Development of thixotropic nanomaterial in fluid flow with gyrotactic microorganisms, activation energy, mixed convection

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Development of thixotropic nanomaterial in fluid flow with gyrotactic microorganisms, activation energy, mixed convection M. Ijaz Khan , Fazal Haq , Sohail A. Khan , T. Hayat , M. Imran Khan PII: DOI: Reference:

S0169-2607(19)31833-4 https://doi.org/10.1016/j.cmpb.2019.105186 COMM 105186

To appear in:

Computer Methods and Programs in Biomedicine

Received date: Revised date: Accepted date:

18 October 2019 29 October 2019 3 November 2019

Please cite this article as: M. Ijaz Khan , Fazal Haq , Sohail A. Khan , T. Hayat , M. Imran Khan , Development of thixotropic nanomaterial in fluid flow with gyrotactic microorganisms, activation energy, mixed convection, Computer Methods and Programs in Biomedicine (2019), doi: https://doi.org/10.1016/j.cmpb.2019.105186

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Highlights 

MHD flow of thixotropic nanomaterial with gyrotactic microorganisms is considered.



Brownian and thermophoretic diffusion effects are accounted.



Bio-convective flow is addressed.



Binary chemical reaction with activation energy is discussed.

Development of thixotropic nanomaterial in fluid flow with gyrotactic microorganisms, activation energy, mixed convection M. Ijaz Khan1,*, Fazal Haq2, Sohail A. Khan1, T. Hayat1,3 and M. Imran Khan4 1

Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan 2

3

Karakoram International University, Hunza Campus, Hunza, 15700, Pakistan

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University P. O. Box 80207, Jeddah 21589, Saudi Arabia 4

Heriot Watt University, Edinburgh Campus, Edinburgh EH14 4AS, United Kingdom

Abstract: Background: In this article, impact of gyrotactic microorganisms on nonlinear mixed convective MHD flow of thixotropic nanoliquids is addressed. Effects of Brownian motion and thermophoresis diffusion are considered. Characteristics of heat and mass transfer are analyzed with activation energy, Joule heating and binary chemical reaction. Nonlinear PDE's are reduced to ordinary equation by using suitable transformations.

Method: For convergent series solution the given system is solved by the implementation of the homotopic analysis technique (HAM).

Results: Influences of different flow controlling variables on the velocity, microorganisms, concentration and temperature are examined through graphs. Surface drag force, density number, Sherwood number and gradient of temperature are examined versus different flow parameters through graphs. For larger thixotropic fluid parameters the velocity field boosts up. For rising values of Hartmann number the velocity and temperature have opposite behaviors.

Keywords: Thixotropic nanofluid; Magnetic field; Gyrotactic microorganisms; Nonlinear

mixed convection; Activation energy; Joule heating. Numenclature

u, v velocity components  ms 1 

  ,   thixotropic fluid parameters

x, y Cartesian coordinates  s 

H a Hartmann number

T temperature  K 

 mixed convection parameter

T ambient temperature  K 

 t thermal convection parameters

Tw wall temperature  K 

 c concentration convection parameters

C concentration

N  buoyancy ratio parameter

C ambient concentration

Gr temperature Grashof number

Cw wall concentration

Pr Prandtl number

0 , 1 material constant

Nt thermophoresis parameter

c p specific heat  jkg 1K 1 

Nb Brownian motion parameter

k thermal conductivity Wm1 K 1 

Ec Eckert number

DT thermophoresis coefficient  kg 1m1s 1K 1 

Sc Schmidt number

DB

  chemical reaction parameter

Brownian

movement

coefficient

 kg 1m1s 1  N microorganism concentration

E activation energy parameter

N  ambient microorganism concentration

 temperature difference parameter

N w wall microorganism concentration

Gr  concentration Grashof number

 electric conductivity  kg 1m3 s 3 A2 

Pe bioconvection Peclet number

 kinematic viscosity  m2 s 1 

 microorganisms difference parameter

 density  kgm3 

Lb bioconvection Lewis number

k r reaction rate constant

C f skin friction coefficient

Ea activation energy  J 

 w shear stress  Nm2 

K Boltzmann constant  JK 1 

Nu x Nusselt number

Dm diffusivity of microorganism  m²s 1 

qw heat flux Wm2 

wc maximum cell speed of microorganism

Shx Sherwood number

 ms 1  n1 fitted rate constant

jw mass flux

 dimensionless variable

Nnx density number

1 , 2 thermal expansion coefficient  K 1 

qn density flux

3 , 4 concentration expansion coefficient

Re x local Reynolds number

1: Introduction Now a days study of non-Newtonian thixotropic nanofluids such as tomato ketchup, gels, colloids, honey and clays attracts researchers and scientist due to their vast applications in the fields of industry and engineering like fracturing of plastic and ceramic products, coating of wires, fiber technology, crystal growth, paper and printing and cooling of microelectronics. Two dimensional steady thixotropic or anti thixotropic liquid in a channel is highlighted by Pritchard

