Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a

15 Pages·2017·0.97 MB·English

Aeronautics and Aerospace Open Access Journal
Research Article
Open Access
Finite element analysis of flow and heat transfer
of dissipative Casson-Carreau nanofluid over a
stretching sheet embedded in a porous medium
Abstract Volume 2 Issue 5 - 2018
In this paper, combined influences of thermal radiation, inclined magnetic field and MG Sobamowo, AA Yinusa, AA Oluwo, SI
temperature-dependent internal heat generation on unsteady two-dimensional flow
Alozie
and heat transfer analysis of dissipative Casson-Carreau nanofluid over a stretching
Department of Mechanical Engineering, University of Lagos,
sheet embedded in a porous medium is investigated. Similarity transformations are
Nigeria
used to reduce the developed systems of governing partial differential equations to
nonlinear third and second orders ordinary differential equations which are solved
Correspondence: MG Sobamowo, Department of Mechanical
using finite element method. In the study, kerosene is used as the base fluid which is
Engineering, University of Lagos, Akoka, Lagos, Nigeria, Tel
embedded with the silver (Ag) and copper (Cu) nanoparticles. Also, effects of other 2347034717417, Email [email protected]
pertinent parameters on the flow and heat transfer characteristics of the Casson-
Carreau nanofluids are investigated and discussed. From the results, it is established Received: September 18, 2018 | Published: October 08, 2018
temperature field and the thermal boundary layers of Ag-Kerosene nanofluid are
highly effective when compared with the Cu-Kerosene nanofluid. Heat transfer rate is
enhanced by increasing in power-law index and unsteadiness parameter. Skin friction
coefficient and local Nusselt number can be reduced by magnetic field parameter and
they can be enhanced by increasing the aligned angle. Friction factor is depreciated
and the rate of heat transfer increases by increasing the Weissenberg number. A very
good agreement is established between the results of the present study and the previous
results. The results of present analysis can help in expanding the understanding of the
thermo-fluidic behaviour of the Casson-Carreau nanofluid over a stretching sheet as
applied in manufacturing industries and production engineering.
Keywords: MHD, nanofluid, non-uniform heat source/sink, Carreau fluid, thermal
radiation and free convection, finite element method
Introduction In an early study, MHD fluid flow over a surface that is susceptible
to stretching was critically studied by Anderson et al.2,3 In their studies,
The recent studies in the past few decades have shown that the effect of transient variables on the film size2 and magnetic influence
study of Non-Newtonian have attracted tremendous attractions in on the fluid flow characteristics were explored numerically.3 Few
the study of fluid dynamics. The flow applications of non-Newtonian years later, Chen4 investigated the fluid film that obeys power-law
fluids are evident in polymer devolatization and processing, wire flow of unsteady thermal-stretching sheet while Dandapat5,6 focused
and fiber coating, heat exchangers, extrusion process, chemical on the effect of changing viscosity as well as thermo-capillarity on the
processing equipment, etc. Also, combining heat transfer with the heat transfer rate of film flow over a sheet susceptible to stretching.
