Copyright © 2023 by American Scientific Publishers
Journal of Nanofluids
Vol. 12, pp. 1074–1081, 2023
(www.aspbs.com/jon)
All rights reserved.
Printed in the United States of America
Impact of Non-Similar Modeling for Thermal
Transport Analysis of Mixed Convective Flows of
Nanofluids Over Vertically Permeable Surface
Muzamil Hussain1, 2, ∗ , Wafa Khan1 , Umer Farooq1, ∗ , and Raheela Razzaq1
1
ARTICLE
2
Department of Mathematics, COMSATS University Islamabad, Park Road Chak Shahzad Islamabad, 44000, Pakistan
Department of Mathematics, University of the Poonch Rawalakot, Rawalakot, 12350, Pakistan
In the current article, non-similar model is developed for mixed convective boundary layer flow over a permeable
vertical surface immersed in nanofluid. The flow is initiated due to the plate stretching in vertical direction and
by natural means such as buoyancy. The governing dimensional equations are converted to non-dimensional
equations through characteristic dimensions. Furthermore the non-similar modeling is done by choosing X
as non-similarity variable and X Y as pseudo-similarity variable. The non-similar partial differential system
(PDS) is then solved by using local non-similarity method via bvp4c. The heat and mass transfer analysis are
carried out by studying local Nusselt and Sherwood numbers in tabular form for some important parameters
involved in the non-similar flow. The concentration, velocity and temperature profiles are graphically represented for various dimensionless number such as Prandtl number (Pr), Brownian motion Nb , Lewis number
Le and thermophoresis Nt . Reversed flow is observed for the velocity profile as non-similar variable is varied. Enhancement in thermal profile is witnessed for Nb Nt and reduction in temperature is observed for Pr.
182.191.214.121
On:Le
Sat,
Oct 2023
Concentration is reduced for IP:
different
values of Pr
Nb 21Finally
this 08:06:46
article intends to develop an intuitive
Copyright:
American
Scientific
understanding of non-similar models
by emphasizing
the
physicalPublishers
arguments. The authors developed the nonDeliverednon-similar
by Ingentastructure by employing the local non-similarity
similar transformations and tackled the dimensionless
technique. To the best of authors’ observations, no such study is yet published in literature. This study may be
valuable for the researchers investigating towards industrial nanofluid applications, notably in geophysical and
geothermal systems, heat exchangers, solar water heaters, biomedicine, and many other fields.
KEYWORDS: Non-Similar Modeling, Local Non-Similarity Via Bvp4c, Mixed Convection.
1. INTRODUCTION
The revolutionary concept of momentum and thermal
boundary-layer theory1–9 has been emerging significantly
and is utilized in many engineering applications of heat
transfer. Fluid flow over solid surfaces often appear
in practice, and it is responsible for various physical
aspect such as friction force acting on the automobiles,
steam and hot water pipes, the cooling of metal or
plastic sheets, and extruded wires. Numerous attempts
have been made to conduct studies due to its enormous
importance.10–15 Krishna et al.16 studied the flow of an
electrically conducting fluid through over a semi-infinite
vertical stretching surface with the considerations of magnetic field, thermal radiation, thermophoresis, and convective boundary conditions. In boundary layer analysis,
∗
Authors to whom correspondence should be addressed.
Emails: muzamil@upr.edu.pk, umer_farooq@comsats.edu.pk
Received: 26 May 2022
Accepted: 7 August 2022
1074
J. Nanofluids 2023, Vol. 12, No. 4
the non-dimensionalization of the governing equations can
be done using similarity or non-similarity approaches. In
similar flow phenomenon the dimensionless numbers does
not vary along the flow direction. However in non-similar
flows the basic flow quantities change in the flow direction. In that case we are forced to transform governing
PDS into dimensionless PDS by introducing two new independent dimensionless variables, called the non-similarity
and pseudo-similarity variables. It should be noticed finding such a variable, considering it prevails is more an art
than science, and it requires to have good mathematical
understanding of the problem. Many researchers have been
working on the similarity boundary-layer flows due to its
simplicity. Numerous articles have been published investigating the similarity boundary layer flows.7–9 However
in non-similarity flow such type of similarity is lost.18–22
The flows which occur naturally and are widely used in
our daily lives are the non-similar boundary-layer flows.
