https://doi.org/10.1007/s12043-019-1809-0
Entropy generation optimisation in the nanofluid flow of a second grade fluid with nonlinear thermal radiation
TASAWAR HAYAT1,2, MEHREEN KANWAL1, SUMAIRA QAYYUM1,∗, M IJAZ KHAN1 and AHMED ALSAEDI2
1Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan
2Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
∗Corresponding author. E-mail: sumaira@math.qau.edu.pk
MS received 12 February 2019; revised 13 March 2019; accepted 14 March 2019
Abstract. The flow of a second grade fluid by a rotating stretched disk is considered. Brownian motion and thermophoresis characterise the nanofluid. Entropy generation in the presence of heat generation/absorption, Joule heating and nonlinear thermal radiation is discussed. Homotopic convergent solutions are developed. The behaviour of velocities (radial, axial, tangential), temperature, entropy generation, Bejan number, Nusselt number, skin friction and concentration is evaluated. The radial, axial and tangential velocities increase for larger viscoelastic parameters while the opposite trend is noted for temperature. Concentration decreases when Schmidt number and Brownian diffusion increase. Entropy generation increases when the Bejan number increase while the opposite is true for the Brinkman number and the magnetic parameter.
Keywords. Buongiorno model; entropy generation; heat generation/absorption; Joule heating; nonlinear thermal radiation; second grade fluid.
PACS Nos 47.10.A; 47.15.G; 47.27.Ak 1. Introduction
The requirements for the development in the heat transfer rate cannot be achieved by ordinary fluids like water, kerosene oil, ethylene glycol, etc. Several exper- iments have been carried out by the researchers for improving heat exchange. Many techniques have been proposed in this direction by enhancing micrometre- sized particles for the thermal conductivity of con- vectional fluids. The major drawbacks in heat trans- fer components and high-pressure drops are blockage and erosion. To overcome such issues, the idea of nanofluids is introduced. Nanofluids is a suspension of particles of size 1–100 nm in base fluids. It is very effective as the suspension of these particles can enhance the thermal conductivity of base fluids and thus is useful in increasing the heat transfer rate. The enhancement of the thermophysical properties of con- ventional fluids using nanoparticles suspension is first examined by Choi [1]. Nanofluids have many applica- tions in fields such as nanocryosurgery, environment engineering, chemical industry, heat control systems, heat exchangers, energy storage, power production,
refrigeration process, etc. Due to these noteworthy applications, many researchers have already worked on this topic [1–10]. Many materials in nature have diverse properties. All such materials cannot be handled by the Navier–Stokes theory. These materials are viscoelastic.
Food stuff, care products, ketchup, shampoo, many fuel and oils are a few examples of such fluids. Many models like Maxwell, Williamson, Sisko, Jeffrey, Oldroyd-B, Burgers, generalised Burgers, etc. are developed for describing these fluids. Some contributions in this direc- tion have already been made [11–19].
The irreversibility process in the system is called entropy. In thermodynamics, the transfer of heat is related to the minimum change of entropy. To enhance the ability of machines, entropy generation minimi- sation (EGM) is utilised. Some applications of EGM include spin moment, internal molecular friction, kinetic energy and vibration. This type of loss of energy cannot be regained without extra work. That is why entropy is called the measure of irreversibility through heat transfer, mass transfer or viscous dissipation.
Several scientists used this process of minimisation in many systems like natural convection, fuel cells, 0123456789().: V,-vol
cooling by evaporation, gas turbines, etc. Qayyumet al [20] analysed entropy generation in radiative and Von Karman’s swirling flow with Soret and Dufour effects.
Berdichevsky [21] studied the effect of crystal plastic- ity in the presence of entropy. Khan et al[22] worked on disorderedness of the system for nanofluid flow after considering Arrhenius activation energy. An increase in the efficiency of thermal power plants through entropy generation is examined by Haseli [23]. Hayatet al[24]
discussed the entropy generation of a flow with non- linear thermal radiation. The generation of entropy in a gaseous phosphorus dimer is discussed by Jia et al [25]. The simulation of entropy generation by a similar method can be seen in [25] and Gibbs free energy and enthalpy generation in nitrogen monoxide and gaseous phosphorus dimer can be seen in refs [26–28].
This paper examines entropy generation optimisa- tion for the flow of a second-grade nanofluid. Nonlinear thermal radiation, heat generation/absorption and Joule heating in formulation are considered. The relevant non- linear problems are computed for the convergent series solutions by the homotopy analysis method [7,29–35].
The effects of sundry variables on velocity, temperature, Bejan number, entropy generation, skin friction coeffi- cients and concentration are examined.
