Heat Transfer on a Chemically Reacting Non-Newtonian Casson Fluid Over a Vertically Stretched Magnetized Surface
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Copyright © 2022 by American Scientific Publishers Journal of Nanofluids
All rights reserved. Vol. 11, pp. 629–645, 2022
Printed in the United States of America (www.aspbs.com/jon)
Heat Transfer on a Chemically Reacting
Non-Newtonian Casson Fluid Over a Vertically
Stretched Magnetized Surface
Golbert Aloliga1, ∗ , Yakubu Ibrahim Seini2 , and Rabiu Musah2
1
Faculty of Mathematical Sciences, CK. Tedam University of Technology and Applied Sciences, Navrongo, 00233, Upper
East Region, Ghana
2
School of Engineering, University for Development Studies, Tamale, Nyankpala Campus, 00233, Northern Region, Ghana
An extensive investigation into heat transfer through Casson fluid on a stretched magnetized surface with pres-
ence of chemical reactants has been conducted. The magnetic strength influence at the plate surface and
within the body of the fluid has been analysed as well as effects of radiation and convection fields are consid-
ered. The methods of similarity analysis have been used to transform the multivariable dependent equations
modelling the flow to a single variable dependent equation. The emerged dimensionless parameters describ-
ing the flow have been presented numerically. The effects of magnetization of the surface along with the bulk
fluid are presented in tables and graphs. It is evident that magnetizing the surface enhances the temperature
ARTICLE
distribution near the surface. Similar results can be seen with the coefficient of wall resistance, and the mass
and transfer rate on the magnitised plate. From the study, it is recommended that surface magnetization can
influence flow kinematics involving Casson fluids for efficient control.
IP: 5.10.31.211 On: Fri, 30 Dec 2022 01:13:17
Copyright:
KEYWORDS: Casson Fluids, Non-Newtonian American
Fluids, Scientific
Magnitised Surface,Publishers
Thermal Diffusion, Control Parameters.
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1. INTRODUCTION a single constitutive relation to represent its rheological
The advances in science and technology have led properties. Several researchers explored this model under
researchers to explore various ways by which heat can varied rheological conditions. Many engineering processes
be transferred during the colling of industrial products to take place with high temperature and radiative conditions
achieve desired characteristics. Heat transfer innvolving and are important in the design of pertinent equipment.4
non-Newtonian fluids have numerous industrial applica- Under steady conditions, flow on sliding surfaces with
tions and play essential roles in modern industry prac- transverse magnetic field significantly affects the heat
tice. Industrial fluids serve as media through which heat transfer rates5 with thermal radiation contributing to low-
transport occur. For some technological applications such ering the viscosity of fluids.6 7 In exploring for sub-ground
as cooling processes of micro-ships or heating, solar water reservoirs in agriculture,8 convective flow of non-
energy recovery, open-flow switching, simulating reser- Newtonian fluids often important. Different approaches are
voirs and nuclear reactors, among others. Non-Newtonian used in the analysis of non-Newtonian fluids. The similar-
fluids such as honey, blood, grease and oil are classified ity analysis approach has been widely used in investigating
as Jeffrey fluids, Maxwell fluids, 2nd and 3rd grade fluids, the flow of Casson fluids over vertical porous surfaces.9 A
Casson fluids and viscoelastic fluids.1 2 time varying heat transfer of Caason fluid on moving sur-
Casson fluids belong to a class of special type of non- faces in parallel free streams have been analysed with the
Newtonian fluids defined by Makinde and Mhoneb3 as a use of the homotopy analysis method.10 Industrial applica-
liquid with shear-thinning at zero rate of shear with yield tions involving wire drawing and fiber coating uses thin-
stress. Seddeek4 noted that the thinning and the thickening film stretching which require regulated cooling to ensure
of a is largely determined by its viscosity. The nonlinear- the quality of the penultimate product.11–13
ity and the complex nature of Casson fluids do not allow A liquid hydromagnetic Casson fluid with transient con-
ductivity and radiation has been reported by El-Aziz and
∗
Afify.14 Other studies15–17 explored the transport behav-
Author to whom correspondence should be addressed.
Email: aloligagolbert@gmail.com ior of different fluids with non-Newtonian characteris-
Received: 30 September 2021 tics in process industries which underscore the existence
Accepted: 21 October 2021 and uniqueness of the solution under varying parameters.
