Framing the effects of solar radiation on magneto-hydrodynamics bioconvection nanofluid flow in presence of gyrotactic microorganisms

    Framing the Effects of Solar Radiation on Magneto-hydrodynamics Bioconvection Nanofluid Flow in Presence of Gyrotactic Microorganisms Nilankush Acharya, Kalidas Das, Prabir Kumar Kundu PII: DOI: Reference:

S0167-7322(16)31318-6 doi: 10.1016/j.molliq.2016.07.023 MOLLIQ 6037

To appear in:

Journal of Molecular Liquids

Received date: Accepted date:

25 May 2016 6 July 2016

Please cite this article as: Nilankush Acharya, Kalidas Das, Prabir Kumar Kundu, Framing the Effects of Solar Radiation on Magneto-hydrodynamics Bioconvection Nanofluid Flow in Presence of Gyrotactic Microorganisms, Journal of Molecular Liquids (2016), doi: 10.1016/j.molliq.2016.07.023

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Nilankush Acharya

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Framing the Effects of Solar Radiation on Magnetohydrodynamics Bioconvection Nanofluid Flow in Presence of Gyrotactic Microorganisms

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Dept of Mathematics, M.A.Chandra Mohan High School ,Malda, PIN-732121, India. Email: [email protected], Mob no. +919474469850

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Kalidas Das

Dept. of Mathematics, A.B.N.Seal College, Cooch Behar, PIN-736101, West Bengal, India,

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Email: [email protected], Mob no. +919748603199

Prabir Kumar Kundu Dept. of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India,

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Email: [email protected], Mob no. +91943315434

Abstract: The present article investigates the effects of solar radiation on hydromagnetic bioconvection of water-based nanofluid flow in presence of gyrotactic microorganisms past a permeable surface. The flow analysis has been considered under the effect of surface slip

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condition. The resulting partial differential equations take the form of ordinary differential equation by employing suitable transformation and then solved numerically using RK-4 method together with shooting technique. A parametric study on the flow characteristics are presented through tables and graphs coupled with required discussion and physical implication. A comparative study has been taken into account between previously published literature and present work to show the efficiency of our investigation and it shows excellent accord. Keywords: Bioconvection; Gyrotactic microorganisms; Nanofluid; Solar radiation; Magnetic field; Slip condition; permeable surface. 2010 Mathematics Subject Classification: 76W05

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ACCEPTED MANUSCRIPT 1. Introduction : Far from being newly revealed phenomenon, Bioconvection is the most ubiquitous event that

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includes the macroscopic movement of upwardly swimming motile microorganisms such as

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Bacillus subtilis, Chlamydomonas nivalis, Oxyatic bacteria etc. Bioconvection occurs because they are little denser than water and can swim upwardly which results too dense

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unstable situation due to the gathering of microorganisms at the upper surface of water. But the basis of this upswimming observable fact may be different for some species. Because they respond to certain factors such as gravity, light, chemical reaction, oxygen gradient

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which tends them to swim in a particular direction. These factors are called “ taxes” like Gravitaxis, Gyrotaxis, Phototaxis etc etc. Gravitaxis refers to the swimming opposite to

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gravity and Gyrotaxis is the swimming determined by the equilibrium of torques due to viscous forces from shear flows and gravity. Phototaxis is due to the movement toward or away from light. A tremendous reflection on bioconvection patterns can be found in the

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article developed by Childrees et al.[1] and Kesseler et al.[2]. Childrees demonstrates the

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gravitactic phenomenon by invoking Navier-Stokes equation for the fluid and diffusion convection equation for microorganisms concentration. Earlier in 1974,

a report on

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bioconvection patterns in terms of Rayleigh-Taylor instability was addressed by Plesset and Winet [3] . Gradually day by day considerable efforts have been devoted by the researchers to avail it more easier and reliable [4-8].

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Recently nanofluid draws a remarkable attention due to its highly efficient heat transfer mechanism. It was coined by Choi [9]. Nanofluid refers to the suspension of nanoparticle (150 nm) into the base fluid. There are some basic principle which defers us to assume nanoparticle and motile microorganisms to be same in aqueous suspension. Microorganisms move due to taxes but nanoparticles move due to Brownian motion, Thermophoresis introduced by the flow of the base fluid. We know that science is blessed by so many advanced technologies. So it will be a great tribute to science if we coalesce nanofluid with bioconvection which occurs only when nanoparticle concentration is small. There is some major use of nanofluid in bioconvection in the form of bio-microsystem [10], biosensors [11], microdevices to asscess nanoparticle toxicity [12]. Kuznetsov and Avramenko [13] was the first to initiate bioconvection between small particles and gyrotactic microorganisms which was extended by several investigations [14-18]. Hydromagnetic bioconvection of nanofluid

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ACCEPTED MANUSCRIPT flow over a permeable vertical plate due to gyrotactic microorganism can be traced out in [19]. Khan and Makinde [20] authenticate the bioconvection due to gyrotactic microorganism over a convectively heated stretching sheet. Kuznetsov and Nield [21] discussed the Cheng-

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Minkowycz problem for natural convective boundary layer nanofluid flow through a porous

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medium. Recently mixed convection flow of both nanoparticles and gyrotactic microorganisms was studied by Xu and Pop [22]. Kuznetsov [23] analysed thermo-

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bioconvection in a permeable surface in presence of oxytactic microorganisms. The impact of soret effects on bioconvection of nanofluid including gyrotactic microorganisms was investigated by Shaw et al.[24]. Nanofluid bioconvection in presence of gyrotactic

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microorganisms and chemical reaction in a porous medium was addressed by Das et al.[25]. Very recently excellent modification of bioconvection MHD nanofluid flow past an upper

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surface of paraboloid of revolution can be found in [26-27].