Boerner et al.19 observed these types of flows may occur
2169-432X/2023/12/1074/008
doi:10.1166/jon.2023.1985
Hussain et al.
Impact of Non-Similar Modeling for Thermal Transport Analysis of Mixed Convective Flows
J. Nanofluids, 12, 1074–1081, 2023
1075
ARTICLE
due to the presence of variation in dimensional velocity,
Recently many researchers have been doing analysis
to investigate the flow due to natural convection over
transverse curvature or due to mass transfer onto the sheet.
a vertical surface. Numerous similarity solution are calIn majority of the physical situations it may happen
culated such as Eichhorn30 investigated the effect of
that similarity approach does not work for boundary layer
linear thermal stratification over a vertical iso-thermal
equations. Mathematically it is complicated to solve nonplate, and solutions were obtained by series expanlinear PDS. The two approaches used to find out the
sion up to four terms. Non-linear thermal stratification
solution of non-similar flows are numerical and analythas been studied by Fujii et al.,31 reconsidering the
ical. Some of the research studies based on numerical
problem of Eichhorn.30 Later, Eichhorn’s problem was
techniques can be found in Refs. [23–29]. One can comreconsidered by Chen and Eichhorn,32 by using method
pute numerical solutions for infinite number of discretized
of local non-similarity. Implicit finite-difference scheme,
points, by converting the infinite domain to finite domain.
local non-similarity method and perturbation were used by
Due to this, error or uncertainty in solutions may arise.
Venkatachala and Nath33 to calculate the solution of isoTo overcome this issue analytical methods are used for
thermal wall in linearly stratified atmosphere. The appliinfinite domain. Unfortunately applying analytical methcation concerning the effect of suction and blowing in
ods, it is difficult to get accurate and valid approximation
natural convection flows are seeking more attention of
for all variables. As perturbation method depends on small
researchers because of the safe removal or addition of heat
parameters which give invalid results for these variables.
22
through a porous surface. Shooting method technique was
Cimpean et al. combined perturbation and analytical
applied by Kao34 to find solution for forced convection
method which was used to investigate non-similar thermal
flow along a flat plate with random injection or suction at
boundary layer flow by applying the method of local sim21
20
the wall. Raees et al.35 developed the non-similar model
ilarity Massoudi and Sparrow et al. In this method the
for the mixed convective, magnetized flow of second-grade
non-similar terms are assumed to be very small and are
nanofluid over an exponentially stretching surface with the
neglected, i.e., = 0. Due to this assumption the PDEs
assumption of buoyancy effects in terms of concentration
become ODEs but the result obtained are of “uncertain
20
21
and temperature differences in the x-momentum equation.
accuracy” as given by Sparrow et al. Massoudi invesCui et al.36 developed a non-similar mathematical model
tigated that this is only valid for small value of . Lately
IP: 182.191.214.121
On:
Sat,
Oct 2023
17 19 20
to 21
examine
the 08:06:46
melting aspects in the radiative flow of
a new method was introduced by Sparrow
et al.
Copyright:
American
Scientific
Publishers
21
nanofluid
over
a
vertically heated surface with the conseknown as ‘method of local non-similarity.’ Massoudi
Delivered by Ingenta
quences
of
chemical
reactions, magnetic field, and buoyapplied the method of non-similarity over a wedge by
ancy
forces.
differentiating the equation by non-dimensional variable
In this article, a vertical flat plate is installed in a quies, which results in two additional auxiliary equations.
cent
fluid. The simultaneous forced and unforced flow is
These complicated system of ODEs i.e., two original and
initiated
due to the stretching in vertical direction and by
two auxiliary momentum and thermal equations have been
means
of
buoyancy, respectively. Considering the imporsolved by numerical techniques.
tance
of
non-similarity
transformations, highly effective
The existing literature in non-similar flow is much less
non-similar
modeling
is
proposed for mixed convective
than the similar flows. Non-similar flows are important
boundary
layer
flow
of
nanofluid.