J. Nanofluids 2022, Vol. 11, No. 5 2169-432X/2022/11/629/017 doi:10.1166/jon.2022.1877 629
Heat Transfer on a Chemically Reacting Non-Newtonian Casson Fluid Over a Vertically Stretched Magnetized Surface Aloliga et al.
field and convective heating. They concluded that the Cas-
son fluid reduces to a Newtotian fluid with large enough
values of the Casson parameter and there obeys the New-
tons law of viscosity.
In a related study, Akolade et al.45 discussed the two dif-
fusive effects in different geometries. In their pinoneering
works, Aloliga et al.46 analysed the effects of magnetized
surfaces on flow kinematics and obtained interesting
results on the flow of Casson fluids on magnetized plane
surfaces. Dogonchi et al.47 outlined the impact of nanopar-
ticles in cavities of an inclined magnetic plates and con-
Fig. 1. Diagrammatic representation of the flow. cluded that, the magnetic field strength diretly controls the
heat generation. Ismael et al.48 analised the entropy trans-
fer with saturated nanoparticles in permeable medium.
Jawad et al.18 outlined the relevance of Casson fluids in Seini et al.49 analysed the flow dynamics Casson fluids
pharmaceutical, chemical and cosmetic industries. on exponentially stretching porous surfaces and concluded
The applications of Casson fluids with hybrid nano- that the quality of engineering products depended largely
size particles in science and engineering has been widely on the rate of cooling. Mohammad et al.50 studied nanoflu-
reported, Refs. [19–21]. Rawi et al.22 observed that the ids surrounded by squared hollow spaces with convection
temperature profiles in Casson fluid were enhanced due whilst Salva and Chamkha51 presented results of laminar
to increases in fluid volume. Some researchers23–25 have flow with convection and entropy generation of nanoflu-
reported both empirical and theoretical results for Casson ids and concluded that nanoparticles slow down fluids in
ARTICLE
fluids with nano-size particles as it has great relevance motion.
in industry. In conjunction with these exemplified appli- Chamkha and Abdul-Rahim52 investigated the linear
cations, literature is still abound with the analysis of the stratified stagnation point flows with an applied external
dynamics of fluid organization in science and engineering magnetic field with heat generation and absorption. The
fields.26–30 Ram-Reddy and NaveenIP: 31 5.10.31.211 On: Fri, 30 Dec 2022 01:13:17
analysed the convec- simulation of fluid dynamic characteristics and heat dis-
Copyright: American Scientific Publishers
tive non-Newtonian fluids with magnetic effects. Delivered bytribution
Ingenta of various shapes has been reported by Menni
In thermal science and welding processes, heat injec- et al.53 with Ramesh et al.54 discussing the heat trans-
tors and sinks32–34 are included in the process architec- fer of aluminum alloy and magnetite graphene oxide of
ture to address the thermo-diffusion and diffusion-thermo porous cylindrical sheet with heat supply or sink. Sahin
phenomena in a semi-infinite permeable channel whose et al.55 presented a 3-D model of chemically reactive fluids
walls were contracting or expanding, with heat source/sink along two parallel surfaces in a channel with transverse
effects. The transient flow of Casson fluids in chemically magnetic effects. Sahin et al.56 further obtained analyti-
reactive media on surfaces with magnetic effects under cal solutions for a convective flow of MHD in a chem-
convective boundary conditions has been reported.35 ically reactive fluid of the mixed kind over a vertically
Some researchers36–38 theoretically and numerically infinite porous surface. Joaquín et al.57 applied an explicit
explored heat and mass transport phenomena in various finite difference method to explain the steady-state numer-
media. Hybrid nanofluids greatly impacts on the heat abil- ical solutions for the velocity field. Krishna et al.58–59
39
ity of the fluid to transport heat from the surface, Ali. extensively discussed the heat transfer effects with radi-
The combination of different nano-size particles signifi- ation taking into account the hall current in fluids of
cantly enhances the heat transfer coefficient. The process non-Newtonian behavior through porous media. Similar
cycle is better controlled when the amount of heat is varied studies by Chamkha et al.60 determined the radiation and
40 41
in the fluid as the sheet is being stretched. Abbas et al. shape factor effects of nano-size particles in natural con-
extended the model presented by Yamada–Ota and Xue vective flow. The effects of micropolar fluids on infinitely
to include micropolar fluid towards the stagnation point. vertical porous plates with inclined magnetic field has
Muhammad et al.42 used similarity analysis approach to been discussed.61 Dogonchi et al.62 examined the impact
examine the impact of heat transfer during a chemical of various parameters on the flow current with mecha-
reaction process. They obtained solutions of dual nature nisms of heat transfer in nanofluids through parallel disks
due to the rheological impact of hydromagnetic Casson in the process of during suction or blowing with mag-
fluid. The Keller-box method as adopted43 to study the netism. Dogonchia et al.63 investigated the role of Brown-
thin-film characteristics of flow. The diminishing effects ian motion in a thermally conductive nanofluid in porous
for temperature fields were detected except for the volume media whilst Dogonchi et al.64 analysed the entropy gen-
44
fraction nanoparticles. Timothy et al. investigated vari- eration in Fe3 O4 -water nanoliquid in a porous enclosure
ous parameter effects on Casson nanofluids with magnetic surrounded by two squared cylinders. Govindarajan et al.65
630 J. Nanofluids, 11, 629–645, 2022
Aloliga et al. Heat Transfer on a Chemically Reacting Non-Newtonian Casson Fluid Over a Vertically Stretched Magnetized Surface
Table I. Values of f 0 and − 0 compared to previous published results for varying Pr.
