Research Article Heat and Mass Transfer in Three-Dimensional Flow of an Oldroyd-B Nanofluid with Gyrotactic Micro-Organisms

Home , Lewis number

Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 6790420, 15 pages https://doi.org/10.1155/2018/6790420

M. Sulaiman,1,2 Aamir Ali ,1 and S. Islam 2

1 Department of Mathematics, COMSATS University Islamabad, Attock Campus, Attock, 43600, Pakistan 2Department of Mathematics, Abdul Wali Khan University, Mardan, Khyber Pakhtunkhwa, 23200, Pakistan

Correspondence should be addressed to Aamir Ali; aamir [email protected]

Received 10 December 2017; Revised 28 February 2018; Accepted 29 April 2018; Published 6 September 2018

Academic Editor: Efstratios Tzirtzilakis

Copyright © 2018 M. Sulaiman et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Tis paper discusses the three-dimensional fow of the gyrotactic bioconvection of an Oldroyd-B nanofuid over a stretching surface. Teory of microorganisms is utilized to stabilize the suspended nanoparticles through bioconvection induced by the efects of buoyancy forces. Analytic solution for the governing nonlinear equations is obtained by using homotopy analysis method (HAM). Te efects of involved parameters on velocity, temperature, nanoparticles concentration, and density of motile microorganisms are discussed graphically. Te local Nusselt, Sherwood, and motile microorganisms numbers are also analyzed graphically. Several known results have been pointed out as the particular cases of the present analysis. It is found that the non-Newtonian fuid parameters, i.e., relaxation time parameter �1 and retardation time parameter �2, have opposite efects on the velocity profle. Te velocity of the fuid and boundary layer thickness decreases for increasing relaxation time while it decreases for increasing retardation time efects.

1. Introduction non-Newtonian fuid models [2–10]. However, in practical applications sometimes, the fow is three-dimensional. In In recent years the signifcance of the study of non- view of such motivation, researchers have studied the three- Newtonian fuids has remarkably increased due to its appli- dimensional fow for diferent fow geometries (see ref. [11– cations in engineering, industry, and biological sciences, e.g., 17]). in material processing, food, and cosmetic industries [1]. Te problem of heat and mass transfer analysis over a Tese applications generally involve complex geometries and stretched boundary layer is of great interest to researchers due require that the response of the fuid to given temperature to its engineering and industrial applications like glass fber, conditions is controlled. Te basic equations that governs manufacture and drawing of plastics and rubber sheets, paper the fow of non-Newtonian fuids are highly nonlinear. Te production, cooling of continuous stripes and an infnite mathematical description of a non-Newtonian fuid requires metallic sheet, the continuous casting, polymer extrusion a constitutive equation that determines the rheological prop- process, heat treated materials travelling on conveyer belts, erties of the fuid. Due to the complexity, the aforementioned food processing, and many others. Sakiadis [18] has done the fuids are generally divided into three types, namely, the pioneer work in the analysis of the boundary layer fow on diferential type, the rate type, and the integral type. Te a moving surface, which was later extended by Crane [19] simplest example of rate type fuid is Maxwell fuid model by considering the linearly stretching surface. Rashad and which narrates the properties of relaxation time. But it cannot Kabeir [20] studied the heat and mass transfer in transient predict the retardation time efects. Te Oldroyd-B fuid fow by mixed convection boundary layer over a stretching is one subclass of the rate type fuid which describes the sheet embedded in a porous medium with chemically reactive properties of both the relaxation time and the retardation species. Sandeep and Sulochana [21] studied the general- time. Tere are several investigations in this direction for two- ized model for momentum and heat transfer behavior of dimensional fow confgurations with diferent features of the Jefrey, Maxwell, and Oldroyd-B nanofuids over a stretching 2 Mathematical Problems in Engineering

→∞,→ ∞,→ ∞ →0, v → 0 Nanoparticles and micro-organisms

= w = w = w

v=v(y) =by  = () =  Figure 1: Geometry of the problem.

