Journal of King Saud University – Engineering Sciences xxx (xxxx) xxx

Contents lists available at ScienceDirect

Journal of King Saud University – Engineering Sciences

journal homepage: www.sciencedirect.com

Original article

Dusty nanofluid flow with bioconvection past a vertical stretching

surface
Debasish Dey ⇑, Barbie Chutia
Department of Mathematics, Dibrugarh University, Assam 786004, India

a r t i c l e i n f o a b s t r a c t

Article history: The problem of two phase bio-convective nano fluid flow past a vertical stretching flat surface in presence
Received 22 April 2020 of volume fraction of the dust particles has been investigated. Governing partial differential equations of
Accepted 5 November 2020 the problem for both fluid and dust phases are transformed into ordinary differential equations using
Available online xxxx
suitable similarity transformations. The resulting non-linear differential equations are solved numerically
using MATLAB built-in bvp4c solver scheme. The velocity, temperature of fluid motion and concentration
Keywords: of micro-organisms are presented graphically for various flow parameters. Coefficient of local skin fric-
Dusty fluid
tion, local Nusselt number and local density number of the micro-organisms are calculated and presented
Volume fraction
Bio-convection
in tables for various parameters.
Nano-particles Ó 2020 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an
Stretching sheet open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction mophoresis and Brownian motion. Unsteady MHD nanofluid flow

over wavy trapezoidal channel has been investigated by Job et al.
The problems of two phase fluid flow involving dust particles (2016). Umavathi and Mohite (2016) have studied numerical solu-
have gained attentions of various researchers due to its extensive tions of nanofluid flow through porous surface. Gireesha et al.
applications in the fields of engineering, fluidized beds, gas purifi- (2017) have investigated the numerical solutions of MHD two phase
cation, sedimentation process, exhaust nozzle and many more. The nano liquid in presence of Hall effect. Kuznetsov (2010) has identi-
boundary layer equations for laminar dusty fluid flow were first fied the topic of bioconvection involving nanoparticles.
formulated by Saffman (1962). The dusty gas flow through plane Bioconvection is the process in which micro-organisms swim
parallel plates neglecting volume fraction has been studied by upwardly and consequently density gradient is produced. In order
Michael and Miller (1966). Investigations on visco-elastic dusty to attain this process, different types of micro-organisms such as
fluid with volume fraction past a channel have been done by oxytactic, negative gravitaxis and gyrotactic are required. The
Gupta and Gupta (1990). Dey (2016a, 2016b) has analysed the mechanics of bio-convection has many applications like microbial
effects of volume fraction on MHD dusty fluid flow. enhanced oil recovery (MEOR), model oil, sedimentary basins etc.
Nanoparticles can be defined as the particles having size ranges Various bio-convection problems have been solved by (Kuznetsov
between 1 and 100 nm. The study of nanofluid nowadays has a great and Nield, 2011; Kuznetsov and Avramenko, 2004). Siddiqa et al.
impact in research field of fluid dynamics as it has huge importance (2016b) have studied the effect of Rayleigh number on the bio-
in medicine, biomedical, rocket launching etc. Choi (1995) has dis- convective nanofluid flow over a wavy vertical cone. Begum et al.
covered the flow problems of nanofluids. Saleem and Nadeem (2017) have carried out the study of two phase nanofluid flow con-
(2015) have carried out the study of viscous nanofluid flow over a taining dust particles and gyrotactic micro-organisms past a verti-
rotating cone with or without slip effects, viscous dissipation, ther- cal surface. Numerical analysis of nanofluid flow involving
gyrotactic micro-organisms over a vertical cylinder has been stud-
ied by Dey (2018). Rehman et al. (2017) have investigated the
⇑ Corresponding author.
effect of bio-convective Schmidt number on the density of micro-
E-mail addresses: debasish41092@gmail.com, debasishdey1@dibru.ac.in (D.
Dey), chutia.barbie10@gmail.com (B. Chutia). organisms in nanofluid flow. Recently, Nadeem et al. (2019) have
Peer review under responsibility of King Saud University. discussed an unsteady bio-convective nanofluid flow past a
stretching sheet.
The main aim of this paper is to study the volume fraction
effects on two phase nanofluid flow past a vertical stretching sheet
Production and hosting by Elsevier in presence of gyrotactic micro-organisms. To the best of our

https://doi.org/10.1016/j.jksues.2020.11.001
1018-3639/Ó 2020 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article as: D. Dey and B. Chutia, Dusty nanofluid flow with bioconvection past a vertical stretching surface, Journal of King Saud University –
Engineering Sciences, https://doi.org/10.1016/j.jksues.2020.11.001
D. Dey and B. Chutia Journal of King Saud University – Engineering Sciences xxx (xxxx) xxx