concept of stretching flow is vital in the afore-mentioned areas of Meanwhile, Wang7 developed an analytical or close form solution
applications. Such processes have great affinity for cooling rates for momentum and energy (heat transfer) of film flow over a surface
and stretching simultaneously.1 Consequently, in the past few years, susceptible to stretching. Also, Chen8 and Sajid et al.9 investigated
research efforts have been directed towards the analysis of this very the motion characteristics involving non-Newtonian thin film over
important phenomenon of wide areas of applications. Moreover, a transient stretching surface considering viscous dissipation using
the promising significance of magnetohydrodynamics (MHD) HAM and HPM. After a year, Dandapat et al.,10 presented the analysis
fluid behavior in concerned applications such as in blood flow using a two-dimensional flow over a transient sheet that is capable
still provokes the continuous studies and interests of researchers. of stretching while in the same year, impact of power-law index was
Additionally, the incorporation of radiation through thermal analysis carried out by Abbasbandy et al.11 Santra & Dandapat12 numerically
is vital in technology involving solar energy, space vehicles, investigated the same considering a transient horizontal elongating
systems with propulsion, plasma physics in the flow structure of sheet. A numerical approach was also used by Sajid et al.13 to analyze
atomic plants, combustion processes, internal combustion engines, the micropolar film flow over an inclined plate, moving belt and
ship compressors, solar radiations and in chemical processes and vertical cylinder. A year later, Noor & Hashim14 investigated the
space ship with high temperature level re-entry aerodynamics. influence of magnetic parameter and thermocapillarity on an unsteady
Furthermore, there are various engineering and industrial applications liquid film flow over similar sheet while Dandapat & Chakraborty15
of magnetohydrodynamics (MHD) fluid behavior such as in the study and Dandapat & Singh16 presented the thin film flow analysis over a
of the growth of crystals, blood flow etc. Therefore, the influences non-linear stretching surface with the effect of transverse magnetic
of external factors such as thermal radiation and magnetic field on field. Heat transfer characteristics of the thin film flows considering
the thermofluid problem concerning Newtonian and Non-Newtonian the different channels have also been analyzed by Abdel-Rahman,17
fluid have been widely analyzed in recent times. Khan et al.18 Liu et al.19 and Vajaravelu et al.20 Meanwhile, Liu &
Submit Manuscript Aeron Aero Open Access J. 2018;2(5):294‒308. 294
| http://medcraveonline.com © 2018 Sobamowo et al. This is an open access article distributed under the terms of the Creative Commons Attribution License,
which permits unrestricted use, distribution, and build upon your work non-commercially.
Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a stretching Copyright:
sheet embedded in a porous medium ©2018 Sobamowo et al. 295
Megahad21 used HPM to analyze the thermofluidic effect of thin film a conducting and heat generating Casson and Carreau nanofluids over
with internal heat generation and changing heat flux while Seddeek a sheet susceptible to stretching extreme by liquid film of uniform
as well as other recent works22-35 investigated the impacts of thermal size (thickness) h(t) as represented schematically in Figure 1. The
radiation and changing viscosity on magnetohydrodynamics in stretching velocity along x-axis is U(x,t) and y–axis is perpendicular
unforced convection fluid flow over a flat plate that is semi-infinite. to it with dissipation and volume fraction considered.
Other researchers went on to present their view on the interesting
topic by adapting nanofluid into the generic three dimensional fluid
model with radiation effect.36-48 Makinde & Animasaun49 investigated
the effect of cross diffusion on MHD bioconvection flow over
a horizontal surface. In another study, Makinde & Animasaun50
presented the MHD nanofluid on bioconvection flow of a paraboloid
revolution with nonlinear thermal radiation and chemical reaction
while Sandeep,51 Reddy et al.52 and Ali et al.53 studied the heat transfer
behaviour of MHD flows. Maity et al.54 analyzed thermocapillary
flow of a thin Nanoliquid film over an unsteady stretching sheet.
Furthermore, different studies on the flow and heat transfer behaviour
as well as entropy generation for different non-Newtonian fluids
under difference internal and external conditions.55-72 The above
reviewed studies have been the consequent of the various industrial
and engineering applications of non-Newtonian fluids. Among the
classes of non-Newtonian fluids, Carreau fluid which its rheological Figure 1 Flow geometry of the problem.
