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Numerically framing the features of second order velocity slip in mixed convective flow of Sisko nanomaterial considering gyrotactic microorganisms
International Journal of Heat and Mass Transfer 112 (2017) 521–532
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerically framing the features of second order velocity slip in mixed convective flow of Sisko nanomaterial considering gyrotactic microorganisms Shahid Farooq a,⇑, Tasawar Hayat a,b, Ahmed Alsaedi b, Bashir Ahmad b a b
Department of Mathematics, Quaid-I-Azam University, 45320 Islamabad 44000, Pakistan Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 13 April 2017 Received in revised form 29 April 2017 Accepted 2 May 2017
Keywords: Sisko nanomaterial Mixed and bio convection Compliant wall properties Second order velocity slip Heat, mass and motile microorganism density Newtonian heat, mass and motile microorganism density
a b s t r a c t Here peristalsis of Sisko nanoliquid with gyrotactic microorganism in a curved channel is investigated. Channel boundaries comprises the wall properties and second order slip conditions for velocity. Consideration of Newtonian heat, mass and gyrotactic microorganisms aspects characterizes the heat, mass and motile density transfer processes. Flow formulation is established utilizing constitutive relations of Sisko fluid. Lubrication theory is employed for the simplification of governing expressions. Further numerical solution is carried out for velocity, temperature, concentration and motile density gyrotactic microorganisms. The numerical solution is justified through graphical results. Graphical discussion determined that velocity has opposite behavior for first and second order velocity slip parameters. Interestingly Sisko fluid parameter has opposite impact on velocity for both shear thinning and shear thickening cases. It is seen that temperature enhances for Newtonian heating whereas concentration and gyrotactic microorganism reduces for Newtonian mass and Newtonian gyrotactic microorganisms. Ó 2017 Published by Elsevier Ltd.
1. Introduction During the past few decades the nanofluids have become very popular among the recent scientists, engineers, mathematicians, medical specialists working in material science, electronics systems, mechanical and health care branches. Commonly nanofluids with nanometer-size solid particles (compared with nanofluid particles of millimeter or micrometer size) have many rheological properties, better stability and significantly higher thermal conductivity efficient. The surprising mechanical, electrical, optical, and thermal characteristics of nanomaterials have developed them most aspired after materials of the existing situation. By definition the nanofluids are the dilute suspensions of nano-sized particles and fibers immersed in materials. In view of such facts Choi [1] was the first who used the terminology of nanofluids in 1995. The nanoliquids as a whole change the thermal ability of such compounds which commences e.g. density, viscosity, diffusivity and thermal conductivity. The thermal conductivity is very important for all the aforementioned physical quantities. Mostly the new ⇑ Corresponding author. E-mail (S. Farooq).
addresses:
[email protected],
[email protected]
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.05.005 0017-9310/Ó 2017 Published by Elsevier Ltd.
researchers have used nanoparticles generally synthesize of oxide ceramics, nonmetals, metal carbides, nitrides, metals (Al, Cu, Au), metal oxides (Al2O3, TiO2, SiO2, ZnO2), single wall or multiple wall carbon nanotubes (i.e. SWCNT or MWCNT) to form nanofluids. Ordinary liquid has weaker thermal conductivity. This weaker conductivity can be improved greatly with the use of such nanoparticles. There are numerous combinations of nanoparticles e.g. various combination of nanoparticles with/without surfactant molecules can be distributed into ordinary liquids like water, ethylene or propylene glycol and other alternative lubricants. In this direction, the characteristics of nanoparticles Brownian motion factor in base liquid is quite significant. A large quantity of heat is created in microelectro mechanical and heat exchangers procedures to control the system performance. Thermal conductivity of ordinary liquids is exceeded by insertion of nanoparticles just to cool down such industrial activities. The nanoparticles have achieved great significance in several biological and engineering processes such as medicine, solar cells, plasma and laser cutting, catalysts, electronics, materials manufacturing, glass industry, in cooling of turbine blades processes and in several other industrial processes. A considerable research on nanofluids have been communicated by the researchers. Some important studies covering
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the aspects of Brownian diffusion and thermophoresis can be mentioned (for detail see [2–10]). Microorganism particles are significantly useful in the production of several commercial and industrial products e.g. bio fuel made from waste, fertilizers, ethanol, etc. The motile microorganisms are self-urged which enlarges the denseness of ordinary liquids by swimming toward a particular direction within the material in attraction to motivate like gravity, oxygen, daylight whereas nanoparticles cannot swim. Gyrotactic microorganisms are also used during the production of biodiesle and hydrogen, an important sustainable energy source and in water treatment plants. Due to such fact it is necessary to analyze the swimming patterns and mass transfer characteristics of microorganisms so that demands of the organisms seen more useful, profitable and significant for the bright future of the whole mankind. Bioconvection is the development of distinct types of random liquids sequences at the microscopic level because of instinctive swimming of automotive microorganisms which are included in the water and in the other materials denser than water. Those microorganisms swimming is influenced by the reasonable procedures such as oxygen for respiration, searching for nutrient, upgrade light input for photosynthesis. The arbitrary movement of nanoparticles is created due to the thermophoretic and Brownian diffusion features and are transported by the drift of base fluid. Nanofluids have extensive application in micro fluidic tool as they are very supportive for mass transport improvement and induce mixing particularly in micro volumes [11]. Kuznetsov [12] employed the idea