Present work Present work
Seini et al. 49
Sharmila and Kaleeswari 8
Ms = 0, Mb > 0 Ms , Mb > 0
Pr f 0 − 0 f 0 − 0 f 0 − 0 f 0 − 0
0.72 0.6238107 0.8077915 0.6238107 0.8077915 0.6238107 0.8077915 0.59233 0.756791
1 0.6554146 0.9547582 0.6554146 0.9547582 0.6554146 0.9547582 0.63528 0.854768
4 0.7448565 1.3464869 0.7448565 1.3464869 0.7448565 1.3464869 0.68494 0.946412
5 0.7610054 1.3515818 0.7610054 1.3515818 0.7610054 1.3515818 0.70101 0.951582
Notes: See Salva and Chamkha51 for Ms = Mb condition.
Table II. Coefficient of skin friction [f (0)], Nusselt [− (0)] and Sherwood numbers [− 0] for various values of controlling parameters when
Ms = 0.
Pr Ec Ra n Sc fw Gr Gm Mb Q f 0 − 0 − 0
0.72 1 1 0.1 1 0.7 0.8 0.1 1 1 1
0 645385 0.1 −1 214199 0 115461
0.74 0 646128 −1 244304 0 114956
0.76 0 646883 −1 274313 0 114478
2.0 0 993078 −1 968827 −1 861661
4.0 2 971807 −9 279889 −19 00840
2.0 0 782053 −2 177870 0 178309
4.0 1 331489 −7 878106 0 334443
0.3 0 658108 −1 684910 0 109848
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0.6 0 723989 −4 374277 0 103435
3.0 0 102674 −0 012085 0 513018
4.0 −0 04405 −0 167469 0 653373
0.8 0 678597 −1 198673 0 114348
IP:
0.95.10.31.211 On: Fri, 30 Dec 2022 01:13:17 0 707781 −1 186332 0 113178
Copyright:
0.9 American Scientific Publishers 0 631239 −1 184346 0 125944
1.0 Delivered by Ingenta 0 619318 1 159905 −0 134893
0.2 0 645385 −1 214199 0 115461
0.3 0 645385 −1 214199 0 115461
2.0 1 385187 −2 807578 0 314724
3.0 2 736934 −7 481800 0 516677
2.0 0 927423 −1 592533 0 179520
3.0 1 188084 0 228102 −1 992608
2.0 1 783630 −6 665921 0 403644
3.0 2 551194 −11 86510 0 512212
0.6 0 814691 −2 509954 0 182272
0.8 0 914405 −3 309556 0 215139
theoretically investigated the consequence of the move- analised the flow of Oldroyd-B-type liquid in a Cattaneo–
ment of mass with the chemical transmission on MHD Christov model with heat flux and variable thickness.
natural convective glide of dissipative and radiative liquid Hazarika et al.71 reported that, heat generation and Soret
through the endless perpendicular plate. Krishna et al.66 number for Cu, Ag and Fe3 O4 played a significant role
observed that the magnetic field and the angle of inclina- in thermophoresis and viscous dissipation of nanoflu-
tion retards the circulation velocity. Seyyed et al.67 exam- ids. Dogonchi et al.72 investigated the effects of thermo-
ined the characteristics of entropy generation for natural physical factors and shape of the cavity in a study on
convection flow of nanofluids in a wavy-hexagonal perme- entropy generation of nano liquids in a crown cavities.