surface with nonuniform heat source/sink. Afer that, several In present study, we have entered a new concept in the investigations have been done by considering diferent fow problem of boundary layer fow passing a stretching sheet. We confgurations [22–28]. have considered the three-dimensional bioconvection fow of With the advent of nanoscience, nanofuids have become an Oldroyd-B nanofuid which contains self-impelled motile a focus of attention in the study of fuid fow in the presence gyrotactic microorganisms, over a bidirectional stretching of nanoparticles. Nanofuids can be used to increase the surface.TisworkisanextensionoftheworkofW.A. thermal conductivity of the fuids and are stable fuids having Khan et al. [16] by considering the gyrotactic microorganisms much better writing, dispersion, and spreading properties in the three-dimensional fow of Oldroyd-B nanofuid. Te on solid surfaces. Choi and Eastman [29] initially presented problem is formulated in terms of governing equations of the idea of suspension of nanoparticles in base fuids. Later, motion with appropriate boundary conditions, which are Buongiorno [30] extended the concept by considering Brow- then transformed into a system of ordinary diferential nian motion and thermophoresis movement of nanoparticles equations using similarity transformations. Te analytical in view of application in hybrid power engine, thermal solution is found using Homotopy Analysis Methods (HAM). management, heat exchanger, domestic refrigerators, etc. Te efects of involved physical parameters on the fow felds Tey have enhanced heat conduction power as compared to are analyzed with the help of graphs. base fuids and are useful in cancer therapy and medicine. With the presentation of a reliable model of nanofuid 2. Problem Formulation given by Buongiorno [30], nanofuids have become a subject of great interest for researchers in the last few decades We have considered the steady, incompressible three- [31–34]. Recently, Nabwey et al. [35] studied the group dimensional fow of an Oldroyed-B nanofuid containing method analysis of mixed convection stagnation-point fow gyrotactic microorganisms past a stretching surface. Te sur- of non-Newtonian nanofuid over a vertical stretching surface is maintained at constant temperature ��, the nanopar- face. ticlevolumefraction��, and the number of motile microor- Bioconvection is a phenomenon that is used to describe ganisms ��; the ambient values of temperature, nanoparticle the unstructured pattern and instability produced by the volume fraction, and number of motile microorganisms are microorganisms, which are swimming to the upper part of denoted by �∞, �∞,and�∞, respectively, while the reference a fuid and has a lesser density. Due to swimming upwards, values are by �0, �0,and�0, respectively. Te microorganisms these involved microorganisms such as gyrotactic microor- are taken into the nanofuid to stabilize the nanoparticles due ganisms like algae tend to concentrate in the upper part of the to bioconvection. Te mechanical analog for the Oldroyd- fuid layer, causing unstable top heavy density stratifcation B model and the geometry of the present analysis is shown (see [36]). Nanofuid bioconvection phenomenon describes in Figure 1. Te boundary layer form of the continuity, the unstructured density stratifcation and formation of momentum, energy, nanoparticle concentration, and density patterns due to synchronized interaction of the nanoparticles, of microorganisms for three-dimensional Oldroyd-B fuid is thicker self-propelled microorganisms and buoyancy forces. as follows ([16, 37, 50]): Brownian motion and thermophoresis efect produce the motion of these particles, which is independent of the move- �� V �� + + =0, (1) ment of motile microorganisms. Tat is why the combined �� interface of bioconvection and nanofuids becomes signif- 2 2 2 icant for microfuidic appliances. Uddin et al. studied the �� 2 � � 2 � � 2 � � � + V +� +�1 (� + V +� computational investigation of Stefan blowing and multiple- �� 2 �� 2 ��2 slip efects on buoyancy-driven bioconvection nanofuid fow �2� �2� �2� �2� with microorganisms [37] and numerical solutions for gyro- +2�V +2V� +2�� )=] [ tactic bioconvection in nanofuid saturated porous media �� 2 with Stefan blowing and multiple slip efects [38]. Diferent �3� �3� �3� �� 2� types of microorganisms, theoretical biconvection models, +� (� + V +� − have been studied [39–51]. 2 ��2 �� 2 ��3 �� 2 Mathematical Problems in Engineering 3

�� 2V �� 2� 1 � � � − − )] + [� �(1−� )(� 1 and 2 are the relaxation time and retardation time, � is 2 2 � ∞ �� the density of nanofuid, �� is the density of nanoparticles, �� is the density of microorganisms particles, � is the gravity, −�∞)−(�� −��)(�−�∞)−�(�� −��)(� �=(��)�/(��)� is the heat capacity ratio, �� and �� are the stretching velocities, �, � > 0 are the stretching rate, and � ,� ,� −�∞)] g, 1 1 1 are the dimensional constant. Let us introduce the following transformations: (2) �V �V �V �2V �2V �2V � � + V +� +� (�2 + V2 +�2 �=√ �, �� 1 ��2 �� 2 �� 2 ]

� �2V �2V �2V �2V �=�� (�) , ⋅+2�V +2V� +2�� )=] [ �� 2 V =�� (�) , (3) �3� �3� �3� �� 2� +� (� + V +� − �=−√�] [� (�) + � (�)] , 2 2 2 3 2 �� (8) �−�∞ �� 2V �� 2� �(�)= , − − )] , �� −�0 �� 2 �� 2 �−� �(�)= ∞ , �� 2� �� −� � + V +� =� +�[� ( ) � 0 � 2 � �� −� (4) �(�)= ∞ , 2 � �� −�0 + � ( ) ], �∞ �� Te continuity (1) is identically satisfed and (2) to (6) takes �� 2� � �2� � + V +� =� + � ( ), the following forms: � 2 2 (5) �� ∞ �� +(�+�)�� −��2 �� 2� � + V +� + 0 [ (� )] = � , �� Δ� �� 2 (6) � �� 2 �� +�1 [2 (� + �) � � −(�+�) � ] (9) and the boundary conditions are as follows: �� V +�2 [2 (� +� )� −(�+�)� ]