of bioconvection of nanoliquid with gyrotactic microorganisms. Analysis of magneto Jeffrey nanoliquid considering gyrotactic microorganism is reported by Bhatti et al. [13]. Few studies comprising the characteristics of gyrotactic microorganism with nanofluid can be stated through [14–18] and several studies therein. Peristalsis is an inherent property of many industrial and biopyhsical mechanisms. Nodoubt, peristaltic mechanism is due to propagating wave along the boundaries of flexible channel/tube. By owing the importance of peristalsis in physiology and industry, the physiologists and scientists consider it as a major mechanism for the transportation of several physiological and industrial materials. Mainly mechanism of peristalsis is experienced in various biological processes like food swallowing via oesophagus and gastrointestinal tract, movement of chyme in the gastrointestinal tract, spermatozoa movement in ductus afferents of male reproductive tract, ovum movement in the fallopian tube, urine motion from kidney to bladder, embryo transportation through uterus and several others. Many advanced medical devices such as dialysis machine, open heart bypass machine, B.P device hose, infusion and finger pumping devices have been made on the theory of peristalsis pumping. In nuclear industry peristalsis is used for the transportation of noxious, sterile, corrosive and many waste materials. After the initial experimental and theoretical attempts presented by Latham [19] and Shapiro et al. [20], several researches on the peristalsis have been listed with and without nanomaterials in literature (for detail see [21–30]). In past mostly researches on peristaltic transport have been studied in planner or straight channel. However in many realistic situations the flows in conduits and ducts are curvature dependent. Sato et al. [31] filled this void with the primary presentation on peristaltic motion of viscous liquid in a curved geometry. Later on several researchers [32–40] focused their attention on peristaltic motion with and without nanomaterial in a curved channel. Our motivation here is to model the peristaltic transport of Sisko nanomaterial in the presence of motile gyrotactic microorganisms. Curved channel walls comprises Newtonian effects for
heat, mass and motile density gyrotactic microorganisms respectively. Momentum equation is studied in presence mixed of convection. Further viscous dissipation effects are absent. The flow governing expressions are simplified through large wavelength and small Reynolds number. Coupled non-linear systems are solved numerically via shooting procedure a well-known built-in technique in Mathematica 8 computational software. The obtained numerical results for velocity, temperature, concentration and gyrotactic microorganisms are analyzed through graphical discussion. 2. Problem formulation Here we examined the peristalsis of an incompressible nonNewtonian (Sisko) nanomaterial with gyrotactic microorganisms in a curved channel having radius R and uniform width 2a0 twisted in a semi-circle with center at O. The axial and radial directions are denoted by x and r respectively. The velocity components v 2 and v 1 are along axial and radial directions simultaneously. Aspects of Brownian and thermophoresis are present in heat and mass equations. Moreover viscous dissipation and thermal radiation features are absent. The channel boundaries are subject to constant temperature, concentration and volume fraction motile microorganisms T 0 ; C 0 and N 0 respectively. Characteristics of compliant walls are also discussed. The second order velocity slip conditions are employed. Further Newtonian type heating, concentration and gyrotactic microorganisms effects at channel boundaries are also studied. Fluid motion is due to the propagation of flexible walls along xdirection with constant speed c. Mathematically such walls expressed as (see Fig. 1):
ðx; tÞ ¼ a0 a1 sin h
2p ðx ct Þ : kc
ð1Þ
Here a0 denotes the half width of channel, a1 the amplitude, kc and stress tensor s the wavelength and t the time. The velocity v for Sisko liquid are given by [36,41,42]:
v ¼ ðv 1 ðx; r; tÞ; v 2 ðx; r; tÞ; 0Þ; h
n1
i
s ¼ a þ bðc_ Þ 2 A1 ;
ð2Þ ð3Þ
In Eq. (3) a; b and n designated the constants of Sisko material. Here we discuss both shear thinning (i.e. n < 1) and thickening (i.e. n > 1) cases. One can achieve the stress tensor for generalized power law model by choosing a ¼ 0. Furthermore it can be reduced to Newtonian fluid model by taking a ¼ b ¼ 0. Also A1 the deformation rate of tensor and c_ the second invariant strain tensor are
þ ðrv ÞT A1 ¼ rv 1 2
c_ ¼ trðA1 Þ2 ;
ð4Þ ð5Þ
The flow governing equations are [45–49]:
@ @ v 2 fðR þ r Þv 1 g þ R ¼ 0; @r @ x
q
@ v 1 @ v R v @ v 1 v 22 1 1 þ 2 2 þ v @r R þ r @ x R þ r @t rx @p 1 @ R @s sxx rr g þ ¼ þ ; fðR þ r Þs @r R þ r @r R þ r @ x R þ r
ð6Þ
ð7Þ
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Fig. 1. Physical diagram.
q
@ v 2 @ v R v @ v 2 v 1 v 2 1 2 þ 2 þ v @r R þ r @ x R þ x @t o R @p 1 @ n 2 rx ¼ þ 0 ðR þ r Þ s 2 R þ x @ x R þ r @r xx R @s þ qbT g ðT T 0 Þ þ qbC g ðC C 0 Þ þ R þ r @ x
h
h
ð8Þ
" # @T @T R v 2 @T j @2T 1 @T R2 @2T þ v 1 þ ¼ þ þ @r R þ r @ x ðqC Þf @r 2 R þ r @r ðR þ r Þ2 @ x2 @t
s DT Tm
ðrT rT Þ; ð9Þ
" # @C @C R v 2 @C @2C 1 @C R2 @ 2 C þ v1 þ ¼ DB þ þ @r R þ r @ x @r 2 R þ r @r ðR þ r Þ2 @ x2 @t 2 3 DT 4@ 2 T 1 @T R2 @2T 5 þ þ þ 2 ; T m @R2 R þ R @R R þ R2 @ X ð10Þ
" # @N @N R v 2 @N @2N 1 @N R2 @2N 1 þ þ ¼ D þ v þ n @r R þ r @ x @r 2 R þ r @r ðR þ r Þ2 @ x2 @t b WC @ @C @ @C þ : N N þ @r @ x @r C 0 @r ð11Þ
In above expressions (6)–(11) the quantities q; p; bT ; bC ; C f ; s ; j; DB ; DT ; T m ; Dn ; b ; W C denote the density, pressure, thermal expansion coefficient, mass expansion coefficient, specific heat of basefluid, ratio of nano and base materials heat capacities, thermal conductivity constant, Brownian diffusion coefficient, thermophoretic diffusion coefficient, motile microorganism diffusion coefficient, chemotaxis number and maximum cell swimming speed respectively. Also (T,C and N) are the temperature, concentration and volume fraction of motile microorganisms. The xx ; s rx and s rr are given as stress components s
h
n1
sxx ¼ 2 a þ bðc_ Þ 2
i R @ v v 1 2 þ ; r þ R @ x r þ R
i@ v
ð12Þ
2
@r
n1
srr ¼ 2 a þ bðc_ Þ 2
ðN N0 ÞcDq;
þ s DB ðrC rT Þ þ
n1
srx ¼ a þ bðc_ Þ 2
þ
R @ v 1 v 2 ; r þ R @ x r þ R
ð13Þ
i@ v 1 : @r
ð14Þ
The dimensional form of compliant wall [23,36,43,44,50] and second order velocity slip boundary constraints are