Nomenclature


a Constant, Pe ¼ bw mo
Dmo Pecelet number,
b Chemotaxis constants. qmC
Pr ¼ j p Prandtl number,
cp ; cs Specific heat of fluid & dust particles respectively,

Cf x Local Friction factor, Rb ¼ qcbðqð1C
m qÞðN w N 1 Þ
1 ÞðT w T 1 Þ
Bio-convective Rayleigh number,
Cw; C1 Concentration of nanoparticles at the wall and free
stream, Rex Local Reynolds number,


DB Brownian diffusion coefficient, m
Sc ¼ Dmo Schmidt number,
Dmo Coefficient of mass diffusivity, Shx Local Sherwood number,
DT Thermophoretic diffusion coefficient, Tw ; T1 Temperature at the wall and at a large distance respec-
e ¼ 1/
1
Volume fraction parameter, tively,
1
u2
T, Tp Temperature of fluid and dust particles respectively,
Ec ¼ ðTw T01 ÞCp Eckert number, (u; vÞ Velocity components of fluid phase,
ðup ; vp Þ Velocity components of dust phase,
F; g Dimensionless fluid & dust velocities,
Uw Stretching velocity,
g1
Acceleration due to gravity, W mo Maximum cell swimming speed.
Lb ¼ Dmmo
Bio-convective Schmidt number, Greek symbols:


L0 ¼ 1
Fluid-particle interaction parameter for temperature, b Volume expansion coefficient of fluid,
asT
g Similarity variable,

 h; hp Dimensionless fluid & dust temperature,
q
l1 ¼ qp Relative density,

 j Thermal Conductivity,
l2 ¼ ccps Relative specific heat,

g 1 bð1C 1 ÞðT w T 1 Þ
k ¼ Mixed convection parameter,

 aU w
l3 ¼ NwNN 1
1
Micro-organism’s concentration difference parame- M Dynamic viscosity of the fluid,
ter. m Kinematic viscosity of the fluid,
m
Mass of the dust particle,
q; qp Density of the fluid & dust particles resp.,
Mc ¼ as1m Fluid-particle interaction parameter for velocity, qm ; qnp Density of micro-organisms & nanoparticles respec-
tively,
Nw ; N1 Concentration of microorganisms at the wall and free s Shear Stress,
stream sm , sT Momentum & thermal equilibrium time,
Nux Local Nusselt number, / Dimensionless concentration of the fluid,
 
ðqnp qÞðC w C 1 Þ /1 Volume fraction,
Nr ¼ qbð1C 1 ÞðT w T 1 Þ Buoyancy ratio parameter,

 v Dimensionless micro-organisms concentration.
Nt ¼ sDT ðmTTw1T 1 Þ Thermophoresis parameter,
Cp
nð¼ Cs ¼ l12 Þ Ratio of specific heats of fluid and dust particles

 respectively.
Nb ¼ sDB ðCmw C 1 Þ Brownian motion parameter,

knowledge, not enough works are done on fluid flow with simulta- The governing equations of the problem following the refer-
neous effects of micro-organisms and volume fraction of dust par- ences (Siddiqa et al., 2016a; Makinde and Chinyoka, 2010;
ticles. In the methodology, governing boundary layer equations are Begum et al., 2017; Mahdy and Hoshoudy, 2019; Attia et al.,
transformed into ordinary differential equations using similarity 2011; Gireesha et al., 2010) are given below:
transformations. Then, the resulting differential equations are Equation of continuity for fluid phase:
solved using MATLAB built-in Bvp4c solver scheme and results
are presented through graphs and tables.