�=�� (�) =�� =��, +�(�−��−��)=0, V = V (�) = � =��, � � �� +(�+�)�� −��2

�=0, � �� 2 �� +�1 [2 (� + �) � � −(�+�) � ] (10) �=�� =�0 +�1�, �� V +�2 [2 (� +� )� −(�+�)� ]=0, �=�� =�0 +�1�, 2 �� +��(�+�)�� −��+�� +� (��) =0, �=�� =�0 +�1�, � � (11) �=0, � at (7) �� +��(�+�)�� −��+ � �� =0, � (12) ��→0, � �� +��(�+�)�� −�� V �→ 0 , (13) −��[�� + (�+Ω) ��]=0, ��→�∞, ��→� , ∞ and the boundary conditions (7) is reduced to the following:

��→�∞ � (0) =0, as ��→∞, � (0) =0, in which �, V,and� are the velocity components in the �, � �,and� directions, respectively, ] is the kinematic viscosity, � (0) =1, 4 Mathematical Problems in Engineering

� � � (0) =�, Ω= ∞ , �� −�0 � (0) =1, � �= . � (0) =1, � � (0) =1, (15)

�� (∞) =0, Te nondimensional ordinary diferential (9)-(13) subject to the boundary conditions (14) are solved analytically by using �� (∞) =0, homotopy analysis method (HAM). HAM technique was frst introduced by Liao [52] in his Ph.D. thesis. He devised a � (∞) �→ 0 , parameter that controls convergence of the series solution. Te method has been successfully applied in the literature � (∞) �→ 0 , to compute analytic solution of various problems ([53– � (∞) �→ 0 , 56] and references therein). Te details of the method are given in the appendix. Homotopy analysis method ensures (14) the convergence of derived series solution. Te auxiliary parameter ℎ plays a vital role in controlling and adjusting the convergence region of series solutions. We have plotted the ℎ− where prime is the diferentiation with respect to �.Further- curves shown in Figure 2 and found that the admissible values more, �1,�2, �, ��, ��, ��, ��, ��, ��, ��, ��, Ω,and� of auxiliary parameters ℎ�, ℎ�, ℎ�, ℎ� and ℎ� are −1.45 ≤ ℎ� ≤ are the Deborah number, the mixed convection parameter, −0.05, −1.25 ≤ ℎ� ≤−0.1, −1.45 ≤ ℎ� ≤ −0.05, −1.3 ≤ ℎ� ≤ the buoyancy ratio parameter, the bioconvection Rayleigh −0.15,and−1.5 ≤ ℎ� ≤−0.2, respectively. Te expressions number, the Prandtl number, the Brownian motion param- for local Nusselt number, local Sherwood number, and local eter, the thermophoresis parameter, the Lewis number, the density number of the motile microorganisms are defned as bioconvection Lewis number, the bioconvection Peclet num- follows: ber, the microorganisms concentration diference parameter, �� and stretching ratio parameter which are the nondimensional �� = � , parameters and are defned as follows: �(�� −�0) �� ℎ = � , � =� �, (16) 1 1 �� (�� −�0) � =� �, �� 2 2 �� = � , �� (�� −�0) �� (1 − � )(� −�)(�3/]2) �� = � ∞ � 0 = � , 2 2 2 2 where �� (� /] ) �� =(−� ) , (� −� )(� −� ) � �� = � � � 0 , �=0 � (1 − � )(� −�)� � ∞ � 0 �� =(−�� ) , (17) �(� −� )(� −� )� �� =0 �� = � � � 0 , � (1 − � )(� −�)� �� ∞ � 0 � =(−� ) . � � �� ] �=0 �� = , �� Te dimensionless forms of local Nusselt number, local Sherwood number, and local motile microorganism are as (� −� ) � 0 follows �� =�� , �� −1/2 � �� =−� (0) , (� −�) � =�� 0 , −1/2 � � � �ℎ�� =−� (0) , (18) ��∞ ] ��−1/2 =−�� (0) , �� = , � �� where �� =��/] is the Reynolds number. ] �� = , �� 0 3. Results and Discussion

�� = 0 , Tis section explores the results of three-dimensional fow � �0 of an Oldroyed-B nanofuid over a stretching sheet in the Mathematical Problems in Engineering 5

 = .,  = .,  = .,  = Nr = Rb =.  = .,  = .,  = .,  = Nr = Rb =. −1.0 −0.4

−1.1 −0.5

−0.6

−1.2 (0) (0)   g f −0.7 −1.3 −0.8

−1.4 −0.9

−1.5 −1.0 −0.5 0.0 −1.5 −1.0 −0.5 0.0  g (a) (b) Nt = 0.3, Nb = 0.5, Le = 1, Pr = 1.2 Nt = 0.3, Nb = 0.5, Le = 1, Pr = 1.2

−1.0 −1.0

−1.2 (0) (0) −1.5  

 −1.4 ϕ

−1.6 −2.0

−1.8 −1.5 −1.0 −0.5 0.0 −1.5 −1.0 −0.5 0.0  ϕ (c) (d) Nt = 0.3, Nb = 0.5, Lb = 1.2, Pe = .,Ω= .