Solar radiation is the most non-conventional, non-polluting source of energy. It is the energy

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in the form of radiant light and heat from sun. It is an important source of renewable energy.

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Modern science and advanced technology is much indebted to solar radiation due to its huge application in the form of solar thermal electricity, solar photovoltaic cells, solar heating,

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artificial photosynthesis etc. Nanoparticles in nanofluid absorb solar radiation significantly because of small size as compared to the wavelength of de Broglie wave. Therefore nanoparticles also offer the promising quality of enhancing the radiative properties of liquids, leading to an increase in the efficiency of direct absorption of solar collectors. Hunt [28] was

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the first to introduce the concept of absorbing nanoparticles using nanoparticles. Investigation on this ground can be found in [29-30]. Flow analysis and heat transfer mechanism in the above mentioned investigations are confined only with no-slip boundary conditions. No-slip assumption has been carried out and utilized in modelling several flow problems. But Navier [31] was the first who pointed out that the velocity along the tangential direction of the fluid is proportional to tangential stress and the constant of proportionality is known as velocity slip parameter. Investigation regarding velocity slip parameter can be found in [32-34]. Recently the effects of solar radiation on cu-water nanofluid over a stretching sheet with surface slip can be found in Das et al.[35]. Anbuchezhian et al.[36] reported the MHD effects on nanofluid due to solar radiation. Non-aligned MHD stagnation point flow of variable viscosity nanofluids past a stretching sheet with radiative heat was addressed by [37]. In recent times MHD flow of variable viscosity nanofluid over a radially stretching convective surface with radiative heat has been proposed by [38]. [3]

ACCEPTED MANUSCRIPT To the author’s knowledge no studies considering the fact developed in this article have far been communicated. Keeping this in mind and motivated by the above investigations the present article is devoted to deal with a novel type of MHD nanofluid flow containing both

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nanoparticles and gyrotactic microorganisms in presence of solar radiation. The flow analysis

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has been performed on the basis of transforming governing partial differential equations in the form of ordinary differential equations and solving numerically using RK-4 method

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together with shooting criteria.

2. Mathematical formulation :

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2.1. Governing equations:

An electrically conducting steady laminar nanofluid flow in two dimensional frame

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containing nanoparticles with water as a base fluid and gyrotactic microorganisms in presence of solar radiation over a vertical permeable plate has been considered. The co-

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ordinate axes are chosen assuming x-axis along the plate and y-axis normal to the plate as

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shown in Figure 1. We have assumed magnetic Reynolds number to be small enough in order to neglect the induced magnetic field. Here we can neglect the viscous dissipation effect

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because viscous dissipation effect on heat transfer mechanism is considerable for high velocity profile i.e. for the case of flow converting laminar to turbulence or for highly viscous flow or flow with moderate velocity with small wall to fluid temperature differences. Moreover neglecting rapidly alternating magnetic field together with induced magnetic field

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we have avoided the possibility of joule heating due to magnetic field. It is assumed that the nanoparticle suspension is stable and do not agglomerate in the fluid. It is also assumed that the swimming direction of microorganisms and their swimming velocity are not affected due to the presence of nanoparticle. This is a justified assumption if the concentration of the nanoparticles is less than 1% generally, otherwise a large suspension viscosity will appear which will suppress the bioconvection. In addition we have assumed that the porous medium is in thermal equilibrium with the fluid and solar radiation is inclined normal to the plate.

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Figure 1: Physical model of the problem

Now under the above mentioned assumption together with the Oberbeck-Boussinesq

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approximation, the governing equations unfolding the conservation of mass, momentum, thermal energy, nanoparticle volume fraction and microorganisms can be represented, based on the model proposed by Khan et al.[18] in the following form

u v  0 x y

p  2u   2   B02u   f x y

(1)

 u u   u  v   u y  k  x

 g 1     f   T  T   (  p   f  )(C C )  Cm p =0 y





m

  f   

(2)

(3)

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ACCEPTED MANUSCRIPT 2   T T  2T 1 qrad  C T DT  T    u v   2    DB     x y y y y T  y        c p  f y

C C  2C D  2T v  DB 2  T x y y T y 2

u

Cm C bWc    C    2Cm v m  C  D   m m  x y  Cw  C   y  y   y 2

(5)

(6)

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u

(4)

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where u, v are the velocity components along the x and y-axis respectively, T is temperature, C is the nanoparticle concentration, Cm is the concentration of microorganisms,  is the