" # 2 R @3 @3 @ h s þ m þ d 1 ðr þ R Þ @ x3 @ x@t 2 @t@ x o @s 1 @ n xx 2 ¼ ðr þ R Þ srx þ q 2 @ r @ x ðr þ R Þ @ v 2 @ v R v 2 @ v 2 v 1 v 2 1 2 þ þ v þ qbT g ðT T 0 Þ r þ R @ x @r R þ r @t þ qbC g ðC C 0 Þ ðN N0 ÞcDq; at r ¼ h;
ð15Þ
v 2 a0 srx b0
rx @s ¼ 0; @r
at r ¼ h;
ð16Þ
v 2 þ a0 srx þ b0
rx @s ¼ 0; @r
at r ¼ h:
ð17Þ
Similarly Newtonian heating, mass and motile density gyrotactic microorganisms boundary constraints are
T ¼ T 0 ; C ¼ C 0 ; N ¼ N0 ;
at r ¼ h;
@T @C @N ¼ c1 T; ¼ c2 C; ¼ c3 N; @r @r @r
ð18Þ at r ¼ h:
ð19Þ
Dimensionless variables are
d¼
x r a0 ; x¼ ; r¼ ; kc kc a0
ct t¼ ; kc
sij ¼
v1 ¼
v 1 c
;
v2 ¼
v 2 c
; g¼
h a2 p ; p¼ 0 ; a0 clkc
a0 T T0 C C0 N N0 R s; h ¼ ; /¼ ; U¼ ; k¼ ; c l ıj T0 C0 N0 a0
Re ¼
qca0 qb ga2 T 0 qb ga2 C 0 N0 cDq ; Gt ¼ T 0 ; Gc ¼ C 0 ; Rb ¼ ; l cl cl l
Pr ¼
lc p s D B C 0 s D T T 0 b WC sa3 ; Nb ¼ ; Nt ¼ ; Pe ¼ ; E1 ¼ 3 0 ; j m mT m Dn k c lc
E2 ¼
mca30 k3c l
3
; E3 ¼
da0
k3c l
; b ¼
n1 b c ; a a0
ð20Þ
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v1 ¼ d
k @w ; r þ k @x
v2 ¼
@w ; @r
ð21Þ
where d the wave number, k the curvature parameter, b the Sisko fluid constant, p the dimensionless pressure, Gt the thermal Grashof number, Gc the mass Grashof number, Rb the bioconvection Ralyeigh parameter, Pr the ratio of momentum diffusivity to thermal diffusivity (i.e. Prandtl number), N b the Brownian motion quantity, Nt the thermophoretic motion quantity, P e the Peclet number, E1 ; E2 ; E3 the compliant wall parameters and w the stream function. After using dimensionless quantities (21), large wavelength ðd ! 0Þ and low Reynolds number ðRe ! 0Þ approximations Eq. (6) vanished identically and other expressions are reduced to the forms:
c_ ¼
@w
@2w þ @r @r 2 r þ k
!2 :
ð32Þ
Results for straight/planner channel can be retained by taking curvature parameter larger (i.e. k ! 1). Also present study can be reduced for Newtonian case by putting dimensionless Sisko material parameter zero (i.e. b ¼ 0). 3. Methodology The analytical or exact solution of the resulting non-linear and coupled equations seems difficult. Thus numerical solution of the resulting problem has been developed through numerical approach called Shooting technique. Further this numerical solution is justified via graphical discussion for each flow quantity.
dp ¼ 0; dr
ð22Þ
o k dp 1 @ n 2 ¼ ðr þ kÞ srx þ Gt h þ Gc / Rb U; 2 r þ k dx ðr þ kÞ @r
ð23Þ
4. Discussion
ð24Þ
Our key intention here is to interpret the graphical results of velocity, temperature, concentration and motile gyrotactic microorganisms for different embedded dimensionless variables. This section is further divided into five subsections for of better understanding.
2 ! @ h 1 @h @h @/ @h þ Pr N þ N þ ¼ 0; t b @r 2 r þ k @r @r @r @r 2
@2/ 1 @/ Nb @ 2 h 1 @h þ þ þ 2 @r r þ k @r Nt @r2 r þ k @r
! ¼ 0;
ð25Þ
! @2U 1 @U @ U @/ @2/ þ U 2 ¼ 0: þ Pe @r @r 2 r þ k @r @r @r
ð26Þ
Eq. (22) indicates that the pressure p along radial direction is constant. Here h; / and U are the dimensionless temperature, concentration and motile density profiles respectively. Eliminating pressure from Eqs. (22) and (23) through cross differentiation we get
" # 2 o @ k @ n kM @w 2 ðr þ kÞ srx : 0¼ 1 @r ðr þ kÞ @r @r ð1 þ m2 Þðr þ kÞ
ð27Þ Dimensionless boundary conditions are
" # k @3 @3 @2 þ E g E1 3 þ E2 3 ðr þ kÞ @x @t@x @x@t 2 o 1 @ n 2 ðr þ kÞ srx þ Gt h þ Gc / Rb U; ¼ 2 @r ðr þ k Þ
at r ¼ g; ð28Þ
@w @ srx asrx b ð29Þ ¼ 0; h ¼ 0; / ¼ 0; U ¼ 0; at r ¼ g; @r @r @w @ srx @h @/ þ asrx þ b þ c1 ðh þ 1Þ ¼ 0; þ c2 ð/ þ 1Þ ¼ 0; ¼ 0; @r @r @r @r @U ð30Þ þ c3 ðU þ 1Þ ¼ 0; at r ¼ þg; @r
g ¼ 1 þ sin ½2pðx tÞ:
0 0 Here a ¼ aa0 and b ¼ ab0 are the dimensionless first and second order velocity slip numbers respectively. Here c1 ¼ c1 a0 , c2 ¼ c2 a0 and c3 ¼ c3 a0 are Newtonian heating, mass and motile density gyrotactic microorganisms dimensionless numbers and
srx
h
¼ 1 þ b ðc_ Þ
n1 2
i
! @w @2w @r ; 2þ @r rþk
ð31Þ
4.1. Velocity This subsection contains the graphical discussion of velocity. Here velocity profile is discussed for both shear thinning and shear thickening properties of Sisko nanomaterial. Thus Figs. 2–9 are made to see the reaction of velocity for first order velocity slip parameter a, second order velocity slip parameter b, curvature parameter k, Sisko material constant b , compliant wall numbers E1 ; E2 ; E3 , thermal Grashof number Gt , mass Grashof number Gc and bioconvection Ralyeigh parameter Rb . Part ðiÞ of each of these figures describes the shear thinning behavior whereas part ðiiÞ of each of these figures describes the shear thickening behavior of Sisko material. Fig. 2(i) and (ii) shows the velocity behavior for first order velocity slip parameter a. It is seen through this figure that velocity profile reduces for both shear thinning and thickening characteristics of Sisko material when first order velocity slip parameter is increased. It is evident form Fig. 3(i) and (ii) fluid velocity has opposite behavior for second order velocity slip parameter b when compared with a. Impact of curvature parameter k on velocity is witnessed through Fig. 4(i) and (ii). This figure determine that symmetry of velocity profile is disturbed at center of channel due to curvature. It is worthmentioning that velocity has revers behavior at both channel walls. Fig. 5(i) and (ii) provides the action of Sisko material parameter b on velocity. This figure depicts that velocity enhances for shear thinning (i.e. n < 1) behavior and it decreases for shear thickening (i.e. n > 1) behavior when Sisko material parameter b is increased. Variation in velocity for different values of compliant wall properties (E1 ; E2 and E3 ) are disclosed in Fig. 6(i) and (ii). Clearly it is noticed that velocity attains its maximum value when wall tension E1 and mass characterizing E2 are enhanced. The velocity decays for larger wall damping E3 parameter. Impact of thermal Grashof number Gt on velocity has been visualized through Fig. 7(i) and (ii). This figure depicts that velocity becomes larger when impact of thermal buoyancy forces are increased (i.e. larger thermal Grashof number). It is also noticed that effects of Gt on velocity profile are more prominent for shear thinning case (i.e. n < 1) than for shear thickening case (i.e. n > 1). Velocity profile has similar results for concentration Grashof number Gc as for thermal Grashof number Gt (see Fig. 8(i) and (ii)). Physically it describes that thermal and concentration buoyancy forces provide small resistance in the flow field and thus
S. Farooq et al. / International Journal of Heat and Mass Transfer 112 (2017) 521–532
Fig. 2. (i and ii): Velocity
v2
via a.