expressions were first introduced by Carreau,73 is one of the non-
Using [85],
Newtonian fluids that its model is substantial for gooey, high and low
shear rates.74 On account of this headway, it has profited in numerous τ=τ +µσ (1)
0
innovative and assembling streams.74-87 Owing to these applications,
or
different studies have been carried out to explore the characteristics
of Carreau liquid in flow under different conditions. Kumar et al.40
  p  
applied numerical scheme to analyze the thermofluidic behaviour of a τ=2µ + y e , π>π 
liquid film capable of conducting electricity. The fluid is based on the   B 2π ij c
structure of liquid phase and interactive behaviour of solid of a two-
(2)
phase suspension. It is able to capture complex rheological properties   p  
of a fluid, unlike other simplified models like the power law88 and =2µ + y e , π <π
B ij c
second, third or fourth-grade models.89,90   2π  
c
Casson fluids are Jelly, honey, protein, Human blood and fruit
Also employing the tensor given as [95]
juices. Concentrated fluids like sauces, honey, juices, blood, and
printing inks can be well described using this model. The effect of  (n−1) 
( )2
magnetohydrodynamic Casson fluid flow on a laterally positioned τ =η 1+ Γγ γ  (3)
ij 0 ij
elongating sheet was explained by Nadeem et al.91 The review studies  2 
have been analyzed using approximate analytical, semi-analytical and
numerical methods. Among the numerical methods, the numerical τ is the extra tensor, η is the zero shear rate viscosity,Γ is the
ij o
solutions of FEM represent efficient ways of obtaining solutions time constant, n is the power-law index and γ is defined as
to nonlinear problems even with complexities in the boundary
conditions and geometries.92-94 Therefore, using FEM, this work 1 1
presents numerical investigations of the combined influences of γ = ∑i∑jγijγji = Π
2 2
thermal radiation, inclined magnetic field and internal heat generation
that is temperature-dependent on unsteady two-dimensional flow and
where Π is the second invariant strain tensor.
heat transfer analysis of dissipative Casson-Carreau nanofluid over
a stretching sheet embedded in a porous medium. Using kerosene Following the assumptions, the equations for continuity and
as the base fluid embedded with the silver (Ag) and copper (Cu) motion for the flow analysis of Carreau and Casson fluids are
nanoparticles, the effects of other pertinent parameters on flow and
heat transfer characteristics of the nanofluids are investigated and ∂u ∂v
+ =0
discussed. The analysis of the stretched flows with heat transfer is ∂x ∂y (4)
very significant in controlling the qualities of the end products in
mgraenatu fdaecptuernidnegn caensd omn etthael fsotrremtcinhgin gp raoncdes sceoso. liSnugc hra tperso. cTehssee sr ehsualvtes ρnf∂∂ut +u∂∂ux+v∂∂uy=µnf1+β11+3(n−21)Γ2∂∂uy2∂∂y2u2 −σBo2ucos2γ−µKnfu
of present analysis can help in expanding the understanding of the
(5)
thermo-fluidic behaviour of the Casson-Carreau nanofluid over a

stretching sheet as applied in manufacturing industries and production ( ) ∂u ∂T ∂T  ∂2T ∂u2 ∂q
engineering. ρC  +u +v =k +µ   +q′′′− r
p nf  ∂t ∂x ∂y  nf ∂y2 nf ∂y ∂y
Problem formulation
(6)
From the transient, two-dimensional boundary layer fluid flow of where
Citation: Sobamowo MG, Yinusa AA, OluwoAA, et al. Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a
stretching sheet embedded in a porous medium. Aeron Aero Open Access J. 2018;2(5):294‒308. DOI: 10.15406/aaoaj.2018.02.00064
Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a stretching Copyright:
sheet embedded in a porous medium ©2018 Sobamowo et al. 296
of the stretching sheet varies with respect to distance x-from the slit as
( )
ρ =ρ 1−φ +ρφ
nf f s  
(7a) 2
 bx 
(ρC ) =(ρC ) (1−φ)+(ρC )φ Ts =To −Tref  ( )3  (14)
p nf p f p (7b) 2v 1−at 2 
f
 σ   And the stretching velocity varies with respect to x as
 3 s −1φ 
 σf   bx
σnf =σf 1+ σ  σ   U = (1−at) (15)
  s +2φ− s −1φ
 σ  σ   On introducing the following stream functions
f f
(7c)
∂ψ ∂ψ
u = , v = (16)
µ
µ = f (8) ∂y ∂x
nf ( )2.5
1−φ And the similarity variables
k =k ks +2kf −2φ(kf −ks) u=(1−bxat) f′(η,t), v=−(bνf)−12(1−at)−12 f(η,t),
nf f ( )
 ks +2kf +φ kf −ks  (9) η=(b/ν )12(1−at)−12 y, T =T −T (bx2/2v )(1−at)−23θ(η)
f o ref f (17)
∂q 4σ∗ ∂T4 16σ∗T3 ∂2T Substituting Eq. (16) and (17) into Eq. (5), (6) and (7), we have
r =− ≅ − s (using Roseland’s a partially coupled third and second orders ordinary differential
∂y 3k∗ ∂y 3k∗ ∂y2 equation
approximation) (10)
Assuming no slip condition, the appropriate boundary conditions 1+ 1f′′′1+3(n−1)We(f′′)2+B B Sf′+ηf′′+ ff′′−(f′)2
are given as  β  2  1 2  2  
u =Uw, v =0, T =Ts at y =0 (11a) −Ha2fc′os2γ− 1 f′=0
Da
∂u ∂T (18a)
=0, =0, y =h (11b)
∂y ∂y B 1+4Rθ′′+EcPr(f′′)2+(A∗f′+B∗θ)−BPr{S((ηθ′+3θ)+2f′θ−fθ′)}=0
3  3  B 4 2
It should be stated at this juncture that the formulated mathematical 1
model is implicitly in the domain x≥0. In other to annul discontinuities (18b)
as a result of surface effects, a further assumption of smooth surface
where
is made. Likewise, the effect of shear due to the interfacial quiescent
atmosphere i.e. is removed. The shear stress based on the Newton’s We2= b3x2Γ2 , Pr=µcp, Ha2=σnfBo2, Ec= Uw2 , S=α, R=4σ∗T03
law of viscosityτ=µ∂u as well as the heat flux q′′=−k∂T  vf(1−at)3 kf ρfb cp(Ts−(T0)) b k∗kf
disappear when the adia∂byatic free surface condition is considere∂dy (at B1=(1−φ)2.5, B2=1−φ+φρρsf , B3=kknff , B4=1−φ+φ(ρρccpp)sf , Da=hKo
y=h).
(19)
It should be noted that And the boundary conditions become
1  1 η=0, f =0, f′=1, θ=0
dh αβ vf 2 αβ vf 2
v= =−  ( ) , y=h(t)=−∫  ( ) dt (12) Sβ
dt 2 b 1−αt   2 b 1−αt   η=β, f = , f′′=0, θ′=0 (20)
 
2
The above boundary conditions are in line with the works of Method of solution: finite element method
Kumar et al.,56 and Abel et al.,76
Equations (18a) and (18b) are systems of coupled non-linear
The non-uniform heat generation/absorption q′′′ is taken as
ordinary differential equations which are to be solved along side with
boundary conditions in Eq. (20). The exact analytical solutions for
k U
q′′′= f w A∗(T −T ) f '+B∗(T −T ) these non-linear equations are not possible. Therefore, in order to
s o s o
xν solve the equations, recourse is made to a numerical method. In this
f (13)
work, finite element method is applied to analyze the systems of the
coupled nonlinear equations. The variational and the finite element
where T is the ambient temperature and the surface temperature T
o s formulation of Eqs. (18a) and (18b) are given as follows:
Citation: Sobamowo MG, Yinusa AA, OluwoAA, et al. Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a
stretching sheet embedded in a porous medium. Aeron Aero Open Access J. 2018;2(5):294‒308. DOI: 10.15406/aaoaj.2018.02.00064
Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a stretching Copyright:
sheet embedded in a porous medium ©2018 Sobamowo et al. 297
In order to reduce the system of the nonlinear equation, we let The Galerkin finite element formulation may be obtained from
Eq. (24)-(25) by substituting the finite element approximations of the
g = f′
form:
(21)
2 2 2
The system of equation (18a) and (18b) thus reduces to f = ∑ Njfj, g = ∑ Njgj, θ= ∑ Njθj.