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porosity of the porous medium, k is the permeability of the porous medium,  is the viscosity of the nanofluid and microorganisms,  is the electrical conductivity, β is the volumetric thermal expansion coefficient of base fluid,   is the average volume of a microorganism, 

 c  p

f

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is the thermal diffusivity of the nanofluid,  is the thermal conductivity,     c p  s

is the ratio of the effective heat capacity between nanoparticle material and the base fluid,

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 f is the density of base fluid,  p is the nanoparticle density, DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, Dm is the diffusivity of the microorganisms, Wc is the maximum cell swimming speed and the subscript  denotes the

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 is the radiation heat transfer per unit area. Using corresponding values at far field. qrad    Rosseland approximation we have qrad

4 1 T 4 . where  1 is the Stefan-Boltzmann 3k  y

constant and k  is the mean absorption coefficient. The boundary conditions for the problem under the study are given by:

u  , v  vw , C  Cw , T  Tw , Cm  Cmw at y  0  y  u  0, v  0, C  C , T  T , Cm  C m  as y    uN

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(7)

ACCEPTED MANUSCRIPT where N is the Navier slip coefficient, vw is the velocity of suction/injection at the wall, Tw is the constant wall temperature, Cw and Cmw are the nanoparticle volume fraction and the

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density of motile microorganisms at the wall respectively.

 1     p   f   m   f    u u  2u  B 2  v  f 2  0 u  f u     f   g T  T     g(C C )  Cm g     x y y f k f  f   f   

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u

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Now by eliminating pressure terms from (2) and (3) we have

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(8)

Now we introduce the following dimensionless functions f , ,  and  in order to convert the governing partial differential equations into ordinary differential equations with regard to

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similarity variables  as follows:

T  T C C  y 1/4 Rax ,  = Ra1/4 ,  ( )= , x f ( ),  ( )= x Tw  T Cw  C   C  Cm    ( )= m  Cmw  Cm 

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

where  (x, y) is the stream function defined by u 

1     g T f  f

  ,v   and satisfies Eq. (1), y x

x3 is the local Rayleigh number.

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Rax 

(9)

Now, substituting (9) into the Equations. (4)-(6) and (8), we get the nonlinear ordinary differential equations as follows: 1 f ''' .Pr 1  3 ff '' 2 f '2   Mf '   Br  Rb  Da 1 f '  0 4

 ''

(10)

3 4  3 f  ' Nb ' ' Nt '2  N  CT       0  4 3 

(11)

3 4

(12)

 '' Le. f  '

Nt . ''  0 Nb

3 4

 '' Lb. f  ' Pe.  '  '  ''  b      0

(13)

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ACCEPTED MANUSCRIPT And the boundary conditions in (7) transforms as

f (0)  f w , f '(0)   f ''(0),  (0)=1,  (0)=1,  (0)=1,   f '()  0,  ()  0,  ()  0,  ()  0 

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is the bioconvection Lewis

f bWc  is the Prandtl number, Le  is the traditional Lewis number, Pe   DB Dn



f

  f   Cw

 f 1    Tw

is the buoyancy ratio parameter,

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is the bioconvection Peclet number, Br 

Cm  nw  is the bioconvection constant, Rb  is the bioconvection  f 1    Tw Cmw

Rayleigh number, N 

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b 

Dn

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number, Pr 



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where prime denotes differentiation with respect to  , Lb 

(14)

4 1T3  DB Cw is the solar radiation parameter , Nb  is the    cp  k

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f

 B02 x 2  DT Tw is the thermophoresis parameter, M   f Ra1/2  T x

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Brownian motion parameter, Nt 

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is the modified magnetic parameter, CT 

xvw NRa1/4 x is the velocity slip parameter and f w   is the x  Ra1/4 x

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the inverse Darcy number,  

 x2 T is the temperature ratio, Da 1  is Tw  T kRa1/2 x

suction / injection parameter. It should be noted that f w  0 corresponds to an impermeable surface, f w  0 to suction and f w  0 to injection of the fluid through a permeable surface.

2.2. Physical quantities: Now the physical quantities are skin friction coefficient, Nusselt number, Sherwood number, density number of motile microorganisms which can be defined respectively as follows:

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ACCEPTED MANUSCRIPT      4     T  4 1  T  x  Nu x   k      k Tw  T    y  y 0 3k   y  y 0     C  x  Shx       Cw  C   y  y 0    Cm  x Nnx       Cmw  Cm   y  y 0   u 

(15)

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f    y  y 0 Cf   fU 2

In terms of dimensionless variables reduced skin friction coefficient, reduced Nusselt

respectively as follows:

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C fr   Rax 1/4 Re x Pr  .C f  f   0 

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number, reduced Sherwood number, reduced density number of motile microorganism are

(16)

(17)

Shr  Rax 1/4 Shx     0 

(18)

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3  4N Nur  Rax 1/4 Nux   1  CT    0       0   3  

Nnr  Rax 1/4 Nnx      0 

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U .x

(19)

where Re x 

f

is the local Reynolds number.

