20150106

Appl. Math. Mech. -Engl. Ed., 36(1), 69–80 (2015)DOI 10.1007/s10483-015-1896-9c©Shanghai University and Springer-Verlag

Berlin Heidelberg 2015

Applied Mathematicsand Mechanics(English Edition)

Flow of Oldroyd-B fluid with nanoparticles and thermal radiation∗

T. HAYAT1,2, T. HUSSAIN3, S. A. SHEHZAD4, A. ALSAEDI2

(1. Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan;

2. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science,

King Abdulaziz University, Jeddah 21589, Saudi Arabia;

3. Department of Mathematics, Faculty of Computing, Mohammad Ali Jinnah University,

Islamabad 44000, Pakistan;

4. Department of Mathematics, COMSATS Institute of Information Technology,

Sahiwal 57000, Pakistan)

Abstract The two-dimensional boundary layer flow of an Oldroyd-B fluid in the pres-ence of nanoparticles is investigated. Convective heat and mass conditions are consideredin the presence of thermal radiation and heat generation. The Brownian motion and ther-mophoresis effects are retained. The nonlinear partial differential equations are reducedinto the ordinary differential equation (ODE) systems. The resulting ODE systems aresolved for the series solutions. The results are analyzed for various physical parameters ofinterest. Numerical values of the local Nusselt and Sherwood numbers are also computedand analyzed.

Key words nanoparticle, Oldroyd-B fluid, heat generation, convective condition

Chinese Library Classification O3732010 Mathematics Subject Classification 76A05

1 Introduction

A working fluid is involved in many engineering and industrial processes which is used totransfer energy/heat from one position to another position. The enhancement in energy transferhas been a serious issue for a long time. Nanofluid is a better candidate in this regard. Ad-vancement in nanoparticles is acting as a new heat transfer medium which introduced new andhigh potentials. The common working fluids involved in industry and engineering processeshave less thermal conductivity in comparison to metal and metal oxides. An enhancementin heat transfer performance can be obtained by adding high conductivity materials in basefluids[1–2]. The betterment in the performance of thermal conductivity without causing a pres-sure drop is the major advantage of nanofluids. As a consequence, the performance of variousheat transfer devices is increased, which leads to the larger capacity of operating systems. Suchfluids are looking to be very interesting in transportation, nuclear reactors, solar collectors,car radiators, chillers, micro-electro mechanical systems, cooling of electronic devices, and incooling/heating of energy conversion. Having such in view, Oztop and Abu-Nada[3] discussedthe buoyancy driven flow of nanofluid filled in an enclosure. A numerical solution was pre-sented. Turkyilmazoglu[4] presented a study to analyze the magnetohydrodynamic (MHD) flow

∗ Received Jan. 12, 2014 / Revised Jun. 18, 2014Project supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah,Saudi Arabia (No. 37-130-35-HiCi)Corresponding author S. A. SHEHZAD, Professor, Ph.D., E-mail: ali [email protected]

70 T. HAYAT, T. HUSSAIN, S. A. SHEHZAD, and A. ALSAEDI

of nanofluid with heat and mass transfer in the presence of thermal slip boundary condition.Both exact and analytical solutions were given. The boundary flow of nanofluid over a verti-cal plate with convective thermal condition was examined by Rashidi et al.[5]. They used thedifferential transform method to obtain the results of the velocity, the temperature, and the vol-umetric fraction of nanoparticles. The effect of double-stratification was numerically analyzedon the boundary layer flow of nanofluid over a vertical flat plate by Ibrahim and Makinde[6].Moradi et al.[7] studied the Jeffery-Hamel flow of nanofluid with viscous dissipation. Hatamiand Ganji[8] carried out an analysis to examine the sodium alginate (SA) TiO2 non-Newtoniannanofluid passing through a porous medium between two coaxial cylinders. Sheikholeslami etal.[9] investigated the flow of nanofluid in a semi-porous channel. They computed the solutionexpressions using the least square and Galerkin methods. The effects of the buoyancy force,convective heating, the Brownian motion, and thermophoresis in the MHD stagnation pointflow of nanofluid were examined by Makinde et al.[10]. They obtained the numerical solutionby the Runge-Kutta fourth-order scheme.