0.5

0.0 (0) 

 −0.5

−1.0

−1.5 −1.5 −1.0 −0.5 0.0 N (e) Figure 2: ℎ−curves for (a) �,(b)�,(c)�,(d)�,and(e)�. presence of gyrotactic micr-organisms. Te analytic results layer will be thinner which is noted. Te similar behavior for � for the problem are presented graphically to study the �1 and �2 canbeobservedforthevelocitycomponent� in � behavior of some physical parameters on the velocities � , Figure 4. � � ,temperature�, concentration of nanoparticles �,and Te infuence of stretching ratio parameter � on the � concentration of microorganisms �.InFigure3,theefectsof velocity component � isshowninFigure5.Itisobservedthat � Deborah numbers �1 and �2 on the velocity component � are when the values of � are increased the velocity at the wall, examined for fxed values of other parameters. Te opposite �=0is increased. behavior of velocity profle is observed for increasing values In Figure 6, the impact of (a) bioconvection Rayleigh of �1 and �2. Nonzero values of �1 and �2 correspond to number �� and (b) buoyancy ratio parameter �� on the � elastic efects which retards the fow and hence the boundary velocity component � is shown. Te graphical results show 6 Mathematical Problems in Engineering

 = 0.5, 2 = 0.5, ．Ｎ = ．＜ = 0.1,  = ．Ｌ = ２＜ = 0.3  = 0.5, 1 = 0.3, ．Ｎ = ．＜ = 0.1,  = ．Ｌ = ２＜ = 0.3 1.0 1.0

0.8 0.8

0.6 0.6 f f  = 0, 0.5, 1, 1.5 0.4 1 0.4 2 = 0, 0.5, 1, 1.5

0.2 0.2

0.0 0.0 0123456 0 1234 576  

(a) (b)

� Figure 3: Infuence of (a) �1 (b) �2 on � .

 = 0.5, 2 = 0.1, ．Ｎ = ．＜ = 0.1,  = ．Ｌ = ２＜ = 0.3  = 0.5, 1 = 0.4, ．Ｎ = ．＜ = 0.1,  = ．Ｌ = ２＜ = 0.3 0.5 0.5

0.4 0.4

0.3 0.3  g  0.2 g 0.2  = 0, 0.5, 1, 1.5 1 = 0, 0.5, 1, 1.5 2

0.1 0.1

0.0 0.0 0123456 0 123456  

(a) (b)

� Figure 4: Infuence of (a) �1 (b) �2 on � .

1 = 0.3, 2 = 0.4, ．Ｎ = ．＜ = 0.1 Te combined efects of the thermophoresis parameter 1.5 �� and Brownian motion parameter �� on the temperature profle are shown in Figure 7(a). Te temperature profle increases when we increase the values of �� and ��,since 1.0 increasing the magnitude of Brownian motion on the particles and thermal difusivity of the nanoparticles will acceler-  g ate the temperature of the nanofuid. Te impact of Prandtl �� 0.5 number on temperature profle is shown in Figure 7(b).  = 0, 0.5, 1, 1.5 Te increasing values of Prandtle number decrease the temperature of the system. Figure 7(b) clearly depicts that the larger value of Prandtl number �� corresponds to the 0.0 lower heat and thermal boundary layer thickness. Te Prandtl 012345number indeed plays an essential role in heat transfer; it  controls the relative thickness of the thermal boundary � layer. In order to characterize the mixed convection fow, Figure 5: Infuence of stretching ratio � on velocity component � . it is also important to investigate the efect of the Prandtl number on the temperature profle. We observe that the temperature decreases with increasing Prandtl number. Tis may be explained by the fact that increasing the Prandtl that when the values of �� and �� are increasing, a rapid number is equivalent to reducing the thermal difusivity, decrease in the velocity profle is observed. which characterizes the rate at which heat is conducted. Mathematical Problems in Engineering 7

 = 0.5, 1 = 0.3, 2 = 0.6,  = 0.7, ．Ｌ = 0.4  = 0.5, 1 = 0.3, 2 = 0.6,  = 0.9, Rb = 0.5 1.0 1.0

0.8 0.8

0.6 0.6 f f 0.4 0.4 Rb = 0, 0.8, 1.6, 2.4 Nr = 0, 0.4, 0.8, 1.2

0.2 0.2

0.0 0.0 0 123456 0123456  

(a) (b)

� Figure 6: Infuence of (a) �� (b) �� on � .