2. Mathematical formulation

We have considered a steady, incompressible, bio-convective

dusty nanofluid flow with volume fraction of dust particles past
a vertical stretching sheet as shown in Fig. 1. The fluid flow system
is influenced by gyrotactic micro-organisms. Water may be consid-
ered as the base fluid in the present study as gyrotactic types of
micro-organisms survive in water only under natural conditions.
Some of the assumptions taken into considerations are listed
below:

Motions of dust particles and nanoparticles occur due to ther-

mophoresis and Brownian motion respectively.

Gyrotactic micro-organisms are automotive in nature.

The velocity of micro-organisms is not affected by nanofluid.

Dust particles are assumed to be spherical in shape, uniformly
sized and distributed equally in nanofluid. Fig. 1. Flow configuration of the problem.

2
D. Dey and B. Chutia Journal of King Saud University – Engineering Sciences xxx (xxxx) xxx

 
@u @ v @N
þ ¼0 ð1Þ Localmotilemicroorganismsfluxatwallðqn Þ ¼ Dm
@x @y @y y¼0

Equation of momentum for fluid phase:


ð1  u1 Þq u @u
@x
þ v @u
@y
¼ ð1  u1 Þ 3. Numerical solutions
h i
l @@y2u þ qg 1 bð1  C 1 ÞðT  T 1 Þ  g 1 ðqnp  qÞðC  C 1 Þ  g 1 cðqm  qÞðN  N1 Þ
2

We have introduced ‘similarity transformations’ to convert the

q
þ smp ðup  uÞ above governing equations into ordinary differential equations
ð2Þ and these are given as follows:
pffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi
Equation of energy for fluid phase:
g ¼ amy,w ¼ amxFðgÞ,wp ¼ amxgðgÞ,hðgÞ ¼ TTT 1
w T 1
,-
"  2 # hp ðgÞ ¼ T wp T11 ,uðgÞ ¼ CCC
T T 1
,up ðgÞ ¼ C wp C11 ,
C C
vðgÞ ¼ NNN 1
,
@T @T j @2T @C @T DT @T qp c s w C 1 w N 1

u þv ¼ þ s D þ þ vp ðgÞ ¼ Np N1
@x @y qcp @y2
B
@y @y T 1 @y sT qcp Nw N1

where, w be the stream function andu ¼ @w

@y
; v ¼  @w
@x
, then sub-
 ðT p  TÞ ð3Þ stituting the value ofw, we have
pffiffiffiffiffiffi pffiffiffiffiffiffi
Equation of nanoparticles’ concentration for fluid phase: u ¼ axF ; v ¼  amF; up ¼ axg ; v p ¼ amg
0 0

!
@C @C @ 2 C DT @ 2 T
u þv ¼ DB 2 þ ð4Þ By substituting the above transformations, Eqs. (2), (3), (4), (5),
@x @y @y T 1 @y2 (7) and (8) become:
0 00 000 0
Equation of micro-organisms’ concentration for fluid phase: F 2  FF ¼ F þ kðh  Nr/  RbvÞ þ eMcl1 ðF  g 0 Þ ð10Þ
    
@N @N bW mo @ @C @ @C
þv
0 00 002 0 0 0
u þ N þ N PrFh þ h þ EcF þ PrNbh / þ PrNth 2 þ l1 l2 L0 Prðhp  hÞ ð11Þ
@x @y C w  C 1 @y @y @x @x
!
@2N 00 Nt 0 0 0
¼ Dmo ð5Þ / þ h þ ScF/ ¼ 0 ð12Þ
@y2 Nb

0 0 
Equation of continuity for dust phase:
v þ LbF v ¼ Pe v / þ ðv þ l3 Þ/
00 0 00
ð13Þ
@up @ v p
þ ¼0 ð6Þ
@x @y
0 
00 
0
ðg 2  gg Þ ¼ Mc F  g þ /1 F þ kðh  Nr/  RbvÞ
00 0
ð14Þ
Equation of momentum for dust phase:


qp up @u@xp þ v p @u@yp ¼ sqmp ðup  uÞ þ u1 0

h i ghp þ nðL0 hp  L0 hÞ ¼ 0 ð15Þ

l @2 u
@y2
þ qg 1 bð1  C 1 ÞðT  T 1 Þ  g 1 ðqnp  qÞðC  C 1 Þ  g 1 cðqm  qÞðN  N 1 Þ
Using the similarity transformation, the corresponding bound-
ð7Þ ary conditions are as follows:
Equation of energy for dust phase: 0 0
f ¼ 1; F ¼ 0; h ¼ / ¼ v ¼ 1 : g ¼ 0andF ¼ F ¼ g ¼ g ¼ 0; h
0

