Heat Transfer in a Magnetohydrodynamic Boundary Layer Flow of a Non-Newtonian Casson Fluid Over an Exponentially Stretching Magnetized Surface
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Copyright © 2021 by American Scientific Publishers Journal of Nanofluids
All rights reserved. Vol. 10, pp. 172–185, 2021
Printed in the United States of America (www.aspbs.com/jon)
Heat Transfer in a Magnetohydrodynamic Boundary
Layer Flow of a Non-Newtonian Casson Fluid Over
an Exponentially Stretching Magnetized Surface
Golbert Aloliga1, ∗ , Yakubu Ibrahim Seini2 , and Rabiu Musah2
1
Department of Mathematics, St. Vincent College of Education, Yendi, 00233, Northern Region, Ghana
2
School of Engineering, University for Development Studies, Nyankpala Campus, 00233, Northern Region, Ghana
In this current paper, an investigation has been conducted on the magnetohydrodynamic boundary layer flow
of non-Newtonian Casson fluids on magnetized sheet with an exponentially stretching sheet. The similarity
approach has been used to transform the governing models for Casson fluid to ordinary differential equations.
We presented numerical results for momentum, energy and concentration equation parameters. Effects of the
magnetized sheet and varying all the emerged parameters on the flow of Casson fluid with respect to the
friction between the fluid and the surface, temperature and concentration are presented in tables. As a result of
the induced magnetization of the sheet, the thickness of the thermal boundary layer has been enhanced. This
behaviour brings a considerable reduction to the heat transfer. The induced magnetized sheet has a similar
ARTICLE
influence on the skin friction, Nusselt number and the Sherwood number. We however proposed incorporation
of magnetized surfaces in MHD flows for controlling the flow rate of the fluid and heat transfer characteristics.
IP: 192.168.39.151
KEYWORDS: Chemical Reaction, Magnetized Plate,On: Wed, 13 Oct Casson
Non-Newtonian 2021 19:58:31
Fluid, Convective Boundary Condition,
Internal Heat Generation. Copyright: American Scientific Publishers
Delivered by Ingenta
1. INTRODUCTION increases heat transfer thereby increasing its viscosity.
Research on transport equations for Casson fluids have Many research findings have been reported on the impact
gained the awareness of many investigators because of of viscosity, thermal transmission, and conductivity on
the numerous usage in the manufacturing sector. Works the boundary layer of nanofluids, Meyers et al.4 and
involving coating and polymer extraction, petroleum Anyakoha.5
refinery, aerodynamic heating, and hot rolling, involves The study of Casson fluid is another area of intense sci-
non-Newtonian Casson fluids Cortell.1 Fluids exhibiting entific research. Nadeem et al.6 employed the Adomian
non-Newtonian fluid characteristics include mud, blood, decomposition method to analyze the Casson fluid flowing
polymer solution, and paint among others. Because of their on an exponentially shrinking surface. They discovered a
unique properties, sometimes it is always difficult to find process to reduce the Newtonian problem when the viscous
a generalized mathematical representation to adequately parameter approaches infinity. Afify and Bazid7 extended
describe all transport charactereistics of the fluid. One clas- the problem to include variable viscosity with a significant
sical type of fluid that portrays this behavior among oth- observation of how the increment in the viscous parameter
ers, is the Casson fluid. According to Dash et al.,2 Casson gradually reduces heat transfer and a decrease in viscos-
fluid is considered to be a shear-thinning fluid with immea- ity enhanced the skin resistance of the exterior part of the
surable viscosities at a zero rates of shear. These shears sheet. Bagai and Nishad8 used the shooting technique to
produce stresses up to a stage where fluid flows cannot inspect the effect of temperature distribution on the bound-
occur. aries of fluid flowing on a plane sheet immersed in a per-
A study on thermal conduction and temperature- meable medium saturated by nanoparticles. They observed
dependent viscosity was conducted by Attia.3 Some reveal- that, the viscous parameter had tremendously affected the
ing outcomes showed that, an increase in fluid velocity rate of heat transfers between the surface and the non-
Newtonian fluid.
∗
Casson fluids as a matter of importance, typi-
Author to whom correspondence should be addressed.