The boundary layer flow of non-Newtonian fluid over a surface has attained considerableattention from the recent investigators due to its broad applications in the industrial and engi-neering processes. More specifically, the flow and heat transfer of non-Newtonian fluids have keyimportance in power engineering, petroleum production, polymer solutions, paper production,etc. The fluid model under consideration is an Oldroyd-B fluid, which falls into the categoryof rate type non-Newtonian fluid. It exhibits the characteristics of the relaxation time and theretardation time. Very little has been said yet about the boundary layer flow of the Oldroyd-Bfluid. Sajid et al.[11] initiated the boundary layer stagnation point flow of an Oldroyd-B fluidtowards a stretching surface. They presented the results numerically. Hayat et al.[12] extendedthis analysis for the three-dimensional flow of an Oldroyd-B fluid with a convective boundarycondition. The three-dimensional flow of an Oldroyd-B fluid with variable thermal conductiv-ity and heat generation/absorption was recently addressed by Shehzad et al.[13]. Moreover, theimportance of thermal radiation is prevalent in the industrial and space technological processesat very high temperature. Examples of such processes are glass production and furnace design,comical flight aerodynamic rocket, space craft re-entry, plasma physics, propulsion systems, etc.Further, the heat generation/absorption plays a vital role in disassociating fluids in packed-bedreactors, storage of food stuffs, and heat removal from nuclear fuel debris, underground disposalof radioactive waste material, and many others[14–18].

The present investigation deals with the two-dimensional boundary layer flow of an Oldroyd-B fluid with nanoparticles. Mathematical formulation consists of convective heat and mass con-ditions. Besides these, the contributions due to thermal radiation and heat generation/absorpt-ion are also taken into account. The governing nonlinear problems are computed for the se-ries solutions through the homotopy analysis method (HAM)[19–25]. Discussion reflecting theinterpretation of sundry parameters is made. Important conclusions are presented.

2 Mathematical formulation

An incompressible flow of an Oldroyd-B nanofluid over a stretching sheet is considered. Thefluid is assumed to be incompressible. We assume that the surface heated by a hot fluid has thetemperature Tf and the concentration Cf . The heat and mass transfer coefficients are denotedby h1 and h2, respectively. The effects of the Brownian motion and thermophoresis are presen-ted. We also consider heat and mass transfer in the presence of thermal radiation and heat gen-eration. The governing boundary layer equations for the present problem are given as follows:

∂u

∂x+

∂v

∂y= 0, (1)

u∂u

∂x+v

∂u

∂y+λ1

(

u2 ∂2u

∂x2+v2 ∂2u

∂y2+2uv

∂2u

∂x∂y

)

Flow of Oldroyd-B fluid with nanoparticles and thermal radiation 71

= ν(∂2u

∂y2+ λ2

(

u∂3u

∂x∂y2+v

∂3u

∂y3− ∂u

∂x

∂2u

∂y2− ∂u

∂y

∂2u

∂y2

))

, (2)

u∂T

∂x+ v

∂T

∂y= α

∂2T

∂y2+ τ

(

DB∂C

∂y

∂T

∂y+

DT

T∞

(∂T

∂y

)2)

− 1

(ρc)f

∂qr

∂y+

Q

(ρc)f(T − T∞), (3)

u∂C

∂x+ v

∂C

∂y= DB

∂2C

∂y2+

DT

T∞

∂2T

∂y2. (4)

The boundary conditions for the considered flow analysis are

u = uw(x) = cx, v = 0, −k∂T

∂y= h1(Tf − T ), −DB

∂C

∂y= h2(Cf − C) at y = 0, (5)

u → 0, T → T∞, C → C∞ as y → ∞, (6)

where u and v are the velocity components in the x- and y-directions, ν is the kinematic viscosity,λ1 and λ2 are the relaxation time and the retardation time, respectively, ρf is the density of

fluid, α is the thermal diffusivity, τ (=(ρc)p(ρc)f

) is the ratio of the nanoparticle heat capacity and

the base fluid heat capacity, qr is the radiative heat flux, Q is the heat source/sink parameter,DB is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, k is thethermal conductivity, uw is the stretching velocity at the wall, and T∞ and C∞ are the ambienttemperature and the concentration, respectively.