The non-similar PDEs
both in a theoretical sense as well as in practical terms.
is
numerically
simulated
using
local
non-similarity via
Researchers have considered studying natural convection
bvp4c. Finally we discuss the simultaneous natural and
due to its importance in engineering and nature. With natforced convection through graph and tabular results. This
ural convection, fluid flow is not driven by an external
study indicates the effectiveness of non-similar modeling
force, but by variations in density. Gravity is responsible
for naturally convective boundary layer flows.
for such types of flows. Forced convection is characterized
by the presence of an external source, such as a fan or
pump. Mixed convection flows are boundary layer flows in
2. MATHEMATICAL FORMULATION
which forced and natural convection are both significant.
Consider the two-dimensional steady, laminar boundary
In oceans the convection process occur due to the preslayer flow along a vertical permeable isothermal surface
ence of salt water being heavier than fresh water, so the
with isothermal surrounding as shown in Figure 1. The
salty water layer on the top of the fresh water causes natconsidered cartesian coordinate system is such that that the
ural convection. Natural convection can be observed in
y-axis is measured normal to the surface and the x-axis is
rising fumes of fire, oceanic current, and sea wind. In
taken along the direction of the stretching sheet. Accordengineering, natural convection is visualized in formation
ing to Boussinesq approximation for buoyancy driven flow
of very small structures (i.e., 1 × 10−6 during cooling of
that is natural convection, it states that the variation in denmolten metal and fluid flowing around the shrouded heatsity is significant in buoyancy term. The flow is triggered
dissipation fins. Natural convection is used on small scale
by the plate stretching in a vertical direction and natural
for cooling computer chips.
processes including buoyancy force. Flow is retained in
Impact of Non-Similar Modeling for Thermal Transport Analysis of Mixed Convective Flows
Hussain et al.
describe the thermal diffusivity, f is the fluid density.
Non-dimensionalized equations can be obtained by applying following suitable characteristic dimensions,
y
Y= l
x
X= l
=
Pr =
T = Tw − T + T U=
ul
V=
vl
(6)
C = Cw − C + C
Where is the characteristic length.
By applying the above-mentioned dimensionless variable, the dimensionless continuity, momentum, thermal
and concentration equations are as follow,
ARTICLE
Fig. 1. Represents the physical model for the problem.
y > 0 and uw = bx n is the stretching velocity of the surface across the x-axis (b is constant). The governing model
for laminar boundary-layer equations describing continuity, momentum, energy and concentration of nanoparticles