 = 0.5,  = 0.3,  = 0.6 1 2 , Pr = 1.2, Le = 0.2  = 0.5, 1 = 0.3, 2 = 0.6, Nt = 0.4, Nt = 0.7, Le = 0.2 1.0 1.0

0.8 0.8

0.6 0.6   0.4 Nb = Nt = 0.1, 0.5, 0.9, 1.4 0.4 Pr = 0, 0.6, 1.2, 1.8

0.2 0.2

0.0 0.0 0 12345 0123456 7  

(a) (b)

Figure 7: Infuence of (a) �� and �� (b) �� on �.

To analyze the concentration � of the nanoparticles, the Te efects of Peclet number �� on � are shown in values of Brownian difusion parameter and thermophoresis Figure 10, which shows the buoyancy parameter is found difusion parameter are varied for fxed values of other to be more pronounced for a fuid with greater values of parameters as shown in Figure 8(a); for larger values of �� bioconvection Peclet´ number ��. �� and ��, the concentration profle � increases. Increase in Numerical values for −� (0) and −� (0) are compared the values of �� and �� refers to increase in number of with the existing literature in the absence of non-Newtonian nanoparticles. Tus higher concentration of nanoparticles efects are shown in Table 1, and an excellent agreement has corresponds to higher Brownian and thermophoresis difu- been found, which validate the present results. Te numerical sion which can also be seen from Figure 8(a). Te efect of values of local Nusselt number, local Sherwood number, Lewis number can be seen in Figure 8(b). If we increase the and local density of motile microorganisms are presented value of Lewis number, the concentration of the nanoparticle in Table 2 for diferent values of �, �, ��, ��,andΩ.We decreases. can observe that the magnitude of local Nusselt number Te infuence of dimensionless bioconvection Lewis is increased by increasing � and � while it decreased by number �� on concentration of microorganisms � is shown increasing �� and ��. In other words, heat fux at the in Figure 9(a). Te graphs illustrate a rapid decline in the surface is a decreasing function of nanoparticle properties. profle, because the bioconvection Lewis number opposes In other words, heat fux at the surface reduces in the the motion of the fuid. Te relation of microorganisms presence of nanoparticles and can be controlled by varying concentration diference parameter Ω and motile density of the quantity and quality of nanoparticles. Similarly, the the microorganisms � is discussed in Figure 9(b). For higher magnitude of local Sherwood number and local density of values of Ω, a decrease noticed in the value of �. motile microorganisms is increasing by increasing values 8 Mathematical Problems in Engineering

 = 0.3, 1 = 0.6, 2 = 0.3, Pr = 1.2, Sc = 0.22  = 0.3, 1 = 0.4, 2 = 0.2, Pr = 1.2 1.0 1.0

0.8 0.8

0.6 0.6   Le = 0.5, 1, 1.5, 2 0.4 Nb = Nt = 0.1, 0.5, 0.9, 1.3 0.4

0.2 0.2

0.0 0.0 0123456 0123456  

(a) (b)

Figure 8: Infuence of (a) �� and �� (b) �� on �.

Nt = 0.3, Nb = 0.6, Pe = 0.9, Ω = 0.1, Le = 0.2 Nt = 0.3, Nb = 0.6, Pe = 0.9, Lb = 1.2, Le = 0.5 1.0 1.0

0.8 0.8

0.6 0.6 N N 0.4 Lb = 0, 0.5, 1, 1.5 0.4 Ω = 0, 0.3, 0.6, 0.9

0.2 0.2

0.0 0.0 0 123456 0 1234 5  

(a) (b)

Figure 9: Infuence of (a) �� and �� (b) Ω on �.

Nt = 0.3, Nb = 0.6, Lb = 0.3, Ω = 0.1, Le = 0.4 thermophoretic difusion and Brownian difusion parameter 1.0 for all values of other parameters.

0.8 4. Concluding Remarks 0.6 In this paper, we have discussed the three-dimensional bio- N convection fow of an Oldroyd-B nanofuid over a stretching Pe = 0, 0.5, 1, 1.5 0.4 surface with gyrotactic microorganisms. Analytic technique is applied to investigate the problem and series solution is 0.2 computed by using homotopy analysis method. Te velocity, temperature, concentration, and microorganisms profles are 0.0 plotted to analyze the efects of various physical parameters. 0 12345Te main observations are as follows:  (i) Te Deborah number �1 decreases the velocity pro- Figure 10: Infuence of �� on �. fle, while �2 increases the velocity profle. � (ii) Te velocity component � decreases for increasing values of �� and ��. of �, �,and�� while it decreases for increasing ��.In other words, local Sherwood number and local density of (iii) Te stretching ratio parameter � increases the velocity � motile microorganisms are decreasing function of both the component � . Mathematical Problems in Engineering 9