able enclosure. Silpi and Sahin68 illustrated the influence Takhar et al.73 examined a rotating fluid on a moving
of micropolar fluids on solid sphere surrounded by porous surface within a magnetic field whilst Mohamad et al.74
media. studied the natural convection flow of CuO-water based
Krishna et al.69 employed the perturbation methods to nanofluid in a wavy cavity of rectangukar shape in porous
investigate the impact of various parameters on a tran- media whilst Modather et al.75 analytically examined the
sient double-diffusive natural convective rotating micro- micropolar fluid flowing over a vertical porous surface.
polar fluid over a vertically directed semi-infinite plate Takhar et al.76 discussed a 3-D boundary layer problem
acted on by a uniformly distributed normal magnetic of an impulsively moving stretching surface. Takhar et al.77
field with convective boundary conditions. Venkata et al.70 then examined the flow along a vertically moving cylinder.
J. Nanofluids, 11, 629–645, 2022 631
Heat Transfer on a Chemically Reacting Non-Newtonian Casson Fluid Over a Vertically Stretched Magnetized Surface Aloliga et al.
Table III. Skin friction coefficient [f (0)], Nusselt [− (0)] and Sherwood numbers [− 0] for various values of control parameters.
Pr Ec Ra N Sc Fw Gr Gm Mb Ms Q f 0 − 0 − 0
0.72 1 1 0.1 1 0.7 0.8 0.1 1 1 1 0.1 0.1 0 751515 −1 23269 0 078503
0.74 0 752317 −1 26334 0 077978
0.76 0 753132 −1 29390 0 077481
1.5 0 869484 −1 48167 0 584334
2.0 1 129744 −2 12077 1 999420
2.0 0 8936270 −2 33160 0 146137
3.5 1 272399 −6 20475 0 266423
0.3 0 765090 −1 71321 0 072679
0.6 0 834688 −4 49475 0 066273
3.0 0 214998 0 058388 0 461586
4.0 0 074432 0 254214 0 597063
0.8 0 788863 −1 21881 0 078590
0.9 0 821606 −1 20781 0 078412
0.9 0 737120 −1 19935 0 084897
1.0 0 724996 −1 17202 0 089867
0.2 0 751515 −1 23269 0 078503
0.3 0 751515 −1 23269 0 078503
2.0 1 515884 −2 99184 0 291902
2.5 2 1013810 −4 85187 0 395607
2.0 1 041894 −1 65474 0 148234
3.0 1 3088610 −2 09472 0 200205
1.5 1 250192 −3 30624 0 247477
2 1 897331 −6 83761 0 381914
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0.4 1 063408 −1 47478 −0 03530
0.5 1 165721 −1 62096 −0 07368
0.6 0 929755 −2 59613 0 152122
0.8 1 035268 −3 44444 0 188019
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78 Copyright: American Scientific Publishers
on con- bymoving
Chamkha et al. investigated the radiation effectsDelivered Ingentasurface in nanofluids exposed to a uniformly act-
vective flow over vertical surface. Najiyah et al.79 rob- ing transverse magnetic field. Krishna et al.81 highlighted
tained numerical solutions for Cu–Al2 O3 -water nanofluids the impact of radiation on hybrid Casson nanofluid over an
inside two parrale plates and Krishna et al.80 computa- accelerated vertically moving porous surface with surface
tionally explored the flow dynamics across a vertically
Fig. 2. Velocity profile for varying values of Mb . Fig. 3. Velocity profile for varying values of Ms .
632 J. Nanofluids, 11, 629–645, 2022
Aloliga et al. Heat Transfer on a Chemically Reacting Non-Newtonian Casson Fluid Over a Vertically Stretched Magnetized Surface
Fig. 4. Velocity profile for varying values of .
Fig. 6. Velocity profile for varying values of Pr.
ARTICLE
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Fig. 7. Velocity profile for varying values of n.