j=1 j=1 j=1 (27)
1+β11+3(n−1)2We(g′)2g′′+B1B2Sg+η2g′+ fg′−(g)2 is Feoqru athl e tGo alethrkei n bfiansiitse/ sehlaepmee/inntt efropromlautliaotnio nf, utnhcet iwone.i ghTt hfeurnecftoiroen,
w=w=w=N(i=1,2), where N are the basis/shape/interpolation
1 2 3 i i
−Ha2gcos2γ− 1 f′=0 functions when considering the linear element (ηe,ηe+1)which are
Da defined as follows:
(22a)
η −η η−η
{ } N = e+1 , N = e , η ≤η≤η
B31+34Rθ′′+EBcPr(g′)2+(A∗g+B∗θ)−B4Pr S2((ηθ′+3θ)+2gθ−fθ′) =0 1 ηe+1−ηe 2 ηe+1−ηe e e+1 (28)
1
Therefore, the equivalent Galerkin finite element formulations of
(22b)
Eqs. (24)-(26) are
And the corresponding boundary conditions become
∫ηe+1N [f′−g]dη=0
η=0, f =0, g =1, θ=0 ηe i
(29)
η=β, f = S2β, g′=0, θ′=0 (23) ∫ηe+1N 1+β11+3(n−1)2We(g′)2g′′+B1B2Sg+η2g′+fg′−(g)2dη=0
ηe i 
oveTr hae vtyapriiactailo ntw foor-mnosd aasls olicniaetaerd ewleimthe tnhte E(ηqs., (η21), )(2 2aare) agnidv e(2n2 bas) −Ha2gcos2γ−D1ag 
e e+1
follows: (30)
∫ηηee+1w1[f′−g]dη=0 (24) ∫ηηee+1NiB31+43Rθ′′+EBcP1r(g′)2+(A∗g+B∗θ)−B4PrS2((ηθ′+3θ)+2gθ−fθ′)dη=0
(31)
1+11+3(n−1)We(g′)2g′′+B B Sg+ηg′+fg′−(g)2 I n c o r p o r a t i n g t h e b o u n d a r y c o n d i t i o n s d i r e c t l y i n t h e s t rong
∫ηe+1w  β 2  1 2  2  dη=0 forms as presented Eqs. (30) and (32) is a daunting task. Also, the
ηe 2  requirement on continuity of field variables is much stronger in its
−Ha2gcos2γ−D1ag  present strong forms. In order to overcome the difficulties, weak
formulations are preferred. Indisputably, the weak formulations help
(25) to reduce the order of continuity needed for elements selected i.e. it
∫ηηee+1w3B31+34Rθ′′+EBcPr(g′)2+(A∗g+B∗θ)−B4PrS2((ηθ′+3θ)+2gθ−fθ′)dη=0 wfuinllc rtieodnusc)e f uthnec ctioonntsi nthueitrye breyq aulilroewmienngt sth oen u tshee o afp eparsoyx-itmo-actoionns t(rourc bt aasnids
1 implement polynomials. Moreover, weak formulation automatically
(26)
enforces natural boundary conditions.
where w, w and w are the weight functions or variational in f, g
1 2 3 The weak formulations of Eqs. (30) and (31) are
and θ, respectively.
1+β1 Nig′ηηee+1 −∫ηηee+1g′N′dη+(n−21)WeNi(g′)3ηηee+1 −∫ηηee+1(g′)3 N′dη
+∫ηe+1N B B Sg+ηg′+ fg′−(g)2−Ha2gcos2γ− 1 gdη=0 (32)
ηe i  1 2  2   Da 
 
EcPr(g′)2 +(A∗g+B∗θ) 
B 1+ 4RNθ′ηe+1 −∫ηe+1θ′N′dη+∫ηe+1N  B1 dη=0
3 3  i ηe ηe  ηe i  S (( ) ) (33)
−B Pr ηθ′+3θ +2gθ− fθ′ 
 4 2 
Note that Eq. (29) has already been expressed in a weak form.