Email: aloligagolbert@gmail.com cally behave like elastic solids Animasaun.9 Jawali and
Received: 13 April 2021 Chamkha10 conducted a study on the consequence of the
Accepted: 29 April 2021 thickness and thermal conduction on viscous convection
172 J. Nanofluids 2021, Vol. 10, No. 2 2169-432X/2021/10/172/014 doi:10.1166/jon.2021.1777
Aloliga et al. Heat Transfer in a Magnetohydrodynamic Boundary Layer Flow
fluid. An important observation was the generation of efficient thermal runaway processes such as the develop-
heat from the lubricant that influences the thickness and ment of magnetized surfaces. Chamkha19 numerically ana-
the rate of conduction of heat on the plate. The molec- lyzed the convective Casson fluids flow on a perpendicular
ular properties of fluid thus deforms across the bound- plate in a thermally permeable system by means of Hall
ary layer during heating. These studies re-affirm the claim effects. Chamkha20 extended the study to include solar
that thermal conductivity and viscosity of fluids should heating with normal convection in a homogeneous leaky
not always be assumed constant because the temperature system. Takhar et al.21 deliberated on nanofluid flow on
can influence the flow characteristics of the fluid. Saidulu a vertically moving plate with heat and mass transfer and
and Ventakata11 further analyzed the effect of boundary observed that, there was high transmission of heat when
slip of MHD Casson fluid with thermal emission and the fluid was in motion on the plate. Takhar et al.22 con-
chemical response using the Keller box method. Again, sidered the MHD boundary layer flow of non-Newtonian
Mahdy12 examined the effects of the stretching plate fluid on an uneven multi-dimensional spontaneous stretch-
of the unsteady MHD Casson fluid flows with blowing ing plate. Chamkhar et al.23 analyzed the energy reactions
13
and suction. Ibukun et al. employed spectral relaxation on free stream flows of a partiali-infinite erected sheet
with linearized-Rosseland radiation term to investigate the with heat and mass transmission. Etwire et al.24 investi-
unsteadiness of Casson fluid on a stretching sheet subject gated the impact of thermophoretic transport of oil-based
to the slip boundary conditions of the fluid. They real- nanofluid containing Al2 O3 nanoparticles over an exponen-
ized that, the unsteadiness of the fluid had a direct impact tially stretching porous surface in the vicinity of activation
on the fluid heat and its velocity. Similarly, Animasaun energy.
et al.14 used the homotopy analysis method to investigate From the magnetic point of view, Modather et al.25
Casson fluid flow with varied thermo-physical properties investigated a paper on the systematic study of mass
on an exponentially stretched surface with suction. Mag- and heat transport of micropolar fluid on a vertical leaky
ARTICLE
sheet within an absorbent solution. Reddy et al.26 studied
netohydrodynamic Casson fluid flow on a stretching sur-
magnetohydrodynamic boundary layer of non-Newtonian
face with heat and chemical reaction was conducted by
15 nanofluid with particles of copper, water and gold on
Gireesha et al. Their results showed clearly that, whilst
turning disk in a permeable medium saturated with chem-
the Casson parameter was used IP: 192.168.39.151
to control On:
the velocity of Wed, 13 Oct 2021 19:58:31 27
ical reaction. Younes et al. researched on magnetohy-
the flow, the radiation values thickensCopyright:
the thermalAmerican
bound- Scientific Publishers
Delivered bydrodynamic
Ingenta fluid flow and thermal radiation of some base
ary layer. Arshad et al.16 examined the sodium-Casson
chemical solution discreted from aluminum oxide and
nanofluid flow through porous surface using Laplace trans-
nanoparticles. Kumar et al.28 used the concept of Cattaneo-
form method with Darcian porous medium and the lin-
Christov heat diffusion to explain the effects of a slopping
earized Rosseland radiation term. Whilst Gbadeyan et al.17
magnetic plate of Reiner-Philippoff fluid. Kumar et al.29
considered the similarity transformation method to inves- observed the effect of induced magnetization and the emis-
tigate Casson nanofluids flow over perpendicular plane sion of electromagnetic energy of the convective flow of
surface with slip boundary condition and convective heat non-Newtonian fluids. Thameem et al.30 studied the stag-
transfer. The collective effects of nonlinear radiation, ther- nation point of diamond-ethylene and glycol nanofluids
mal conduction, viscosity, and the porous medium on the on a wedge with induced magnetic field with heat trans-
boundary layer were discussed as well. mission. Krishna and Chamkha31 32 on the other hand
Amos et al.18 used Chebyshev collocation spectral used the perturbation technique to analyzed the impact of
approach to analyze mass and convective heating of Cas- Hall and ion slip of a rotating boundary layer flow of
son fluid with thermal conductivity, and unpredictable MHD nanofluid on inestimable vertical sheet entrenched
thickness. It is seen severally in literature that, both the in spongy solution.