are as follows;35–37
Continuity Equation
U Y
+
=0
(7)
X Y
2 U
U
U
U
+V
= Pr
+ GrP r 2 + Gr ∗ P r 2 (8)
X
Y
Y 2
2 2
+V
=
U
+PrNb
+PrNt
X
Y
Y 2
Y Y
Y
(9)
2
1 2 Nt +V
=
(10)
U
+
X
Y
Le Y 2
Nb Y
The boundary conditions are
u v
U = aX n V = 0 = 1 = 1 as Y = 0
+
=
0 182.191.214.121 On:
(1) Sat, 21 Oct 2023 08:06:46
IP:
x y
U → Publishers
0 → 0 → 0 as Y → (11)
Copyright: American Scientific
Delivered
by
Ingenta
x-Momentum Equation
where a = bn+1 / Nt is thermophoresis, Nb is Brown 2 ian motion, Gr ∗ and Gr are Grashof number’s and Le is
u
u
u
=
u +v
T
−
T
+
g
C
−
C
+
g
the Lewis number, they are defined as
t
c
x
y
y 2
(2)
g Cw − C l3 C
Tw − T Dt
∗
=
Nt =
Gr
Thermal energy Equation
v2
T
2 2
gl3 Tw − T T
Cw − C DB
T
T
C T
T
DT T
Gr =
Nb =
+v
=
+
D
u
+
B
v2
x
y
y 2
y y
T y
(3)
1
LePr =
a
=−
Nano-particles Concentration equation
DB
2Gr 1/2 Pr
2 C
C
DT 2 T
C
2.1. Non-Similarity Model
+v
= DB
+
(4)
u
x
y
y 2
T y 2
Non-similar model is constructed by defining the following
transformations
The boundary conditions are
1/4
Gr
= Gr 1/4 Pr 43/4 f = X = Y
u = bx n v = 0 C = Cw T = Tw at y = 0
4
u → 0 v → 0 C → C T → T as y → T = Tw −T +T C = Cw −C +C
(5)
Where
In this model u and v are x and y- velocity compoT = Tw − K C = Cw − K
nents, T is the temperature, is the ratio between nanoparticle heat capacity and original heat capacity, T is the
thermal enlargement coefficient, textitg is the acceleration due to gravity, DB is the Brownian diffusion coefficient,
the kinematic viscosity, DT describe coefficient
of thermophoresis diffusion, C is the volume fraction of
the nanoparticles, T and C are represents the quiescent fluid temperature and concentration respectively,
1076
The chain rule can now be used to link parameters in
x y plane to parameters in plane
f
= Gr1/2 Pr41/2 Y
1/4
Gr
f
f
V =−
=−
−
+ 3f (12)
Pr 4
X
4
U =
J. Nanofluids, 12, 1074–1081, 2023
Hussain et al.
Impact of Non-Similar Modeling for Thermal Transport Analysis of Mixed Convective Flows
T = Tw − K, C = Cw − K and K is constant. is
the stream function, X is the non-similarity variable,
X Y as pseudo-similarity variable represent the dimensionless temperature and concentration,
respectively.
Substituting Eq. (11) into Eqs. (6)–(10) we get the nondimensionalized governing equations,
2
2 f
f
3 f
+
3
f
−
2
+ + N
3
2
2 f f
f f
− 2
(13)
= 4
2 2 + Pr 3f
+ Nb
+ Nt
2
f f
−
= 4Pr
(14)
f
= 0
f
= 0
0 = 1
2 d
d d
d2
d
=0
+ Nb
+ Nt
+ Pr 3f
2
d
d
d d
d
(20)
(21)
(22)
With boundary conditions
(15)
f 0 = 0
df
− n−1/2
0 = 3/2 1/4 d
2 Gr
df
= 0
d
f
0 = 1 = 0 0 = 1 = 0 (23)
0 = n+1/4 IP: 182.191.214.121 On: Sat, 21 Oct 2023 08:06:46
Now these
are the ordinary differential equations for
Copyright: American Scientific
Publishers
Delivered byfunction
Ingentaof f with respect to .