et al. [1]. Qayyum et al. [2] studied the thixotropic flow of nanofluid with binary chemical reaction with Newtonian mass and heat conditions. Characteristics of thixotropic nanofluid flow near boundary with MHD and mixed convection under the effect of stratifications phenomenon is illustrated by Hayat et al. [3]. Effect of magnetohydrodynamic in mixed convective thixotropic flow of nanoliquids with variable thickness due to a stretchable sheet is illustrated by Hayat et al. [4]. Effect of Joule heating and solar radiation on mixed convection thixotropic nanomaterials flow over a flat sheet is examined by Hayat et al. [5]. The term bioconvection can be defined as pattern development in suspensions of microorganisms, for instance algae and bacteria. Bioconvection is formed by collectives swimming of microorganisms. The density of base liquid is increased due to self-propelled microorganisms by swimming in a particular direction. Now a days gyrotactic microorganisms attracted numerous investigators because of its numerous applications in the field of biotechnology and industry like biofuel, bio microsystems, ethanol and fertilizers etc. Impact of gyrotactic microorganisms on nanofluid due to a vertical wall is presented by Khan et al. [6]. The features of nanofluid in presence of gyrotactic microorganisms is illustrated by Kuznetsov [7]. Khan et al. [8] discussed the characteristics of Maxwell nanofluid with stratification phenomenon with magnetic field and gyrotactic microorganisms. Impact of gyrotactic microorganisms and radiation effects on Magneto-Burgers nanomaterial flow is discussed by Khan et al. [9]. Heat transfer behavior of magnetohydrodynamic flow of an Oldroyd-B nanoliquids under the influence of gyrotactic microorganisms is highlighted by Waqas et al. [10]. Magnetohydrodynamic (MHD) has vital significance in the fields of medical, engineering, Physics, Chemistry and metallurgy such as cancer therapy, asthma treatment, removal of cancers with hyperthermia, gastric medications, magnetic cell separation, optical modulators, magnetic

resonance imaging, thinning of copper wires and optical switches. Venkateswarlu et al. [11] examined dissipation and melting effect on magnetohydrodynamic flow of nanofluid by a continuously moving surface. Soret and Dufour effects of MHD flow by a curved surface is reported by Hayat et al. [12]. An incompressible steady magnetohydrodynamic flow of nanoliquid with Hall effects and variable viscosity for heat transfer is highlighted by Evcin at al. [13]. The impact of MHD and radiation effects on Casson nanofluid flow is illustrated by Khan et al. [14]. Some advancements made by researchers are highlighted in Refs. [15-21]. The above-mentioned surveys witness that no effort has been made to investigate the impact of gyrotactic microorganisms and activation energy in thixotropic nanofluid flow with nonlinear mixed convection. Therefore our key objective is to study the characteristics of MHD flow of thixotropic nanomaterial with gyrotactic microorganisms and activation energy under the influence nonlinear mixed convection. Homotopy analysis technique (HAM) is implemented to obtain the series solution. Influences of various flow controlling variables on microorganisms, temperature, velocity and concentration are graphically examined. Surface drag force, Sherwood number, density number and gradient of temperature are examined versus different flow parameters through graphs.

2: Problem formulation Here we scrutinized the magnetohydrodynamic mixed convective flow of thixotropic nanoliquid with gyrotactic microorganisms by a stretchable surface. Flow is generated by a stretching surface. Furthermore thermophoresis effect Brownian diffusion are also considered. Moreover binary chemical reaction with Arrhenius activation energy is executed at the surface. Magnetic field  B0  is exerted in y  direction. The physical flow diagram is presented in Fig. 1.