The radiative heat flux qr using Rosseland’s approximation gives

qr =4σ

3k∗

∂T 4

∂y, (7)

in which σ is the Stefan-Boltzmann constant, and k∗ is the mean absorption coefficient. Thetemperature difference within the flow is assumed in such a manner that T 4 can be written inthe linear combination of temperature. By expanding T 4 about T∞ in terms of Taylor’s seriesand neglecting higher order terms, we have

T 4 ∼= 4T 3∞

T − 3T 4∞

(8)

and

∂qr

∂y= −16σT 3

∞

3k∗

∂2T

∂y2. (9)

Now, Eq. (3) reduces to

u∂T

∂x+v

∂T

∂y=α

∂2T

∂y2+τ

(

DB∂C

∂y

∂T

∂y+

DT

T∞

(∂T

∂y

)2)

+1

(ρc)f

16σT 3∞

3k∗

∂2T

∂y2+

Q

(ρc)f(T − T∞). (10)

Use the following transformations:

u = cxf ′(η), v = −√

cνf(η), η = y

√

c

ν, θ(η) =

T − T∞

Tf − T∞

, φ(η) =C − C∞

Cf − C∞

. (11)

Then, Eqs. (2), (4)–(6), and (10) become

f ′′′ + ff ′′ − f ′2 + β1(f2f ′′′ − 2ff ′f ′′) + β2(ff ′′′′ − f ′′2) = 0, (12)

(

1 +4

3Tr

)

θ′′ + Prfθ′ + PrNbθ′φ′ + PrNtθ′2 + PrSθ = 0, (13)

φ′′ + Scfφ′ + (Nt/Nb) θ′′ = 0, (14)

f = 0, f ′ = 1, θ′ = −Bi1(1 − θ(0)), φ′ = −Bi2(1 − φ(0)) at η = 0, (15)

f ′ → 0, θ → 0, φ → 0 as η → ∞, (16)


where β1 = λ1c is the Deborah number with respect to the relaxation time, β2 = λ2c is theDeborah number with respect to the retardation time, Pr = ν/α is the Prandtl number, Tr =4σT 3

∞/(kk∗) is the radiation parameter, Sc = ν/DB is the Schmidt number, Nb = (ρc)pDB(Cf−

C∞)/((ρc)fν) is the Brownian motion parameter, Nt = (ρc)pDT(Tf − T∞)/((ρc)fνT∞) is the

thermophoresis parameter, and Bi1 (= (h1/k)√

ν/a) and Bi2 (= (h2/DB)√

ν/a) are the Biotnumbers. The local Nusselt number and the local Sherwood number are given by

Nux =xqw

k(Tf − T∞), Shx =

xqm

DB(Cf − C∞), (17)

where qw is the surface heat flux, and qm is the surface mass flux. The local Nusselt and localSherwood numbers in dimensionless forms are given below:

Nux/Re1/2x = −

(

1 +4

3Tr

)

θ′(0), Shx/Re1/2x = −φ′(0), (18)

where Rex = uw(x)x/ν is the local Reynolds number.

3 Homotopy analysis solutions

By choosing a set of base functions[19]

{ηk exp(−nη), k > 0, n > 0}, (19)

the functions f, θ, and φ can be expressed as follows:

fm(η) =

∞∑

n=0

∞∑

k=0

akm,nηk exp(−nη), (20)

θm(η) =∞∑

n=0

∞∑

k=0

bkm,nηk exp(−nη), (21)

φm(η) =

∞∑

n=0

∞∑

k=0

ckm,nηk exp(−nη), (22)

in which akm,n, bk

m,n, and ckm,n are the coefficients. The initial guesses and auxiliary linear

operators are selected in the following forms:

f0(η) = 1 − exp(−η), θ0(η) =Bi1 exp(−η)

1 + Bi1, φ0(η) =

Bi2 exp(−η)

1 + Bi2, (23)

L(f) = f ′′′ − f ′, L(θ) = θ′′ − θ, L(φ) = φ′′ − φ (24)

subject to the properties

L(f)(C1 + C2eη + C3e

−η) = 0, L(θ)(C4eη + C5e

−η) = 0, L(φ)(C6eη + C7e

−η) = 0, (25)

where Ci (i = 1, 2, · · · , 7) are the arbitrary constants. The zeroth order problems are definedas follows[20–25]:

(1 − q)L(f)(

f(η; q) − f0(η))

= q~fNf

[

f(η; q)]