= 0
0 = 1
= 0
(16)
The local Sherwood number Sh, friction coefficient Cf
and the local Nusselt number Nu are defined as follows
2
ljw
Cf = w2 Sh =
uc
DB Cw − C lqw
Nu =
k Tw − T Where
2
d3f
d2f
df
+3 2f −2
+ + N = 0
3
d
d
d
d 2 Nt d 2 d
=0
+
+ 3LePrf
2
2
d
Nb d
d
Subject to the dimensionless boundary conditions
3f 0 + 4
2.3. First Level of Truncation
In first level of truncation neglecting / in
Eqs. (14)–(16). We will get the following ordinary differential system:
C
u
jw = −DB
y y=0
y y=0
T
qw = −k
y y=0
(17)
w =
(18)
Using Eq. (11) in Eq. (18), the friction coefficient, the Nusselt number and Sherwood number takes form as follows:
2Gr 3/4 41/4 2 f
0 √
2
Rex
1/4
Gr
0 Sh = −
4
1/4
Gr
Nu = −
0
4
Cf =
J. Nanofluids, 12, 1074–1081, 2023
(19)
2.4. Second Level of Truncation
For second level of truncation, we will introduce the following new functions
r =
f t =
s =
(24)
Now the Eqs. (20)–(23) after introducing the new functions are transformed as
2
d3f
d2f
df
+
3
f
−
2
+ + N
d3
d2
d
df dr d 2 f
= 4
−
r
(25)
d d d2
2 d2
d
d d
d
+ Nb
+ Nt
+ Pr 3f
2
d
d
d d
d
d
df
s−
r
(26)
= 4Pr
d
d
d 2 Nt d 2 df
d
d
= 4LePr
t−
r
+
+3LePrf
d2 Nb d2
d
d
d
(27)
1077
ARTICLE
Nt 2 2 +
+
+ 3LePrf
2
2
Nb f f
−
= 4LePr
2.2. Local Non-Similarity Method
Sparrow and Yu20 and recently Farooq et al.38 used the
method of non-similarity to solve different non-similar
boundary value problems. Formulation for the local similarity and local non-similarity equations are as follows.
Impact of Non-Similar Modeling for Thermal Transport Analysis of Mixed Convective Flows
Hussain et al.
According to the boundary conditions
3f 0 + 4 r 0 = 0
df
− n−1/2
0 = 3/2 1/4 d
2 Gr
df
= 0
d
0 = 1
= 0
0 = 1
= 0
(28)
ARTICLE
Now taking partial derivatives of Eqs. (25)–(28) with
respect to , we get
d3r
d 2f
d2r
dr df
+7 2r +s+N t
+ 2f −8
3
d
d
d d
d
Fig. 2. Shows the effect of = 1 5 10 on f (), for fixed values of
2
2
n
= 0, N = 1 = 1/4, Nb = Nt = 01, Le = 3, Gr = 30 and Pr = 0.7.
dr
d r
= 4
− 2r
(29)
d
d
d2s
df
d
ds
+7 r
+ Pr 3 f − 4s
2
d
d
d
d
d
ds d dt d
+Nb
+
+ 2Nt s
d d d d
d
ds
dr
= 4Pr r
−s
(30)
d
d
IP: 182.191.214.121 On: Sat, 21 Oct 2023 08:06:46
dt
dfAmerican Scientific Publishers
Nt d 2 s
d Copyright:
d2t
Delivered
by Ingenta
+
3
f
−
4
t
+
+
LePr
7r
d2 Nb d2
d
d
d
dr
dt
= 4LePr
t−r
(31)
d
d
related to the boundary conditions
− n−1/2
r
0 = n − 1/2 3/2 1/4 2 Gr
r
= 0
s 0 = 1
s = 0
r 0 = 0
Fig. 3. Shows the effect of Pr = 072 3 68 on () for fixed values
of n = 0, N = 1 = 1/4, Nb = Nt = 0.1, Le = 3, Gr = 30 and = 1.
t 0 = 1
t = 0
(32)
The
terms
r /,
s /,
t / and their derivatives with respect to are
neglected.
3. RESULTS AND DISCUSSION
The numerical approximations of Eqs. (20)–(32) through
local non-similarity methods are graphically expressed via
MATLAB based algorithm bvp4c. Graphs have been plotted for Nb , Nt , Le and Pr for f and .
The Figure 2 graph represents the relationship between
the velocity and the non-similar variable . Reversed flow
is observed, the velocity is initially increasing near the
wall and then starts decreasing for greater value of and finally velocity profile approaches to zero. Figure 3
1078
Fig. 4. Shows the effect of Nb = 0.1, 0.5, 1 on (), for fixed values of
n = 0, Pr = 10, N = 1, = 1/4, Nt = 0.1, Le = 3, Gr = 30 and = 1.
J. Nanofluids, 12, 1074–1081, 2023
Hussain et al.
