Waves in Random and Complex Media
ISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/twrm20
Thermal aspects of radiation in Casson fluid with
nonlinear stretching surface: non-similar solutions
Muavia Mansoor, M. Shoaib Kamran, Qazi Mahmood Ul-Hassan &
Muhammad Irfan
To cite this article: Muavia Mansoor, M. Shoaib Kamran, Qazi Mahmood Ul-Hassan & Muhammad
Irfan (2022): Thermal aspects of radiation in Casson fluid with nonlinear stretching surface: nonsimilar solutions, Waves in Random and Complex Media, DOI: 10.1080/17455030.2022.2071502
To link to this article: https://doi.org/10.1080/17455030.2022.2071502
Published online: 12 May 2022.
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WAVES IN RANDOM AND COMPLEX MEDIA
https://doi.org/10.1080/17455030.2022.2071502
Thermal aspects of radiation in Casson fluid with nonlinear
stretching surface: non-similar solutions
Muavia Mansoora , M. Shoaib Kamrana , Qazi Mahmood Ul-Hassana and Muhammad
Irfanb
a Department of Mathematics, University of Wah, Wah, Pakistan; b Department of Mathematical Sciences,
Federal Urdu University of Arts Science and Technology, Islamabad, Pakistan
ABSTRACT
ARTICLE HISTORY
A mathematical model is established to peruse the effects of viscous
dissipation and thermal radiation on non-Newtonian, incompressible, steady, mixed convective flow next to a nonlinear vertical
stretching surface. The Rosseland approximation is used in the
energy equation to describe the radiative heat flux. Non-similarity
transformation has been used to convert coupled partial differential
equations (PDEs) into dimensionless PDEs. For dealing with nonsimilar problems, the non-similarity method is more effective as compared to similarity transformation. First, this method utilizes transformations to reduce governing equations into dimensionless systems
of PDEs, and further, it utilizes different truncation levels to get a
detailed set of PDEs. MATLAB built-in solver bvp4c is used to solve
obtained dimensionless PDEs. Parameters describing the nature of
flow like local Sherwood number, local Nusselt number and local skin
friction coefficient are explored in detail after finding the numerical solution. The effects of Casson number, Radiation number, Eckert
number, Schmidt number, Prandtl number and Grashof number for
heat and mass transfer on concentration, temperature and velocity
profiles are specified with the aid of graphs.
Received 27 December 2021
Accepted 22 April 2022
Nomenclature
α
β
βT
βC
D
σ∗
k∗
Ec
k1
ν
l
thermal diffusivity
Casson parameter
coefficients of thermal expansion
coefficients of concentration expansion
mass diffusivity
Stefan-Boltzman
absorption coefficient
Eckert number
chemical reaction rate
kinematic viscosity
reference length
CONTACT Muavia Mansoor
muavia.mansoor@uow.edu.pk
© 2022 Informa UK Limited, trading as Taylor & Francis Group
KEYWORDS
Casson fluid; thermal
radiation; viscous dissipation;
mixed convection; Matlab
solver bvp4c
2
M. MANSOOR ET AL.
Nux
Shx
T∞
Sc
U0
σ
Pr
Cf
Cp
GrC
qr
Grl
Re
k
Rd
Tw
γ
local Nusselt number
local Sherwood number
fluid temperature at free stream
Schmidt number
reference velocity
electrical conductivity
Prandtl number
skin coefficient friction
specific heat
Olutal Grashof number
radiative heat flux
thermal Grashof number
Reynolds number
thermal conductivity
radiation parameter
ratio of velocities
fluid temperature surface
rate of Chemical reaction parameter
Introduction
Non-Newtonian fluids have numerous applications in our daily life such as in textile,
polymer industries, food making, paper industries and agriculture sector to find out below
groundwater storage. Examples of these fluids are blood, honey, oils, and greases. As we
know that there is a nonlinear relation between stress and strain for non-Newtonian fluid
that’s why it is impossible to describe all characteristics of non-Newtonian fluid in a single equation. As a result, many non-Newtonian fluid models [1–5] have been developed
depending on different physical parameters. The most famous among these fluids is the
Casson fluid.