Substituting Eq. (27) into Eqs. (24), (32) and (33), we have
Citation: Sobamowo MG, Yinusa AA, OluwoAA, et al. Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a
stretching sheet embedded in a porous medium. Aeron Aero Open Access J. 2018;2(5):294‒308. DOI: 10.15406/aaoaj.2018.02.00064
Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a stretching Copyright:
sheet embedded in a porous medium ©2018 Sobamowo et al. 298
∫ηηee+1 N i  i∑ =2 1 N 'j f j − j∑2 = 1 N j g j  d η = 0 (34)
1+ 1N ∑2 N'g ηe+1 −∫ηe+1∑2 N'g N'dη+(n−1)WeN (∑2 N'g )3ηe+1 −∫ηe+1(∑2 N'g )3N'dη
 β ii=1 j jηe ηe i=1 j j i  2  i i=1 j j ηe ηe i=1 j j i 
    2 η2 '   
  S∑N g + ∑N g    (35)
+∫ηe+1N B B  i=1 j j 2i=1 j j −Ha2∑2 N g cos2γ− 1 ∑2 N g dη=0
ηe i 1 2+(∑2 N f ⋅∑2 N'g )−(∑2 N g )2 i=1 j j Dai=1 j j
   i=1 j j i=1 j j i=1 j j  
B 1+ 4RN ∑2 N'θ'ηe+1 −∫ηe+1∑2 N'θN'dη
3 3  ij=1 j jηe ηe i=1 j j i 
EcPr(∑2 N'g )2 +(A∗(∑2 N g )+B∗∑2 Nθ) 
j j j j j j
 B1 i=1 i=1 j=1 
+∫ηe+1N   ( 2 ' 2 ) (2 2 )dη=0
ηe i    η∑ Nθ +3∑ Nθ +2 ∑N g ⋅∑ Nθ  (36)
S j j j j j j j j
−B Pr  j=1 j=1 i=1 j=1 
4
 2−f ∑2 N'θ 
   jj=1 j j 
( ) ( )
A linearized analysis of the above equations can be performed if Eqs. 35 and 36 are linerarized by
incorporating the functions f and g, which are assumed to be known. Therefore, we arrived at
1+ 1N ∑2 N'g ηe+1 −∫ηe+1∑2 N'g N'dη+(n−1)WeN g2∑2 N'g ηe+1 −∫ηe+1N'g2∑2 N'g dη
 β ii=1 j jηe ηe i=1 j j i  2  i i=1 j jηe ηe i i=1 j j 
    2 η 2 '   
+∫ηe+1N B B Si∑=1Njgj + 2i∑=1Njgj −Ha2∑2 N g cos2γ− 1 ∑2 N g dη=0 (37)
ηe i  1 2+(fji∑=21N'jgj)−gi∑=21Njgj i=1 j j Dai=1 j j
B 1+ 4RN ∑2 N'θ'ηe+1 −∫ηe+1∑2 N'θN'dη
3 3  ij=1 j jηe ηe i=1 j j i 
 ( ( ) ) 
EcPr g∑2 N'g + A∗ ∑2 N g +B∗∑2 Nθ 
j j j j j j
 B1 i=1 i=1 j=1 
+∫ηe+1N   ( 2 ' 2 ) ( 2 )dη=0 (38)
ηe i  S  η∑ Njθj +3∑ Njθj +2 gj ∑ Njθj 
−B Pr  j=1 j=1 j=1 
4
 2−f ∑2 N'θ 
   j j=1 j j 
The finite element model of the equations in matrix form is given as follow
 11  12  13{f } {S(1)}
K  K  K     
K21 K22 K23{g }={S(2)}
K31 K32 K33{θ} {S(3)} (39)
 
Citation: Sobamowo MG, Yinusa AA, OluwoAA, et al. Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a
stretching sheet embedded in a porous medium. Aeron Aero Open Access J. 2018;2(5):294‒308. DOI: 10.15406/aaoaj.2018.02.00064
Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a stretching Copyright:
sheet embedded in a porous medium ©2018 Sobamowo et al. 299
 mn  m
where K  and S  (m, n=1, 2, 3) are defined as follows:
2×2 2×1
K11 =1+ 1 ∫ηe+1N N'dη=0, K12 =−1+ 1 ∫ηe+1N N dη=0, K13 =0, S(1) =0
ij  βηe i j ij  βηe i j ij
K21 =0
ij
Ki2j2 =−1+β1 ∫ηηee+1N'jNi'dη−(n−21)We∫ηηee+1Ni'N'jg2dη
+∫ηe+1N B B SN +ηN' +(f N')−gN −Ha2N cos2γ− 1 N dη
ηe i  1 2 j 2 j j j j j Da j
K23 =0
ij
S(2) =1+ 1 N N'g ηe+1 +(n−1)WeN N'g2g ηe+1
 β i j jηe  2  i j jηe 
K31 =0
ij
 
K32 = ∫ηe+1N EcPr gN' +(A∗N )dη
ij ηe i  B j j 
1
  ( ' ) 
Ki3j3 =−B31+ 43R∫ηηee+1N'jNi'dη+∫ηηee+1Ni (B∗Nj)−B4PrS2+η2N(gjj+N3jN)−j fjNjdη
S(3) = B 1+ 4RN ∑2 N'θ'ηe+1
3 3  i j=1 j jηe 
with
2 2
f = ∑ N f , g = ∑ N g
j j j j
j=1 j=1
The element matrix given by Eq. (39) is 6×6 order and it’s domain
is sectioned into 1200 line elements. Thus, a 3603 × 3603 order
matrix is obtained after assembly. The remaining system of equations
is solved numerically after using the boundary conditions. It should
be noted that if Eq. (25) and (26) for the nonlinear forms of Eqs. (35)
and (36), the developed finite element equations is nonlinear. The
nonlinear algebraic equations so obtained are modified by imposition
of boundary conditions. The set of equations were solved with the aid
of MATLAB and the convergence was conditioned to be:
∑φp −φp−1 ≤10−4 (40)
i
Figure 2A Effects of radiation parameter on the temperature profile of Ag-
kerosene casson-carreau nanofluid.