The governing equation are

u u   0, x y

(1)

 

u 



2  v yv2      2 2 3 3 2 2  B 0  uy u xuy 2   yu3  ux yu2  yv yu2 u  f   g 1 (T  T )  2 (T  T ) 2  3 (C  C )   4 (C  C ) 2 ,  

(2)

2 2  T T k  2T  C T DT  T     Bo 2 u v    D   u ,  B    x y   c p  y 2  y  y T  y  c        f   f

(3)

u ux  v uy  



 2u y 2

 



 6  0

u y

2

 2u y 2

  4  1  uy 

 2u y 2

 2u xy



n1

 T   Ea C C  2C D  2T u v  DB 2  T 2  kr2  C  C    e T , x y y T y  T  u

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

(4)

(5)

with u  ax, v  0, T  Tw , C  Cw , N  N w at y  0,   u  0, T  T , C  C as y  , 

(6)

in which u, v indicate the velocity components in x  and y  direction respectively,  kinematic viscosity,  electric conductivity,  density of fluid, 1 , 2 linear and nonlinear thermal expansion coefficients, 3 ,  4 are linear and nonlinear concentration expansion coefficients, 0 and 1 material constants,  c p  specific heat, k thermal conductivity, DT f coefficient of themophoretic diffusion, DB

coefficient of Brownian motion diffusion, T

ambient temperature, C ambient concentration, N  ambient microorganisms concentration,

kr2 reaction rate, Tw wall temperature, Cw wall concentration, N w wall concentration of microorganisms, Ea activation energy,

  T T

n1

E

e

 kTa

modified Arrhenius function, K Boltzmann

constant, Wc maximum cell speed of microorganisms, n1 fitted constant and Dm diffusivity of

microorganisms. Considering

  a y, u  axf ', v   a f ,     TT TT ,   CC CC ,   NN NN ,  

w





(7)





w



w

One can get





f   ff   f     f  f    f f 2

2

2

f   f

4



 ff f

2

 ff

2

 H f    1  t    N  1   c    0, 2 a



f iv      

(8)

 '' Pr Nb  ' Pr Nt   Pr EcH a2 f '2  Pr f    0, 2

 ''

(9)

n1 Nt  E     Scf  '   Sc 1     exp    0,  Nb 1    

(10)

 '' Lbf  ' Pe  ' '       ''  0,

(11)

f  0   0, f   0   1,   0   1,   0   1,   0   1, at   0   f      0,      0,      0,      0, as   , 

(12)



in which Pr     signifies Prandtl number,  



parameter/Hartmann number,   



Gr   Re

2 x

 thermal diffusivity, H    magnetic

 and   

6 0 a3 x 2



k cp



2

4 1a 4 x 2

 2





buoyancy forces, t  12 Tw  T 



convection parameters, Gr 





g 1 Tw T 





2



ax 2





number in terms of concentration, N   Gr Gr 



3  Cw C  1 Tw T 



ac p

 are thixotropic fluid parameters,

 mixed convection parameter, Re    Reynolds number, Gr   2 x

 B02

2 a

g 2  Cw C 

2

 Grashof

ratio of concentration to thermal

and c  43  Cw  C 



are thermal and concentration

 Grashof number in terms of temperature, 

Eckert number, Nt   DT (TTwT ) thermophoresis parameter, Nb 

 DB  Cw C  





Ec 

uw2 c p Tw T 



Brownian motion







parameter,    TwTT , temperature difference parameter,   



parameter, E 



 



Nw N w  N

Ea kT





activation energy parameter, Pe  bWc Dm



K r2 a



bioconvection Peclet number,



microorganisms concentration difference parameter, Lb 



Lewis number and Sc 

 DB

chemical reaction

vf Dm



bioconvection

 Schmidt number.

3: Engineering quantities Gradient of velocity  Cf x  , Nusselt  Nux  , Sherwood  Shx  and density  Nnx  numbers are defined as: w 1 u2 w 2

  xqw Nu x  k f Tw T  ,   xj Shx  DB CwwC  ,   xq Nnx  Dm  N wn N  ,   Cf x 

,

(13)

where





qw   k f

u y

jw   D

C B y

qn   Dm



 ,  y 0   , y 0   , y 0   ,  y 0

2  w     20  uy   uy

  

N y

(14)

Dimensionless version are



Cf x Re0.5 x 2 f



 0   3



f

3

Nu x Re x 0.5    0  , 

Shx Re

0.5 x

   0  ,

Nnx Re x 0.5  





 0 ,

 ,

0

     

(15)



here Re x 

ax 2 v

 denotes local Reynolds number.