Casson fluid was founded in 1959 for the very first time while studying the flow of oil suspension printing ink type [6]. By applying small stress, Casson fluid performs like an elastic
solid. And by applying a large amount of stress, Casson fluid flows. So, this fluid behaves
like solid as well as liquid. As we know that human blood contains different elements like
protein and red blood cells in plasma, so it is also an example of Casson fluid. Casson fluid
model presents flow properties of blood when the shear rate is low. Other examples are
jelly, honey, and sauce. Many researchers have been interested in studying Casson fluid
due to its numerous applications in the food sector and biological treatments. Oka was
the first one who examined the flow of Casson fluid through a tube [7]. T. Hayat et al. [8]
considered boundary layer flow of Casson fluid with mass transfer by using similarity transformation. Mustafa et al. also discussed Casson fluid with heat transfer in moving plates
and applied the Homotopy analysis method to find out analytical solutions and observed
effects of physical parameters [9]. Mukhopadhyay investigated the effects of blowing for
Casson fluid and calculated numerical solution [10]. Pramanik studied the effects of heat
flux and thermal radiation on Casson fluid [11]. Seini et al. investigated the effect of a
chemical reaction and magnetic effect for Casson fluid [12] and used the Newton Raphson
method to solve transformed ordinary differential equations and presented results with
WAVES IN RANDOM AND COMPLEX MEDIA
3
help of graphs. Recently T Anwar et al. [13] analyzed Casson fluid over an infinite plate with
effects of thermal conductivity and heat mass transfer and used suitable dimensionless variables and solutions were calculated with the help of Laplace transform. The main purpose of
this research was to observe ramped temperature and velocity conditions on Casson fluid.
When fluid is heated, it rises, leaves a low-pressure region behind and cool fluid rushes in.
In this way, cool fluid is heated. This motion is due to temperature difference which results
in density difference and this process is called convection. Free convection is due to density
variation while forced convection is due to some external agent like a blower for gas. The
relative contribution of forced and free convection in the total rate of convective heat transfer is determined by the ratio of Grashof number and square of Reynolds number. If this ratio
is less, then forced convection dominates, and if this ratio is greater, then free convection
dominates. If this ratio is in between, then relative contribution of both convections must
be considered for the calculation of total convective heat transfer. T. Hayat et al. examined
mixed convection flow of Casson nanofluid with chemical reaction, magnetic effect and in
the presence of thermal radiation [14]. Izadi et al. also presented mixed convection flow
with heat transfer of carbon nanotubes [15]. M. Mushtaq et al. [16] explored the mixed convection flow of second-grade viscoelastic fluid and calculated numerical solutions. M. J. H.
Munshi et al. [17] examined mixed convection of nanofluid in a porous medium with heat
effects.
Thermal conductivity is a physical property that depends upon temperature. Thermal
conductivity is defined as the rate of heat transfer through a unit length of material per
unit temperature difference. According to Batchelor and Animasaun, thermal conductivity varies with variation in temperature [18,19] Large amount of work has been done that
explains the effect of thermal conductivity on flowing fluids. Chaim et al. observed thermal
conductivity in boundary layer flow of 2D Newtonian fluid in a porous medium [20]. Attia
[21] presented unsteady flow between two parallel plates with variable thermal conductivity and viscosity. Jawad et al. used the Attia model for observing both thermal conductivity
and viscosity of flow in vertical channels. He observed that heat transfer and fluid flow
decrease with an increase in thermal conductivity [22]. Gbadeyan et al. [23] investigated the
effects of thermal conductivity on viscous, incompressible Casson nanofluid. After solving
the system of equations, the effects of different physical parameters like Sherwood number
and Nusselt number were observed and plotted graphically. T. Hayat et al. [24] investigated
mixed convection flow over a stretching surface with variable thermal conductivity and
observed the effects of governing parameters. T. Hayat also discussed Casson fluid over
a stretching sheet with the effect of thermal conductivity and presented a series of solution
[25]. Mondal and Pal [26] investigated the effects of thermal conductivity and temperaturedependent viscosity in MHD strained fluid flow. More detail about current studies can be
reported in references [27–31].
In fluid flow, viscous dissipation is a resistive force, and it leads to a permanent change of
shear stresses into heat energy in contiguous layers and is acknowledged as a dissipation
function. Brinkman was the first one who observed heat generation by viscous dissipation
[32]. Wu and Cheng examined viscous dissipation effects for horizontal plate channels in
which the lower plate is heated, and the upper plate is cooled [33]. Orhan et al. [34] discussed the viscous dissipation effect for Poiseuille flow and fully developed flow is studied.
Oliveria and Pinho [35] observed forced convection flow in channel flows with the addition
4
M. MANSOOR ET AL.
of the viscous dissipation effect. It was observed that due to viscous dissipation, fluid elasticity increases. Ajayi et al. [36] focused on the viscous dissipation effect for Casson fluid
over the melting surface, a term indicating viscous dissipation is included in the energy
equation. Similarity transformation was applied to convert partial differential equations
into systems of ordinary differential equations. Noor et al. [37] presented Casson nanofluid
model with effects of viscous dissipation and chemical reaction and Buongiorno’s model
was considered to observe these effects. It was examined that viscous dissipation increases
fluid temperature and heat transfer. After obtaining a solution comparison was made with
existing results. Dada et al. [38] examined Williamson fluid under effects of viscous dissipation, chemical reaction and radiation in a porous medium with the addition of variable
thermal conductivity and viscosity. Solutions were obtained by using suitable transformation and then by using the Homotopy analysis method to find out analytical solutions. After
obtaining solutions, the effects of involved parameters are analyzed. Yang et al. [39] also
studied Casson fluid model with the viscous dissipation effect and analyzed the viscoelastic effect in porous medium and calculated a numerical solution. For more information, see
references [40–42].