 
@T p @T p qp cp
qp cs up þ vp ¼ ðT p  TÞ ð8Þ ¼/¼v!0:g!1 ð16Þ
@x @y sT
Boundary conditions are:
9 4. Results and discussion
u ¼ ax; v ¼ 0; T ¼ T w ; C ¼ C w >
>
>
=
N ¼ Nw as y ¼ 0 In this section, we have solved the ordinary differential Eqs.
ð9Þ
u ¼ v ¼ 0; up ¼ v p ¼ 0; T ! T 1 >
>
>
(10)–(15) using MATLAB built-in Bvp4c solver scheme with the
; transformed boundary conditions in Eq. (16). The effects of various
C ! C 1 ; N ! N 1 ; as y ! 1
The quantities such as coefficient of local skin friction, local
Nusselt number, local Schmidt number and local density number
of micro-organisms have importance in different types of engi-
neering problems which are defined as:
sw pffiffiffiffiffiffiffi 00 q x pffiffiffiffiffiffiffi 0
Cf ¼ ¼ Rex C f ) F ð0Þ; Nux ¼ w )  Rex h ð0Þ; Shx
qU w
2 jDT
xqm pffiffiffiffiffiffiffi 0 xqn pffiffiffiffiffiffiffi 0
¼ )  Rex / ð0Þ; Nnx ¼ )  Rex v ð0Þ
D B DC Dm DN
 
@u
wherelocalwallskinfrictionðsw Þ ¼ l ;
@y y¼0
 
@T
Localheatfluxatwallðqw Þ ¼ j ;
@y y¼0
 
@C
Localmassfluxatwallðqm Þ ¼ DB ;
@y y¼0
Fig. 2. Velocity profiles of fluid phase for primary and secondary flows via Nb.

3
D. Dey and B. Chutia Journal of King Saud University – Engineering Sciences xxx (xxxx) xxx

flow parameters on velocity, temperature, coefficient of skin fric- Fig. 2 shows the effect of Brownian motion parameter Nb on the
tion, rate of heat transfer and local density number of motile velocity profile of the fluid phase for primary and secondary flows.
microorganisms are represented. Some of the parameters are con- The parameter Nb represents the Brownian diffusion parameter
sidered as Pr ¼ 1; Mc ¼ 0:6; Ec ¼ 0:1; Sc ¼ 0:5; Pe ¼ 0:1; e ¼ 2; and its positive value signifies that the concentration of fluid at
/1 ¼ 0:1; Lb ¼ 0:1; L0 ¼ 2; l1 ¼ 0:1; l2 ¼ 0:1; l3 ¼ 0:1: the surface is greater than free stream concentration. The primary
stream of fluid experiences a critical point in the vicinity of the sur-
face. Further, fluid motion (primary) experiences acceleration with
the Brownian diffusion in the neighbourhood of the surface and it
changes its nature beyond the critical point. The secondary motion
also experiences acceleration with the Brownian diffusion.
Fig. 3 shows the effect of Brownian motion parameter Nb on
temperature profile of the fluid phase. Brownian diffusion helps
to reduce the temperature in the vicinity of the surface but beyond
the critical point, there is an increment in temperature with Nb
until it reaches the thermal equilibrium.
We have observed that with increasing values ofNb, thermal
boundary layer thickness near the wall diminishes while it thick-
ens in the free stream region. From the Fig. 3, it is seen the dimen-
sionless temperature falls down with the displacement variable.
The dimensionless temperature tends to its thermal equilibrium
(zero value of dimensionless temperature) stage in the region
g > 5.

Fig. 3. Temperature profiles of fluid phase via Nb.

Fig. 6. Concentration of micro-organisms via Rb (Bio-convective Rayleigh number).

Fig. 4. Temperature profile of fluid via Rb (Bio-convective Rayleigh number).

Fig. 5. Concentration of nanoparticles via Nb (Brownian motion parameter). Fig. 7. Concentration of micro-organisms via Nb (Brownian motion parameter).