3. Numerical experiment : 3.1: Numerical method: The required ordinary differential equations as we represent in (10)-(13) are highly nonlinear in nature, hence they cannot be solved analytically. So these equations together with boundary conditions in (14) have been solved by using Runge-Kutta-Fehlberg method in conjunction with shooting method. The computations were done by using a computer algebra software MAPLE-17. The software uses shooting method with Runge-Kutta-Fehlberg

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ACCEPTED MANUSCRIPT method. The boundary conditions at    are replaced by those at    . The inner iteration is done with the convergence criterion of 106 in all cases.

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3.2: Testing of the code:

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To test the efficiency of our present work we have computed the values of  (0) for

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various values of Prandlt number assuming the absence of velocity slip parameter, bioconvection Rayleigh number, porosity parameter, magnetic field parameter, solar radiation parameter, bioconvection Lewis number. Also we have taken the values of Nt, Nb, Br, Le as

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Nt  Nb  Br  105 , Le  10 and we have tabulated those values of  (0) in Table-1. We have checked the values of  (0) with Khan et al.[18], Das et.al.[23] and we see that the

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values are in very good agreement which fortifies the justification of our current work.

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Table-1 Comparison of  (0) for various values of Pr Khan et al. [18]

Das et al. [23]

Present investigation

1.0

0.40135

0.401452

0.40145361

0.46903

0.469315

0.46931620

0.49260

0.492529

0.49252822

0.49878

0.498650

0.49865112

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100.0

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10.0

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Pr

1000.0

4. Results and discussions : This section includes the parametric study of the flow region. We have represented the whole discussion through tables and graphs in order to inspect the effects of various parameters on nanofluid temperature, nanoparticle concentration, nanofluid velocity and microorganism concentration. We have assigned the values of the parameters in the simulation as Pr = 6.2, M = 1.0, Br = 0.1, Rb = 0.3, Nb = 0.2, Nt = 0.5, Pe = 1.0, Le = 2.0, Lb = 0.3, C = 0.1, σ = 0.5, fw = 0.5, N= 0.0 or 1.3, Da-1=0.5 and   0.5 unless otherwise specified.

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ACCEPTED MANUSCRIPT 4.1. Effect of Brownian motion parameter Nb : Figure 2 demonstrates the impact of Brownian motion parameter Nb on nanofluid temperature in presence and absence of solar radiation. We can easily seen from the figure

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that the nanofluid temperature increases as Nb increases both in the case of presence and

increases. This phenomenon can be

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and decreases asymptotically as the co-ordinate 

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absence of solar radiation. Temperature possesses high value near the boundary layer region

elucidates as, the on growing values of Nb aid to increase the velocity of the Brownian motion of nanoparticle and water molecules. Hence the kinetic energy both in molecular and

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nanoparticle level enhances which will convert into an increase in nanofluid temperature. From the theory of physics we are familiar with the fact that

1 2 3 mv  K B .T where K B is 2 2

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the Boltzmann’s constant, T is the absolute temperature, v is the velocity. It shows a everlasting connection between kinetic energy and temperature which justifies our explanation. The effect is prominent within the region 0    3.5 (not precisely determined)

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in both the cases. In addition the temperature of the nanofluid is less in absence of solar

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radiation as compared to the temperature in presence of solar radiation. Because the radiative solar energy leads to increase the Brownian motion more effective in presence of solar

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radiation than in absence of solar radiation. Figure 3 indicates that an increase in Nb tends to decrease the concentration of nanoparticle for both the cases. Since the presence of solar radiation has a positive impact on faster

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Brownian motion, hence we see that the concentration profile assumes high value in absence of solar radiation. Moreover the effect is significant for 1.5    4.5 (not precisely determined) in both cases i.e when N = 0.0 and N = 1.3.

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Figure 2 : Effect of Nb on temperature of nanofluid

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ACCEPTED MANUSCRIPT Figure 3 : Effect of Nb on nanoparticle concentration

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4.2. Effect of thermophoresis parameter Nt:

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In Figure 4 the effect of thermophoresis parameter Nt on nanofluid temerature profile has

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been portrayed. It shows that the influence of Nt increases the temperature of nanofluid significantly both in presence and absence of solar radiation. Subsequently the thickness of the thermal boundary layer increases. The phenomenon describes the fact that the thermophoretic force due to temperature gradient creates a fast flow away from the surface.

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Consequently more heated fluid is flowed away from the surface. So the temperature increases. An important observation can be made out that the temperature in absence of solar

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radiation is on the lower side as compared to the presence of solar radiation. Variation on nanoparticle concentration  for various values of Nt is shown in Figure 5. It

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reveals that in presence of solar radiation the curve is decreasing within the region

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0    0.5 (not precisely determined). After that the approach is strictly increasing for 0.5    4.0 (not precisely determined) and here it attains its maximum value at   4.0 . For

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4.0    10.5 the curve again shows a monotonic decreasing behavior. For   10.5 the

profile approaches asymptotically to zero as  increases. But in absence of solar radiation the curve is strictly increasing for 0    0.5 and attains peak value at   4.0 . After that

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smoothly decreases. The effect is more prominent and clear within the region 0    0.5 than in absence of solar radiation. Briefly nanoparticle concentration increases as Nt increases. This can be confirmed from the formula given by GS Mcnab and A. Meisen [39] in 1973 invoking the thermophoretic constant DT as DT  

kbf bf is . where   0.26 2kbf  k p bf

the proportionality factor and bf , bf are viscosity and density of the base fluid respectively. Since increase in Nt corresponds to the enhancement in DT , hence the formula shows that  will automatically increase.