Table 1: Comparison for the velocity gradient for diferent values of � when �1 =�2 =0. −��(0) −��(0) � HPM result [15] Exact result [15] Present result HPM result [15] Exact result [15] Present result 0.0 1.0 1.0 1.0 0 0 0 0.1 1.02025 1.020259 1.020253 0.06684 0.066847 0.066849 0.2 1.03949 1.039495 1.039498 0.14873 0.148736 0.148730 0.3 1.05795 1.05794 1.057959 0.24335 0.243359 0.243360 0.4 1.07578 1.075788 1.075789 0.34920 0.349208 0.349212 0.5 1.09309 1.093095 1.093093 0.46520 0.465204 0.465206 0.6 1.10994 1.109946 1.109944 0.59052 0.590528 0.590527 0.7 1.12639 1.126397 1.126396 0.72453 0.724531 0.724529 0.8 1.14248 1.142488 1.142489 0.86668 0.866682 0.866683 0.9 1.15825 1.158253 1.158256 1.01653 1.016538 1.016541 1.0 1.17372 1.17372 1.173723 1.17372 1.173720 1.173723

−1/2 −1/2 Table 2: Numerical values of local Nusselt number �� , local Sherwood number �ℎ�� , and local density of motile microorganisms −1/2 �� for diferent values of �, �, ��, ��,andΩ. ��Ω−��(0) −��(0) −��(0) 0.0 0.2 0.3 0.3 0.2 0.861666 0.522014 1.024791 0.5 0.929823 0.571493 1.118607 1.0 1.002058 0.634386 1.227815 0.5 0.0 0.3 0.3 0.2 0.918579 0.554933 1.104269 0.5 0.943804 0.592014 1.136527 1.0 0.960169 0.616448 1.158511 0.5 0.3 0.1 0.3 0.2 1.010617 −0.700285 1.129376 0.5 0.864386 0.832249 1.125671 0.9 0.738619 0.997418 1.127016 0.5 0.3 0.3 0 0.2 0.998064 1.124014 1.124014 0.5 0.896472 0.263520 1.125716 1.0 0.812228 −0.390333 1.127555 0.5 0.2 0.3 0.3 0 0.929823 0.571493 1.118607 0.5 0.929823 0.571493 1.118607 1.0 0.929823 0.571493 1.118607

(iv) Te combined efects of Brownian difusion param- Appendix eter �� and thermophoretic difusion parameter �� decrease the temperature and concentration pro- A. Homotopy Analysis Method (HAM) fles. Te initial guesses and auxiliary linear operators for the (v) Te Prandtl number �� and Schmidt number �� also dimensionless momentum, energy, concentration of nano- decrease the temperature and concentration profles. particles, and concentration of motile microorganisms equations are denoted by �0,�0,�0,�0,�0 and L�, L�, L�, L�, L� (vi) Te concentration of microorganism � has decreas- and are defned as follows: ing behavior for increasing ��, ��,and��. �0 (�) = 1 − exp (−�) , (vii) Te local Nusselt number and local Sherwood number are decreasing functions of nanoparticle proper- �0 (�) = � [1 − exp (−�)] ties. �0 (�) = exp (−�) , (A.1) (viii) Te density of motile microorganisms is an increasing �0 (�) = exp (−�) , function of bioconvection Lewis number and bioconvection Peclet number. �0 (�) = exp (−�) , 10 Mathematical Problems in Engineering

̂� and � (∞, �) = 0, �3� �� (0,�)=0,̂ L = − , � ��3 �� ̂� (0, �) = �, �3� �� L = − , �̂� (∞, �) = 0, � ��3 �� ̂ �2� �(0,�)=1, L = −�, � 2 (A.2) �� (∞,�)=0,̂ �2� L = −�, �(0,�)=1,̂ � ��2 �(∞,�)=0,̂ �2� L = −�, � ��2 �(0,�)=1,̂ with �(∞,�)=0,̂ (A.5) L� [�1 +�2 exp �+�3 exp −�] = 0,

L� [�4 +�5 exp �+�6 exp −�] = 0, and L [� �+� −�] = 0, � 7 exp 8 exp (A.3) ̂ ̂ ̂ ̂ N� [�(�,�),�(�,�),̂ �(�,�),�(�,�),�(�,�)] L� [�9 exp �+�10 exp −�] = 0, �3�(�,�)̂ �2�(�,�)̂ = +(�(�,�)+̂ �(�,�))̂ 3 2 L� [�11 exp �+�12 exp −�] = 0, �� ̂ 2 where �� (� = 1 − 12) are the arbitrary constant. Te zeroth ��(�,�) −( ) +� [2 (�(�,�)̂ and �-th order deformation problems are as follows. �� 1