4
D. Dey and B. Chutia Journal of King Saud University – Engineering Sciences xxx (xxxx) xxx

Fig. 4 shows the effect of bio-convective Rayleigh number ðRbÞ particles helps to accumulate more micro-organism especially in
on temperature profile of the fluid phase. The parameter Rbrepre- the neighbourhood of the surface.
sents the convection due to the movement of microorganisms and From Fig. 8, we have seen the concentration profile of micro-
its positive value reflects that concentration of microorganism at organisms in the fluid for various values of thermophoresis param-
the surface is greater than concentration of microorganism at free eter (Nt). Concentration gradient due to temperature flux is known
stream region. This concentration difference generates the bio- as thermophoresis and it has a significant effect on the concentra-
convection. Bio-convection parameter reduces the temperature of tion of microorganism. The theromophoresis parameter reduces
the fluid motion until it reaches the thermal equilibrium. the concentration of microorganism. Physically, it interprets that
Fig. 5 shows the effect of Brownian motion parameter Nb on the with the temperature change in the medium, concentration fol-
concentration profile of nanoparticles. Concentration or mass accu- lows reduction across the flow.
mulation is higher in the vicinity of the surface and Brownian dif- Fig. 9 shows the effect of volume fraction parameter (e) on the
fusion also helps to accumulate more mass until the fluid reaches concentration profile of micro-organisms. The dimensionless
to critical point. parameter e is directly proportional to volume fraction/ and it sig-
Fig. 6 illustrates the effect of bio-convective Rayleigh number nifies the amount of dust particles in the base fluid. The figure
(RbÞ on the concentration profile of micro-organisms. The bio- depicts that with the increase of volume fraction of dust particles,
convective parameter enhances the concentration of the microor- accumulation of microorganism is declining over the entire flow.
ganism throughout the flow region. Also, the figure reveals the fact Table 1 represents the influence of micro-organisms concentra-
that accumulation of microorganism is more at the surface and tion difference parameter (l3 ) and bio-convective Schmidt number
gradually it reduces as we move away from the surface. (Lb) on the local density number of motile micro-organisms. It is
Fig. 7 shows the effect of Brownian motion parameter Nb on the observed that magnitude of local density number of micro-
concentration profile of micro-organisms. Mass boundary layer of organisms is reduced for high iterative values of l3 andLb. As we
micro-organisms is enlarged with flaring values of Nb. Micro- know that Lb / D1mo , we can say that higher values of Lb leads to
organism concentration is enhanced by Brownian diffusion param- devaluation of diffusivity of micro-organisms into the fluid and it
eter. Physically it may be interpreted that the movement of nano- leads to the local density number of micro-organisms to slow
down.

Table 1
The local density number of motile microorganisms varies with l3 andLb when
Mc ¼ 0:6; Pr ¼ 0:2; Nb ¼ 0:1; Ec ¼ 0:1; Sc ¼ 0:22; Pe ¼ 0:1; e ¼ 2; c ¼ 2; l1 ¼ 0:1; l2 ¼
0:1; L0 ¼ 2; e1 ¼ 1:2; L ¼ 1; k ¼ 0:1; Rb ¼ 0:1; Nr ¼ 0:

Lb v ð0Þ
0
l3

0.1 2.6425
0.2 0.1 1.0654
0.3 0.9707
0.1 2.6425
0.1 0.3 1.1763
0.5 0.8346

Table 2
Local skin friction coefficient k; Nr; Rb when Mc ¼ 0:6; Pr ¼ 1; Nb ¼ 0:1;
Ec ¼ 0:1; Sc ¼ 0:5; Pe ¼ 0:1; e ¼ 2; c ¼ 2; l1 ¼ 0:1; l2 ¼ 0:1; L0 ¼ 2; e1 ¼ 1:2; Rb ¼ 0:1;
Nr ¼ 0:2.
Fig. 8. Concentration of micro-organisms via Nt.
k Nr Rb 00
F ð0Þ
0.1 1.9204
0.2 0.2 0.1 1.6703
0.3 0.9146
0.2 1.9204
0.1 0.4 0.1 2.5330
0.6 9.0510
0.1 1.9204
0.1 0.2 0.2 2.5939
0.3 4.1076

Table 3
Local Nusselt number varies with Pr when Mc ¼ 0:6; Nb ¼ 0:1; Ec ¼ 0:1; Sc ¼ 0:5;
Pe ¼ 0:1; e ¼ 2; c ¼ 2; l1 ¼ 0:1; l2 ¼ 0:1; L0 ¼ 2; e1 ¼ 1:2; L ¼ 1; Rb ¼ 0:1; Nr ¼ 0:2:

Pr 0
h ð0Þ
0.2 4.9089
0.4 2.9780
0.6 0.4396

Fig. 9. Concentration of micro-organisms via e (volume fraction parameter).