Fig. 3. (i and ii): Velocity
v2
via b.
Fig. 4. (i and ii): Velocity
v2
via k.
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Fig. 5. (i and ii): Velocity
Fig. 6. (i and ii): Velocity
v2
v2
via b .
via wall properties.
Fig. 7. (i and ii): Velocity
v2
via Gt .
S. Farooq et al. / International Journal of Heat and Mass Transfer 112 (2017) 521–532
Fig. 8. (i and ii): Velocity
v2
via Gc .
Fig. 9. (i and ii): Velocity
v2
via Rb .
Fig. 10. Temperature h via k.
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velocity is enhanced. Efficiency of bioconvective Rayleigh number Rb on velocity is sketched in Fig. 9(i) and (ii). This figure witnesses that bioconvective Rayleigh number has dominating role for velocity profile (i.e. velocity profile decreases for higher Rb ). Finally through all these figures one can easily concludes that velocity profile is maximum for shear thinning case (i.e. n < 1) and it is minimum for shear thickening case (i.e. n > 1). 4.2. Temperature Figs. 10–14 visualize the temperature variation for curvature parameter k, Newtonian heating parameter c1 , Prandtl number Pr, thermophoretic parameter N t and Brownian motion parameter N b . Fig. 10 is sketched to view the impact of curvature parameter k on temperature. This figure demonstrates that curvature parameter has dominating effects on temperature (i.e. temperature reduces with an increment in k). Also it is noticed that temperature is maximum in curved channel (i.e. small k) when compared with planner channel (i.e. larger k). Behavior of temperature for Newtonian heating parameter c1 can be seen through Fig. 11. This figure reveals that c1 has assisting role for temperature (i.e. h becomes
Fig. 11. Temperature h via c1 .
Fig. 13. Temperature h via N t .
Fig. 14. Temperature h via N b .
Fig. 12. Temperature h via Pr.
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529
larger for higher c1 ). Physically c1 ¼ 0 indicates to the insulated situation, whereas situation in which c1 – 0 corresponds to constant wall temperature. It is evident that larger c1 prompt the rate of heat transport thereby maximize the fluid thermal state. Fig. 12 demonstrates the decreasing behavior of temperature for Prandtl number Pr. Effects of thermophoretic parameter N t on temperature are depicted in Fig. 13. An enhancement in temperature is seen for N t . Higher N t correspond to the difference between reference temperature and wall temperature of system. Increasing behavior of Brownian motion parameter N b on temperature is pointed out in Fig. 14. It is noticed that the role of Brownian motion parameter N b for temperature profile is accelerating. Such situation is developed due to random motion of molecules and temperature profile enhances. 4.3. Concentration Figs. 15–18 are sketched to describe the response of concentration for curvature parameter k, Newtonian mass parameter c2 , Brownian motion parameter N b and thermophoretic parameter N t physically. Fig. 15 pointed out that larger curvature parameter
Fig. 17. Concentration / via N b .
Fig. 18. Concentration / via N t . Fig. 15. Concentration / via k.
rises the concentration. Impact of Newtonian mass parameter c2 on concentration is disclosed in Fig. 16. It is revealed that magnitude of concentration is maximum for c2 . Fig. 17 encloses the behavior of concentration for larger Brownian motion parameter N b . It is observed that for higher N b the collision between macroscopic and random motion of particles nanoliquid enhances and thus concentration decays. Thermophoretic parameter N t behavior on concentration profile is exhibited in Fig. 18. This figure illustrates that concentration enhances due to enhancement in N t . Physically more nanoparticles are thrown away from the heated surface. The nanomaterial volume fraction quantity boosts up. 4.4. Motile density microorganism
Fig. 16. Concentration / via c2 .
To see the influence of curvature parameter k, Newtonian motile density microorganisms parameter c3 and Peclet number Pe on motile density microorganism the Figs. 19–21 are made. Impact of curvature parameter on motile density are visualized through Fig. 19. It is determined that larger curvature effects boosts up the microorganisms and thus motile density rises.
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Fig. 20 portrays the motile density in response to Newtonian motile density parameter c3 . This plot describes that motile density is decreasing quantity of c3 . Behavior of Peclet number on motile density profile is disclosed in Fig. 21. Physically Peclet number is interpreted as the ratio of advective to diffusive transport charge. It is evident through this figure that an enhancement in Peclet number P e leads to decay in the motile density. 4.5. Heat, mass and motile density microorganisms transfer rates
Fig. 19. Concentration U via k.
Bar charts in Figs. 22–25 are made to see the variation in heat and mass transfer rates for thermophoretic N t and Brownian motion N b parameters respectively. Figs. 22 and 23 point out the increasing behavior of heat transfer rate for both N t and N b respectively. This increment in the heat transfer rate is because of larger N t and N b , which leads towards the situation of temperature difference between reference and wall temperatures and also due to random motion of molecules respectively. Opposite response of mass transfer rate is seen for N t and N b respectively (see bar charts prepared in Figs. 24 and 25). Through these figures it is evident that mass transfer rate is minimum for larger N t and it is maximum for higher N b . Decreasing behavior of motile density transfer rate
Fig. 20. Concentration U via c3 . Fig. 22. Heat transfer rate h0 ðgÞ via N t .
Fig. 21. Concentration U via P e .
Fig. 23. Heat transfer rate h0 ðgÞ via N b .
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5. Final remarks We investigated thermphoretic and Brownian motion effects on peristalsis of Sisko nanoliquid with motile gyrotactic microorganism. The important outcomes are as fallows:
Fig. 24. Mass transfer rate /0 ðgÞ via N t .
Velocity have opposite behavior for first order and second order velocity slip parameters. Reverse behavior of velocity is seen for larger Sisko material parameter b in case of shear thinning and shear thickening cases. Velocity enhances for Gt ; Gc and it decays for Rb . Temperature is low for larger Prandtl number Pr and it boosts up for thermophoretic and Brownian motion phenomena. Newtonian heating parameter c1 produces significant increment in temperature profile. It is seen that concentration profile reduces significantly for larger Newtonian mass parameter c1 . Thermophoretic N t and Brownian motion N b parameters have opposite behavior for concentration. Motile density is decreasing quantity of c3 and P e .
References
Fig. 25. Mass transfer rate /0 ðgÞ via N b .