4: Series solution Homotopic technique was proposed by Liao [22] is employed for convergent solution. The initial guesses and linear operators are expressed as:

f 0    1  e  ,  0    e  , 0    e  ,  0    e  ,

      

(16)

 ddf ,   2 £  dd2   ,    d 2 £  d 2   ,   2 £  dd2   ,  

£f 

d3 f

d 3

(17)

with

£ f  A1  A2 e  A3e   ,   £  A4e  A5e   ,   £  A6 e  A7 e   ,   £  A8e  A9e   , 

(18)

where Ai  i  1  9  represents the arbitrary constants.

5: Convergence analysis The auxiliary variables

f

,



,



and



have key role in regulating the convergence region

in homotopy analysis method (HAM). Fig. 2 is plotted for

 curves of velocity, energy,

concentration and microorganisms equations. Approximate range of convergence for velocity, energy,

concentration

and

microorganisms

are

1.1 

f

 0.2,

1.4 



 0.4,

1.6 



 0.2 and 1.6 



 0.2 . From Table 1 it is noticed that 15th , 25th , 30th and 30th

order of approximation are fulfill for convergence of velocity, energy, concentration and microorganisms respectively. Table 2 is highlighted for the validation of results with published ones (Refs. 23, 24).

Table 1: Computations results for various order of approximation when f (0),  (0),  '(0), and

 (0)

when

t  Nb  0.3,

Pr  0.5,

Sc  0.7,

      c  H a       0.1,

Nt     0.2, E  Pe  1.2, Lb  1.5,   n1  1.0. Order of approximations

 f (0)

 (0)

 '(0)

 (0)

01

0.859

0.495

0.806

1.850

05

0.861

0.360

0.628

1.512

10

0.857

0.336

0.562

1.393

15

0.855

0.330

0.543

1.361

20

0.855

0.329

0.538

1.352

25

0.855

0.328

0.536

1.349

30

0.855

0.328

0.535

1.348

35

0.855

0.328

0.535

1.348

40

0.855

0.328

0.535

1.348

6: Validation of analytical results Table 2 is described to confirm the accuracy of present numerical approach. These tables illustrated the comparison of gradients of temperature against  Pr  while all other interesting parameters are set to be zero, with those of Shiaq et al. [23] and Wang [24]. Clearly the results are in good agreement. Table 2: Comparison of gradient of temperature with Shiaq et al. [22] and Wang [23].

Pr

Shiaq et al. [22]

Wang [32]

Recent result

0.07

0.0656

0.0656

0.0639

0.20

0.1691

0.1691

0.1685

0.70

0.4539

0.4539

0.4564

2.00

0.9114

0.9114

0.9118

7.00

1.8954

1.8954

1.8965

20.00

3.3539

3.3539

3.3539

70.00

6.4622

6.4622

6.4639

7: Discussion Influences of different pertinent variables on velocity profile, motile microorganisms,

concentration and temperature are scrutinized in this section. Gradient of velocity, Sherwood number, density number and gradient of temperature are examined versus different interesting parameters through graphs.

7.1: Velocity Salient effect of various interesting parameters like   and    , ( H a ), ( ), ( N  ) on velocity

 f    

is demonstrated in Figs.  3  7  . Figs. (3 and 4) are delineated to analyze the behavior

of thixotropic fluid variables on velocity

 f     .

Here one can noticed that velocity increases

for higher values of   and   . Impact of  Ha  on f    is portrayed in Fig. 5. Clearly velocity decreases for rising values of H a . Physically the Lorentz force increases for higher Hartmann number which causes the reduction in velocity

 f     .

Fig. 6 is sketched to

examined the characteristics of velocity versus N  . It is observed that f    boosts up versus

N  . Influence of  on f    is captured in Fig. 7. Clearly f    increases for higher  . For higher estimation of  increases buoyancy force and consequently velocity  f     boosts up.

( )

( )

( )

( )

( )

7.2: Temperature Figs. (8  11) are sketched to discuss the effect of various pertinent variables like (Pr), ( H a ),

( Nb) and ( Nt ) on temperature ( ( )). Impact of (Pr) on  ( ) is depicted in Fig. 8. It is noticed that for larger Pr temperature ( ( )) decays. Since Prandtl number has inverse relation with thermal diffusivity. Therefore for rising values of Pr decreases thermal diffusivity as a result  ( ) decays. Fig. 9 is delineated to shows the effect of H a on  ( ). One can observe that temperature field upsurges for larger H a . Physically For higher values of  H a  Lorentz force becomes stronger and therefore  ( ) boosts up. Effect of  Nb  on  ( ) is depicted in Fig. 10. Clearly  ( ) increases for larger ( Nb). Influence of ( Nt ) on ( ( )) is displayed in Fig. 11. One can find that via ( Nt ).