Fig. 5. Velocity profile for varying values of Ra.
slip. Tayebi et al.82 analysed the dynamics of Al2 O3 –H2 O species and radiation is available. The bulk and surface
nanofluid surrounded by two cylindrical pipes with mag- magnetizations in radiation has significant impact on the
netic effects. Abderrahim et al.83 gave further views to the growth and development of the boundary layer. To address
thermodynamic irreversibilities in dissipative flow over a this problem, a steady incompressible flow of Casson flu-
porous horizontal surface along with joule heating. ids with surface magnetization is explored. A similarity
From the above detailed literature presented, very lim- transformation approach and a shooting techniques are
ited information about Casson fluids flowing over mag- implemented to obtained numerial and graphical solutions
netized stretching surfaces exposed to chemically reactive to the problem.
J. Nanofluids, 11, 629–645, 2022 633
Heat Transfer on a Chemically Reacting Non-Newtonian Casson Fluid Over a Vertically Stretched Magnetized Surface Aloliga et al.
Fig. 8. Velocity profile for varying values of Gr.
Fig. 10. Velocity profile for varying values of Gm.
ARTICLE
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Fig. 9. Velocity profile for varying values of .
2. MATHEMATICAL MODEL
A 2-D steady flow of electrically conductive Casson fluid
on a magnetized surface with a variable magnetic field
B x = B0 x axn applied normaly to the stretching sur- Fig. 11. Velocity profile for varying values of Sc.
face is considered. (see Fig. 1). Assuming the origin is
fixed and subjected to opposing equal forces acting verti-
cally so as to cause stretching of the surface.
Further assume the surface to be stretched with veloc- Taking the components of velocity in the x- and y-axes
ity uw x = axn , with a and n represent some constants. to respectively be u and v, with T representing fluid tem-
Neglecting the magnetic field induced as compared to the perature whilst and C represent the species concentration.
magnetic field applied and also assume a negligible vis- The governing models of the flow can be obtained similar
cous dissipation. to that of Ismael et al.48 under Boussinesq approximations
634 J. Nanofluids, 11, 629–645, 2022
Aloliga et al. Heat Transfer on a Chemically Reacting Non-Newtonian Casson Fluid Over a Vertically Stretched Magnetized Surface
and boundary-layer assumptions become as (1)–(4). local Grashof and the modified Grashof numbers respec-
tively represented by Gr = Lgt To ex/2L /u20 e2x/L and
u v Gm = Lgt Co ex/2L /e2x/L u20 , = /axn−1 is the
+ =0 (1)
x y reaction rate parameter, Pr = / represents the Prandtl
number, Ra = 4 ∗ T 3
/K represents a parameter for
2 thermal radiation, Ec = a2 x n /cp is the Eckert num-
u u 1 u
u +v = 1+ + gt T − T ber, Q = T / caxn−1 is heat source dimensionless
x y y2
parameter, Mb = B 2 axn+1 /T0 is the magnetization at
B02 the plate and Sc = /D is the Schmidt number.
+ gC C − C + u (2)
The convective boundary conditions are;
When y = 0, = 0 v = 0 C = Cw and T = Tw .
2 Thus,
T T 2
T 1 u qr
u +v = + 1+ −
x y y 2 cp y k y f 0 = 1 − Ms f 0 = 0 0 = 1
T B02 u2 0 = 1 as = 0
+ T − T + (3)
c
f → 0 → 0 → 0 as →
2
C C C (11)
u +v =D − C − C (4)
x y y2
where Ms = B̄02 /a /L is the magnetic field at the
The conditions existing at the boundary are taken as;
surface.
u = uw = U0 B axn V = 0 T = TW
ARTICLE
C = CW as y = 0 4. NUMERICAL PROCEDURE
Equations (8)–(10) are the coupled ordinary differential
u → 0 T → T C → C as y → (5)
equations whilst Eq. (11) is the corresponding boundary
conditions.
IP: 5.10.31.211 On: Fri, 30 Dec 2022These coupled ODEs are observed to be of
01:13:17
3. SIMILARITY ANALYSIS Copyright: American Scientific
third Publishers
higher order and therefore difficult to solved directly.