, (26)

(1 − q)L(θ)(

θ(η; q) − θ0(η))

= q~θNθ

[

f(η; q), θ(η, q), φ(η, q)]

, (27)

(1 − q)L(φ)(

φ(η; q) − θ0(η))

= q~φNφ

[

f(η; q), θ(η, q), φ(η, q)]

, (28)


f(0; q) = 0, f ′(0; q) = 1, θ′(0, q) = −Bi1(1 − θ(0, q)),

φ′(0, q) = −Bi2(1 − φ(0, q)),

f ′(∞; q) = 0, θ(∞, q) = 0, φ(∞, q) = 0,

(29)

Nf [f(η, q)] =∂3f(η, q)

∂η3+

(

f(η, q)∂2f(η, q)

∂η2−

(∂f(η, q)

∂η

)2)

+ β1

(

(

f(η, q))2 ∂3f(η, q)

∂η3− 2f(η, q)

∂f(η, q)

∂η

∂2f(η, q)

∂η2

)

+ β2

(

f(η, q)∂4f(η, q)

∂η4−

(∂2f(η, q)

∂η2

)2)

, (30)

Nθ[θ(η, q), f(η, q), φ(η, q)] =(

1 +4

3Tr

) ∂2θ(η, q)

∂η2+ PrNb

∂θ(η, q)

∂η

∂φ(η, q)

∂η

+ PrNt

(∂θ(η, q)

∂η

)2

+ PrSθ(η, q), (31)

Nφ[φ(η, q), f(η, q), θ(η, q)] =∂2φ(η, q)

∂η2+ Scf(η, q)

∂φ(η, q)

∂η+

Nt

Nb

∂2θ(η, q)

∂η2, (32)

where ~f , ~θ, and ~φ are the non-zero auxiliary parameters, q ∈ [0, 1] is an embedding pa-rameter, and Nf , Nθ, and Nφ are the nonlinear operators. Putting q = 0 and q = 1, onehas

{

f(η; 0) = f0(η), θ(η, 0) = θ0(η), φ(η, 0) = φ0(η),

f(η; 1) = f(η), θ(η, 1) = θ(η), φ(η, 1) = φ(η).(33)

If we increase the value of q from 0 to 1, then f(η, q), θ(η, q), and φ(η, q) vary from f0(η), θ0(η),and φ0(η) to f(η), θ(η), and φ(η). By adopting the Taylor series expansion, we have

f(η, q) = f0(η) +

∞∑

m=1

fm(η)qm, (34)

θ(η, q) = θ0(η) +

∞∑

m=1

θm(η)qm, (35)

φ(η, q) = φ0(η) +

∞∑

m=1

φm(η)qm, (36)

fm(η)=1

m!

∂mf(η; q)

∂ηm

∣

∣

∣

∣

q=0

,

θm(η) =1

m!

∂mθ(η; q)

∂ηm

∣

∣

∣

∣

q=0

,

φm(η)=1

m!

∂mφ(η; q)

∂ηm

∣

∣

∣

∣

q=0

.

(37)

The convergence of the above series highly depends upon the suitable values of ~f , ~θ, and~φ. Considering that ~f , ~θ, and ~φ are selected properly such that Eqs. (34)–(36) converge at


q = 1, we have

f(η) = f0(η) +

∞∑

m=1

fm(η), (38)

θ(η) = θ0(η) +

∞∑

m=1

θm(η), (39)

φ(η) = φ0(η) +

∞∑

m=1

φm(η). (40)

The general solutions can be written as

fm(η) = f∗

m(η) + C1 + C2eη + C3e

−η, (41)

θm(η) = θ∗m(η) + C4eη + C5e

−η, (42)

φm(η) = φ∗

m(η) + C6eη + C7e

−η, (43)

where f∗

m, θ∗m, and φ∗

m(η) are the special solutions.

4 Convergence of homtopy solutions and discussion

The auxiliary parameters ~f , ~θ, and ~φ are important in controlling and adjusting theconvergence region of series solutions. We plot the ~-curves at the 22nd-order of HAM approxi-mations in order to find the appropriate values. It is noted from Fig. 1 that the suitable values of~f , ~θ, and ~φ are −1.45 6 ~f 6 −0.10,−1.25 6 ~θ 6 −0.50, and −1.20 6 ~φ 6 −0.30, respec-tively. The series converge in the whole region of η when ~f =~θ =~φ =−0.8 (see Table 1).