Impact of Non-Similar Modeling for Thermal Transport Analysis of Mixed Convective Flows
Fig. 6. Shows the effect of Pr = 0.72, 3, 7.2 on (), for fixed values
of n = 0, N = 1, = 1/4, Nt = Nb = 0.1, Le = 3, Gr = 30 and = 1.
Fig. 8. Shows the effect of Nb = 0.1, 0.5, 1 on (), for fixed values
of n = 0, Pr = 0.72, N = 1, = 1/4, Nt = Nb = 0.1, Le = 3, Gr = 30
and = 1.
Table I. Comparison of findings for
= 1 and Gr ∗ = Gr = 0.
2f
0 (Cui et al.39 ) when =
2
2f
0
2
Fig. 7. Shows the effect of Le = 1, 3, 5 on (), for fixed values of
n = 0, Pr = 0.72, N = 1, = 1/4, Nt = Nb = 0.1, Gr = 30 and = 1.
J. Nanofluids, 12, 1074–1081, 2023
Parameter
Cui et al.36
Present result
M = 0.0
1.28141466
1.2814134224
1079
ARTICLE
presents the temperature versus . is decreasing function for increasing value of Pr This is due to the fact that
for greater value of Pr the thermal diffusivity is low. The
evolution of temperature is witnessed in Figure 4. For
higher value of Brownian motion Nb , the number of nanoparticles increases, due to which the surface temperature is
increased. Figure 5 depicts that is the increasing function
for different values of Nt . Indeed, when Nt increase, more
heat dispersion occurs, resulting in the greatest amount of
heat transmission. Thermal profiles intensify in this physical phenomenon as thermal layer thicknesses rise. Analysis of Figure 6 for values of Pr reveals that concentration
is reduced as value of Pr is increased. Figure 7 reveals
the effect of Le on concentration profile is decreasing for
greater values of Lewis number. As mass diffusivity is
lower the rate of mass transfer also reduces and heat conFig. 5. Shows the effect of Nt = 0.1, 0.5, 0.9 on (), for fixed values
duction is low. It decreases the concentration profile. The
of n = 0, Pr = 0.72, N = 1, = 1/4, Nb = 0.1, Le = 3, Gr = 30 and
= 1.
reduction in concentration is observed in Figure 8 as the
value of Nb is increased. Enhancing values of Nb disrupt
Brownian motion and hence inhibit the nanoparticles diffusion in flow regime, resulting a decline in the concentration
of the nanoparticle volume fraction.
Table I indicates the comparison of the results with the
published literature Cui et al.39 Table II represent local
Nusselt number, decrease is witnessed as the value of Nb
IP: 182.191.214.121 On: Sat, 21 Oct 2023 08:06:46
Copyright: American Scientific Publishers
Delivered by Ingenta
Impact of Non-Similar Modeling for Thermal Transport Analysis of Mixed Convective Flows
Table II. Nu (Local Nusselt number) for different values of Pr versus
Nb when Nt = 0.1, Le = 10, = 1/4, n = 0 for Pr = 0.7, 6.8, 10 and
Nb = 02, 04, 06.
Pr
Nb
Nu Rex−1/2
ARTICLE
0.7
0.2
0.4
0.6
Hussain et al.
• NuRex−1/2 shows an decreasing pattern against Nb
and Pr.
• ShRex−1/2 increases with the enhancement of Le and
Pr.
0. 5527568893
0. 5084701286
0. 4715422874
LIST OF ABBREVIATIONS
x y X Y Cartesian coordinates
6.8
0.2
0. 4493055200
u v U V Velocity components
0.4
0. 1854729129
Ratio between nanoparticle heat capacity
0.6
0. 0689515419
cp Specific heat
10
0.2
0. 2947142744
f Fluid density
0.4
0. 0715306609
Dynamic viscosity
0.6
0. 0145384118
Stream function
Non-similarity variable
Pseudo-similarity variable
Table III. Sh (Local Sherwood number) for different values of Le verThermal diffusivity
sus Pr when Nb = Nt = 0.1, = 1/4, n = 0 for Le = 0 1 3 and Pr = 0.7,
6.8, 10.