Fig. 26. Micro transfer rate U0 ðgÞ via P e .
for Peclet number P e is noticed (see bar charts made in Fig. 26). Physically larger Peclet number P e significantly maximize the swimming speed of micro cells and thus motile density transfer rate reduces.
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Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Numerically framing the features of second order velocity slip in mixed convective flow of Sisko nanomaterial considering gyrotactic microorganisms Shahid Farooq a,⇑, Tasawar Hayat a,b, Ahmed Alsaedi b, Bashir Ahmad b a b
Department of Mathematics, Quaid-I-Azam University, 45320 Islamabad 44000, Pakistan Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 13 April 2017 Received in revised form 29 April 2017 Accepted 2 May 2017
Keywords: Sisko nanomaterial Mixed and bio convection Compliant wall properties Second order velocity slip Heat, mass and motile microorganism density Newtonian heat, mass and motile microorganism density
a b s t r a c t Here peristalsis of Sisko nanoliquid with gyrotactic microorganism in a curved channel is investigated. Channel boundaries comprises the wall properties and second order slip conditions for velocity. Consideration of Newtonian heat, mass and gyrotactic microorganisms aspects characterizes the heat, mass and motile density transfer processes. Flow formulation is established utilizing constitutive relations of Sisko fluid. Lubrication theory is employed for the simplification of governing expressions. Further numerical solution is carried out for velocity, temperature, concentration and motile density gyrotactic microorganisms. The numerical solution is justified through graphical results. Graphical discussion determined that velocity has opposite behavior for first and second order velocity slip parameters. Interestingly Sisko fluid parameter has opposite impact on velocity for both shear thinning and shear thickening cases. It is seen that temperature enhances for Newtonian heating whereas concentration and gyrotactic microorganism reduces for Newtonian mass and Newtonian gyrotactic microorganisms. Ó 2017 Published by Elsevier Ltd.
1. Introduction During the past few decades the nanofluids have become very popular among the recent scientists, engineers, mathematicians, medical specialists working in material science, electronics systems, mechanical and health care branches. Commonly nanofluids with nanometer-size solid particles (compared with nanofluid particles of millimeter or micrometer size) have many rheological properties, better stability and significantly higher thermal conductivity efficient. The surprising mechanical, electrical, optical, and thermal characteristics of nanomaterials have developed them most aspired after materials of the existing situation. By definition the nanofluids are the dilute suspensions of nano-sized particles and fibers immersed in materials. In view of such facts Choi [1] was the first who used the terminology of nanofluids in 1995. The nanoliquids as a whole change the thermal ability of such compounds which commences e.g. density, viscosity, diffusivity and thermal conductivity. The thermal conductivity is very important for all the aforementioned physical quantities. Mostly the new ⇑ Corresponding author. E-mail (S. Farooq).
addresses:
[email protected],
[email protected]
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.05.005 0017-9310/Ó 2017 Published by Elsevier Ltd.
researchers have used nanoparticles generally synthesize of oxide ceramics, nonmetals, metal carbides, nitrides, metals (Al, Cu, Au), metal oxides (Al2O3, TiO2, SiO2, ZnO2), single wall or multiple wall carbon nanotubes (i.e. SWCNT or MWCNT) to form nanofluids. Ordinary liquid has weaker thermal conductivity. This weaker conductivity can be improved greatly with the use of such nanoparticles. There are numerous combinations of nanoparticles e.g. various combination of nanoparticles with/without surfactant molecules can be distributed into ordinary liquids like water, ethylene or propylene glycol and other alternative lubricants. In this direction, the characteristics of nanoparticles Brownian motion factor in base liquid is quite significant. A large quantity of heat is created in microelectro mechanical and heat exchangers procedures to control the system performance. Thermal conductivity of ordinary liquids is exceeded by insertion of nanoparticles just to cool down such industrial activities. The nanoparticles have achieved great significance in several biological and engineering processes such as medicine, solar cells, plasma and laser cutting, catalysts, electronics, materials manufacturing, glass industry, in cooling of turbine blades processes and in several other industrial processes. A considerable research on nanofluids have been communicated by the researchers. Some important studies covering
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the aspects of Brownian diffusion and thermophoresis can be mentioned (for detail see [2–10]). Microorganism particles are significantly useful in the production of several commercial and industrial products e.g. bio fuel made from waste, fertilizers, ethanol, etc. The motile microorganisms are self-urged which enlarges the denseness of ordinary liquids by swimming toward a particular direction within the material in attraction to motivate like gravity, oxygen, daylight whereas nanoparticles cannot swim. Gyrotactic microorganisms are also used during the production of biodiesle and hydrogen, an important sustainable energy source and in water treatment plants. Due to such fact it is necessary to analyze the swimming patterns and mass transfer characteristics of microorganisms so that demands of the organisms seen more useful, profitable and significant for the bright future of the whole mankind. Bioconvection is the development of distinct types of random liquids sequences at the microscopic level because of instinctive swimming of automotive microorganisms which are included in the water and in the other materials denser than water. Those microorganisms swimming is influenced by the reasonable procedures such as oxygen for respiration, searching for nutrient, upgrade light input for photosynthesis. The arbitrary movement of nanoparticles is created due to the thermophoretic and Brownian diffusion features and are transported by the drift of base fluid. Nanofluids have extensive application in micro fluidic tool as they are very supportive for mass transport improvement and induce mixing particularly in micro volumes [11]. Kuznetsov [12] employed the idea of bioconvection of nanoliquid with gyrotactic microorganisms. Analysis of magneto Jeffrey nanoliquid considering gyrotactic microorganism is reported by Bhatti et al. [13]. Few studies comprising the characteristics of gyrotactic microorganism with nanofluid can be stated through [14–18] and several studies therein. Peristalsis is an inherent property of many industrial and biopyhsical mechanisms. Nodoubt, peristaltic mechanism is due to propagating wave along the boundaries of flexible channel/tube. By owing the importance of peristalsis in physiology and industry, the physiologists and scientists consider it as a major mechanism for the transportation of several physiological and industrial materials. Mainly mechanism of peristalsis is experienced in various biological processes like food swallowing via oesophagus and gastrointestinal tract, movement of chyme in the gastrointestinal tract, spermatozoa movement in ductus afferents of male reproductive tract, ovum movement in the fallopian tube, urine motion from kidney to bladder, embryo transportation through uterus and several others. Many advanced medical devices such as dialysis machine, open heart bypass machine, B.P device hose, infusion and finger pumping devices have been made on the theory of peristalsis pumping. In nuclear industry peristalsis is used for the transportation of noxious, sterile, corrosive and many waste materials. After the initial experimental and theoretical attempts presented by Latham [19] and Shapiro et al. [20], several researches on the peristalsis have been listed with and without nanomaterials in literature (for detail see [21–30]). In past mostly researches on peristaltic transport have been studied in planner or straight channel. However in many realistic situations the flows in conduits and ducts are curvature dependent. Sato et al. [31] filled this void with the primary presentation on peristaltic motion of viscous liquid in a curved geometry. Later on several researchers [32–40] focused their attention on peristaltic motion with and without nanomaterial in a curved channel. Our motivation here is to model the peristaltic transport of Sisko nanomaterial in the presence of motile gyrotactic microorganisms. Curved channel walls comprises Newtonian effects for
heat, mass and motile density gyrotactic microorganisms respectively. Momentum equation is studied in presence mixed of convection. Further viscous dissipation effects are absent. The flow governing expressions are simplified through large wavelength and small Reynolds number. Coupled non-linear systems are solved numerically via shooting procedure a well-known built-in technique in Mathematica 8 computational software. The obtained numerical results for velocity, temperature, concentration and gyrotactic microorganisms are analyzed through graphical discussion. 2. Problem formulation Here we examined the peristalsis of an incompressible nonNewtonian (Sisko) nanomaterial with gyrotactic microorganisms in a curved channel having radius R and uniform width 2a0 twisted in a semi-circle with center at O. The axial and radial directions are denoted by x and r respectively. The velocity components v 2 and v 1 are along axial and radial directions simultaneously. Aspects of Brownian and thermophoresis are present in heat and mass equations. Moreover viscous dissipation and thermal radiation features are absent. The channel boundaries are subject to constant temperature, concentration and volume fraction motile microorganisms T 0 ; C 0 and N 0 respectively. Characteristics of compliant walls are also discussed. The second order velocity slip conditions are employed. Further Newtonian type heating, concentration and gyrotactic microorganisms effects at channel boundaries are also studied. Fluid motion is due to the propagation of flexible walls along xdirection with constant speed c. Mathematically such walls expressed as (see Fig. 1):
ðx; tÞ ¼ a0 a1 sin h
2p ðx ct Þ : kc
ð1Þ
Here a0 denotes the half width of channel, a1 the amplitude, kc and stress tensor s the wavelength and t the time. The velocity v for Sisko liquid are given by [36,41,42]:
v ¼ ðv 1 ðx; r; tÞ; v 2 ðx; r; tÞ; 0Þ; h
n1
i
s ¼ a þ bðc_ Þ 2 A1 ;
ð2Þ ð3Þ
In Eq. (3) a; b and n designated the constants of Sisko material. Here we discuss both shear thinning (i.e. n < 1) and thickening (i.e. n > 1) cases. One can achieve the stress tensor for generalized power law model by choosing a ¼ 0. Furthermore it can be reduced to Newtonian fluid model by taking a ¼ b ¼ 0. Also A1 the deformation rate of tensor and c_ the second invariant strain tensor are
þ ðrv ÞT A1 ¼ rv 1 2
c_ ¼ trðA1 Þ2 ;
ð4Þ ð5Þ
The flow governing equations are [45–49]:
@ @ v 2 fðR þ r Þv 1 g þ R ¼ 0; @r @ x
q
@ v 1 @ v R v @ v 1 v 22 1 1 þ 2 2 þ v @r R þ r @ x R þ r @t rx @p 1 @ R @s sxx rr g þ ¼ þ ; fðR þ r Þs @r R þ r @r R þ r @ x R þ r
ð6Þ
ð7Þ
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Fig. 1. Physical diagram.
q
@ v 2 @ v R v @ v 2 v 1 v 2 1 2 þ 2 þ v @r R þ r @ x R þ x @t o R @p 1 @ n 2 rx ¼ þ 0 ðR þ r Þ s 2 R þ x @ x R þ r @r xx R @s þ qbT g ðT T 0 Þ þ qbC g ðC C 0 Þ þ R þ r @ x
h
h
ð8Þ
" # @T @T R v 2 @T j @2T 1 @T R2 @2T þ v 1 þ ¼ þ þ @r R þ r @ x ðqC Þf @r 2 R þ r @r ðR þ r Þ2 @ x2 @t
s DT Tm
ðrT rT Þ; ð9Þ
" # @C @C R v 2 @C @2C 1 @C R2 @ 2 C þ v1 þ ¼ DB þ þ @r R þ r @ x @r 2 R þ r @r ðR þ r Þ2 @ x2 @t 2 3 DT 4@ 2 T 1 @T R2 @2T 5 þ þ þ 2 ; T m @R2 R þ R @R R þ R2 @ X ð10Þ
" # @N @N R v 2 @N @2N 1 @N R2 @2N 1 þ þ ¼ D þ v þ n @r R þ r @ x @r 2 R þ r @r ðR þ r Þ2 @ x2 @t b WC @ @C @ @C þ : N N þ @r @ x @r C 0 @r ð11Þ
In above expressions (6)–(11) the quantities q; p; bT ; bC ; C f ; s ; j; DB ; DT ; T m ; Dn ; b ; W C denote the density, pressure, thermal expansion coefficient, mass expansion coefficient, specific heat of basefluid, ratio of nano and base materials heat capacities, thermal conductivity constant, Brownian diffusion coefficient, thermophoretic diffusion coefficient, motile microorganism diffusion coefficient, chemotaxis number and maximum cell swimming speed respectively. Also (T,C and N) are the temperature, concentration and volume fraction of motile microorganisms. The xx ; s rx and s rr are given as stress components s
h
n1
sxx ¼ 2 a þ bðc_ Þ 2
i R @ v v 1 2 þ ; r þ R @ x r þ R
i@ v
ð12Þ
2
@r
n1
srr ¼ 2 a þ bðc_ Þ 2
ðN N0 ÞcDq;
þ s DB ðrC rT Þ þ
n1
srx ¼ a þ bðc_ Þ 2
þ
R @ v 1 v 2 ; r þ R @ x r þ R
ð13Þ
i@ v 1 : @r
ð14Þ
The dimensional form of compliant wall [23,36,43,44,50] and second order velocity slip boundary constraints are