Here the concept of MHD radiative flow of Casson fluid under the aspect of convective
heat transport has been reported. The properties of Joule heating and viscous dissipation
are also examined for the heat transport phenomenon. The current article has the following
novelties. Firstly the local non-similar solutions have been achieved. Secondly, the deliberated problem has an essential use in cooling progression for the determination of refining
mechanical possessions of the heated sheet that cannot be refrigerated by the usage of
old-fashioned approaches established on the base fluids. Third, the aspect of radiation is
very significant in industrial activities for the project of trustworthy apparatus, nuclear control plants, vapor turbines and numerous thrust strategies for planes, artillery, satellites
and interplanetary automobiles. Lastly, the magnetic effects have vital uses in physics and
industries. Numerous metallurgical developments comprise the chilling of uninterrupted
floorings or threads by sketching them via a quiescent liquid, and in the procedure of illustration, these floorings are occasionally stretched. In these circumstances, the possessions
of the closing creation are contingent to a great amount on the percentage of chilling by
sketching such strips in an electrically conducting liquid focus to a magnetic field and the
specific desired in the closing creation. Thus, under these novelties, we examined Casson
fluid with properties of radiation, MHD, Joule heating, viscous dissipation and convective
effects. The graphs are plotted and discussed.
Problem formulations
Consider steady, laminar, and incompressible two-dimensional non-Newtonian Casson
fluid flowing over stretched plates under the effects of mixed convection, viscous dissipation, and thermal radiation. The x - axis is taken in the direction of flow and y - axis is
perpendicular to the direction of flow. u and v are components of velocity in x and y direcx
tions. The stretching velocity is Uw (x) = u0 e /l in a positive x direction. Tw and T∞ are
temperatures of fluids at the wall and free stream. Similarly, Cw and C∞ are representing
concentration at the wall and free stream (Figure 1).
WAVES IN RANDOM AND COMPLEX MEDIA
5
Figure 1. Geometry of the problem.
Equations representing Casson fluid [14] under effects of viscous dissipation, thermal
radiation and mixed convection are stated as
ux + vy = 0
(1)
uux + vuy = ν 1 +
1
β
(uTx + vTy ) = α(Tyy ) +
uyy + gβT (T − T∞ ) + gβC (C − C∞ ) −
ν
Cp
1+
1
β
(uy )2 −
σ B0
u
ρ
2
1
σ B0 2 2
(qr )y +
u
ρCp
ρCp
(uCx + vCy ) = D(Cyy )
(2)
(3)
(4)
The equivalent boundary conditions will be
hf
u = Uw (x), v = 0, ∂T
∂y = − kf (Tw − T), C = Cw aty = 0
T → T∞ , u → 0, C → C∞ at y → ∞
(5)
Using Rosseland approximation, the expression for the radiative heat flux qr is given by
qr = −
4σ ∗ 4
T
3k∗ y
(6)
It is assumed that for radiative heat flux, temperature deviation within the flow is very small,
the term T 4 can also be expounded as a linear function of temperature T with the assistance
of Taylor’s series expansion about T∞ and by ignoring all the higher terms, given as
3
4
T4 ∼
T − 3T∞
= 4T∞
By using (6) and (7) into (3), it is obtained
1
16σ ∗ T∞ 3
ν
σ B0 2 u2
2
(uTx + vTy ) = α +
(T
1
+
)
+
)
+
(u
yy
y
3k∗ ρCp
Cp
β
ρCp
(7)
(8)
6
M. MANSOOR ET AL.
Transformation of the governing system of equations
Consider [43] the non-similar transformations, in which we have two independent variables
η is called pseudo similar while ξ is non-similar, from mathematics perspective it is just a
trick or method to reduce the dimensionality of the problem, or it is often used to reduce a
PDEs to PDEs.