Fig. 1 ~-curves for functions f ′′(0), θ′(0), and φ′(0) at 22nd-order of approximations when β1 = β2 =0.3, S = 0.1, P r = 1.0, Sc = 0.8, Nt = 0.2, Nb = 0.3, Bi1 = Bi2 = 0.8, and Tr = 0.4

Table 1 Convergence of homotopy solution for different orders of approximations when β1 = β2 =0.2, S = 0.1, P r = 1.2, Sc = 1.0, Nt = Nb = 0.3, Bi1 = Bi2 = 0.5, Tr = 0.4, and ~f = ~θ =~φ = −0.8

Order of approximation −f ′′(0) −θ′(0) −φ′(0)

01 0.960 000 0.258 07 0.259 2605 0.959 758 0.209 21 0.226 1010 0.959 754 0.195 39 0.219 7220 0.959 754 0.188 59 0.217 9630 0.959 754 0.187 02 0.217 8535 0.959 754 0.186 77 0.217 8540 0.959 754 0.186 77 0.217 8545 0.959 754 0.186 77 0.217 85


We plot Figs. 2−9 to examine the behaviors of the Deborah numbers β1 and β2, the Prandtlnumber Pr, the Biot number Bi1, the thermophoresis and Brownian motion parameters Nt

and Nb, the radiation parameter Tr, and the heat generation parameter S on the dimensionlesstemperature θ(η). The variations in the temperature θ(η) for different values of the Deborahnumber β1 are seen in Fig. 2. Here, one can see that the temperature is lower for smaller valuesof the Deborah number β1 and higher for larger values of the Deborah number. Since theDeborah number β1 is directly proportional to the relaxation time, an increase in the value ofβ1 corresponds to an increase in the relaxation time. Higher relaxation time gives rise to thetemperature and the thermal boundary layer thickness. It is noted from Fig. 3 that the temper-ature and the thermal boundary layer thickness are reduced with an increase in the Deborahnumber β2. This is due to the fact that the Deborah number β2 is directly proportional tothe retardation time. An increase in the retardation time corresponds to lower temperatureand thinner thermal boundary layer. The comparison of Figs. 2 and 3 shows that the Deborahnumbers β1 and β2 have reverse effects on the temperature. Figure 4 depicts that both thetemperature and the thermal boundary layer thickness increase when the values of the Prandtlnumber are smaller. The Prandtl number is the ratio of momentum to thermal diffusivities. Areduction in the value of the Prandtl number implies higher thermal diffusivity and smaller mo-mentum diffusivity. Higher thermal diffusivity and lower momentum diffusivity are responsiblefor an increase in the temperature when the value of the Prandtl number decreases. Figure 5illustrates that the temperature increases for larger Biot number Bi1. It is seen that beyondBi1 = 0.8, the increase in the temperature slows down. An increase in the temperature corre-sponding to the Biot number Bi1 is due to the heat transfer coefficient h1. The definition ofBi1 involves the heat transfer coefficient h1. The heat transfer coefficient increases when thevalue of Bi1 increases. This increase in the heat transfer coefficient leads to an enhancementin the temperature and the thermal boundary layer thickness. Variations in temperature vs.η for different values of the thermophoresis and Brownian motion parameters are examined inFigs. 6 and 7. These figures show that the temperature is enhanced for larger values of thethermophoresis and Brownian motion parameters. Also, we notice that the variations in thetemperature due to the thermophoresis parameter are much more than the variations due tothe Brownian motion parameter. An increase in the radiation parameter leads to an increase inthe temperature and the thermal boundary layer thickness (see Fig. 8). In fact, the increase inthe radiation parameter leads to an enhancement in the fluid temperature. Figure 9 illustratesthe variations in the temperature for various values of the heat generation parameter S. It isfound that the temperature and the thermal boundary layer thickness are increasing functionsof the heat generation parameter.