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Figure 4 : Effect of Nt on nanofluid temperature

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ACCEPTED MANUSCRIPT Figure 5 : Effect of Nt on nanoparticle concentration

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4.3. Effect of solar radiation parameter N:

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Figure 6 indicates that in increase in solar radiation parameter N tends to increase the

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temperature of both nanofluid and regular fluid into the flow field. Since the radiative solar energy enhances the Brownian motion, hence temperature increases as discussed in the section 4.1. in addition the temperature of the regular fluid is less as compared to nanofluid

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because of the fact that the presence of nanoparticles add extra impact on Brownian motion. It is observed from Figure 7 that the influence of N causes to decrease the concentration of

After that the effect of N is negligible.

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the nanofluid and regular fluid. The effect is prominent for both the fluid inside the region.

Figure 8 conveys that the influence of N enhances the nanofluid velocity for both fluids

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because of the same Brownian motion reason as discussed in section 4.1. For 0    0.5 (not

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precisely determined) the curve is increasing and attains maximum value at   0.7 (not precisely determined). After that the curve decreases smoothly within the region and

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approaches asymptotically to zero as increases

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Figure 6 : Effect of N on nanofluid temperature

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Figure 7 : Effect of N on nanoparticle concentration

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Figure 8 : Effect of N nanofluid velocity

4.4. Effect of traditional Lewis number Le: The influence of traditional Lewis number Le on nanofluid temperature is presented in Figure 9 which conveys that nanofluid temperature decreases as Le increases in both the cases when N = 0.0 and N = 1.3. We know that Le 

 DB

, hence increase in Le will

demand decrease in DB . Again from Einstein fluctuation dissipation formula we know that DB 

K B .T where K B is the Boltzmann’s constant, K drag is the drag coefficient and K drag

T is the absolute temperature. Hence decreasing the value of Le will generally decrease

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ACCEPTED MANUSCRIPT DB which will allow to decrease the temperature of the nanofluid. Also the effect on nanofluid in absence of solar radiation is less as compared to presence of solar radiation as discussed in section 4.1.

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Figure 10 exhibits that nanoparticle concentration decreases with the influence of Le. With the more analytic view we can say that the curve shows strictly increasing nature in

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the region 0    1.6 (not precisely determined) and at   1.6 it assumes peak value. After that the curve smoothly decreases and asymptotically tends to zero. This fact is true for both N=0.0 and N=1.3.

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Effect of Le on concentration of microorganism  is portrayed in Figure 11. The influence of Le has a propensity to decrease the concentration of microorganism in both

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cases i.e presence and absence of solar radiation.

Figure 9 : Effect of Le on nanofluid temperature

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Figure 10 : Effect of Le on nanoparticle concentration

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ACCEPTED MANUSCRIPT Figure 11 : Effect of Le on microorganism concentration

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4.5. Effect of bioconvection Peclet number Pe and bioconvection constant  b :

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Figure 12 exhibits that the impact of bioconvection Peclet number Pe is to decrease the

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concentration of microorganism both in presence and absence of solar radiation. The concentration profile is high when N=0.0 as compared to when N=1.3. The influence of

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bioconvection constant  b is identically coincides with that of Pe as shown in Figure 13.

Figure 12 : Effect of Pe on microorganism concentration

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Figure 13 : Effect of  b on microorganism concentration

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4.6. Effect of bioconvection Rayleigh number Rb and buoyancy parameter Br: Figure 14 demonstrates the effect of bioconvection Rayleigh number Rb on nanofluid velocity in presence and absence of solar radiation. It is observed from the figure that the nanofluid velocity reduces as Rb increases within the region 0    2.6 (not precisely determined) in both the cases. But after that i.e for   2.6 the nanofluid velocity increases. The physics behind the phenomenon is that enhancement of Rb corresponds to increase the average volume of microorganism i.e   which creates a destabilization situation for nanofluid flow hence fluid velocity decreases. Decrease in fluid velocity causes to increase velocity gradient at there. Since the mass flow rate is kept conservative hence decrease in fluid velocity will be compensated by the increasing fluid velocity so that mass flow conservation will not be violated. That’s why at   2.6 we found a point of separation and back flow occurs.

[21]

ACCEPTED MANUSCRIPT Figure 15 conveys that the fluid velocity decreases with the increasing of buoyancy ratio parameter Br. It is worth mentioning that for 0    1.7 (not precisely determined) the curve is strictly increasing and attains peak value at   1.7 . After that it starts to decrease

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smoothly. The effect in absence of solar radiation is same but is on lower side as

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compared to presence of solar radiation.