2. Formulation
The flow of a second-grade nanofluid by a stretchable rotating disk is examined. Entropy generation for vis- cous dissipation, Joule heating and nonlinear thermal radiation is also discussed. A magnetic field of constant strength (B0) is exerted in the z-direction. The disk at z=0 rotates at an angular velocity (1) (see figure1).
The stretching velocity of the disk isa(withabeing the stretching rate). The disk and ambient temperature are denoted by Tˆw and Tˆ∞, respectively. The surface and ambient concentrations areCˆwandCˆ∞.
The governing equations in component form are
∂wˆ
∂z +uˆ r +∂uˆ
∂r =0, (1)
∂uˆ
∂zwˆ −vˆ2 r + ∂uˆ
∂ruˆ
= α1
ρf
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎝
∂3uˆ
∂r∂z2uˆ−1 r
∂uˆ
∂z 2
+2∂uˆ
∂r
∂2uˆ
∂z2
∂3uˆ
∂z3wˆ +∂vˆ
∂r
∂2vˆ
∂z2 +∂2uˆ
∂z2
∂wˆ
∂z +∂vˆ
∂z
∂2vˆ
∂r∂z +3∂uˆ
∂z
∂2uˆ
∂r∂z −∂2vˆ
∂z2 ˆ v r
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎠
Figure 1. Flow geometry.
+νf ∂2uˆ
∂z2 − σfB02
ρf u,ˆ (2)
∂vˆ
∂zwˆ +∂vˆ
∂ruˆ +vˆuˆ r
= α1
ρf
⎛
⎜⎜
⎜⎝
∂2vˆ
∂z2 ˆ u r −2∂vˆ
∂z
∂2uˆ
∂r∂z
∂3vˆ
∂r∂z2uˆ +∂3vˆ
∂z3wˆ −1 r
∂uˆ
∂z
∂vˆ
∂z
⎞
⎟⎟
⎟⎠+νf∂2vˆ
∂z2
−σfB02
ρf v,ˆ (3)
ρcp
f
∂Tˆ
∂z wˆ +∂Tˆ
∂r uˆ
= kf ∂2Tˆ
∂z2 +Q∗(Tˆ − ˆT∞) +σfB02(vˆ2+ ˆu2)
−16σ∗ 3k∗
⎡
⎣Tˆ3∂2Tˆ
∂z2 +3Tˆ2 ∂Tˆ
∂z 2⎤
⎦
+ ρcp
s
⎡
⎣DT
T∞ ∂Tˆ
∂z 2
+DB
∂Tˆ
∂z
∂Cˆ
∂z
⎤
⎦, (4)
∂Cˆ
∂r uˆ+ ∂Cˆ
∂z wˆ =DB∂2Cˆ
∂z2 + DT
T∞ ∂2Tˆ
∂z2
, (5)
with boundary conditions ˆ
u =r a, vˆ =r1, wˆ =0,
Cˆ = ˆCw, Tˆ = ˆTw, pˆ →0 at z =0, ˆ
u =0, vˆ =0, Cˆ → ˆC∞
Tˆ = ˆT∞whenz→ ∞. (6)
Here(uˆ,v,ˆ w)ˆ are velocities in the(ˆr,θ,ˆ zˆ) directions of the disk,α1is the material parameter,νf is the kine- matic viscosity,ρf is the density, pˆ is the pressure,kf is the thermal conductivity,cp is the specific heat, Q∗ is the heat generation/absorption coefficient and DB is the coefficient of diffusion species. Considering [18]
ˆ
u =r1f˜(ξ), vˆ =r1g˜(ξ), w˜ = −2h1f˜(ξ), ˆ
ϕ = Cˆ − ˆC∞
Cˆw− ˆC∞, θˆ = Tˆ − ˆT∞ Tˆw− ˆT∞, ˆ
p=ρf1vf