Results and discussion
Grid independency test
For the selected domain, numerical solutions are computed and
grid-independence test is to obtain the results accurately. The necessary Table 1 and Table 2 show the grid refinement studies. The analysis
convergence of the results is achieved with the desired degree of are performed from 300 to 1500 elements in steps on 300 for arbitrary
accuracy. The results with the discussion are illustrated through the values of the thermophysical parameters of using M=2, ϕ=0.3, We=5,
Figures 2‒19 to substantiate the applicability of the present analysis. S=0.6, n=10, A*=3, E=3 and γ=60. The scale of 300 elements show
Citation: Sobamowo MG, Yinusa AA, OluwoAA, et al. Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a
stretching sheet embedded in a porous medium. Aeron Aero Open Access J. 2018;2(5):294‒308. DOI: 10.15406/aaoaj.2018.02.00064
Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a stretching Copyright:
sheet embedded in a porous medium ©2018 Sobamowo et al. 300
a very little difference with the results obtained for the elements 600,
900, 1200 and 1500. A mesh sensitivity exercise was carried out
to ensure grid independence. It is observed that for large values of
number of elements greater than 300, there is no appreciable change
in the results. The convergence of results is depicted in Table 1 &
Table 2. Increasing the element were observed not to after the result
obtained. Hence the grid size of 300-1500 elements is sufficient for
optimum result. However, for parametric studies, 1200 linear elements
is selected and used.
Figure 4 Effect of magnetic field parameter (Hartmann number) on the fluid
velocity distribution.
Figure 2B Effects of radiation parameter on temperature profile of the Cu-
kerosene casson-carreau nanofluid.
Figure 5 Effect of magnetic field parameter (Hartmann number) on the fluid
temperature distribution.
Figure 3A Effects of casson parameter on the velocity profile of Ag-kerosene
casson-carreau nanofluid.
Figure 3B Effects of casson parameter on temperature profile of Cu-
kerosene casson-carreau nanofluid. Figure 6 Effect of unsteadiness parameter on the fluid velocity distribution.
Citation: Sobamowo MG, Yinusa AA, OluwoAA, et al. Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a
stretching sheet embedded in a porous medium. Aeron Aero Open Access J. 2018;2(5):294‒308. DOI: 10.15406/aaoaj.2018.02.00064
Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a stretching Copyright:
sheet embedded in a porous medium ©2018 Sobamowo et al. 301
Figure 7 Effect of unsteadiness parameter on the fluid temperature Figure 10 Effect of aligned angle on the fluid velocity distribution.
distribution.
Figure 8 Effect of Weissenberg number on the fluid velocity distribution. Figure 11 Effect of aligned angle on the fluid temperature distribution.
Figure 9 Effect of Weissenberg number on the fluid temperature distribution. Figure 12 Effect of power-law index on the fluid velocity distribution.
Citation: Sobamowo MG, Yinusa AA, OluwoAA, et al. Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a
stretching sheet embedded in a porous medium. Aeron Aero Open Access J. 2018;2(5):294‒308. DOI: 10.15406/aaoaj.2018.02.00064
Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a stretching Copyright:
sheet embedded in a porous medium ©2018 Sobamowo et al. 302
Figure 13 Effect of power-law index on the fluid temperature distribution.