( )

( )

( )

( )

7.3: Concentration Behavior of ( Sc), ( E ), ( Nb), ( Nt ),    and    on concentration     are displayed in Figs. 12  17  . Fig. 12 is sketched to examine the impact of ( Sc) on     . Clearly for larger

( Sc) the concentration decays. Fig. 13 elucidates the behavior of    versus ( E ). Physically higher activation energy reduces the temperature which leads to slow down chemical reaction, as a result    boosts up. Fig. 14 is delineated to shows the characteristics of    versus  Nb  . Larger  Nb  improves the random motion and collision of fluid nanoparticles which diminishes concentration     . Characteristics of concentration for increasing values of  Nt  is depicted in Fig. 15. An increment occurs in concentration for larger values of  Nt  . Behavior of    on

   is displayed in Fig. 16. One can find that concentration decays against    . Fig. 17 is delineated to examine the characteristic of    versus decreasing functions of   .

  . 

Clearly, concentration is

( )

( )

( )

( )

( )

( )

7.4: Microorganisms Influence of  Pe  ,

 Lb 

and    on concentration of microorganisms    is discussed in

Figs. 18  20  . Fig. 18 shows the variation in    versus  Pe  . As expected    decays via larger Pe. For larger Pe causes a reduction in microorganisms diffusivity and hence    decreases. Salient features of  Lb  on    is plotted in Fig. 19. One can clearly noted that increments in

 Lb 

decays microorganisms concentration. In fact for larger Lb the

microorganisms diffusivity decays and as a result    decays. Fig. 20 is plotted to discuss the characteristics of    against higher . Clearly concentration of microorganisms decreases against . . Physically for larger  improves the microorganisms concentration in ambient fluid as a result decay in density field is observed.

( )

( )

( )

7.5: Engineering quantities Figs. 21-28 are captured to analyze the effect of important variables on engineering quantities interest like Skin friction  Cf x  , local Nusselt number density number

 Nnx  .

 Nux  ,

Sherwood number  Shx  and

Fig. 21 is delineated for surface drag force  Cf x  against thixotropic



fluid parameters    and     . It is observed from Fig. 21 that Cf x Re x

 decays for rising

values of thixotropic fluid parameters. The combine behavior of ( H a ) and    on surface drag

force is displayed in Fig. 22. One cane observe from this figure that  Cf x  boosts up via    and  H a  . Fig. 23 is sketched to discuss the characteristics of heat transfer rate versus higher values of ( H a ) and (Pr). Clearly heat transfer rate decreases for larger H a and Pr . The variation in Nusselt number against higher Pr . and Brownian motion parameter ( Nb) is portrayed in Fig 24. One can observe from Fig. 24 that  Nux  monotonically increases for fixed values of Nb and increasing values of Pr . Fig. 25 is prepared to discuss the effect of mass transfer rate (Sherwood number) for higher Schmidt number ( Sc) and activation energy parameter ( E ). One can easily observe that magnitude of  Shx  upsurges for higher  Sc  and

 E .

The combine effect of  Sc  and Brownian motion parameter ( Nb) is displayed in Fig. 26.

Clearly  Shx  decays against  Sc  and  Nb  . Fig. 27 is delineated to discuss the effects of

 Pe 

and  Lb  on density number. It is observed from this figure that  Nnx  is enhanced versus

 Pe 

and  Lb  . The salient aspects of  Nnx  versus lager values of  Pe  and    is depicted in

Fig. 28.  Nnx  monotonically upsurges for higher  Pe  and    .

8: Conclusions The key observations are: 

Thixotropic fluid parameters and Hartmann number have opposite behavior for f    .



 ( ) increases for larger estimation of  H a  ,  Nb  and  Nt  .



For larger  Pr  the temperature decays.



 ( ) decays for higher estimation of  Sc  .



Concentration increases for rising values of  E  .



Microorganisms concentration decreases versus  Pe  ,  Lb  and    .



Skin friction coefficient decays for higher thixotropic fluid parameters.



Heat transfer rate reduces via  H a  and  Pr  .



Sherwood number enhances against larger  Sc  and  E  .



Magnitude of density number upsurges versus higher bio convection Peclet and Lewis number.

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Conflict of interest

The authors declared that they have no conflict of interest and the paper presents their own work which does not been infringe any third-party rights, especially authorship of any part of the article is an original contribution, not published before and not being under consideration for publication elsewhere.