boundary layer thickness and thermal conduction have Very recently, Krishna et al.33 analyzed the influence of
been investigated and the fluid physical properties change Hall and ion slip of unsteady electromagnetic fluid flows
considerably with temperature. Amos and his colleagues in saturated absorbent medium and Abderrahim et al.34
found that there was an inverse relationship between the investigated a novel physical thermodynamic irreversibil-
velocity and the Casson viscosity parameter. ity within dissipative electric magnetohydrodynamic flu-
Heat carrier fluids like ethylene, glycol, water, mineral ids past a flat Riga plate with suction and joule heating.
oil, etc. are of practical relevance in industrial application Then another time, Ramesh et al.35 discussed heat trans-
such as in chemical production, power generation, air con- fer of aluminum alloy and magnetite graphene oxide of
ditioning, etc. The performances of these fluids depend on porous cylindrical sheet with heat supply or sink. Menni
their ability to transfer heat without changing their prop- et al.36 also considered how to improve the control of
erties. Owing to the high demand for these fluids with energy transmission in smooth air channels and wall-
high thermal conductivities, engineers are now focused on mounted impediments of the flow trail. They all observed
developing techniques that will delay flow processes for in their results that the flow system and the geometry
J. Nanofluids, 10, 172–185, 2021 173
Heat Transfer in a Magnetohydrodynamic Boundary Layer Flow Aloliga et al.
of the canal provided with obstacles increase the ther- nanofluid volume fraction. From Dogonchi et al.,53 the
mal energy between the heated surface and the coolant. controled volume finite element strategy was used to
Menni et al.37 again studied aerodynamics and electro- explore the problem of usual convection in a square and
magnetic radiation of an isochoric non-Newtonian fluid curvy circular heater with the control of magnetic field and
with invariable properties on a two-dimensional flat sur- nanoparticles. Mohamad et al.54 analysis covered about
face using computational fluid dynamics method. Menni five different types of geometries; trapezoidal, triangular,
et al.38 equally investigated nanofluid flow in complex unconventional geometries, circular, and squared. It was
geometries. Menni et al.39 reported another paper on the observed that the performance of the squared shape was
turbulent fluid flows and transmission of heat in solar chan- poor compared with the trapezoidal shape in enhancing the
nels with different forms. Their outcome suggested that the heat transfer phenomenon. The thermal entropy parameter
heat generated from Z-shape is higher than the flat rect- increases as the porosity of the medium increases due to
angular, and the other shapes. Menni et al.40 looked over higher friction in a fluid.
again at the effects of V-baffle and wall-mounted shapes Several engineering and industrial applications require
in a turbulent flow through a waterway. It was config- better heat transfer performance. For example, the cool-
ured that, different shapes of the baffles produce different ing and heating of engines of machines, electronical chips,
rate of heat transfer. Thermal radiation and heat transfer solar panels, insulators of nuclear, are still a major con-
of turbulent fluid flow in staggered shape obstacle in air cern. Heat transfer between a fluid and a heated surface
channel was investigated by Menni et al.41 42 Whilst the depends mainly on the transfer rate of heat. The magnetic
simulation of fluid dynamics characteristics and heat dis- field has a very important role to play in controlling the
tribution of various shapes was analyzed by Menni et al.,43 heat transfer characteristics. Many researchers have tried
he later on Menni et al.44 used the computational fluid to develop some fluid models to provide enhanced perfor-
dynamics method to replicate the steady flows and energy mances. Applying the magnetic field on both the fluid and
ARTICLE
transmit in a solar air channel. Menni et al.45 investigation plate is a new concept of the magnetic field phenomenon
however contained a simulation of a definite heat. Takhar in convective flows. The magnetization of the plate is
et al.46 considered the implicit finite difference method to proposed to address the existing heat transport challenge.
study the transformation of uneven varied fluids flow from The
IP: 192.168.39.151 On: Wed, 13novelty
Oct 2021of this type of MHD boundary layer flow is
19:58:31
a rotation upright conduit with magnetic effect. It was obtained by
Copyright: American Scientific Publishers magnetizing the surface of the plate. There-
exposed from their study that, increasing of magnetic field byfore,
Delivered Ingenta
in this study, considerable effort and interest are made
parameter values reduced the friction between the fluid to investigate the problem.