��(�,�)̂ �2�(�,�)̂ A.1. Zeroth-Order Problem + �(�,�))̂ �� 2

̂ ̂ 3 (A.6) (1 − �) L� [�(�,�)−�0 (�)] = �ℎ�N� [�(�,�), 2 � �(�,�)̂ −(�(�,�)+̂ �(�,�))̂ ] ��3 �(�,�),̂ �(�,�),̂ �(�,�),̂ �(�,�)],̂ �2�(�,�)̂ �2�(�,�)̂ �2�(�,�)̂ (1 − �) L [�(�,�)−�̂ (�)] = �ℎ N [�(�,�),̂ +� [2 ( + ) � 0 � � 2 ��2 ��2 ��2 �(�,�),̂ �(�,�),̂ �(�,�)],̂ �4�(�,�)̂ −(�(�,�)+̂ �(�,�))̂ ]+�[�(�,�)̂ ̂ ̂ ��4 (1 − �) L� [�(�,�)−�0 (�)] = �ℎ�N� [�(�,�), (A.4) �(�,�),̂ �(�,�),̂ �(�,�)],̂ −��(�,�)−��̂ �(�,�)],̂

̂ ̂ N [�(�,�),̂ �(�,�),̂ �(�,�),̂ �(�,�)]̂ (1 − �) L� [�(�,�)−�0 (�)] = �ℎ�N� [�(�,�), � ̂ ̂ �3�(�,�)̂ �2�(�,�)̂ �(�,�),�(�,�),̂ �(�,�)], = +(�(�,�)+̂ �(�,�))̂ ��3 ��2 (1 − �) L�[�(�,�)−�̂ (�)] 0 ��(�,�)̂ 2 −( ) +� [2 (�(�,�)̂ ̂ ̂ ̂ 1 =�ℎ�N� [�(�,�),�(�,�),�(�,�),̂ �(�,�)], �� (�,�)̂ �2�(�,�)̂ with + �(�,�))̂ �� 2 �(0,�)=0,̂ 3 2 � �(�,�)̂ −(�(�,�)+̂ �(�,�))̂ ] �̂� (0, �) = 1, ��3 Mathematical Problems in Engineering 11

�2�(�,�)̂ �2�(�,�)̂ �2�(�,�)̂ �� (∞) =0, +� [2 ( + ) � 2 ��2 ��2 ��2 �� (0) =0, �4�(�,�)̂ −(�(�,�)+̂ �(�,�))̂ ], �� (0) =0, ��4 � � (A.7) �� (∞) =0, N [�(�,�),̂ �(�,�),̂ �(�,�),̂ �(�,�)]̂ � �� (0) =0,

2 � �(�,�)̂ ��(�,�)̂ �� (∞) =0, = +��(�(�,�)+̂ �(�,�))̂ ��2 �� (0) =0, ��(�,�)̂ ��(�,�)̂ (A.8) −��( ) �(�,�)+��.��̂ �� (∞) =0, �� (0) =0, 2 ��(�,�)̂ ��(�,�)̂ ⋅ +��.��( ) , �� (∞) =0, �� (A.12) ̂ ̂ ̂ N� [�(�,�),�(�,�),�(�,�),̂ �(�,�)] and �2�(�,�)̂ ��(�,�)̂ = +��(�(�,�)+̂ �(�,�))̂ 2 �−1 �� (A.9) � �� R� (�) = ��−1 + ∑ (��−1−�� +��−1−�� −��−1−��) ��(�,�)̂ �� 2�(�,�)̂ �=0 −��( ) �(�,�)+��.̂ , 2 �−1 � �� +�1 ∑ ∑ [2 (��−1−��−�� +��−1−��−�� ) ̂ ̂ ̂ �=0 �=0 N� [�(�,�),�(�,�),�(�,�),̂ �(�,�)] �� −(��−1−��−�� +��−1−��−�� +2��−1−��−�� )] �2�(�,�)̂ ��(�,�)̂ � � � (A.13) = +��(�(�,�)+̂ �(�,�))̂ 2 �−1 � �� +�2 ∑ ∑ [2 (��−1−�� +��−1−�� ) ��(�,�)̂ ��(�,�)̂ (A.10) �=0 �=0 ̂ −��( ) �(�,�)−��.[ �� −(��−1−�� +��−1−�� )] + � (��−1 −��−1

