J. Heat Mass Transf. 95 (2016) 116e130. [38] O.A. Beg, V.R. Prasad, B. Vasu, Numerical study of mixed bioconvection in porous media saturated with nanofluid containing oxytactic microorganisms, J. Mech. Med. Biol. 13 (2013), 1350067. [39] J. Buongiorno, Convective transport in nanofluids, J. Heat Transf. 128 (2006) 240e250. [40] H.B. Keller, Numerical methods in boundary-layer theory, Annu. Rev. Fluid Mech. 10 (1978) 417e433. [41] K. Vajravelu, K.V. Prasad, Keller-box Method and its Applications, De Gruyter, 2014. [42] R. Kumar, S. Sood, S.A. Shehzad, M. Sheikhoeslami, Radiative heat transfer study for flow of non-Newtonian nanofluid past a Riga plate with variable thickness, J. Mol. Liq. 248 (2017) 143e152.

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