Figure 14 : Effect of Rb on nanofluid velocity

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Figure 15 : Effect of Br on nanofluid velocity

4.7: Computational and graphical analysis of physical quantities:

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From Table-2 it can be verified that the reduced Nusselt number Nur decreases both in presence and absence of solar radiation as Nb increases. The rate in reduction has been recorder 68.4% in absence of solar radiation and 37.92% for presence of solar radiation. From Table-4 we see that Nb is able to amplify the reduced Sherwood number Shr in both presence and absence of solar radiation. The conclusion from Table-4 can also be verified from Figure 16. It is evident from the figure that comparatively the reduced Sherwood number is slightly higher for the presence of solar radiation. Computational data of Table-4 shows the comparative enhancement is 24.07%. Table-2 implies that Nur decreases in both the cases as Nt increases. For presence of solar radiation reduction rate is found to be 21.02%, while it is 30.02% for absence of solar

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Table -5 shows that reduced Nusselt number Nur increases when N goes high. Same can be

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confirmed from Figure 17. One general observation from Table-5 and Figure 18 is that

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reduced Sherwood number Shr increases as N increases. When solar radiation parameter jumps from 0.4 to 1.2, then the rate of increment has been discovered 15.78%. Again Table-2 shows that Nur decreases with the influence of Le. Table-4 implies that in

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both cases Shr enhances as Le increases. It is observed from Figure 19 that increasing value of Le gives rise to increase the value of the density number of microorganism Nnr

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significantly. Numerically from Table 3 it is by 24.64% on average. From Figure 20 we can conclude that both Pe and  b enhances the Nnr . Comparatively 17.87%

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increment has been noticed for presence of solar radiation with the impact of  b . Figure 21

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communicates that enhancement of Br and Rb lead to decrease the reduced skin friction coefficient.

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Presence of solar radiation dominates the profile as compared to absence of solar radiation.

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Figure 16: Effect of Nb and Nt on Sherwood number

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Figure 17: Effect of Nb and N on Nusselt number

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Figure 18: Effect of Le and N on Sherwood number

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Figure 19: Effect of Le and  b on density number of motile microorganisms

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Figure 20: Effect of Pe and  b on density number of motile microorganisms

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ACCEPTED MANUSCRIPT Figure 21: Effect of Br, Rb and N on reduced skin friction coefficient

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5. Conclusion :

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In this present article the influence of solar radiation on bioconvection containing both

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gyrotactic microorganisms and water based nanofluid through a permeable surface has been analysed. The governing partial differential equations have been transformed into its dimensionless form by introducing similarity transformation and then solved numerically by

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means of a robust programming MAPLE-17 which includes RK-4 method together with shooting criteria. The consequences of arising pertinent parameters on the entire flow province have been taken into account to illustrate the details of the flow. Some major

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conclusions which we have encountered from the whole study are brought to light as follows:  The Brownian motion parameter, thermophoresis parameter, solar radiation parameter

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enhance the temperature of nanofluid while the reverse effect is true for traditional

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Lewis number.

 The concentration of nanoparticle drops off due to positive impact of Brownian

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motion parameter, traditional Lewis number and solar radiation parameter but opposite outcome has been traced out for thermophoresis parameter.  The reflection on the density of motile microorganisms has been found to decrease

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due to the influence of traditional Lewis number, Peclet number and bioconvection constant.

 In addition solar radiation parameter amplifies the velocity profile of nanofluid while buoyancy ratio parameter reduces it. But the most interesting effect is seen for bioconvection Rayleigh number which has both increasing and decreasing nature simultaneously for velocity distribution within a certain province of the flow field.  The reduced Nusselt number reduces for the increasing values of Nb, Nt, Le but it is a increasing function of N. The reduced Sherwood number increases for on growing values of N, Le, Nb but opposite phenomenon is observed for escalating behaviour of Nt.  The positive impact of Le, Pe,  b enhances the value of density number of motile microorganism significantly.

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ACCEPTED MANUSCRIPT Acknowledgement The authors wish to express their cordial thanks to reviewers for valuable suggestions and comments to improve the presentation of this article.

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References

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1. Childress, S.; Levandowsky, M.; Spigel, E.A.: Pattern information in a suspension of swimming micro-organisms. J. Fluid Mech. 69, 591-613 (1975)

2. Pedley, T.J.; Kesseler, J.O.: Bioconvection. Sci Progr. 76, 105-123 (1992) 3.

Plesset, M.S.; Winet, H.: Bioconvection patterns in swimming microorganisms cultures as

NU

an example of Rayleigh Taylor instability. Nature. 248, 441-443 (1974) 4. Geng, P.; Kuznetsov, A.V.: Effect of small solid particles on the development of

5.

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bioconvection plumes. Int Commun Heat Mass Transf. 31, 629-638 (2004) Kuznetsov A.V.: Thermo-bioconvection in a suspension of oxytactic bacteria. Int Commun Heat Mass Transf. 32, 991-999 (2005)

spatial

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Sokolov, A.; Goldstein, R.E.; Feldchtein, F.I.; Aranson, I.S.: Enhanced mixing and instability in concentrated bacterial suspensions. Phys Rev E . 80, 031903 (2009)

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6.