5
D. Dey and B. Chutia Journal of King Saud University – Engineering Sciences xxx (xxxx) xxx

Table 4 References
Comparison of local coefficient of skin friction for various values of k and Nr when
Mc ¼ 0:1; Pr ¼ 0:9; Nb ¼ 0:1; Ec ¼ 0:1; Sc ¼ 0:22; Pe ¼ 0:1; l3 ¼ 0:2; Lb ¼ 0:1: Attia, H.A., Al-Kaisy, A.M.A., Ewis, K.M., 2011. MHD Couette flow and heat transfer of
00 00 a dusty fluid with exponential decaying pressure gradient. Tamkang J. Sci. Eng.
k Nr Present (F ð0Þ) (Rehman et al., 2017) (F ð0Þ) 14 (2), 91–96.
0.1 1.6915 1.6552 Begum, N., Siddiqa, S., Sulaiman, M., Islam, S., Hossain, M.A., Gorla, R.S.R., 2017.
0.2 0.2 1.6649 1.64823 Numerical solutions for gyrotactic bio-convection of dusty nanofluid along a
0.3 1.6400 1.6410 vertical isothermal surface. Int. J. Heat Mass Transf. 113, 229–236.
Choi, S.U.S., 1995. Enhancing thermal conductivity of fluids with nanoparticles.
0.2 1.6915 1.6741
Proc. ASME Int. Mech. Eng. Congress Expo. 66, 99–105.
0.1 0.4 1.6985 1.7131
Dey, D., 2016a. Dusty hydromagnetic Oldryod fluid flow in a horizontal channel
0.6 1.7053 1.7541
with volume fraction and energy dissipation. Int. J. Heat Technol. 34, 415–422.
Dey, D., 2016b. Non-Newtonian effects on hydromagnetic dusty stratified fluid flow
through a porous medium with volume fraction. Proc. Natl. Acad. Sci., India
Section A Phys. Sci. 86, 47–56.
From Table 2, the effects of mixed convection parameter (k), Dey, D. 2018. Modelling and analysis of bio-convective nanofluid flow past a
buoyancy ratio parameter (Nr) and bio-convective Rayleigh num- continuous moving vertical cylinder. In: Emerging Technology in Data Mining
ber (Rb) on the local skin friction coefficient are noticed. We have and Information Security, Advances in Intelligent Systems and Computing, vol.
755, pp. 331–340.
observed that the magnitude of local skin friction coefficient is Gireesha, B.J., Roopa, G.S., Bagewadi, C.S., 2010. Unsteady flow and heat transfer of a
degraded for improving values of kand enhanced for increasing val- dusty fluid through a rectangular channel. Math. Prob. Eng., 2010, p. 17. https://
ues of Nrand Rb. doi.org/10.1155/2010/898720.
Gireesha, B.J., Mahanthesh, B., Krupalakshmi, K.L., 2017. Hall effect on two-phase
Table 3 presents the variation of Prandtl number (Pr) on local radiated flow of magneto-dusty-nanoliquid with irregular heat
Nusselt number. The magnitude of local Nusselt number is dimin- generation/consumption. Results Phys. 7, 4340–4348.
ishing significantly with the rise of Pr. As Pr / j1 , this implies that Gupta, R.K., Gupta, K., 1990. Unsteady flow of a dusty visco-elastic fluid through
channel with volume fraction. Indian J. Pure Appl. Mathematics 21, 677–690.
higher values of thermal diffusivity lead to a push up in the values Job, V.M., Gunakala, S.R., Kumar, B.R., Sivaraj, R., 2016. Time-dependent
of local Nusselt number in absolute sense. hydromagnetic free convection nanofluid flows within a wavy trapezoidal
Table 4 depicts the comparison table of skin friction coefficient enclosure. Appl. Therm. Eng. 115, no. 25.
Kuznetsov, A.V., Avramenko, A.A., 2004. Effect of small particles on the stability of
for varying values of k (Mixed convection parameter) and Nr bioconvection in a suspension of gyrotactic microorganisms in a layer of finite
(Buoyancy ratio parameter). We have compared the values of depth. Int. Commun. Heat Mass Transfer 31, 1–10.
F 00 ð0Þ with previous published data (Rehman et al., 2017) to Kuznetsov, A.V., 2010. The onset of nanofluid bioconvection in a suspension
containing both nanoparticles and gyrotactic microorganisms. Int. Commun.
ensure the valediction of our result. We have found a good agree- Heat Mass Transfer 37, 1421–1425.
ment consideringe ¼ 1; e1 ¼ 1; L ¼ 0; l1 ¼ 0; l2 ¼ 0; L0 ¼ 0; c ¼ 0. Kuznetsov, A.V., Nield, D.A., 2011. Double-diffusive natural convective boundary
Hence, we can conclude that the present MATLAB built-in solver layer flow of a nanofluid past a vertical plate. Int. J. Therm. Sci. 50, 712–717.
Mahdy, A., Hoshoudy, G.A., 2019. Two-phase mixed convection nanofluid flow of a
Bvp4c results are accurate. dusty tangent hyperbolic past a non-linearly stretching sheet. J. Egyptian Math.
Soc. 27 (44).
5. Conclusion Makinde, O.D., Chinyoka, T., 2010. MHD transient flows and heat transfer of dusty
fluid in a channel with variable physical properties and Navier slip condition.
Comput. Math. Appl. 60, 660–669.
Some of the conclusions from the above problem are as follows: Michael, D.H., Miller, D.A., 1966. Plane parallel flow of a dusty gas. Mathematika 13,
97–109.
Nadeem, S., Khan, M.N., Muhammad, N., Ahmad, S., 2019. Mathematical analysis of