2 � �(�,�)̂ −��−1), +(�(�,�)+Ω)̂ ], 2 �� −1 � �� R� (�) = ��−1 + ∑ (��−1−�� +��−1−�� −��−1−��) �=0 where �∈[0,1]represents the embedding parameter and ℎ�, ℎ ℎ ℎ ℎ �−1 � �, �, �,and � are the nonzero auxiliary parameters. � �� +�1 ∑ ∑ [2 (��−1−��−�� +��−1−��−�� ) �=0 �=0 � A.2. th-Order Problem. �� (A.14) −(��−1−��−�� +��−1−��−�� +2��−1−��−�� )] � L� [�� (�) − ��−1 (�)] = ℎ�R� (�) , �−1 � �� +�2 ∑ ∑ [2 (��−1−�� +��−1−�� ) L� [�� (�) − ��−1 (�)] = ℎ�R� (�) , �=0 �=0

� �� −(��−1−�� +��−1−�� )] , L� [�� (�) − ��−1 (�)] = ℎ�R� (�) , (A.11) �−1 L [� (�) − �� (�)] = ℎ R� (�) , � �� −1 � � R� (�) = ��−1 +��∑ (��−1−�� +��−1−�� −��−1−�� =0 � (A.15) L� [�� (�) − ��−1 (�)] = ℎ�R� (�) , � � � � +��−1−�� +��−1−��), with �−1 � �� R� (�) = ��−1 +��∑ (��−1−�� +��−1−�� =0 �� (0) =0, (A.16) �� −�� )+ �� , �� (0) =0, �−1−� � �� −1 12 Mathematical Problems in Engineering

�−1 ∞ � �� ̂ � R� (�) = ��−1 +��∑ (��−1−�� +��−1−�� (�,�)=�0 (�) + ∑ �� (�) � , �=0 �=1

� � � �� (A.17) � � −� �� −��(� � +��−1−�� )) ̂ � �−1−� �−1−� � � 1 � �(�,�)� �� (�) = � . �� ! �� +��Ω��−1, ��=0 (A.19) for �=0and �=1,andwecanwrite Te values of auxiliary parameter are chosen in such a way ̂ that the series (A.19) converge at �=1, i.e., �(�,0)=�0 (�) , ∞ ̂ �(�,1)=�(�), �(�)=�0 (�) + ∑ �� (�) , �=1

�(�,0)=�̂ 0 (�) , ∞ �(�)=� (�) + ∑ � (�) , �(�,1)=�(�),̂ 0 � �=1 �(�,0)=�̂ (�) , ∞ 0 �(�)=� (�) + ∑ � (�) , (A.18) 0 � (A.20) �(�,1)=�(�),̂ �=1 ∞ ̂ �(�,0)=�0 (�) , �(�)=�0 (�) + ∑ �� (�) , �=1 �(�,1)=�(�),̂ ∞ �(�)=� (�) + ∑ � (�) . �(�,0)=�̂ (�) , 0 � , 0 �=1

�(�,1)=�(�),̂ Te general solutions (��,��,��,��,��) of (A.11)-(A.12) in ∗ ∗ ∗ ∗ ∗ terms of special solutions (��,��,��,��,��) are given by ̂ ̂ ̂ the following: when � varies from 0 to 1, �(�, �), �(�,̂ �), �(�, �), �(�, �), ̂ ∗ and �(�, �) vary from the initial solutions �0(�), �0(�), �0(�), �� (�) = �� (�) + �1 +�2 exp (�) + �3 exp (−�) , �0(�),and�0(�) to the fnal solutions �(�), �(�), �(�), �(�) � (�) = �∗ (�) + � +� (�) and �(�), respectively. By Taylor’s series, we have � � 4 5 exp

∞ +�6 exp (−�) , �(�,�)=�̂ (�) + ∑ � (�) ��, (A.21) 0 � � (�) = �∗ (�) + � (�) + � (−�) , �=1 � � 7 exp 8 exp

� � ∗ ̂ � �� (�) = � (�) + �9 exp (�) + �10 exp (−�) , 1 � �(�,�)� � �� (�) = � , �! �� ∗ ��=0 �� (�) = �� (�) + �11exp (�) + �12exp (−�) , ∞ where the constants �� (� = 1 − 12) through the boundary �(�,�)=�̂ (�) + ∑ � (�) ��, 0 � conditions (A.12) are as follows: �=1 � =� =� =� =� =0, � � 2 5 7 9 11 1 � �(�,�)̂ � �� (�) = � , ∗ � �! �� =0 � = � , 3 �� ∞ ��=0 ̂ � �(�,�)=�0 (�) + ∑ �� (�) � , ∗ �1 =−�3 −� (0) , �=1 � � ∗ � � ̂ � �� 1 � �(�,�)� � = � , � (�) = � , 6 �� (A.22) � �! �� =0 ��=0 ∗ ∞ �4 =−�6 −�� (0) , �(�,�)=�̂ (�) + ∑ � (�) ��, 0 � � =−�∗ (0) , �=1 8 � ∗ � ̂ � �10 =−� (0) , 1 � �(�,�)� �� (�) = � , �! �� ∗ ��=0 �12 =−� (0) . Mathematical Problems in Engineering 13

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