and surface of the plate at a certain tangent. The convec-
tion nanofluid surrounded by a square hollow space was
addressed by Mohammad et al.47 The outcome of their 2. MATHEMATICAL MODEL
investigation showed that, the temperature shift increases The mathematical equations of the Casson fluids are
by addition of a mixture of nanoparticles for a conduc- derived based on established fluid dynamic models com-
tion dominant process. The laminar normal convection and prising the; continuity, momentum, energy, and concentra-
entropy production of nanofluid intricate void was studied tion equations. Consider a dissipative steady Casson fluid
48
by Salva and Chamkha. Their results demonstrated that, flows on a nonlinearly exponentially stretching magnetized
the presence of nanoparticles in the fluid usually deterio- plate that co-occur with the plate at y = 0. Assuming the
rates the flow strength. Chamkha and Abdul-Rahim stud- 49 fluids flow is restricted to y > 0 with two equal but oppo-
ied the linear stratified stagnation point flows with heat and site forces acting along the horizontal (x-axis) so that the
mass transport in the presence of an external magnetic field
with temperature generation and absorption. They con-
cluded that, increase in Prandtl number and buoyancy ratio
increase Nusselt values. Seini et al.50 numerically investi-
gated the boundary layer flow of Casson fluid moving on
an exponentially stretching porous surface with radiative
heat transfer and concluded that a highly porous surface
cools faster than one with less porosity.
Ismael et al.51 analyzed the heat and entropy generation
filled with saturated nanoparticles in permeable medium
and revealed that the entropy generation increased faster
than mediums with low conduction. Dogonchi et al.52
studied the nanoparticles with convection in cavities on
an inclined magnetic plate. Their results revealed that
the magnetic field controls the heat which grows using Fig. 1. Schematic flow diagram of a problem.
174 J. Nanofluids, 10, 172–185, 2021
Aloliga et al. Heat Transfer in a Magnetohydrodynamic Boundary Layer Flow
2
Table I. Values of − 0 for some values of Pr for = fw = Ra = C C C
Br = 0. u +v =D − 2 C − C (4)
x y y 2
Kasmani Makinde and Present The formulation of the boundary conditions for the
Pr et al.56 Olanrewaju57 Seini55 results problem55 is given by.
0.72 0.07507 0.07507 0.07507 0.07581
0.73 0.08314 0.08314 0.08314 0.08354 u → 0 T → T C → C as y → (5)
0.74 0.09145 0.09145 0.09145 0.09114
0.75 0.09870 0.09870 0.09870 0.09862
0.76 0.10523 0.10523 0.10523 0.10599 3. SIMILARITY ANALYSIS
We introduce a stream function defined as =
x/2l
2luo e fx/2L = and a dimensionless variable, =
sheet is stretched to keep the origin rigid is shown in y u0 /2vle and noting that the velocity components
Figure 1. relate to the stream function as
Supposing the velocities along the x and y axis are
respectively represented by u and v, with T being the tem- u= and v = (6)
y x x y
perature and C, the concentration of the fluid, then the gov-
erning models of the steady Casson fluid can be obtained Equation (6) simplifies to u = uo ex/L f ,
from.55
u v uo x/2L uo x/2L
+ =0 (1)
x y v=− e f +y e f (7)
2L 2L
u u 1 2 u Equation (1) is automatically satisfied by Eq. (7). Introduc-
u +v = V 1+ + gt T − T
x y y 2 ing the following similarity variables, T = To ex/2L + T ,
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B02 and C = Cw − C + C , Eqs. (2)–(4) reduces to;
+ gC C − C + u (2)
1
1+ f + ff − 2f 2 + Gr + Gm + Mf = 0 (8)
IP: 192.168.39.151 On: Wed, 13 Oct
Copyright:
American 2021 19:58:31
T T 2T V 1 u
2 Scientific Publishers
u +v = + 1 + Delivered by Ingenta
x y y 2 cp y 4 1
1 + Ra + Br 1 + f 2
qr B02 u2 3
− + (3)
k y + Prf − f + Mf 2 = 0 (9)
Table II. Results of skin friction coefficient f (0), Nusselt − (0) and Sherwood numbers − (0) for different values of controlling parameters.