7. Avramenko, A.A.; Kuznetsov, A.V.: Bio-thermal convection caused by combined effects

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of swimming of oxytactic bacteria and inclined temperature gradient in a shallow fluid layer. Int J Numer Methods Heat Fluid Flow. 20, 157-173 (2010) 8. Alloui, Z.; Nguyen, T.H.; Bilgen, E.: Bioconvection of gravitic microorganisms in a

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vertical cylinder. Int Commun Heat Mass Transf. 32, 739-747 (2005) 9. Choi, S.U.S.: Developments and application of Non-Newtonian flows. ASME press. New York USA (1995)

10. Tsai, T.; Liou, D.; Kuo, L.; Chen, P.: Rapid mixing between ferro-nanofluid and water in a semi-active Y-type micromixer. Sensors Actuators A Phys.153, 267-273 (2009) 11. Li,H.; Liu, S.; Dai, Z.; Bao, J.; Yang, Z.: Applications of nanomaterials in electrochemical enzyme biosensors. Sensors. 9, 8547-8561 (2009) 12. Munir, A.; Wang, J.; Zhou, H S.: Dynamics of capturing process of multiple magnetic nanoparticles in a flow through microfluidic bioseparation system. IET Nanobiotechnol. 3, 55-64 (2009) 13. Kuznetsov, A.V.; Avramenko, A.V.: Effect of small particles on the stability of bioconvection in a suspension of gyrotactic microorganisms in a layer of finite depth. Int Commun Heat Mass Transf. 31, 1-10 (2004) [30]

ACCEPTED MANUSCRIPT 14. Kuznetsov, A.V.; Geng, P.: The interaction of bioconvection caused by gyrotactic microorganisms and settling of small solid particles. Int J Numer Methods Heat Fluid Flow. 15, 328-547 (2005)

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15. Kuznetsov,A.V.: The onset of nanofluid bioconvection in a suspension containing both

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nanoparticles and gyrotactic microorganisms. Int Commun Heat Mass Transf. 37, 14211425 (2010)

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16. Kuznetsov, A.V.: Non-oscillatory and oscillatory nanofluid bio-thermal convection in a horizontal layer of finite depth. Eur J Mech B Fluids. 30 (2), 156-165 (2011) 17. Kuznetsov, A.V.: Nanofluid bioconvection in water-based suspensions containing

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nanoparticles oxytactic microorganisms: oscillatory instability. Nanoscale Research Letters.6, 100 (2011)

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18. Khan, W.A.; Makinde, O.D.; Khan, Z.H.: Boundary layer flow of nanofluid containig gyrotactic microorganisms past a vertical plate with navier slip. Int J Heat and Mass Transf. 74, 285-291 (2014)

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19. Mutuku, W.N.; Makinde, O.D.: Hydromagnetic bioconvection of nanofluid over a

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permeable vertical plate due to gyrotactic microorganism. Computers and Fluids. 95, 8897 (2014).

Makinde,O.D.:

MHD

nanofluid

bioconvection

due

to

gyrotactic

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20. Khan,W.A.;

microorganism over a convectively heat stretching sheet. Int J Thermal Science. 81, 118124 (2014).

21. Kuznetsov,A.V.; Nield, D.A.: The Cheng-Minkowycz problem for natural convective

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boundary layer flow in a porous medium saturated by a nanofluid: a revised model. Int. Jou. Heat Mass Transfer. 65, 682-685 (2013) 22. Xu, H.; Pop, I.: Mixed convection flow of a nanofluid over a stretching surface with uniform free stream in the presence of both nanoparticles and gyrotactic microorganisms. Int J Heat and Mass Transf. 75, 610-623 (2014) 23. Kuznetsov, A.V.: The onset of thermo-bioconvection in a shallow fluid saturated porous layer heated from below in a suspension of oxytactic microorganisms. Eur J Mech B Fluids. 25, 223-233 (2006) 24. Shaw, S.; Sibanda, P.; Sutradhar, A.; Murthy, P.V.S.N.: Magnetohydrodynamics and Soret Effects on Bioconvection in a Porous Medium Saturated With a Nanofluid Containing Gyrotactic Microorganisms. J. Heat Transfer.136(5), 052601 (2014)

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ACCEPTED MANUSCRIPT 25. Das, K.; Duari, P.R.; Kundu, P.K.: Nanofluid bioconvection in presence of gyrotactic micriorganisms and chemical reaction in a porous medium. J. Mechanical Science and Technology. 29(11), 1-9 (2015)

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26. Makinde, O. D.; Animasaun, I. L.: Bioconvection in MHD nanofluid flow with non-linear

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thermal radiation and quartic autocatalysis chemical reaction past an upper surface of paraboloid of revolution. Int J Thermal Science. 109, 159-171 (2016).

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27. Makinde, O. D.; Animasaun, I. L.: Thermophoresis and Brownian motion effects on MHD bioconvection of nanofluid with non-linear thermal radiation and quartic chemical reaction past an upper horizontal surface of a paraboloid of revolution. Journal of Molecular

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Liquids. 221, 733-743 (2016).