2x
Uw
T − T∞
C − C∞
ξ = e l ,η = y
andϕ(ξ , η) =
, ψ(x, y) = 2υlUw f (ξ , η), θ(ξ , η) =
2υl
Tw − T∞
Cw − C∞
∂f
υUw
∂f
∂f
u = Uw (ξ , η), v = −
f (ξ , η) + 4ξ (ξ , η) + η (ξ , η)
(9)
∂η
2l
∂ξ
∂η
By applying non-similarity transformations in Equations (2), (4) and (8) diminishes to
(10)–(12)
2
∂f
∂η
2
+ 4ξ
∂ 2 f ∂f
∂f ∂ 2 f
∂f
∂ 2f
1 ∂ 3f
+
M
+ 2θ Grl + 2Grc ϕ
− f 2 − 4ξ
=
1
+
∂ξ ∂η ∂η
∂η
∂ξ ∂η2
∂η
β ∂η3
(10)
Pr −f
Sc −f
∂θ
∂θ ∂f
∂f ∂θ
1
+ 4ξ
− 4ξ
− 1+
∂η
∂ξ ∂η
∂ξ ∂η
β
∂ϕ
∂ϕ ∂f
∂f ∂ϕ
+ 4ξ
− 4ξ
∂η
∂ξ ∂η
∂ξ ∂η
Ec (f ) − J
2
∂f
∂η
2
= θ 1 +
4Rd
3
(11)
=
∂ 2ϕ
∂η2
(12)
Subject to the boundary conditions
∂f
f (ξ , η) = 1, f (ξ , η) = −4ξ ∂ξ
, ϕ(ξ , η) = 1, θ (ξ , η) = −Bi(1 − θ)atη = 0
θ(ξ , η) → 0, f (ξ , η) → 0, ϕ(ξ , η) → 0atη → ∞
4σ ∗ T∞ 3
k1 l
k
kk∗ , α = ρCp , γ = Ue ,
2σ B0 2 l
2σ B0 2 lU0
ρU0 , J = ρCp (Tw −T∞) .
where Pr = υα , Rd =
gβC (Cw −C∞ )l3
,M
υ2
=
Ec =
Uw 2
Cp (Tw −T∞ ) ,
Grl =
gβT (Tw −T∞ )l3
,
υ2
(13)
GrC =
Local similarity method
By using the local similarity method (which remains necessary for using non-similarity transformation), Equations (10)–(12) are assumed to be sufficiently small so that they can be
approximated by zero, so consider ξ ∂(.)
∂ξ = 0
f =
θ =
1
1+
Pr
1+
(2f − ff − 2Grl θ − 2Grc ϕ + Mf )
2
1
β
4Rd
3
ϕ = Sc(−f ϕ )
(14)
1
2
2
−f θ − 1 +
Ec (f ) − J(f )
β
(15)
(16)
WAVES IN RANDOM AND COMPLEX MEDIA
7
subject to the boundary conditions
f (ξ , η) = 1, f (ξ , η) = 0, θ (ξ , η) = −Bi(1 − θ), ϕ(ξ , η) = 1atη = 0
(17)
θ(ξ , η) → 0, f (ξ , η) → 0, ϕ(ξ , η) → 0atη → ∞
Local non-similarity method
Moving toward the local non-similarity results of Equations (10)–(12), where prime derivatives are with respect to η while p, q and g are the derivatives of f , θ and ϕ with respect to
ξ . Hence, by supposing
∂f
∂θ
∂ϕ
∂ 2f
∂f ∂ 2θ
∂θ ∂ 2ϕ
∂ϕ = p,
=
= p ,
= q,
=
= q ,
= g,
=
= g
∂ξ
∂ξ ∂η
∂ξ
∂ξ
∂ξ ∂η
∂ξ
∂ξ
∂ξ ∂η
∂ξ
1
2
(2f + 4ξ p f − ff − 4ξ pf − 2Grl θ − 2Grc ϕ + Mf )
f =
(18)
1
1+ β
Pr
1
2
2
θ =
4ξ qf − f θ − 4ξ pθ − 1 +
(19)
Ec (f ) − J(f )
β
1 + 4R3d
ϕ = Sc(4ξ gf − f ϕ − 4ξ pϕ )
(20)
Subject to the boundary conditions
f (ξ , η) = 1, f (ξ , η) = −4ξ p, θ (ξ , η) = −Bi(1 − θ), ϕ(ξ , η) = 1atη = 0
θ(ξ , η) → 0, f (ξ , η) → 0, ϕ(ξ , η) → 0atη → ∞
(21)
Taking derivative w.r.t ξ for (18)–(20), it is obtained
p =
1
1+
2
Pr
q =
(4f p + 4p f + 4ξ p − 5pf − fp − 4ξ pp − 2Grl q − 2Grc g + Mp ) (22)
1
β
1+
4Rd
3
1
4ξ qp − 5pθ − fq − 4ξ pq + 4qf − 2 1 +
β
Ec f p − 2Jf p
g = Sc(−5pϕ − fg + 4gf + 4ξ gp − 4ξ pg )
(23)
(24)
subject to the boundary conditions
p (ξ , η) = 0, p(ξ , η) = 0, q (ξ , η) = Bi ∗ q, g(ξ , η) = 0atη = 0
q(ξ , η) → 0, p (ξ , η) → 0, g(ξ , η) → 0atη → ∞
The terms
∂g(ξ ,η) ∂p(ξ ,η)
∂ξ , ∂ξ
and
∂q(ξ ,η)
∂ξ
(25)
and their derivatives with respect to η are ignored.