Fig. 2 Influence of β1 on temperature θ(η)vs. η when β2 = 0.4, P r = Sc =1.0, Bi1 = Bi2 = 0.8, Nt = Nb =0.2, Tr = 0.5, and S = 0.2

Fig. 3 Influence of β2 on temperature θ(η) vs.η when β1 = 0.4, P r = Sc = 1.0, Bi1 =Bi2 = 0.8, Nt = Nb = 0.2, Tr = 0.5, andS = 0.2


Fig. 4 Influence of Pr on temperature θ(η)vs. η when β1 = β2 = 0.4, Sc =1.0, Bi1 = Bi2 = 0.8, Nt = Nb = 0.2,Tr = 0.5, and S = 0.2

Fig. 5 Influence of Bi1 on temperature θ(η) vs.η when β1 = β2 = 0.4, P r = Sc =1.0, Bi2 = 0.8, Nt = Nb = 0.2, Tr = 0.5,and S = 0.2

Fig. 6 Influence of Nt on temperature θ(η)vs. η when β1 = β2 = 0.4, P r = Sc =1.0, Bi1 = Bi2 = 0.8, Nb = 0.2, Tr =0.5, and S = 0.2

Fig. 7 Influence of Nb on temperature θ(η) vs.η when β1 = β2 = 0.4, P r = Sc =1.0, Bi1 = Bi2 = 0.8, Nt = 0.2, Tr = 0.5,and S = 0.2

Fig. 8 Influence of Tr on temperature θ(η)vs. η when β1 = β2 = 0.4, P r = Sc =1.0, Bi1 = Bi2 = 0.8, Nt = Nb = 0.2,and S = 0.2

Fig. 9 Influence of S on temperature θ(η) vs.η when β1 = β2 = 0.4, P r = Sc =1.0, Bi1 = Bi2 = 0.8, Nt = Nb = 0.2,and Tr = 0.5

To see the variations in the concentration φ(η) for different values of the Deborah numbersβ1 and β2, the Schmidt number Sc, the Biot number Bi2, the thermophoresis parameter Nt,and the Brownian motion parameter Nb, Figs. 10–15 are presented. Figures 10 and 11 elu-cidate that the Deborah numbers β1 and β2 have reverse effects on the concentration φ(η).The concentration φ(η) increases for higher values of β1, while it reduces for larger values ofβ2. Also, it is analyzed that the influence of β1 and β2 on the temperature θ(η) and the con-


centration φ(η) is qualitatively similar. It is noticed from Fig. 12 that the concentration φ(η)and its related boundary layer thickness are lower for higher values of the Schmidt number.The Schmidt number is dependent on the Brownian diffusion coefficient. An increase in theSchmidt number implies a reduction in the Brownian diffusion coefficient. Such a decrease inthe Brownian diffusion coefficient causes a reduction in the concentration φ(η). The change inthe concentration φ(η) for different values of the Biot number Bi2 is analyzed in Fig. 13. Wecan see that the concentration φ(η) increases when the value of Bi2 increases. The change inthe concentration φ(η) is larger for smaller values of Bi2, while this change is smaller whenwe increase the value of Bi2 from 0.8 onward. Figure 13 is plotted to see the effects of thethermophoresis and Brownian motion parameters on the concentration φ(η). We examine thatthe thermophoresis and Brownian motion parameters have opposite effects on the concentra-tion φ(η). The concentration φ(η) increases with an increase in the thermophoresis parameter,but it decreases for higher values of the Brownian motion parameter. Also, we analyze thatthe decrease in the concentration φ(η) is rapid for Nb = 0.1, 0.3, but it is very small whenNb = 0.5, 0.7, 1.0.

Fig. 10 Influence of β1 on concentrationφ(η) vs. η when β2 = 0.4, P r =Sc = 1.0, Bi1 = Bi2 = 0.8, Nt =Nb = 0.2, Tr = 0.5, and S = 0.2

Fig. 11 Influence of β2 on concentration φ(η)vs. η when β1 = 0.4, P r = Sc =1.0, Bi1 = Bi2 = 0.8, Nt = Nb = 0.2,Tr = 0.5, and S = 0.2

Fig. 12 Influence of Sc on concentrationφ(η) vs. η when β1 = β2 = 0.4, P r =1.0, Bi1 = Bi2 = 0.8, Nt = Nb =0.2, Tr = 0.5, and S = 0.2

Fig. 13 Influence of Bi2 on concentration φ(η)vs. η when β1 = β2 = 0.4, P r = Sc =1.0, Bi1 = 0.8, Nt = Nb = 0.2, Tr =0.5, and S = 0.2