P(ξ)+ 1 2
r2 h2ε
, ξ = z
h, (7)
the continuity equation is satisfied and eqs (2)–(6) are reduced in the dimensionless form as
f˜+We Re
2f˜2+ ˜g2−2f˜f˜(iv)+ ˜ff˜
−Ref˜2−2f˜f˜− ˜g2
−Mf˜=0, (8)
˜
g+We Re
2fg˜−2f˜g˜−3f˜g˜
−Re
2f˜g˜−2f˜g˜
−Mg˜ =0, (9) θ˜+2Re Pr f˜θ˜+Re PrQθ˜+NbPrθ˜ϕ˜
+ NtPrθ˜2+Rd
3(θw−1)θ˜2+ ˜θ2θ˜2(θw−1)2 +2θ˜θ˜2(θw−1)
+ ˜θ+(θw−1)3θ˜3θ˜ +3(θw−1)θ˜θ˜+3(θw−1)2θ˜2θ˜ + MPr Ecf˜2+ ˜g2
=0, (10)
˜
ϕ+2Re Scf˜ϕ˜+ Nt
Nb
θ˜=0 (11)
with
f˜(0)=0, f˜(0)= A, f˜(∞)=0, g˜(0)=1,
˜
g(∞)=0, θ(˜ 0)=1, θ(∞)˜ =0, ϕ(˜ 0)=1,
˜
ϕ(∞)=0, p(0)=0, (12)
where We= α1
ρh2, Re= 1h2
νf , Pr= (ρcp)fνf
kf , A= a
1, Q= Q∗
ρcp1, Sc= υ Dc∗, δ = D∗d
Dc∗, Nb = τDB(Cw−C∞)
υ ,
Nt = τDB(Tw−T∞)
T∞υ , M = σB02 ρ1,
Rd = 16σ∗T∞3 3kf pk∗ , θw = Tw
T∞, Ec= r221
cp(Tw−T∞), (13) in which We, Re, Pr, A, Q, Sc, δ, Nb, Nt, M, Rd, θw and Ec represent the Weissenberg number, Reynolds number, Prandtl number, stretching param- eter, heat generation/absorption parameter, Schmidt number, ratio of diffusion coefficient, Brownian param- eter, thermophoresis parameter, magnetic parameter, radiation parameter, temperature difference and the Eck- ert number, respectively.
We haveCfθ andCf r as skin friction coefficients in the tangential and radial direction, i.e.
Cfθ = τzr
ρ(r1)2 Cf r = τzθ
ρ(r1)2
⎫⎪
⎬
⎪⎭, (14) where shear stressesτzr andτzθ are
τzr =μ ∂wˆ
∂r +∂uˆ
∂z
+α1
+2
∂vˆ
∂r −vˆ r
∂vˆ
∂z +
∂uˆ
∂z +∂wˆ
∂r ∂
∂ruˆ+ ∂
∂zwˆ
+∂wˆ
∂z
∂uˆ
∂z +∂uˆ
∂r
∂wˆ
∂r +3 ∂wˆ
∂r
∂wˆ
∂z +∂uˆ
∂z
∂uˆ
∂r
−α1
∂vˆ
∂r −vˆ r
∂vˆ
∂z +
∂uˆ
∂r +∂wˆ
∂z ∂uˆ
∂z +∂wˆ
∂r
, (15)
τzθ =μ∂vˆ
∂z +α1
∂2v
∂z2wˆ −vˆ r
∂uˆ
∂z +∂vˆ
∂z
∂wˆ
∂z + ∂2vˆ
∂r∂zuˆ +∂uˆ
∂z
∂vˆ
∂r +3uˆ r
∂vˆ
∂z
−α1
∂uˆ
∂z
∂vˆ
∂r +2uˆ r
∂vˆ
∂z +2∂vˆ
∂z
∂wˆ
∂z − vˆ r
∂wˆ
∂r
−vˆ r
∂uˆ
∂z +∂wˆ
∂r
∂vˆ
∂r
. (16)
Skin friction coefficients in the dimensionless form are RerCf r = ˜f(0)
+We Re
3f˜(0)f˜(0)−2f˜(0)f˜(0)
, (17)
RerCfθ = ˜g(0) +We Re
4f˜(0)g˜(0)−2f˜(0)g˜(0)
, (18)
where Rer =r1h/νdepicts the local Reynolds num- ber.