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obtain a simplified solution, we employ the order of
Introducing
√ the stream function defined as =
reduction techniques by letting;
avx f and a dimensionless variable, =
n+1
n−1
y ax /v and noting that the velocity components f = x1 f = x2 f = x3 f = x4 = x5
relate to the stream function as usual in the form;
= x6 = x7 = x8 (12)
u= and v = (6)
y x x y Equations (8)–(10) are then reduced to first-order ODEs as
Equation (6) simplifies to; x1 = x2 (13)
n+1 √ an−1 n−1 x2 = x3 (14)
u = axn f v=− avxn−1 f + yx f (7)
2 2 1 n+1
x3 = x22 − xx3 −Grx5 −Gmx7 −Mb x2
Equation (1) is satisfied identically by Eq. (7). 1+1/ 2
Introducing the similarity variables, T = To + T , and (15)
C = Cw − C + C , Eqs. (2)–(4) transforms into;
1 n+1
x5 = − x1 x6
1 n + 1 1/Pr 1 + 4/3Ra 2
1+ f −f 2 + ff + Gr + Gm + Mb f = 0
2 1
(8) −Ec 1 + x32 − Qx5 − Mb x22 (16)
1 4 n+1 1
n+1
1 − Ra + f + Ec 1 + f 2 x7 = −Sc x1 x8 + x7 (17)
Pr 3 2 2
+Q + Mb f 2 = 0 (9) The boundary conditions in (11) become;
1 n + 1 x2 0 = 1−Ms x1 0 = 0 x5 0 = 1
+ f + = 0 (10)
Sc 2
x7 0 = 1 as = 0
where the number of times a function is differentiated
with respect to is represented by prime symbol(s). The x2 = 0 x5 = 0 x7 = 0 as →
J. Nanofluids, 11, 629–645, 2022 635
Heat Transfer on a Chemically Reacting Non-Newtonian Casson Fluid Over a Vertically Stretched Magnetized Surface Aloliga et al.
Fig. 14. Velocity profile for varying values of Q.
Fig. 12. Velocity profile for varying values of fw.
ARTICLE
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Fig. 15. Temperature profile for varying values of Ms
5. NUMERICAL RESULTS
The numerical scheme employed in this research was val-
idated by compraring numerical results with similar inves-
Fig. 13. Velocity profile for varying values of Ec. tigation in the literature for the skin-friction coefficient
(f 0) and the local Nusselt number − 0 to previ-
ously published works of Seini et al.49 and Mohammad
With the aid of MAPLE–19 software package, numeri- et al.50 (See Table I). This shows a perfect agreement to 7
cal and graphical codes were developed and implemented. places of decimals with available data literature.
A step size of h = 0.001 for convergence criterion of
10−6 was assumed for all cases. The highest value of to 5.1. Skin Friction, Rate of Heat and Mass Transfer
each parameter was known when the values of the uniden- for Zero Surface Magnetization (MS = 0)
tified boundary condition remain unchanged to a final loop Table II depicts numerical results for sellected control
with an error not more than 10−6 . parameter values on the wall friction, the Nusselt number
636 J. Nanofluids, 11, 629–645, 2022
Aloliga et al. Heat Transfer on a Chemically Reacting Non-Newtonian Casson Fluid Over a Vertically Stretched Magnetized Surface
Fig. 16. Temperature profile for varying values of .
ARTICLE
Fig. 17. Temperature profile for varying values of n.
and the Sherwood number with zero surface magnetiza-
tion. Increasing the values of Mb , Gm, Q, and Ec lead
to increase resistance to the flow resulting in reduced flow
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speed. Conversely, the skin friction Copyright:
coefficient American
declines Scientific Publishers
when values of Pr, Ra, n, Sc, , and fw increase Delivered
resulting by Ingenta
to increased momentum of the flow. This is so because
of the cumulative impact of high viscosity over magnetic
force, and suction at the material surface increases the skin
frictional effects thereby impeding flow along the surface.
Furthermore, the impact of thermal diffusion over mass
diffusion, and the Casson parameter of the fluid dimin-
ishes the surface wall friction thereby hastening the flow.
Similarly, increasing the values of Pr, , Gm, Ec, Q, Mb ,
and Ra boost the rate of heat transmission but reduced
with increasing , n, Gm, Sc, and fw. From the results
presented, the Sherwood number is noted to increase with
high values of Pr, , Mb , Sc, and Ra but reduced with high
values of Sc, Ec, Q, Gm, , and Gm.