Solution procedure
Since the non-similarity method only gives the non-dimensional equations which are further needed to be solved using some analytical or numerical method. For this reason,
a Matlab built-in utility bvp4c is used to solve the obtained differential equations from
8
M. MANSOOR ET AL.
non-similarity methods. The Matlab solver bvp4c solves the scalar or system of differential
equations using the three-stage Lobatto IIIa formula. Since the obtained equations contain
two independent variables, to solve these equations, one independent variable was held to
be fixed, and so remaining equations become ordinary differential equations. For solving
equations, using a Matlab solver, it requires a function for differential equations and a function for getting boundary conditions. For giving the information of differential equations
to the MATLAB solver, the conversion of the system of second or higher-order differential
equations into a system of first order ordinary differential equations is required. For this
reason, conversion into a system of first-order differential equations can be expressed as
f = f2 , f = f3
f =
1
1+
1
β
θ = f5
θ =
Pr
1+
(2f2 2 + 4ξ f9f2 − f1 f3 − 4ξ f8 f3 − 2Grl f4 − 2Grc f6 + Mf2 )
1
4ξ f11 f2 − f1 f5 − 4ξ f8 f5 − 1 +
Ec (f3 )2 − Jf2 2
β
4Rd
3
ϕ = f7
ϕ = Sc(4ξ f13 f2 − f1 f7 − 4ξ f8 f7 )
p = f9 , p = f10
p =
1
1+
1
β
q = f12
q =
Pr
1+
4Rd
3
(4f2 f8 + 4f9 f2 + 4ξ f9 2 − 5f8 f3 − f1 f10 − 4ξ f8 f10 − 2Grl f11 − 2Grc f13 + Mf9 )
1
4ξ f11 f9 − 5f8 f5 − f1 f12 − 4ξ f8 f12 + 4f11 f2 − 2 1 +
Ec f3 f8 − 2Jf2 f8
β
g = f14
g = Sc(−5f8 f7 − f1 f14 + 4f13 f2 + 4ξ f13 f9 − 4ξ f8 f14 )
where f = f1 , θ = f4 , ϕ = f6 , p = f8 , q = f11 and g = f13
[f0,2 − 1, f0,1 + 2ξ f8 , f1,2 , f0,5 + Bi ∗ (1 − f4 ), f1,4 , f0,6 − 1, f1,6 , f0,9 , f0,8 , f1,9 , f0,12
− Bi ∗ f11 , f1,11 , f0,13 , f1,13 ]t
where f1 and f0 depicts boundary conditions on right and left end points of the domain,
respectively
Result and discussion
This part mainly explains about velocity profile behavior, temperature and concentration
and sketches have been drawn for these profiles against different parameters. Figures 2–5
endorse the reverberation of distinct parameters for velocity profile. Figures 6–9 evaluate
WAVES IN RANDOM AND COMPLEX MEDIA
9
Figure 2. Velocity profile with variation of M.
Figure 3. Velocity profile with variation of β.
changing trends in the temperature profile. Figure 10 elucidates changing development
in concentration sketches. Figure 2 describes the effect of magnetic parameter M on the
velocity profile. By increasing values for magnetic parameters, velocity profile de-escalates.
Figure 3 explains the velocity distribution for the Casson parameter β. By increasing values of β, the velocity profile depletes. Figures 4 and 5 are plotted to observe the effects of
10
M. MANSOOR ET AL.
Figure 4. Velocity profile with variation of Grashof number for mass transfer Grc .
Figure 5. Velocity profile with variation of Grashof number for heat transfer Grl .
Grashof numbers Grl and Grc on the velocity of the fluid. By enhancing the values of both
Grashof numbers for heat and mass transfer results in an accelerating velocity profile. As
WAVES IN RANDOM AND COMPLEX MEDIA
11
Figure 6. Temperature profile with variation of J.
Figure 7. Observation for Temperature profile with changing values of Eckert number Ec.
Grashof number is the ratio of buoyancy force to restraining (viscous) force, and its increasing values is responsible for lowering the viscosity which in turn enhances the velocity of
12
M. MANSOOR ET AL.
Figure 8. Temperature profile with variation of Prandtl number Pr.
Figure 9. Temperature profile with variation of Radiation number Rd.
the fluid. Figure 6 demonstrates the effects of the Joule heating parameter J for temperature profile. By enhancing values for the Joule heating parameter, the temperature profile
shows increasing behavior.