The heat transfer rate is Nux = hqw
k(Tˆw− ˆT∞). (19) The wall heat flux (qw) is defined as
qw|z=0 = −k∂Tˆ
∂z z=0
− 16σ∗Tˆ3 3k∗
∂Tˆ
∂z z=0
. (20)
The Nusselt number in the dimensionless form is Nux = − ˜θ(0)
1+Rdθw3
. (21)
2.1 Entropy generation
Entropy generation in the nanofluid flow of a second- grade fluid with nonlinear thermal radiation irreversibil- ity, viscous dissipation irreversibility and Joule heating irreversibility is discussed here. The dimensional form is defined as
SG = kf Tˆ∞2
1+16σ∗Tˆ3 3kfk∗
∂Tˆ
∂z 2
+
Tˆ∞ +σf
Tˆ∞B02 ˆ
v2+ ˆu2 + R D
Tˆ∞ ∂Tˆ
∂z
∂Cˆ
∂z
+R D Cˆ∞
∂Cˆ
∂z 2
, (22)
where =μf
∂uˆ
∂z 2
+ ∂vˆ
∂z 2
+α1
−vˆ r
∂vˆ
∂z
∂uˆ
∂z −∂vˆ
∂r
∂vˆ
∂z
∂uˆ
∂z +2∂uˆ
∂r ∂uˆ
∂z 2
+ ˆu∂uˆ
∂z
∂2uˆ
∂r∂z +∂wˆ
∂z ∂uˆ
∂z 2
+ ˆw∂vˆ
∂z
∂2vˆ
∂z2 +vˆ
r ∂uˆ
∂z 2
− ∂vˆ
∂r ∂uˆ
∂z 2
. (23)
The above two equations yield
SG = kf
Tˆ∞2
1+ 16σ∗Tˆ3 3kfk∗
∂Tˆ
∂z 2
+ 1 Tˆ∞
μf
∂uˆ
∂z 2
+ ∂vˆ
∂z 2
+α1
2∂uˆ
∂r ∂uˆ
∂z 2
−vˆ r
∂vˆ
∂z
∂uˆ
∂z −∂vˆ
∂r
∂vˆ
∂z
∂uˆ
∂z + ˆu∂uˆ
∂z
∂2uˆ
∂r∂z
+ ˆw∂vˆ
∂z
∂2vˆ
∂z2 + ∂wˆ
∂z ∂uˆ
∂z 2
−∂vˆ
∂r ∂uˆ
∂z 2
+vˆ r
∂uˆ
∂z 2
+σf
Tˆ∞B02 ˆ u2+ ˆv2
+ R D Cˆ∞
∂Cˆ
∂z 2
+R D Tˆ∞
∂Tˆ
∂z
∂Cˆ
∂z
. (24)
Equation (24) consists of four factors: (i) heat transfer irreversibility, (ii) fluid friction irreversibility, (iii) Joule heating irreversibility and (iv) diffusive irreversibility.
Now, eq. (24) in the dimensionless form is NG =α1
1+Rd
1+ ˜θ
θw−13θ˜2 +Brf˜2+ ˜g2
+We Re Brf˜f˜2−2f˜g˜g˜−2f˜g˜g˜ +BrMf˜2+ ˜g2
+L∗θ˜ϕ˜+L∗α2
α1ϕ˜2, (25) in which the dimensionless parameters are
NG = SGTˆ∞h2 Tˆw− ˆT∞
kf , L∗= RD(Cˆw− ˆC∞)
kf ,
α1 = Tˆw− ˆT∞
Tˆ∞ , α2= Cˆw− ˆC∞ Cˆ∞ , Br= μfr221
kf(Tˆw− ˆT∞) , (26) where NG indicates entropy generation, Br is the Brinkman number,α1is the temperature ratio,α2is the concentration ratio and L∗ is the diffusive parameter.
The Bejan number is defined as
Be= Entropy generation due to heat and mass transfer irreversibility
Total entropy generation , (27)
Be= ⎡ α1[1+Rd(1+ ˜θ(θw−1))3] ˜θ2 +L∗θ˜ϕ˜+L∗(α2/α1)ϕ˜2
⎣α1[1+Rd(1+ ˜θ(θw−1))3] ˜θ2+Br(f˜+ ˜g2)+We Re Br(f˜f˜2−2f˜g˜g˜
−2f˜g˜g˜)+BrM(f˜2+ ˜g2)+L∗θ˜ϕ˜+L∗(α2/α1)ϕ˜2
⎤
⎦
. (28)
3. Homotopic solutions
3.1 Zeroth-order deformation equations
The required linear operators and initial guesses are defined as
f˜0(ξ)=(1−exp(−ξ))A, g˜0(ξ)=exp(−ξ),
θ˜0(ξ)=exp(−ξ), ϕ˜0(ξ)=exp(−ξ), (29) Lf˜(f˜)= ˜f− ˜f, Lg˜(g)˜ = ˜g− ˜g,
Lθ˜(θ)˜ = ˜θ− ˜θ, Lϕ˜(ϕ)˜ = ˜ϕ− ˜ϕ (30) with
Lf˜(f˜)
c1+c2e−ξ +c3eξ
=0, Lg˜(g˜)
c4e−ξ +c5eξ
=0, Lθ˜(θ)˜
c6e−ξ +c7eξ
=0, Lϕ˜(ϕ)˜
c8e−ξ +c9eξ
=0, (31)
whereci (i =1−9)are constants.