" # 2 R @3 @3 @ h s þ m þ d 1 ðr þ R Þ @ x3 @ x@t 2 @t@ x o @s 1 @ n xx 2 ¼ ðr þ R Þ srx þ q 2 @ r @ x ðr þ R Þ @ v 2 @ v R v 2 @ v 2 v 1 v 2 1 2 þ þ v þ qbT g ðT T 0 Þ r þ R @ x @r R þ r @t þ qbC g ðC C 0 Þ ðN N0 ÞcDq; at r ¼ h;
ð15Þ
v 2 a0 srx b0
rx @s ¼ 0; @r
at r ¼ h;
ð16Þ
v 2 þ a0 srx þ b0
rx @s ¼ 0; @r
at r ¼ h:
ð17Þ
Similarly Newtonian heating, mass and motile density gyrotactic microorganisms boundary constraints are
T ¼ T 0 ; C ¼ C 0 ; N ¼ N0 ;
at r ¼ h;
@T @C @N ¼ c1 T; ¼ c2 C; ¼ c3 N; @r @r @r
ð18Þ at r ¼ h:
ð19Þ
Dimensionless variables are
d¼
x r a0 ; x¼ ; r¼ ; kc kc a0
ct t¼ ; kc
sij ¼
v1 ¼
v 1 c
;
v2 ¼
v 2 c
; g¼
h a2 p ; p¼ 0 ; a0 clkc
a0 T T0 C C0 N N0 R s; h ¼ ; /¼ ; U¼ ; k¼ ; c l ıj T0 C0 N0 a0
Re ¼
qca0 qb ga2 T 0 qb ga2 C 0 N0 cDq ; Gt ¼ T 0 ; Gc ¼ C 0 ; Rb ¼ ; l cl cl l
Pr ¼
lc p s D B C 0 s D T T 0 b WC sa3 ; Nb ¼ ; Nt ¼ ; Pe ¼ ; E1 ¼ 3 0 ; j m mT m Dn k c lc
E2 ¼
mca30 k3c l
3
; E3 ¼
da0
k3c l
; b ¼
n1 b c ; a a0
ð20Þ
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v1 ¼ d
k @w ; r þ k @x
v2 ¼
@w ; @r
ð21Þ
where d the wave number, k the curvature parameter, b the Sisko fluid constant, p the dimensionless pressure, Gt the thermal Grashof number, Gc the mass Grashof number, Rb the bioconvection Ralyeigh parameter, Pr the ratio of momentum diffusivity to thermal diffusivity (i.e. Prandtl number), N b the Brownian motion quantity, Nt the thermophoretic motion quantity, P e the Peclet number, E1 ; E2 ; E3 the compliant wall parameters and w the stream function. After using dimensionless quantities (21), large wavelength ðd ! 0Þ and low Reynolds number ðRe ! 0Þ approximations Eq. (6) vanished identically and other expressions are reduced to the forms:
c_ ¼
@w
@2w þ @r @r 2 r þ k
!2 :
ð32Þ
Results for straight/planner channel can be retained by taking curvature parameter larger (i.e. k ! 1). Also present study can be reduced for Newtonian case by putting dimensionless Sisko material parameter zero (i.e. b ¼ 0). 3. Methodology The analytical or exact solution of the resulting non-linear and coupled equations seems difficult. Thus numerical solution of the resulting problem has been developed through numerical approach called Shooting technique. Further this numerical solution is justified via graphical discussion for each flow quantity.
dp ¼ 0; dr
ð22Þ
o k dp 1 @ n 2 ¼ ðr þ kÞ srx þ Gt h þ Gc / Rb U; 2 r þ k dx ðr þ kÞ @r
ð23Þ
4. Discussion
ð24Þ
Our key intention here is to interpret the graphical results of velocity, temperature, concentration and motile gyrotactic microorganisms for different embedded dimensionless variables. This section is further divided into five subsections for of better understanding.
2 ! @ h 1 @h @h @/ @h þ Pr N þ N þ ¼ 0; t b @r 2 r þ k @r @r @r @r 2
@2/ 1 @/ Nb @ 2 h 1 @h þ þ þ 2 @r r þ k @r Nt @r2 r þ k @r
! ¼ 0;
ð25Þ
! @2U 1 @U @ U @/ @2/ þ U 2 ¼ 0: þ Pe @r @r 2 r þ k @r @r @r
ð26Þ
Eq. (22) indicates that the pressure p along radial direction is constant. Here h; / and U are the dimensionless temperature, concentration and motile density profiles respectively. Eliminating pressure from Eqs. (22) and (23) through cross differentiation we get
" # 2 o @ k @ n kM @w 2 ðr þ kÞ srx : 0¼ 1 @r ðr þ kÞ @r @r ð1 þ m2 Þðr þ kÞ
ð27Þ Dimensionless boundary conditions are
" # k @3 @3 @2 þ E g E1 3 þ E2 3 ðr þ kÞ @x @t@x @x@t 2 o 1 @ n 2 ðr þ kÞ srx þ Gt h þ Gc / Rb U; ¼ 2 @r ðr þ k Þ
at r ¼ g; ð28Þ
@w @ srx asrx b ð29Þ ¼ 0; h ¼ 0; / ¼ 0; U ¼ 0; at r ¼ g; @r @r @w @ srx @h @/ þ asrx þ b þ c1 ðh þ 1Þ ¼ 0; þ c2 ð/ þ 1Þ ¼ 0; ¼ 0; @r @r @r @r @U ð30Þ þ c3 ðU þ 1Þ ¼ 0; at r ¼ þg; @r
g ¼ 1 þ sin ½2pðx tÞ:
0 0 Here a ¼ aa0 and b ¼ ab0 are the dimensionless first and second order velocity slip numbers respectively. Here c1 ¼ c1 a0 , c2 ¼ c2 a0 and c3 ¼ c3 a0 are Newtonian heating, mass and motile density gyrotactic microorganisms dimensionless numbers and
srx
h
¼ 1 þ b ðc_ Þ
n1 2
i
! @w @2w @r ; 2þ @r rþk
ð31Þ
4.1. Velocity This subsection contains the graphical discussion of velocity. Here velocity profile is discussed for both shear thinning and shear thickening properties of Sisko nanomaterial. Thus Figs. 2–9 are made to see the reaction of velocity for first order velocity slip parameter a, second order velocity slip parameter b, curvature parameter k, Sisko material constant b , compliant wall numbers E1 ; E2 ; E3 , thermal Grashof number Gt , mass Grashof number Gc and bioconvection Ralyeigh parameter Rb . Part ðiÞ of each of these figures describes the shear thinning behavior whereas part ðiiÞ of each of these figures describes the shear thickening behavior of Sisko material. Fig. 2(i) and (ii) shows the velocity behavior for first order velocity slip parameter a. It is seen through this figure that velocity profile reduces for both shear thinning and thickening characteristics of Sisko material when first order velocity slip parameter is increased. It is evident form Fig. 3(i) and (ii) fluid velocity has opposite behavior for second order velocity slip parameter b when compared with a. Impact of curvature parameter k on velocity is witnessed through Fig. 4(i) and (ii). This figure determine that symmetry of velocity profile is disturbed at center of channel due to curvature. It is worthmentioning that velocity has revers behavior at both channel walls. Fig. 5(i) and (ii) provides the action of Sisko material parameter b on velocity. This figure depicts that velocity enhances for shear thinning (i.e. n < 1) behavior and it decreases for shear thickening (i.e. n > 1) behavior when Sisko material parameter b is increased. Variation in velocity for different values of compliant wall properties (E1 ; E2 and E3 ) are disclosed in Fig. 6(i) and (ii). Clearly it is noticed that velocity attains its maximum value when wall tension E1 and mass characterizing E2 are enhanced. The velocity decays for larger wall damping E3 parameter. Impact of thermal Grashof number Gt on velocity has been visualized through Fig. 7(i) and (ii). This figure depicts that velocity becomes larger when impact of thermal buoyancy forces are increased (i.e. larger thermal Grashof number). It is also noticed that effects of Gt on velocity profile are more prominent for shear thinning case (i.e. n < 1) than for shear thickening case (i.e. n > 1). Velocity profile has similar results for concentration Grashof number Gc as for thermal Grashof number Gt (see Fig. 8(i) and (ii)). Physically it describes that thermal and concentration buoyancy forces provide small resistance in the flow field and thus
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Fig. 2. (i and ii): Velocity
v2
via a.