Pr M Ra Sc fw Gr Gm Br f (0) − (0) − (0)
0.72 0.1 1.0 1.0 0.22 1.0 1.0 1.0 1.0 1.0 0.900370 0.137925 0.161014
4.0 0.815769 0.684342 0.108992
7.0 0.790911 0.100065 0.100065
1.0 0.214660 0.042333 0.309058
1.5 −0.247661 −0.678961 0.449283
1.5 0.292102 −0.015520 0.062432
3.0 0.428038 −0.138718 −0.392618
2.0 0.217153 0.067377 0.321751
3.5 0.220871 0.079321 0.334688
0.24 0.899088 0.151591 0.143293
0.28 0.896934 0.171197 0.115933
2.0 0.900370 0.137925 0.161014
5.0 0.900370 0.137925 0.161015
1.5 1.061424 0.107921 0.211589
2.0 1.225211 0.052294 0.255559
3.0 1.601630 −0.270241 0.246804
5.0 2.189422 −0.718047 0.308662
2.0 1.079183 0.145846 0.168281
4.0 1.204634 0.153205 0.175099
4.0 0.963914 −0.710738 0.249704
6.0 1.084219 −1.571764 0.061649
J. Nanofluids, 10, 172–185, 2021 175
Heat Transfer in a Magnetohydrodynamic Boundary Layer Flow Aloliga et al.
Fig. 2. Velocity profile for varying values of the Prandlt parameter.
Fig. 4. Velocity profile for varying values of Schimit parameter.
+ Scf − f + Sc = 0 (10)
where the prime symbol(s) represent the number of times 4. THE NUMERICAL PROCEDURE
a function is differentiated with respect to and Gr = Equations (8)–(10) are the coupled ordinary differential
ARTICLE
Lgt To ex/2L /u20 e2x/L and Gm = Lgt Co ex/2L /e2x/L u20 equations whilst (11) is the corresponding boundary con-
represent the Grashof and the modified Grashof numbers ditions. These coupled ODEs are observed to be of higher
respectively, Pr = / represents the Prandtl number, Ra = order and therefore difficult to solved directly. To obtain
4 ∗ T
3
/K represents the thermal radiation parame- a simplified solution, we employ the order reduction tech-
IP: 192.168.39.151 On: Wed, 13 Oct 2021 19:58:31
ter, Br = U2 /Tw − T represents the Brinkmann niques byPublishers
Copyright: American Scientific letting;
number and Sc = /D is the Schmidt number. Delivered by Ingenta
Below are the convective boundary conditions; f = x1 f = x2 f = x3 = x4 = x5
When y = 0, = 0 u = 0 v = 0 C = Cw and T = Tw = x6
= x7
Thus, f 0 = M f 0 = 0 0 = 1 0 = 1 as
=0 Equations (8)–(11) are then reduced to first-order ODEs as
x = x2
f = 0 = 0 = 0 as → (11)
x2 = x3
Fig. 3. Velocity profile for varying values of reaction rate parameter. Fig. 5. Velocity profile for varying values of the Suction parameter.
176 J. Nanofluids, 10, 172–185, 2021
Aloliga et al. Heat Transfer in a Magnetohydrodynamic Boundary Layer Flow
Fig. 6. Velocity profile for varying values of the local modified Grash
of parameter.
Fig. 8. Velocity profiles for varying values of Casson parameter.
1
x3 = −x1 x3 +2x22 −Grx4 −Gmx6 −Mx2 By assuming the unspecified initial condition M in
1+1/
ARTICLE
Eq. (12) in the shooting method enables Eq. (13) to be
incorporated numerically. The supposed initial conditions
1 1 are measured by a predetermined constant of the depen-
x5 = −Br 1+ x2 −Prx1 x5 −x2 x4 −Mx22
1+4/3Ra 3IP: 192.168.39.151 dent variables with values of accuracy. A number of iter-
On: Wed, 13 Oct 2021 19:58:31
ations arePublishers
Copyright: American Scientific made with the enhanced values of the initial
7 = −Scx1 x5 −x2 x6 − Scx 6 (12) byconditions
Delivered Ingenta until such a time when no disparity is observed
The boundary conditions become; between the computed and the assumed values. With the
aid of MAPLE-16 software package, a numeric and graph-
x2 0 = M x1 0 = 0 x4 0 = 1 ical codes were developed and implemented. A step size
of h = 0.001 for a convergence criterion of 10−6 for all
x6 0 = 1 as = 0 (13)
the cases was assumed. The highest value of to each
x2 = 0 x4 = 0 x6 = 0 as → parameter was known when the values of the unidentified
Fig. 7. Velocity profile for varying values of the local Grash of
parameter. Fig. 9. Velocity profiles for varying values of Magnetic parameter.