WAVES IN RANDOM AND COMPLEX MEDIA
13
Figure 10. Concentration profile with variation of Schmidt number Sc.
Figure 7 clarifies about effects of Eckert number Ec on temperature profile, and it can
be seen that temperature profile increases with increasing values of the Eckert numberEc.
Figure 8 explicates the results of Prandtl number Pr on temperature profile and it shows
that temperature profile is decreasing function of Prandtl number Pr. As Pr is the ratio of
momentum to thermal diffusivity and is used to calculate the heat transfer between fluid
and surface of the sheet. With the increase of Pr results in lowering the thermal conductivity, consequently, conduction even thermal boundary layer thickness decreases. That’s, the
outcome is a decrease in the temperature profile. Figure 9 shows the curves for radiation
parameter Rd against temperature profile and it is observed that by increasing values for
radiation parameter, temperature profile shows decreasing behavior. Figure 10 expounds
on the concentration profile and the effects of the Schmidt number Sc on the mass transfer profile. Decreasing pattern is observed in this figure with increasing values of Schmidt
number Sc.
Table 1 explains values for local Sherwood numbers, local Nusselt and skin friction coefficient, with respect to changings in different parameters. This table represents an increase
in skin friction coefficient by enhancing the value of parameters Grc and Ec while it depicts
decreasing behavior by enhancing values of Sc, β, PrandRd. Local Nusselt number shows
decreasing behavior by increment in Prandtl number because there is an inverse relation
between Prandtl number and thermal diffusivity so by increasing Prandtl number thermal diffusivity decreases and as a result temperature profile also decreases. Local Nusselt
number also shows decreasing behavior by increasing value of Grashof number for mass
transfer. By enhancing values of Sc, βandRd and decreasing values of PrandEc, the local
Nusselt number shows an increasing behavior. Local Sherwood number shows increasing
14
M. MANSOOR ET AL.
Table 1. Numerical values of Skin friction coefficient, local Nusselt number and local Sherwood number
with Variation of parameters using Grl = 2.1, J = 0.3, M = 0.2, Bi = 0.3.
Grc
Sc
β
Pr
Ec
1+
Rd
2.2
2.3
1
β
f (0)
1.8556
2.0044
2.5478
2.2428
2.0034
1.9854
1.9421
1.9412
1.9497
2.1473
2.1371
2.1286
0.4
0.5
1.1
1.2
1
1.5
0.2
0.3
0.2
0.3
− 1+
4Rd
3
−φ (0)
θ (0)
0.0843
0.0830
0.0758
0.0801
0.0931
0.0932
0.0934
0.0840
0.0748
0.0383
0.0491
0.0602
0.6104
0.6117
0.5002
0.5587
0.5805
0.5811
0.5824
0.5778
0.5752
0.5822
0.5831
0.5839
Table 2. Comparison of numerical values for heat flux with Grl = Grc = Ec = J = M = Bi = 0.
θ(0)
Pr
1
2
3
5
10
Ishak [44]
0.9542
1.4715
1.8619
2.5001
3.6604
Present
Error
S. Pramanik [11]
Present
Error
0.9532
1.4711
1.8694
2.5231
3.6602
1 × 10−3
4 × 10−4
75 × 10−4
23 × 10−3
2 × 10−4
0.9547
1.4714
1.8691
2.5001
3.6603
0.9532
1.4711
1.8694
2.5231
3.6602
15 × 10−4
3 × 10−4
3 × 10−4
23 × 10−3
1 × 10−4
behavior by increasing values of Grc , Sc, β, Ec andRd. Local Sherwood number de-escalates
by increasing values for Prandtl number Pr.
Table 2 illustrates the comparison between the past and present research for evaluating
numerical values of the heat flux with varying Prandtl number Pr. From this Table 2, it can
be seen that the present results are in good agreement with those computed in the past
research.
Conclusion
In this work, we observed the effects of various physical parameters on mixed convective
Casson fluid under the effects of viscous dissipation and thermal radiations. We explored
the Casson fluid model represented by a system of PDEs, and non-similarity transformation
is used to convert given equations into dimensionless PDEs. Transformed PDEs are reflected
as ODEs and explained by Matlab solver Bvp4c. The central observations are stated as
(1) Velocity profile proliferated by means of decreasing the Casson parameter.
(2) Velocity profile proliferated by increasing Grashof numbers of heat and mass transfer.
(3) Velocity profile depleted by increasing values of the magnetic parameter.
(4) Temperature profile depleted by means of enhancing values of Prandtl number.
(5) Temperature profile shows increasing behavior with enhancing values of radiation
parameter and joule heating parameter.
(6) Concentration profile depleted by increasing values of Schmidt number.
WAVES IN RANDOM AND COMPLEX MEDIA
15
Disclosure statement
No potential conflict of interest was reported by the author(s).