Ifq ∈ [0,1] and(h¯f˜, h¯g˜, h¯θ˜ andh¯ϕ˜)are the aux- iliary parameters, then the zeroth-order deformation problems are
(1−q)Lf˜
f˜(ξ,q)− ˜f0(ξ)
=qh¯ f˜Nf˜
f˜(ξ,q), g(ξ,˜ q) , (32) (1−q)Lg˜{ ˜g(ξ,q)− ˜g0(ξ)}
=qh¯g˜Ng˜
[ ˜g(ξ,q), f˜(ξ,q) , (33) (1−q)Lθ˜
θ(ξ,˜ q)− ˜θ0(ξ)
=qh¯θ˜Nθ˜
θ(ξ,˜ q), f˜(ξ,q),g(ξ,˜ q),ϕ(ξ,˜ q) , (34) (1−q)Lϕ˜{ ˜ϕ(ξ,q)− ˜ϕ0(ξ)}
=qh¯ϕ˜Nϕ˜
ϕ(ξ,˜ q), f˜(ξ,q),θ(ξ,˜ q) , (35) f˜(0,q)=0, f˜(0,q)= A, f˜(∞,q)=0, (36)
˜
g(0,q)=1, g˜(0,q)=0, (37) θ(˜ 0,q)=1, θ(∞,˜ q)=0, (38)
˜
ϕ(0,q)=1, ϕ(∞,˜ q)=0 (39)
with operatorsNf˜,Ng˜,Nθ˜andNϕ˜ in the forms Nf˜= ∂3f˜(ξ,q)
∂ξ3 +We Re
⎛
⎝2
∂2f˜(ξ,q)
∂ξ2 2
+
∂g˜(ξ,q)
∂ξ 2
−2f˜(ξ,q)∂4f˜(ξ,q)
∂ξ4 +∂f˜(ξ,q)
∂ξ
∂3f˜(ξ,q)
∂ξ3
⎞
⎠
−Re
⎡
⎣
∂f˜(ξ,q)
∂ξ 2
−2f˜(ξ,q)∂2f˜(ξ,q)
∂ξ2
−(g˜(ξ,q))2
⎤
⎦−M∂f˜(ξ,q)
∂ξ , (40)
Ng˜ = ∂2g˜(ξ,q)
∂ξ2 +We Re
2∂f˜(ξ,q)
∂ξ
∂2g˜(ξ,q)
∂ξ2
−2f˜(ξ,q)∂3g˜(ξ,q)
∂ξ3 −3∂2f˜(ξ,q)
∂ξ2
∂g˜(ξ,q)
∂ξ
−Re
2∂f˜(ξ,q)
∂ξ g˜(ξ,q)−2f˜(ξ,q)∂g(ξ,˜ q)
∂ξ
−Mg(ξ,˜ q), (41)
Nθ˜ = ∂2θ(ξ,˜ q)
∂ξ2 +2Re Prf˜(ξ,q)∂θ(ξ,˜ q)
∂ξ +Re PrQθ(ξ,˜ q)
+NtPr
∂θ(ξ,˜ q)
∂ξ 2
+NbPr∂θ(ξ,˜ q)
∂ξ
∂ϕ(ξ,˜ q)
∂ξ +MPr Ec
⎧⎨
⎩
∂f˜(ξ,q)
∂ξ 2
+(g˜(ξ,q))2
⎫⎬
⎭ +Rd
⎧⎨
⎩3(θw−1)
∂θ(ξ,˜ q)
∂ξ 2
×
1+
θ(ξ,˜ q)2
(θw−1)2+2θ(ξ,˜ q)(θw−1)
+∂2θ(ξ,˜ q)
∂ξ2
$
1+(θw−1)3(θ(ξ,˜ q))3
+3(θw−1)2(θ(ξ,˜ q))2+3(θw−1)θ(ξ,˜ q)% , (42) Nϕ˜ = ∂2ϕ(ξ,˜ q)
∂ξ2 +2Sc Re f˜(ξ,q)∂ϕ(ξ,˜ q)
∂ξ +Nt
Nb
∂2θ(ξ,˜ q)
∂ξ2 . (43)
We now expand f˜(ξ,q), g˜(ξ,q), θ(ξ,˜ q) and
˜
ϕ(ξ,q)by using the Taylor series aboutq =0 as f˜(ξ,q)= ˜f0(ξ)+
&∞ m=1
f˜m(ξ)qm; f˜m(ξ)= 1
m
∂mf˜
∂qm
q=0
, (44)
˜
g(ξ,q)= ˜g0(ξ)+
&∞ m=1
˜
gm(ξ)qm;
˜
gm(ξ)= 1 m!
∂mg˜
∂qm
q=0
, (45)
θ(ξ,˜ q)= ˜θ0(ξ)+
&∞ m=1
θ˜m(ξ)qm; θ˜m(ξ)= 1
m!
∂mθ˜
∂qm
q=0
, (46)
˜
ϕ(ξ,q)= ˜ϕ0(ξ)+
&∞ m=1
˜
ϕm(ξ)qm;
˜
ϕm(ξ)= 1 m!