5.2. Coefficient of Skin Friction and Nusselt and
Sherwood Numbers with Surface Magnetization
(Ms > 0) and the Bulk Magnetization (Mb > 0)
Table III presents results of various parameter on sur-
face and bulk magnetization of the fluid on the wall shear
Fig. 18. Temperature profile for varying values of Pr.
stress, the local Nusselt number and the local Sherwood
number. The results show an increasing skin friction coef-
ficient with increasing values of Mb , Ms , , Gm, Gr, Ec, Q, resistance to the flow. Similarly, the rate of heat transfer
, Sc and fw; and decreasing trends when values of Pr, Ra rises with Pr, , Ra, Ec, Q, Gm, Gr, and Mb but dimin-
and n are increased. The results confirm the view that the ishes with values of , n, Sc, and Gm. The results further
cumulative impact of high viscosity over magnetic force revealed that the rate of mass transfer surges with Pr, ,
(the Lorenz force), as well as the surface suction have a Ms , and Ra are increased; and decreases when values of
tendency to increase the local skin friction and hence a Ec, Q, n, Sc, Gr, Gm, , and are increased.
J. Nanofluids, 11, 629–645, 2022 637
Heat Transfer on a Chemically Reacting Non-Newtonian Casson Fluid Over a Vertically Stretched Magnetized Surface Aloliga et al.
Fig. 19. Temperature profile for varying values of Ra.
Fig. 21. Temperature profile for varying values of Gm.
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Fig. 20. Temperature profile for varying values of Gr.
6. GRAPHICAL RESULTS
6.1. Velocity Profiles Fig. 22. Temperature profile for varying values of .
The graphical illustrations represented in Figures 2–14
depicts the velocity profiles for various controlling param- Increase in Pr means an increase in kinematic viscosity
eters. Figure 2 represent the effects of the magnet at the which leads to decreased velocity field. It is evident that
surface while Figure 3 represents the magnetic effect of high values of Pr lead to reduce velocity profiles (Fig. 6).
the bulk fluid. The intensity of the magnetic field tends to The reaction rate parameter () is the only variable
diminish the velocity field as it acts to oppose the flow coupling the governing equations. Increasing imparts
due to the induced Lorenz force. Increasing Ms leads to on the Grashof number which leads to accelerating the
increased values of the wall surface resistance. (Table II) fluid thereby increasing the velocity boundary layer thick-
which supports this observation. Thus, an increased in wall ness (Fig. 9). Increased the buoyancy parameter enhances
shear stress results in reduced velocity within the fluid. convective flow which leads to reduced frictional effects
638 J. Nanofluids, 11, 629–645, 2022Aloliga et al. Heat Transfer on a Chemically Reacting Non-Newtonian Casson Fluid Over a Vertically Stretched Magnetized Surface
Fig. 23. Temperature profile for varying values of Sc.
ARTICLE
Fig. 25. Temperature profile for varying values of Ec.
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Fig. 24. Temperature profile for varying values of fw.
thereby increasing the flow velocity (Fig. 8). This explains
the observed phenomena of increased velocity profile due
to increased radiation parameter (Ra) as in Figure 5. It is
noted here that, the buoyancy parameters (Fig. 10) con- Fig. 26. Temperature profile for varying values of Mb .
tribute to achieving better flow kinematics.
Figure 4 presents the consequence of the Casson param- cause a reduction to the flow velocity. It is attributable to
eter on the velocity profile. It can be seen that the Cas- the fact that suction can be used to increase flow resis-
son parameter infinitesimally affects the velocity field due tance leading to retardation of flow kinematics whilst heat
to viscous and elastic effects. The suction parameter (fw) absorption reduces the molecular activity leading to a
Figure 12 and heat absorption parameter (Q) Figure 14 reduced speed.
J. Nanofluids, 11, 629–645, 2022 639Heat Transfer on a Chemically Reacting Non-Newtonian Casson Fluid Over a Vertically Stretched Magnetized Surface Aloliga et al.
Fig. 27. Temperature profile for varying values of Q.
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Fig. 29. Concentration profile for varying values of fw.
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Fig. 28. Concentration profile for varying values of Pr.
6.2. Temperature Profiles
Figures 15–27 depicts graphical results for various control
parameters on temperature distribution near the surface of Fig. 30. Concentration profile for varying values of .
the plate. Figure 15 displayed the impact of the bulk mag-
netic parameter (Mb ) on the temperature distribution. The and the Prandtl number (Pr) on the temperatre distribution.
magnetic parameter tends to thicken the thermal bound- The Prandtl number tends to diminish the thermal bound-
ary layer near the wall. Similar observations are made ary layer thickness and therefore convection becomes more
(Fig. 16) for Casson parameter. Figures 17 and 18 demon- dominant than heat diffusion. The decrease in the tem-
starte respectively the impact of the index parameter (n) perature field is attributed to that a large Prandtl number
640 J. Nanofluids, 11, 629–645, 2022Aloliga et al. Heat Transfer on a Chemically Reacting Non-Newtonian Casson Fluid Over a Vertically Stretched Magnetized Surface
ARTICLE
Fig. 33. Concentration profile for varying values of Ra.