Fig. 3. (i and ii): Velocity
v2
via b.
Fig. 4. (i and ii): Velocity
v2
via k.
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Fig. 5. (i and ii): Velocity
Fig. 6. (i and ii): Velocity
v2
v2
via b .
via wall properties.
Fig. 7. (i and ii): Velocity
v2
via Gt .
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Fig. 8. (i and ii): Velocity
v2
via Gc .
Fig. 9. (i and ii): Velocity
v2
via Rb .
Fig. 10. Temperature h via k.
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velocity is enhanced. Efficiency of bioconvective Rayleigh number Rb on velocity is sketched in Fig. 9(i) and (ii). This figure witnesses that bioconvective Rayleigh number has dominating role for velocity profile (i.e. velocity profile decreases for higher Rb ). Finally through all these figures one can easily concludes that velocity profile is maximum for shear thinning case (i.e. n < 1) and it is minimum for shear thickening case (i.e. n > 1). 4.2. Temperature Figs. 10–14 visualize the temperature variation for curvature parameter k, Newtonian heating parameter c1 , Prandtl number Pr, thermophoretic parameter N t and Brownian motion parameter N b . Fig. 10 is sketched to view the impact of curvature parameter k on temperature. This figure demonstrates that curvature parameter has dominating effects on temperature (i.e. temperature reduces with an increment in k). Also it is noticed that temperature is maximum in curved channel (i.e. small k) when compared with planner channel (i.e. larger k). Behavior of temperature for Newtonian heating parameter c1 can be seen through Fig. 11. This figure reveals that c1 has assisting role for temperature (i.e. h becomes
Fig. 11. Temperature h via c1 .
Fig. 13. Temperature h via N t .
Fig. 14. Temperature h via N b .
Fig. 12. Temperature h via Pr.
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larger for higher c1 ). Physically c1 ¼ 0 indicates to the insulated situation, whereas situation in which c1 – 0 corresponds to constant wall temperature. It is evident that larger c1 prompt the rate of heat transport thereby maximize the fluid thermal state. Fig. 12 demonstrates the decreasing behavior of temperature for Prandtl number Pr. Effects of thermophoretic parameter N t on temperature are depicted in Fig. 13. An enhancement in temperature is seen for N t . Higher N t correspond to the difference between reference temperature and wall temperature of system. Increasing behavior of Brownian motion parameter N b on temperature is pointed out in Fig. 14. It is noticed that the role of Brownian motion parameter N b for temperature profile is accelerating. Such situation is developed due to random motion of molecules and temperature profile enhances. 4.3. Concentration Figs. 15–18 are sketched to describe the response of concentration for curvature parameter k, Newtonian mass parameter c2 , Brownian motion parameter N b and thermophoretic parameter N t physically. Fig. 15 pointed out that larger curvature parameter
Fig. 17. Concentration / via N b .
Fig. 18. Concentration / via N t . Fig. 15. Concentration / via k.
rises the concentration. Impact of Newtonian mass parameter c2 on concentration is disclosed in Fig. 16. It is revealed that magnitude of concentration is maximum for c2 . Fig. 17 encloses the behavior of concentration for larger Brownian motion parameter N b . It is observed that for higher N b the collision between macroscopic and random motion of particles nanoliquid enhances and thus concentration decays. Thermophoretic parameter N t behavior on concentration profile is exhibited in Fig. 18. This figure illustrates that concentration enhances due to enhancement in N t . Physically more nanoparticles are thrown away from the heated surface. The nanomaterial volume fraction quantity boosts up. 4.4. Motile density microorganism
Fig. 16. Concentration / via c2 .
To see the influence of curvature parameter k, Newtonian motile density microorganisms parameter c3 and Peclet number Pe on motile density microorganism the Figs. 19–21 are made. Impact of curvature parameter on motile density are visualized through Fig. 19. It is determined that larger curvature effects boosts up the microorganisms and thus motile density rises.
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Fig. 20 portrays the motile density in response to Newtonian motile density parameter c3 . This plot describes that motile density is decreasing quantity of c3 . Behavior of Peclet number on motile density profile is disclosed in Fig. 21. Physically Peclet number is interpreted as the ratio of advective to diffusive transport charge. It is evident through this figure that an enhancement in Peclet number P e leads to decay in the motile density. 4.5. Heat, mass and motile density microorganisms transfer rates
Fig. 19. Concentration U via k.
Bar charts in Figs. 22–25 are made to see the variation in heat and mass transfer rates for thermophoretic N t and Brownian motion N b parameters respectively. Figs. 22 and 23 point out the increasing behavior of heat transfer rate for both N t and N b respectively. This increment in the heat transfer rate is because of larger N t and N b , which leads towards the situation of temperature difference between reference and wall temperatures and also due to random motion of molecules respectively. Opposite response of mass transfer rate is seen for N t and N b respectively (see bar charts prepared in Figs. 24 and 25). Through these figures it is evident that mass transfer rate is minimum for larger N t and it is maximum for higher N b . Decreasing behavior of motile density transfer rate
Fig. 20. Concentration U via c3 . Fig. 22. Heat transfer rate h0 ðgÞ via N t .
Fig. 21. Concentration U via P e .
Fig. 23. Heat transfer rate h0 ðgÞ via N b .
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5. Final remarks We investigated thermphoretic and Brownian motion effects on peristalsis of Sisko nanoliquid with motile gyrotactic microorganism. The important outcomes are as fallows:
Fig. 24. Mass transfer rate /0 ðgÞ via N t .
Velocity have opposite behavior for first order and second order velocity slip parameters. Reverse behavior of velocity is seen for larger Sisko material parameter b in case of shear thinning and shear thickening cases. Velocity enhances for Gt ; Gc and it decays for Rb . Temperature is low for larger Prandtl number Pr and it boosts up for thermophoretic and Brownian motion phenomena. Newtonian heating parameter c1 produces significant increment in temperature profile. It is seen that concentration profile reduces significantly for larger Newtonian mass parameter c1 . Thermophoretic N t and Brownian motion N b parameters have opposite behavior for concentration. Motile density is decreasing quantity of c3 and P e .
References
Fig. 25. Mass transfer rate /0 ðgÞ via N b .
Fig. 26. Micro transfer rate U0 ðgÞ via P e .
for Peclet number P e is noticed (see bar charts made in Fig. 26). Physically larger Peclet number P e significantly maximize the swimming speed of micro cells and thus motile density transfer rate reduces.
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