J. Nanofluids, 10, 172–185, 2021 177
Heat Transfer in a Magnetohydrodynamic Boundary Layer Flow Aloliga et al.
Fig. 12. Temperature profile for varying values of the Prandlt
parameter.
Fig. 10. Velocity profile for different values of Brinkmann parameter.
5.1. The Skin Friction, Rate of Heat and Mass
boundary conditions remain unchanged to a final loop with Transfers
an error not more than 10−6 .
ARTICLE
Table II displays results of the effects of varying param-
eters from the transformed model on the skin friction,
5. NUMERICAL RESULTS Nusselt, and Sherwood numbers respectively. It is clearly
The validation of results and theIP:
numerical technique have
192.168.39.151 seen
On: Wed, from2021
13 Oct the 19:58:31
table that, increasing values of Casson
been achieved by comparing resultsCopyright:
of the study with Scientific Publishers (Gm ), the local Grashof (Gr ), and
American (), modified Grashof
previous published results in literature for values of heat bythe
Delivered Brinkmann (Br) parameters enhance the skin friction
Ingenta
transfer at the boundareis of the plate, − 0 for non- cooefficien whilst increasing values of the Magnetic field
Newtonian Casson fluid. The comparison with the avail- parameter (M) reduces the friction between the surface and
able published results of Refs. [55–57] are made and the fluid meduim. The plastic viscosity of the fluid reduces
obtainable in Table I. Evidently, the results conforms with when the Casson parameter increases, thereby reducing the
the earlier findings assuring the robustness of the numeri- speed of the fluid and consequently increasing the skin
cal scheme employed.
Fig. 13. Temperature profile for varying values of the Suction
Fig. 11. Velocity profile for varying values of Radiation parameter. parameter.
178 J. Nanofluids, 10, 172–185, 2021
Aloliga et al. Heat Transfer in a Magnetohydrodynamic Boundary Layer Flow
mass transfer is decreased with increasing values of reac-
tion rate parameter (), the local modified Grashof (Gm )
parameter, the local modified Grashof (Gm ) parameter, and
the magnetic parameter.
To consider the estimated solutions, calculations have
been done using the technique explained in the prior
section.
6. GRAPHICAL RESULTS
The graphical illustrations of the results are presented
in Figures 2–28, the parameters (Ra = 0.1, Br = 0.1, =
0.1, Sc = 0.22, fw = 1, Gr = 1, Gm = 1, M = 1, Pr = 0.72
Fig. 14. Temperature profile for varying values of the local modified
Gras of parameter.
ARTICLE
friction on the plate surface. The Prandtl (Pr), the local
Grashof (Gr ), the local modified Grashof (Gm ) and the
suction (fw) parameters tend to increase the heat transfer
across the fluid. Increasing the values of (Pr) means an
IP: 192.168.39.151
increase in viscosity overheats diffusion On: Wed, 13 Oct 2021 19:58:31
hence the increase
Copyright: American Scientific Publishers
in heat transfer observed. Increasing values of the radia- by Ingenta
Delivered
tion (Ra), magnetic (M), and the Brinkmann (Br) param-
eters also reduce the rate of heat transfer due to radiation
and viscous dissipation. Moreover, increase in reaction rate
parameter ( ), Schmidt number (Sc) and wall suction (fw)
cause a decrease in mass transfer of the fluid. The rate of
Fig. 16. Temperature profiles for varying values of magnetic parameter.
Fig. 15. Temperature profile for varying values of the local Gras of
parameter. Fig. 17. Temperature profiles for varying values of Casson parameter.
J. Nanofluids, 10, 172–185, 2021 179
Heat Transfer in a Magnetohydrodynamic Boundary Layer Flow Aloliga et al.