∂mϕ˜
∂qm
q=0
. (47)
3.2 mth-order deformation equations Themth-order problems are
Lf˜f˜m(ξ)−χmf˜m−1(ξ)
= ¯hf˜Rm˜
f(ξ), (48)
Lg˜
˜
gm(ξ)−χmg˜m−1(ξ)
= ¯hg˜Rmg˜(ξ), (49) Lθ˜θ˜m(ξ)−χmθ˜m−1(ξ)
= ¯hθ˜Rm˜
θ(ξ), (50)
Lϕ˜
˜
ϕm(ξ)−χmϕ˜m−1(ξ)
= ¯hϕ˜Rmϕ˜(ξ), (51) f˜m(0)= ∂f˜m(0)
∂ξ = ∂f˜m(∞)
∂ξ =0, θ(˜ 0)= ˜θ(∞)=0,
˜
gm(0)= ˜gm(∞)=0,
˜
ϕ(0)= ˜ϕ(∞)=0, (52)
where the functions Rm˜
f(ξ),Rmg˜(ξ), Rm˜
θ(ξ) and Rmϕ˜(ξ) are
Rm˜
f(ξ)= ˜fm−1 +We Re
2
m&−1 k=0
f˜m−1−kf˜k+
m&−1 k=0
˜
gm −1−kg˜k
−2
m&−1 k=0
f˜m−1−kf˜kiv+
m&−1 k=0
f˜m−1−kf˜k
−Re m−1
&
k=0
f˜m−1−k f˜k−
m&−1 k=0
f˜m−1−kf˜k
−
m−1&
k=0
˜
gm−1−kg˜k
−Mf˜m−1, (53)
Rmg˜(ξ)= ˜gm−1+We Re
2
m&−1 k=0
fm−1−kg˜k
−2
m&−1 k=0
f˜m−1−kg˜k−3
m&−1 k=0
f˜m−1−k g˜k
−Re
2
m−1&
k=0
f˜m−1−kg˜k−2
m−1&
k=0
f˜m−1−kg˜k
−Mg˜m−1, (54)
Rm˜
θ(ξ)= ˜θm−1 +2Re Pr
m&−1 k=0
f˜m−1−kθ˜k
+Re PrQ
m&−1 k=0
θ˜m−1+NbPr
m&−1 k=0
θ˜m−1−kϕ˜k
+NtPr
m&−1 k=0
θ˜2m−1
+MPr Ec
m−1&
k=0
f˜2m−1−k+gm2−1−k
+Rd
3(θw−1) θ˜2m−1
+
m−1&
k=0
θ˜m2−1−kθ˜2k(θw−1)2
+2
m&−1 k=0
θ˜m−1−kθ˜2k(θw−1)
+ ˜θm−1
+(θw−1)3
m&−1 k=0
θ˜m3−1−kθ˜k
+3(θw−1)
m&−1 k=0
θ˜m−1−kθ˜k
+3
m&−1 k=0
(θw−1)2θ˜m−1−k2 θ˜k
, (55)
Rmϕ˜(ξ)= ˜ϕm−1 +2Re Sc
m&−1 k=0
f˜m−1−Kϕ˜K + Nt
Nb
θ˜m−1 , (56) χm =
'0, m≤1,
1, m>1. (57)
The general solutions can be written as
f˜m(ξ)= ˜fm∗(ξ)+c1+c2e−ξ +c3eξ, (58)
˜
gm(ξ)= ˜gm∗(ξ)+c4e−ξ +c5eξ, (59) θ˜m(ξ)= ˜θm∗(ξ)+c6e−ξ +c7eξ, (60)
˜
ϕm(ξ)= ˜ϕm∗(ξ)+c8e−ξ +c9eξ, (61) where the constantsci(i =1−9)by using the boundary conditions (52) have the values
c1= −c2− ˜fm∗(0), c2= ∂f˜m∗(0)
∂ξ , c4= −∂g˜m∗(0)
∂ξ , c6 = −∂θ˜m∗(0)
∂ξ , c8= −∂ϕ˜m∗(0)
∂ξ , c3 =c5 =c7=c9=0. (62)
4. Convergence analysis
With the help of auxiliary parameters h¯f˜, h¯g˜,h¯θ˜ and
¯
hϕ˜, we can regulate the convergence region. By using the homotopy analysis method (HAM), we can solve the system of equations. Figure 2 shows the h-curves¯ at the 25th order of deformation. Convergence regions for these parameters are −2 ≤ ¯hf˜ ≤ −1, −2 ≤
¯
hg˜ ≤ −0.1,−2 ≤ ¯hθ˜ ≤ −0.5 and −1.5 ≤ ¯hϕ˜ ≤
−0.5. Table1demonstrates the convergence order for f˜(0),g˜(0),θ˜(0) and ϕ˜(0) which converges at the 12th, 11th, 28th and 28th order of approximations, respectively.