Nb Brownian motion parameter
Nt Thermophoresis parameter
Le
Pr
Sh Rex−1/2
T Temperature
1
0.7
0.2605638598
C Concentration
6.8
0.9783897308
c Concentration enlargement coefficient
10
1.2384501277
t Thermal enlargement coefficient
2
0.7
0.5040372873
Effective kinematic viscosity
6.8
1.3654301346
T
10
1.6390564205
w Wall temperature
T
Free stream temperature
3
0.7
0.6574700638
Cw Wall
concentration
IP: 182.191.214.121
On: Sat, 21 Oct 2023
08:06:46
6.8
1.5954038540
Copyright:
American Scientific Publishers
10
1.8724351691
C Free stream concentration
Delivered by Ingenta D Coefficient of Brownian motion diffusion
B
DT Coefficient of thermophoresis diffusion
increases. Table III depicts the increase in Sherwood num Dimensionless temperature
ber as value of Pr is increasing.
Dimensionless concentration
g Gravitational acceleration
l Length
4. CONCLUSION
Gr Heat transfer Grashof
Non-similar model is proposed and analyzed for mixed
Gr ∗ Mass transfer Grashof
convective boundary layer nanofluid flow over a permeable
N Ratio between heat transfer Grashof to
vertical surface. The flow is initiated by natural processes
mass transfer Grashof
including buoyancy phenomenon and plate stretching in a
Pr Prandtl number
vertical direction. The governing system is converted to
Cfx Local skin friction coefficient
non-dimensional structure through appropriate transformaNu Local Nusselt number
tions. The non-similar partial differential system (PDS) is
Sh Local Sherwood number
then tackled by employing local non-similarity technique
Re Reynold number
via bvp4c. The assessments regarding heat and mass transfer are conducted by tabulating the local Nusselt and SherReferences and Notes
wood numbers against some significant non-similar flow
1. L. Howarth, Proceedings of the Royal Society of London.
parameters. Graphic representations of the concentration,
Series A, Mathematical and Physical Sciences 547 (1938), DOI:
velocity, and temperature profiles are shown in associated
10.1098/rspa.1938.0037.
with various dimensionless emerging factors. The main
2. M. Van Dyke, J. Fluid Mech. 14, 161 (1962a).
observation drawn from our research are follow:
3. M. Van Dyke, J. Fluid Mech. 14, 481 (1962b).
• Reversed flow is observed for the velocity profile as
4. M. Van Dyke, J. Fluid Mech. 19, 145 (1964).
5. M. Van Dyke, Annual Review of Fluid Mechanics 1, 265 (1969).
non-similar variable is varied
6. M. Van Dyke, NASA STI/Recon Technical Report A 75, 46926
• Enhancement in thermal profile is witnessed for Nb , Nt
(1975).
and reduction in temperature is observed for Pr.
7. I. Tani, Annual Review of Fluid Mechanics 9, 87 (1977).
• Concentration is reduced for different values of Nb , Pr,
8. H. Schlichting and K. Gersten, Boundary-Layer Theory. Verlag
Berlin Heidelberg, Springer (2016).
Le.
1080
J. Nanofluids, 12, 1074–1081, 2023
Hussain et al.
Impact of Non-Similar Modeling for Thermal Transport Analysis of Mixed Convective Flows
24. P. W. Duck, S. R. Stow, and M. R. Dhanak, J. Fluid Mech. 400, 125
(1999).
25. A. K. Sahu, M. N. Mathur, P. Chaturani, and S. S. Bharatiya, Acta
Mechanica 142, 119 (2000).
26. N. Banu and D. A. S. Rees, International Communications in Heat
and Mass Transfer 27, 775 (2000).
27. M.-I. Char, J.-D. Lin, and H.-T. Chen, International Journal of Engineering Science 39, 897 (2001).
28. W. T. Cheng and H. T. Lin, International Journal of Engineering
Science 40, 531 (2002).