References
[1] Djukic DS. Hiemenz magnetic flow of power-law fluids. Eur J Mech B Fluids. 2002;21(3):317–324.
[2] Dorier C, Tichy J. Behavior of a Bingham-like viscous fluid in lubrication flows. J Non-Newtonian
Fluid Mech. 1992;45:291–310.
[3] Wilkinson W. The drainage of a Maxwell liquid down a vertical plate. Chem Eng J. 1970;1:255–257.
[4] Xiao-Feng Z, Lei G. Effect of multipolar interaction on the effective thermal conductivity of
nanofluids. Chin Phys. 2007;16:2028–2032.
[5] Zhi-Wen C, Jin-Xia L, Gui-Jin Y, et al. Borehole guided waves in a non-Newtonian (Maxwell) fluidsaturated porous medium. Chin Phys B. 2010;19(8):084301.
[6] Casson N. A flow equation for pigment-oil suspensions of the printing ink type. Fluid Dyn Res.
1992;9(1–3):133–141.
[7] Oka S. An approach to α unified theory of the flow behavior of time-independent non-Newtonian
suspensions. Jpn J Appl Phys. 1971;10(3):287–291.
[8] Bhattacharyya K, Hayat T, Alsaedi A. Analytic solution for magnetohydrodynamic boundary layer
flow of Casson fluid over a stretching/shrinking sheet with wall mass transfer. Chin Phys B.
2013;22(2):024702.
[9] Mustafa M, Hayat T, Pop I, et al. Unsteady boundary layer flow of a Casson fluid due to an
impulsively started moving flat plate. Heat Transfer – Asian Research. 2011;40(6):563–576.
[10] Mukhopadhyay S. Effects of thermal radiation on Casson fluid flow and heat transfer over an
unsteady stretching surface subjected to suction/blowing. Chin Phys B. 2013;22(11):114702.
[11] Pramanik S. Casson fluid flow and heat transfer past an exponentially porous stretching surface
in presence of thermal radiation. Ain Shams Eng J. 2014;5(1):205–212.
[12] Arthur EM, Seini IY, Bortteir LB. Analysis of Casson fluid flow over a vertical porous surface
with chemical reaction in the presence of magnetic field. J Appl Math Phys. 2015;03:713–723.
https://doi.org/10.4236/jamp.2015.36085
[13] Anwar T, Kumam P, Watthayu W. Unsteady MHD natural convection flow of Casson fluid
incorporating thermal radiative flux and heat injection/suction mechanism under variable wall
conditions. Sci Rep. 2021;11(1):1–15.
[14] Hayat T, Ashraf MB, Shehzad S, et al. Mixed convection flow of Casson nanofluid over a stretching sheet with convectively heated chemical reaction and heat source/sink. J Appl Fluid Mech.
2015;8:803–813. https://doi.org/10.18869/acadpub.jafm.67.223.22995
[15] Izadi M, Hashemi Pour S, Karimdoost Yasuri A, et al. Mixed convection of a nanofluid in a threedimensional channel. J Therm Anal Calorim. 2019;136(6):2461–2475.
[16] Hayat T, Mustafa M, Pop I. Heat and mass transfer for Soret and Dufour’s effect on mixed convection boundary layer flow over a stretching vertical surface in a porous medium filled with a
viscoelastic fluid. Commun Nonlinear Sci Numer Simul. 2010;15(5):1183–1196.
[17] Munshi MJH, Jahan N, Mostafa G. Mixed convection heat transfer of nanofluid in a lid-driven
porous medium square enclosure with pairs of heat source-sinks. Am J Eng Res. 2019;8:
59–70.
[18] Animasaun I. Effects of thermophoresis, variable viscosity and thermal conductivity on free convective heat and mass transfer of non-Darcian MHD dissipative Casson fluid flow with suction
and nth order of chemical reaction. J Niger Math Soc. 2015;34(1):11–31.
[19] Batchelor G. An Introduction to Fluid Dynamics (Cambridge Mathematical Library). Cambridge:
Cambridge University Press; 2000.
[20] Chiam T. Heat transfer in a fluid with variable thermal conductivity over a linearly stretching
sheet. Acta Mech. 1998;129(1–2):63–72.
[21] Attia HA. Unsteady hydromagnetic channel flow of dusty fluid with temperature dependent
viscosity and thermal conductivity. Heat Mass Transfer. 2006;42(9):779–787.
16
M. MANSOOR ET AL.
[22] Umavathi JC, Chamkha A, Mohiuddin S. Combined effect of variable viscosity and thermal conductivity on free convection flow of a viscous fluid in a vertical channel. Int J Numer Methods
Heat Fluid Flow. 2016;26(1):18–39.