Figure 2. hcurve at f˜(0),g˜(0),θ˜(0)andϕ˜(0).
Table 1. Solution convergence occurs when Re = 0.3, We = 0.01,Rd = 0.01, θw = 0.1,A = 0.4, Pr = 1.5,Q = 0.01,Ec = 0.4,Sc = 1,Nb = 0.3, Nt =0.01,M =0.2.
Order of approximation
− ˜f(0) − ˜g(0) − ˜θ(0) − ˜ϕ(0)
1 0.05554 0.5800 0.5713 0.5233
12 0.09494 0.6942 0.2365 0.3021
17 0.09494 0.6942 0.2277 0.2939
24 0.09494 0.6942 0.2235 0.2890
27 0.09494 0.6942 0.2228 0.2879
28 0.09494 0.6942 0.2226 0.2877
5. Results and discussion
In this section, the behaviour of influential variables on velocity, temperature, concentration, coefficients of skin friction and the Nusselt number is analysed. In figures3–23, we fixed We = 0.3, Re = 0.9 = 0.9
=0.7,Ec =0.4,Q =0.7,Nt =0.3,Nb =0.3,Rd = 0.5, θw =0.2, Sc=1 and M =0.5.
5.1 Dimensionless velocities
Figures 3–10 illustrate the velocities of various parameters. In figures3and4, the effects of viscoelas- tic parameter(ξ)
when We=0,0.4,0.8,1.2
on axial f˜(ξ)
and radial (f˜(ξ)) velocities are shown. We know that the Weissenberg number (We) is inversely proportional to the fluid viscosity because of which the motion of the fluid increases with larger We. The effect of the Reynolds number Re on f˜(ξ),g˜(ξ)
is shown in figures 5 and 6. Here, for increasing val- ues of the Reynolds number (Re)(Re=0,0.5,1,1.5), (f˜(ξ),g˜(ξ))decreases because inertial forces become stronger for higher Re. The behaviour of velocities (f˜(ξ),g˜(ξ)) for the stretching parameter (A) is
Figure 3. Graph of f(ξ)against We.
Figure 4. Graph of f(ξ)against We.
Figure 5. Graph of f(ξ)against Re.
Figure 6. Graph ofg(ξ)against Re.
Figure 7. Graph of f(ξ)againstA.
Figure 8. Graph ofg(ξ)againstA.
Figure 9. Graph of f(ξ)againstM.
discussed in figures 7 and 8. For larger values of A(A=0, 0.2, 0.4, 0.6), momentum boundary layer and f˜(ξ)rise for a longer stretching rate (see figure7).
On the other hand, the reverse trend is noticed forg˜(ξ) because the angular velocity (1) reduces. Figures 9 and 10 show the effect of magnetic parameter M on f˜(ξ),g˜(ξ)
. We know that the Lorentz force is related to the magnetic field which causes resistance to the flow, and sof˜(ξ), g˜(ξ)
reduces forM.
5.2 Temperature
The analysis of temperature distribution θ(ξ)˜ against different parameters is deliberated in figures11–16. The
Figure 10. Graph ofg(ξ)againstM.
Figure 11. Graph ofθ(ξ)against Pr.
Figure 12. Graph ofθ(ξ)againstQ.
Figure 13. Graph ofθ(ξ)againstθw.
Figure 14. Graph ofθ(ξ)againstNb.
Figure 15. Graph ofθ(ξ)againstM.
Figure 16. Graph ofθ(ξ)againstNt.
behaviour of Pr with θ(ξ)˜ is discussed in figure 11.
As Pr is inversely proportional to thermal diffusivity, for large values of Pr (Pr = 0.4,0.8,1.2,1.6), the temperature of the fluid decreases. The effect ofQon the temperature is shown in figure12. Clearly,θ(ξ)˜ boosts up for larger Q (Q =0,0.4,0.8,1.2) because heat generation/absorption coefficient increases. Figure 13 demonstrates the behaviour of θ(ξ)˜ against θw. The thermal state of the fluid enhances by increasing θw (θw = 1.1,1.3,1.5,1.7) due to which the tempera- ture is enhanced. Figure 14 illustrates the effect of Nb on θ(ξ)˜ . Temperature increases for higher Nb. Figure 15 shows the behaviour of the magnetic field
Figure 17. Graph ofϕ(ξ)against Sc.
Figure 18. Graph ofϕ(ξ)againstNt.