The enhancing values of Brownian diffusion coefficient boost up bio-convective micropolar nanofluid. J. Comput. Des. Eng. 6, 233–242.
the velocity profile for primary flow of the fluid phase near the Rehman, K.U., Malik, A.A., Tahir, M., Malik, M.Y., 2017. Undersized description on
wall but reverse trend occurs at free stream. motile gyrotactic micro-organisms individualities in MHD stratified water-
based Newtonian nanofluid. Results Phys. 8, 981–987.

Slow diffusivity of micro-organisms and high Brownian diffu- Saffman, P.G., 1962. On the stability of laminar flow of a dusty gas. J. Fluid Mech. 13,
sivity decline the concentration profile of the micro-organisms. 120–128.

Local skin friction coefficient and local Nusselt number is deval- Saleem, S., Nadeem, S., 2015. Theoretical analysis of slip flow on a rotating cone
with viscous dissipation effects. J. Hydrodyn. 27, 616–623.
uating with the enhancement of mixed convection parameter Siddiqa, S., Abrar, M.N., Hossain, M.A., Awais, M., 2016. Dynamics of two-phase
and Prandtl number. Dusty fluid flow along a wavy surface. Int. J. Nonlinear Sci. Numer. Simul.

Magnitude of local density number of micro-organisms is decel- https://doi.org/10.1515/Ijnsns-2015-0044.
Siddiqa, S., Gul-e-Hina, Begum, N., Saleem, S., Hossain, M.A., Gorla, R.S.R., 2016.
erated with increasing values of bio-convective Schmidt num-
Numerical solutions of nanofluid bioconvection due to gyrotactic
ber i.e. lower diffusivity of micro-organisms. microorganisms along a vertical wavy cone. Int. J. Heat Mass Transfer, 101,
pp. 608–613.
Umavathi, J.C., Mohite, M.B., 2016. Convective transport in a porous medium layer
saturated with a Maxwell nanofluid. J. King Saud Univ. Eng. Sci. 28 (1), 56–68.
Declaration of Competing Interest

The authors declare that they have no known competing finan-

cial interests or personal relationships that could have appeared
to influence the work reported in this paper.