28. Hunt, A.J.: Small particle heat exchangers. Report LBL-78421 for the US Department of

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energy. Lawrence Berkeley Laboratory (1978)

29. Mufuoglu, A.; Bilen, E.: Heat transfer in inclined rectangular receivers for concentrated solar radiation. Int. Commun. Heat Mass Transf. 35, 551-556 (2008)

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30. Kandasamy, R.; Muhaimin, I.; Khamis, A.B.; Roslan, R.B.: Unsteady Heimenz flow of

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Cu-nanofluid over a porous wedge in the presence of thermal stratification due to solar energy radiation: Lie group transformation. Int. J. Therm. Sci. 65, 196-205 (2013)

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31. Navier , C.L.M.H.: Memoire sur les lois du mouvement des fluids. Mem.Acad.R. Sci.Inst. Fr. 6, 389-440 (1823)

32. Hina, S.; Hayat, T.; Alsaedi, A.: Slip effects on MHD peristaltic motion with heat and mass transfer. Arab. J. Sci. Eng. 39(2), 593-603 (2014)

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33. Martin, M.J.; Boyd, I.D.: Momentum and heat transfer in a laminar boundary layer with slip flow. J. Thermophys. Heat Transf. 20(4), 710-719 (2006) 34. Martin, M.J.; Boyd, I.D.: Blasius boundary layer solution with slip flow conditions. Presented at the 22nd Rarefield Gas Dynamics Symposium Sydney Australia (2000) 35. Das, K.; Duari, P.R.; Kundu, P.K.: Solar radiation effect on Cu-Water nanofluid flow over a stretching sheet with surface slip and temperature jump. Arab. J. Sci. Eng. 39, 90159023 (2014) 36. Anbuchezhian, N.; Srinivasan, K.; Chandrasekaran, K.; Kandasamy, R.: Magneto hydrodynamic effects on natural convection flow of a nanofluid in the presence of heat source due to solar energy. Meccanica. 48, 307-321 (2013) 37. Khan,W.A.; Makinde,O.D.; Khan, Z.H.: Non-aligned MHD stagnation point flow of variable viscosity nanofluids past a stretching sheet with radiative heat. Int J Heat Mass Trnafer. 96, 525-534 (2016). [32]

ACCEPTED MANUSCRIPT 38. Makinde, O.D.; Mabood, F.; Khan, W.A.; Tshehla, M.S.: MHD flow of a variable viscosity nanofluid over a radially stretching convective surface with radiative heat. Journal of Molecular Liquids. 219, 624-630 (2016).

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39. McNab, GS.; Meisen, A.: Thermophoresis in liquids. Jounal of colloid and interface

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science. 44(2), 339-346 (1973)

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Nt

Le

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Nb

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Table-2 Effects of Nb , Nt , Le on Nur

0.5

5.0

1.2

-

-

3.0

-

5.4

-

0.2

0.5

-

1.5

-

2.5

-

3.5

-

0.474317

0.153656

0.358252

-

0.039901

0.214836

-

0.005205

0.108973

-

0.279628

0.474316

-

0.193143

0.395183

-

0.146735

0.338461

-

0.115517

0.296172

0.5

0.2

0.250347

0.406005

-

2.0

0.240899

0.392021

-

6.0

0.233246

0.387995

-

10.0

0.227084

0.384630

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0.2

N = 1.3

0.279176

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N = 0.0

Nur

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Pe

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b

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Table-3 Effects of  b , Pe , Le on Nnr

0.2

2.0

1.0

-

-

1.5

-

2.0

-

0.5

0.0

-

0.2

-

0.4

-

0.6

-

0.418458

0.374594

0.440153

-

0.389527

0.461900

-

0.404497

0.483701

-

0.312696

0.350508

-

0.407372

0.487446

-

0.504679

0.628386

-

0.604475

0.773036

0.2

0.2

0.478452

0.571899

-

2.0

0.635881

0.716003

-

8.0

1.078461

1.158389

-

12.0

1.366818

1.450639

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0.5

N = 1.3

0.359697

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N = 0.0

Nnr

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Table-4 Effects of Nb , Nt , Le on Shr

0.5

2.0

2.4

-

-

3.6

-

4.8

-

0.2

2.5

-

3.0

-

3.5

-

4.0

-

0.436530

0.536131

-

0.457614

0.544976

-

0.468466

0.550889

-

0.874754

0.299607

-

0.852631

0.214811

-

0.826865

0.113989

-

0.813193

0.107118

0.5

0.4

2.054815

0.083606

-

0.5

2.261187

0.251180

-

0.6

2.476618

0.422561

-

0.7

2.702059

0.597974

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0.518774

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0.379203

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Table-5 Effects of N on Nur , Shr , Nnr N

Nur

Shr

Nnr

0.0

0.252103

0.202735

0.362759

0.4

0.295703

0.425637

0.426372

0.8

0.331194

0.516974

0.449734

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0.569263

0.465246

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1.2

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

The concentration of nanoparticle reduces on behalf of solar radiation parameter.



Density of motile microorganisms lessens for Peclet number.



Reduced Nusselt number increasess for solar radiation parameter.



Reduced Sherwood number enhances as Lewis number increases.

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

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