Table 1 is computed to see the convergent values of −f ′′(0),−θ′(0), and −φ′(0) at differentorders of HAM approximations. It is seen that the value of −f ′′(0) converges from the 10th-order of approximations, while it converges from 35th-order of deformations for −θ′(0) and−φ′(0). It is concluded from the tabular values that 35th-order deformations are sufficientfor the convergent solutions. Table 2 is computed to examine the numerical values of thelocal Nusselt and Sherwood numbers for various values of Nt, Nb, Tr, P r, Bi1, and Bi2 when


β1 = β2 = S = 0.1. It is seen that the values of the local Nusselt and Sherwood numbers arereduced when the value of Nt increases. The value of the local Sherwood number increases byincreasing the value of the Brownian motion parameter.

Fig. 14 Influence of Nt on concentrationφ(η) vs. η when β1 = β2 = 0.4,P r = Sc = 1.0, Bi1 = Bi2 = 0.8,Nb = 0.2, Tr = 0.5, and S = 0.2

Fig. 15 Influence of Nb on concentration φ(η)vs. η when β1 = β2 = 0.4, P r = Sc =1.0, Bi1 = Bi2 = 0.8, Nt = 0.2, Tr = 0.5,and S=0.2

Table 2 Numerical values of local Nusselt number −Nu/Re1/2

x and −Sh/Re1/2

x for different values ofNt, Nb, Tr, S, Bi1, and Bi2 when β1 = 0.1 = β2 = S and Sc = 1.0

Nt Nb Tr Pr Bi1 Bi2 Nu/Re1/2

x Sh/Re1/2

x

0.1 0.3 0.4 1.2 0.5 0.5 0.308 34 0.250 48

0.4 0.3 0.4 1.2 0.5 0.5 0.274 29 0.203 40

0.6 0.3 0.4 1.2 0.5 0.5 0.253 05 0.178 52

0.3 0.1 0.4 1.2 0.5 0.5 0.292 83 0.107 66

0.3 0.4 0.4 1.2 0.5 0.5 0.281 39 0.231 41

0.3 0.6 0.4 1.2 0.5 0.5 0.273 66 0.245 15

0.3 0.3 0.0 1.2 0.5 0.5 0.221 97 0.205 46

0.3 0.3 0.5 1.2 0.5 0.5 0.296 72 0.220 20

0.3 0.3 0.8 1.2 0.5 0.5 0.328 94 0.226 74

0.3 0.3 0.4 0.8 0.5 0.5 0.238 33 0.227 91

0.3 0.3 0.4 1.5 0.5 0.5 0.315 49 0.211 14

0.3 0.3 0.4 2.0 0.5 0.5 0.349 57 0.203 21

0.3 0.3 0.4 1.2 0.2 0.5 0.183 91 0.236 29

0.3 0.3 0.4 1.2 0.7 0.5 0.318 66 0.211 44

0.3 0.3 0.4 1.2 1.0 0.5 0.350 23 0.205 69

0.3 0.3 0.4 1.2 0.5 0.2 0.289 98 0.119 75

0.3 0.3 0.4 1.2 0.5 0.7 0.284 13 0.257 83

0.3 0.3 0.4 1.2 0.5 1.0 0.282 44 0.299 37

5 Conclusions

We investigate the two-dimensional boundary layer flow of an Oldroyd-B nanofluid withconvective heat and mass conditions. The analysis is carried out with thermal radiation andheat generation. This study has the following key observations:

(i) An increase in the value of β1 leads to an increase in the temperature θ(η) and theconcentration φ(η), but a reverse behavior is noted for higher values of β2.


(ii) The temperature θ(η) and the concentration φ(η) are increasing functions of the Biotnumbers Bi1 and Bi2.

(iii) Larger values of the thermophoresis and Brownian motion parameters give rise to thetemperature θ(η).

(iv) The temperature θ(η) increases when we increase the value of the radiation parameterTr.

(v) The temperature θ(η) is an increasing function of the heat generation parameter S.(vi) The concentration φ(η) is reduced for higher values of the Schmidt number Sc.(vii) The influence of the thermophoresis and Brownian motion on the concentration φ(η)

is quite reverse.

Acknowledgements The authors acknowledge with thanks to the Deanship of Scientific Research

(DSR) for technical and financial support.

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