[23] Gbadeyan J, Titiloye E, Adeosun A. Effect of variable thermal conductivity and viscosity on Casson
nanofluid flow with convective heating and velocity slip. Heliyon. 2020;6(1):e03076.
[24] Hayat T, Shehzad S, Qasim M, et al. Mixed convection flow by a porous sheet with variable thermal
conductivity and convective boundary condition. Braz J Chem Eng. 2014;31(1):109–117.
[25] Hayat T, Farooq M, Iqbal Z. Stretched flow of Casson fluid with variable thermal conductivity.
Walailak J Sci Technol. 2013;10:181–190.
[26] Pal D, Mondal H. Effects of temperature-dependent viscosity and variable thermal conductivity
on MHD non-Darcy mixed convective diffusion of species over a stretching sheet. J Egypt Math
Soc. 2014;22(1):123–133.
[27] Azam M. Bioconvection and nonlinear thermal extrusion in development of chemically reactive Sutterby nano-material due to gyrotactic microorganisms. Int Commun Heat Mass Transfer.
2022;130:105820.
[28] Azam M, Abbas Z. Recent progress in Arrhenius activation energy for radiative heat transport of
cross nanofluid over a melting wedge. Propuls Power Res. 2021;10(4):383–395.
[29] Azam M, Mabood F, Khan M. Bioconvection and activation energy dynamisms on radiative Sutterby melting nanomaterial with gyrotactic microorganism. Case Stud Therm Eng.
2022;30:101749.
[30] Azam M, Xu T, Khan M. Numerical simulation for variable thermal properties and heat
source/sink in flow of cross nanofluid over a moving cylinder. Int Commun Heat Mass Transfer.
2020;118:104832.
[31] Azam M, Xu T, Mabood F, et al. Non-linear radiative bioconvection flow of cross nanomaterial with gyrotatic microorganisms and activation energy. Int Commun Heat Mass Transfer.
2021;127:105530.
[32] Brinkman H. Heat effects in capillary flow I. Appl Sci Res. 1951;2(1):120–124.
[33] Cheng KC, Wu R-S. Viscous dissipation effects on convective instability and heat transfer in plane
Poiseuille flow heated from below. Appl Sci Res. 1976;32(4):327–346.
[34] AydIn O, Avci M. Viscous-dissipation effects on the heat transfer in a Poiseuille flow. Appl Energy.
2006;83(5):495–512.
[35] Pinho F, Oliveira P. Analysis of forced convection in pipes and channels with the simplified
Phan–Thien–Tanner fluid. Int J Heat Mass Transfer. 2000;43(13):2273–2287.
[36] Ajayi TM, Omowaye AJ, Animasaun IL. Viscous dissipation effects on the motion of Casson fluid
over an upper horizontal thermally stratified melting surface of a paraboloid of revolution:
boundary layer analysis. J Appl Math. 2017;2017:1697135.
[37] Noor NAM, Shafie S, Admon MA. Effects of viscous dissipation and chemical reaction on MHD
squeezing flow of Casson nanofluid between parallel plates in a porous medium with slip
boundary condition. Europ Phys J Plus. 2020;135(10):855.
[38] Onwubuoya C, Dada MS. Soret, viscous dissipation, and thermal radiation effects on MHD free
convective flow of Williamson liquid with variable viscosity and thermal conductivity. Heat
Transf. 2021;50(4):4039–4061.
[39] Yang D, Yasir M, Hamid A. Thermal transport analysis in stagnation-point flow of Casson
nanofluid over a shrinking surface with viscous dissipation. Waves Random Complex Media.
2021: 1–15.
[40] Ahmad B, Iqbal Z. Framing the performance of variation in resistance on viscous dissipative transport of ferro fluid with homogeneous and heterogeneous reactions. J Mol Liq.
2017;241:904–911.
[41] Ahmad B, Iqbal Z, Maraj E, et al. Utilization of elastic deformation on Cu–Ag nanoscale particles mixed in hydrogen oxide with unique features of heat generation/absorption: closed form
outcomes. Arab J Sci Eng. 2019;44(6):5949–5960.
[42] Akbar N, Iqbal Z, Ahmad B, et al. Mechanistic investigation for shape factor analysis of
SiO2 /MoS2 –ethylene glycol inside a vertical channel influenced by oscillatory temperature
gradient. Can J Phys. 2019;97(9):950–958.
WAVES IN RANDOM AND COMPLEX MEDIA
17
[43] Muhaimin I, Fazlul K, Mohamad R, et al. Similarity and nonsimilarity solutions on flow and heat
transfer over a wedge with power law stream conditions. Int J Innov Mech Eng Adv Mater.
2015;1(1):5–12.
[44] Ishak A. MHD boundary layer flow due to an exponentially stretching sheet with radiation effect.
Sains Malays. 2011;40:391–395.