29. C.-H. Chen, International Journal of Engineering Science 42, 699
(2004).
30. R. Eichhorn, Progress in Heat and Mass Transfer 2, 41 (1969).
31. T. Fujii, M. Takeuchi, and I. Morioka, Laminar boundary layer of
free convection in a temperature stratified environment, International
Heat Transfer Conference Digital Library (1974).
32. C. C. Chen and R. Eichhorn, J. Heat Transfer 98, 446 (1976).
33. B. J. Venkatachala and G. Nath, Int. J. Heat Mass Transfer 24, 1848
(1981).
34. T.-T. Kao, J. Heat Transfer 97, 484 (1975).
35. A. Raees, U. Farooq, M. Hussain, W. A. Khan, and F. B. Farooq,
Communications in Theoretical Physics 73, 065801 (2021).
36. J. Cui, M. Safeer, U. Farooq, M. E. Ahmed Rabie, and
T. Muhammad, AIP Advances 11, 085018 (2021).
37. M. Waqas, M. Farooq, M. I. Khan, A. Alsaedi, T. Hayat, and
T. Yasmeen, Int. J. Heat Mass Transfer 102, 766 (2016).
38. U. Farooq, M. A. Ijaz, M. I. Khan, S. S. P. M. Isa, and D. C.
Lu, International Communications in Heat and Mass Transfer 119,
104955 (2020).
39. J. Cui, R. Razzaq, U. Farooq, W. A. Khan, F. B. Farooq, and
T. Muhammad, Alexandria Engineering Journal 61, 4253 (2022).
IP: 182.191.214.121 On: Sat, 21 Oct 2023 08:06:46
Copyright: American Scientific Publishers
Delivered by Ingenta
J. Nanofluids, 12, 1074–1081, 2023
1081
ARTICLE
9. I. J. Sobey, Introduction to Interactive Boundary Layer Theory,
Oxford University Press on Demand (2000), Vol. 3.
10. M. VeeraKrishna and A. J. Chamkha, Physics of Fluids 30, 053101
(2018).
11. M. V. Krishna, M. G. Reddy, and A. J. Chamkha, International
Journal of Fluid Mechanics Research 46, 1 (2019).
12. M. V. Krishna, P. V. S. Anand, and A. J. Chamkha, Special Topics
and Reviews in Porous Media: an International Journal 10, 203
(2019).
13. M. V. Krishna and A. J. Chamkha, International Journal of Ambient
Energy 1 (2021), DOI: 10.1080/01430750.2021.1946146.
14. M. V. Krishna and A. J. Chamkha, International Communications in
Heat and Mass Transfer 113, 104494 (2020).
15. M. V. Krishna, N. A. Ahammad, and A. J. Chamkha, Case Studies
in Thermal Engineering 27, 101229 (2021).
16. M. V. Krishna, K. Jyothi, and A. J. Chamkha, Journal of Porous
Media 23, 751 (2020).
17. D. J. Wanous and E. M. Sparrow, J. Heat Transfer 87, 317 (1965).
18. D. Catherall, K. Stewartson, and P. G. Williams, Proceedings of the
Royal Society of London. Series A. Mathematical and Physical Sciences 284, 370 (1965).
19. C. J. Boerner, H. Quack, and E. M. Sparrow, AIAA Journal 8, 1936
(1970).
20. E. M. Sparrow and H. S. Yu, Local Non-Similarity Thermal
Boundary-Layer Solutions 393 (1971), DOI: 10.1115/1.3449827.
21. M. Massoudi, International Journal of Non-Linear Mechanics 36,
961 (2001).
22. D. Cimpean, J. H. Merkin, I. Pop, and D. B. Ingham, Transport in
Porous Media 64, 393 (2006).
23. R. S. R. Gorla, K. Shanmugam, and M. Kumari, Heat Mass Transfer.
33, 281 (1998).