Fig. 18. Temperature profiles for different values of Brinkmann Fig. 20. Temperature profiles for varying values of Schimit parameter.
parameter.
to satisfy the far-field boundary condition. It can observed
and = 1) are kept unchanged throughout the analy-
ARTICLE
that the joint effect of fw and decrease the velocity. This
sis. Figures 2–11 depicts the velocity profiles for vary- is because increasing fw represent a high degree of suc-
ing parameters of Prandtl (Pr), the reaction rate ( ), the tion which causes resistance to the flows and increasing
Schmidt (Sc), the Suction (fw), the modified Grashof means increasing the plastic viscosity of the Casson fluid
IP: 192.168.39.151 On: Wed, 13 Oct 2021 19:58:31
(Gm), the Grashof (Gr), Casson (),Copyright:
the magnetic field and hence
American Scientific a reduction in the fluid velocity. Figures 6 and 7
Publishers
(M), the Brinkmann (Br), and the radiation (Ra), param- byrealized
Delivered Ingentatwo components restricted to the laminar bound-
eters respectively. Figures 4–8 illustrate the influence of ary layer. Increasing both Gr and Gm parameters lead to
the suction (fw), the modified Grashof parameter (Gm), a corresponding increase in velocity close to the plate and
the Grashof parameter (Gr), the Casson parameter (), decreasing them lead to a corresponding decrease of veloc-
and the magnetic parameter on the velocity profiles f . ity away from the plate. Also, a special type of force has
The velocity of the fluid is highly negligible at the sur- been produced due to the application of the magnetic field
face of the plate and increases to the free stream value on the Casson fluid called the Lorentz force. This force
Fig. 19. Temperature profile for varying values of radiation parameter. Fig. 21. Temperature profiles for varying reaction rate parameter.
180 J. Nanofluids, 10, 172–185, 2021
Aloliga et al. Heat Transfer in a Magnetohydrodynamic Boundary Layer Flow
causes a decline in velocity of the fluid flow within the
boundary layer.
7. TEMPERATURE PROFILES
Figures 12, 13, 16–19 are graphs showing the effects
of Prandtl (Pr), suction (fw), Magnetic field (M), Cas-
son (), Brinkman (Br), and radiation (Ra) parameters on
the temperature profile (). Figures 12 and 13 represent
the effect of Pr and fw on the temperature profiles. It
is observed from the graphs that increasing values of Pr
and fw cause a significant reduction in the thermal bound-
ary layer thickness. Increasing the values of Pr could be
the possible cause of growth in the diffusion of momen-
tum at the expense of thermal energy diffusion. Again,
Fig. 24. Concentration profile for varying values of the local modified
Gras of parameter.
ARTICLE
the increase in magnetic parameter (M) has seen a mon-
umental growth in the distribution of temperature across
the plate (see Fig. 16). The Lorentz force produces a class
of resistance in the fluid on the velocity profiles. The fric-
IP: 192.168.39.151 On: Wed, 13 Oct 2021 19:58:31
tion produced
Copyright: American Scientific from the force, consequently produces heat
Publishers
Delivered bythat could increase the temperature distribution in the fluid.
Ingenta
There are corresponding rising numbers of , Ra and Br
(see Figs. 17–19) respectively with the thickness of ther-
mal boundary layer. This could be owing to a correspond-
ing surge in thermal radiation and viscous dissipation.
Fig. 22. Concentration profile for varying values of the Prandlt
parameter.
Fig. 23. Concentration profile for varying values of the Suction Fig. 25. Concentration profile for varying values of the local Gras of
parameter. parameter.
J. Nanofluids, 10, 172–185, 2021 181Heat Transfer in a Magnetohydrodynamic Boundary Layer Flow Aloliga et al.
Fig. 26. Concentration profiles for varying values of magnetic Fig. 28. Concentration profiles for different values of Brinkmann
parameter. parameter.
leading to increased chemical activity, Figure 29. Figure 30
8. CONCENTRATION PROFILES dipicts the effects of the Sc number on the concentration
ARTICLE
Figures 23, 27, 30 and 31 represent the effects of suction boundary layer. Practically, high values of the Schmidt
(fw), Casson (), Schmidt (Sc) and the chemical reaction number leads to increasing momentum diffusion more than
mass diffusion and consequently reduces the concentration
( ) parameters on concentration () profiles respectively.
IP: 192.168.39.151 On: profile.
Wed, 13 OctWhen is zero at a poin in the flow, chemical
2021 19:58:31
In Figure 27, it is observed that increasing the Casson
Copyright: American Scientific Publishers
reaction cannot takes place. Again, increasing values of
parameter () thickens the concentration at bounding sur- by Ingenta
Delivered indicates a significant enhancement in chemical reaction
face. It is instructive to note in see Figures 27 and 28 that
which causes a reduction in concentration. The reaction
the Casson parameter () reacts oppositely on the velocity
rate parameter ( ) varies directly with the concentration
and concentration profiles respectively.
boundary layer thickness as can be observed (see Fig. 31).