Figure 16 Effect of non-uniform heat source/sink parameter (A*) on the fluid
temperature distribution.
Figure 14 Effect of nanoparticle volume fractions on the fluid velocity
distribution. Figure 17 Effect of non-uniform heat source/sink parameter (B*) on the fluid
temperature distribution.
Figure 15 Effect of nanoparticle volume fractions on the fluid temperature Figure 18 Effect of Eckert number on the fluid temperature distribution.
distribution.
Citation: Sobamowo MG, Yinusa AA, OluwoAA, et al. Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a
stretching sheet embedded in a porous medium. Aeron Aero Open Access J. 2018;2(5):294‒308. DOI: 10.15406/aaoaj.2018.02.00064
Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a stretching Copyright:
sheet embedded in a porous medium ©2018 Sobamowo et al. 303
Table 2 Convergence of finite element results for Ag-kerosene
No. of elements fll(0) θl(0)
300 -0.841595 -3.090644
600 -0.841593 -3.090643
900 -0.841593 -3.090642
1200 -0.841593 -3.090642
1500 -0.841593 -3.090642

Figure 19 Variation of film thickness h with time t for different values of .
Code validation
Table 1 Convergence of finite element results for fll(0) and l(0) for Cϕu-
To verify the functionability and reliability of the present numerical
kerosene
θ code, result comparism with other numerical method using RK-4 with
boundary value problem shooting method is adopted for the nonlinear
No. of elements fll(0) θl(0)
equations (14a) and (14b) as presented by Kumar et al.74 Adequate and
300 -0.800675 3.183500 comprehensive reports are depicted in Table 3 and Table 4. The Tables
600 -0.800673 3.183501 show the comparison of the results of numerical methods (NM) and
that of FEM. FEM as presented in the present study gives a splendid
900 -0.800673 3.183502 agreement with the results of the numerical method (NM) using
1200 -0.800673 3.183502 Runge-Kutta coupled with Newton method as presented by Kumar et
al.74 The high accuracy established by FEM validates and represents
1500 -0.800673 3.183502
a bench mark in generating solution to similar nonlinear problems.
Table 3 Physical parameter values of f′′(0)and−θ′(0)for Cu-Kerosene nanofluid
NM[74] FEM NM[74] FEM
M ϕ We S n A* E γ f′′(0) f′′(0) −θ′(0) −θ′(0)
1 -0.800673 -0.800673 3.183502 3.183502
2 -0.951051 -0.951050 3.137925 3.137923
3 -1.077238 -1.077238 3.097322 3.097320
0.1 -0.951051 -0.951050 3.137925 3.137923
0.2 -0.926769 -0.926768 2.900338 2.900336
0.3 -0.843920 -0.843921 2.683437 2.683437
1 -0.865479 -0.865478 3.155764 3.155763
3 -0.611938 -0.611938 3.218581 3.218581
5 -0.484571 -0.484571 3.252867 3.252868
0.2 -1.090240 -1.090238 3.094797 3.094797
0.4 -1.002314 -1.002312 3.125135 3.125137
0.6 -0.894041 -0.894040 3.149859 3.149861
1 -0.995049 -0.995047 3.129665 3.129667
5 -0.796797 -0.796798 3.171534 3.171534
10 -0.700307 -0.700307 3.195477 3.195475
1 -0.951051 -0.951050 3.002623 3.002622
2 -0.951051 -0.951050 2.833496 2.833495
3 -0.951051 -0.951050 2.664369 2.664368
1 -0.951051 -0.951050 2.534955 2.534956
2 -0.951051 -0.951050 1.864989 1.864989
3 -0.951051 -0.951050 1.195023 1.195024
Citation: Sobamowo MG, Yinusa AA, OluwoAA, et al. Finite element analysis of flow and heat transfer of dissipative Casson-Carreau nanofluid over a
stretching sheet embedded in a porous medium. Aeron Aero Open Access J. 2018;2(5):294‒308. DOI: 10.15406/aaoaj.2018.02.00064