Figure 19. Graph ofϕ(ξ)againstNb.
(M) on temperature distribution. An increase in M (M = 0,0.5,1,1.5) gives rise toθ(ξ)˜ . It is for larger Lorentz force. In figure16, bothθ(ξ)˜ and the thermal layer thickness are increased forNt =0,0.5,1,1.5. An enhancement in Nt results in a stronger thermophoretic force due to which nanoparticles are transferred from warm to cold regions, and henceθ(ξ)˜ rises.
5.3 Concentration
The behaviour of the concentration is portrayed in figures 17–19. The influence of the Schmidt number (Sc)onϕ(ξ)˜ is described in figure17. Large values of Sc (Sc = 0,0.3,0.6,0.9) decrease the concentration.
Figure 20. Graph of Nuxagainst We.
Figure 21. Graph of NuxagainstNt.
In fact, higher values of Sc result in low molecular diffusivity. The impacts of Nt and Nb on ϕ(ξ)˜ are shown in figures18 and19. With an enhancement of Nt = 0,0.5,1,1.5, thermophoresis force rises. Such force tends to move nanoparticles from warm to cold regions and hence ϕ(ξ)˜ rises. Moreover, the concen- tration layer thickness is also enhanced for larger Nt. The higher the values ofNb(Nb=0.3, 0.6, 0.9, 1.2), the smoother is the distribution of nanoparticles concen- tration in the fluid system, which eventually decreases
˜ ϕ(ξ).
5.4 Nusselt number
Figures 20 and 21 demonstrate the effect of the vis- coelastic parameter(We)and the thermophoresis param- eter(Nt)on the rate of heat transfer. These figures show that the Nusselt number increases for higher values of We while a reverse behaviour is noticed forNt.
5.5 Skin friction coefficients
Figures22and23indicate the effects of Aand We on the skin friction coefficients. Here, the magnitude of the surface drag force in radial and tangential directions is more for larger Aand We.
Figure 22. Graph ofC fr Rer andC fθ Reθ against We.
Figure 23. Graph ofC fr Rer andC fθReθagainst A.
Figure 24. Graph ofNGagainst Br.
5.6 Entropy generation and Bejan number
Figures24–33illustrate the trends ofNGand Be for dif- ferent parameters. Figures24and25show the behaviour ofNGand Be on the Brinkman number(Br). Br is asso- ciated with the heat transfer from a disk to the flow of a viscous fluid. Figure 24 indicates more entropy generation rate for larger Br because by dissipation, the conduction rate is slowly created. Figure25shows the behaviour of Br on Be as entropy generation is more for large Br. It means viscosity is dominant over heat transfer irreversibility, and hence Be decreases.
Figure 25. Graph of Be against Br.
Figure 26. Graph ofNGagainstM.
Figure 27. Graph of Be againstM.
In figures 26 and 27, the influence of the magnetic variable (M) on NG and Be is noticed. In figure 26, the entropy generation rate is addressed for large val- ues of M. The entropy generation rate is enhanced for large M because drag force is higher for larger M.
Figure 27 shows the decaying behaviour of Be for a larger magnetic variableM. Figures28and29describe the behaviour of the diffusion parameter (L∗) on NG
and Be. It is noticed that for L∗, both NG and Be are increasing functions. The diffusion rate of nanopar- ticles enhances for larger L∗. That is why the total entropy of the system and Be are enhanced. Figures30 and 31 show the impact of θw on NG and Be. Here, both NG and Be are increasing functions of θw. As
Figure 28. Graph ofNG againstL∗.
Figure 29. Graph of Be againstL∗.
Figure 30. Graph ofNGagainstθw.
Figure 31. Graph of Be againstθw.
Figure 32. Graph ofNGagainst We.
Figure 33. Graph of Be against We.
the disk is heated, for larger values of the temperature ratio(θw), the disorderedness near the disk is high and so NG increases (see figure30). Here, the irreversibility of the mass and heat transfer prevail over the irreversibility of the friction of the fluid for higherθw, and so Be also rises (see figure31). Trends ofNG and Be vs. We are displayed in figures 32 and33. Disorderedness in the system is more for larger We (see figure32). Be decays for increasing We (see figure33).
6. Conclusions
Major findings of this study are:
• For larger viscoelastic parameter(We), the velocities (radial(f˜(ξ)), axial(f˜(ξ))and tangential (g˜(ξ))) are increased.
• Temperature (θ(ξ))˜ against θw, Nt, M and Q en- hances.
• Concentration reduces for A,Nb,Re and Sc.
• Entropy generation (NG) enhances for We and Br while the opposite trend is noticed against We.
• Both NG and Be are enhanced for larger L∗ and θw.