The concentration profile increases with increasing radi-
Figure 23 portrays how the increasing values of fw
ation parameter (Ra) near the magnitized surface due to the
affect the decaying process of the concentration boundary
presence of the Lorenz force which acts to dekay the flow
layer thickness.
Fig. 29. Concentration profile for varying values of Radiation
Fig. 27. Concentration profiles for varying values of Casson parameter. parameter.
182 J. Nanofluids, 10, 172–185, 2021Aloliga et al. Heat Transfer in a Magnetohydrodynamic Boundary Layer Flow
(ii) Prandtl number and suction parameter both con-
tributes to reduction in the rate of heat transfer.
(iii) A strong magnetization of the surface delays the flow
across the surface leading to thickening of the thermal
boundary layer thickness.
(iv) The skin friction coefficient is reduced by increasing
the reaction rate parameter ().
NOMENCLATURE
u v w Velocity components along x, y and z axes (m/s)
B0 Applied magnetic field (Wb m−2 )
Cp Specific heat (J kg −1 k−1 )
T Time (s)
Tw Wall temperature (K)
T Temperature of the Casson fluid (K)
U0 Characteristic velocity (m s−1 )
C Concentration (kg m−3 )
Fig. 30. Concentration profiles for varying values of Schimit parameter.
C Concentration in the free stream (kgm−3 )
G Acceleration due to gravity (m s−1 )
K Permeability of porous medium (m2 )
T Temperature of the Casson fluid (K)
ARTICLE
Ra Radiation parameter
f () Similarity function
f () Dimensionless velocity
IP: 192.168.39.151 On: Wed, 13 ()OctDimensionless
2021 19:58:31 temperature
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Dimensionless concentration
Delivered by Ingenta
qr Radiation flux distribution in fluid, W/m2
Gr Local thermal Grashof number
Nu Nusselt number
Sh Sherwood number
Gm Local modified Grashof number
K Thermal conductivity of the fluid (W m−1 k−1 )
Kp Thermal conductivity of plate
Pr Prandtl number
Sc Schmidt number
q Volumetric heat generation
l Characteristic length (m)
Fig. 31. Concentration profiles for varying reaction rate parameter. M The magnetic parameter
Br The Brinkmann parameter
9. CONCLUSIONS Greek Symbols
A steady boundary layer flow of a magnetohydrodynamic Casson parameter
Casson fluid on exponentially stretching magnetized plate Fluid density
has been studied. The partial differential governing the Similarity variable
flow was modelled and transformed to ordinary differen-
Internal heat generation parameter
tial equations. A reduction of order was made and the
Electrical conductivity of the base fluid (m2 s−1 )
resulting first–order odes were solved numerically using
Thermal diffusivity
the fourth-order Runge-Kuta algorithm in a Maple 19 soft-
Kinematic viscosity (m2 /s)
ware package.58 For low magnetic Reynolds number, and
Stream function, (m2 /s)
in the absence of electric field, the following conclusions
Fluid viscosity (kg m−1 s−1 )
can be made:
(i) The magnetization of the plate led to a significant t The thermal coefficients
reduction of the flow speed inside the boundary layer. c Concentration expansion coefficients.
J. Nanofluids, 10, 172–185, 2021 183Heat Transfer in a Magnetohydrodynamic Boundary Layer Flow Aloliga et al.
Funding 26. P. S. Reddy, P. Sreedevi, and A. J. Chamkha, Powder Technology
They did not receive any financial assistance from any- 307, 46 (2016).
where for publication of this paper. 27. M. Younes, A. J. Chamkha, and M. Nicola, International Jour-
nal of Numerical Methods for Heat and Fluid Flow 30 (2019),
DOI: 10.1108/HFF-10–0739.
Recommendations 28. K. G. Kumar, M. G. Reddy, M. V. V. N. L. Sudharani, S. A. Shehzad,
In the design of plate-fin heat exchangers, engineered mag- and A. J. Chamkha, Physica A, Statistical Mechanics and Its Appli-
netized surfaces should be incorporated to improve the cations, Elsevier 541 (2019), DOI: 10.1016/j.physa.2019.123330.
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