Fig. 31. Concentration profile for varying values of n.
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Fig. 34. Concentration profile for varying values of Sc.
show similar trends of decreasing thermal boundary layer
Fig. 32. Concentration profile for varying values of . thickness. The Schmidt number (Sc) had no influence on
the temperature distribution (Fig. 23). Suction (fw) how-
as it acts to deteriorate the thermal conductivity and thin- ever reduces the temperature distribution (Fig. 24) for
ner boundary layer contributes to a decline in the thermal obvious reasons.
boundary layer thickness. In Figure 25, the Eckert number acts to decrease the
The radiation parameter (Fig. 19), the buoyancy parame- temperature distribution on the surface whilst the bulk
ters (Figs. 20 and 21) and reaction rate parameter (Fig. 22) magnetic parameter is presented in Figure 26 and shows
J. Nanofluids, 11, 629–645, 2022 641Heat Transfer on a Chemically Reacting Non-Newtonian Casson Fluid Over a Vertically Stretched Magnetized Surface Aloliga et al.
Fig. 37. Concentration profile for varying values of Ec.
Fig. 35. Concentration profile for varying values of Gr.
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Fig. 38. Concentration profile for varying values of M s .
Fig. 36. Concentration profile for varying values of Gm.
suction (Fig. 29) and Casson (Fig. 30) parameters tend to
an increasing temperature distribution within the boundary increase species concentration near the surface.
layer. In Figure 27, the heat absorption parameter tends to The chemical species concentration within the bound-
decrease the thermal boundary layer thickness. ary layer region increases with reaction rate parameter
shown in Figure 32 and decreases with increasing radia-
6.3. Concentration Profiles tion parameter given in Figure 33, Schmidt number shown
The species concentration profiles for various control in Figure 34, thermal and modified Grashof numbers of
parameters are depicted in Figures 28–40. The Prandtl Figures 35 and 36 respectively. The Eckert number pre-
number (Pr) tends to reduce the species concentration sented in Figure 37 shows similar trends as the Grashof
within the boundary as illustrated in Figure 28, where as numbers for species concentration.
642 J. Nanofluids, 11, 629–645, 2022Aloliga et al. Heat Transfer on a Chemically Reacting Non-Newtonian Casson Fluid Over a Vertically Stretched Magnetized Surface
7. CONCLUSIONS
A 2-D steady and chemically reacting Casson fluid flow-
ing on a magnetized plate with convective boundary con-
ditions have been investigated. The following conclusions
are made:
(i) Surface magnetization can be employed to control flow
kinematics.
(ii) The rate of heat transfer can be altered with Casson
fluids.
(iii) The bulk and surface magnetic fields impacts on the
velocity, thermal and concentration boundary layers and
can be adopted for flow control.
(iv) Chemical reaction diminishes the species concentra-
tion near the magnetised plate.
NOMENCLATURE
u v w Components of velocity along the Cartesian
axes (m/s)
B0 Transverse magnetic field (Wb/m2 )
t Time (s)
Tw Wall temperature (K)
ARTICLE
Fig. 39. Concentration profile for varying values of Mb . U0 Characteristic velocity (m/s)
C Concentration (kg/m3)
g Acceleration due to gravity (m/s2 )
T Temperature of the Casson fluid (K)
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Dimensionless similarity variable
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f
Delivered by Ingenta () Dimensionless similarity function
() Dimensionless temperature
qr Radiation flux distribution in fluid (W/m2 )
Nu Nusselt number
Sh Sherwood number
k Thermal conductivity of the fluid (W/m/K )
Pr Prandtl number
q Volumetric heat generation (w)
Ms The magnetic parameter at the surface
Mb The magnetic parameter in the bulk fluid
Ec The Eckert parameter
Greek Symbols
Casson parameter
Fluid density
Similarity variable
Internal heat generation parameter
Fig. 40. Concentration profile for varying values of Q. Electrical conductivity of base fluid (m2 /s)
Thermal diffusivity
Kinematic viscosity (m2 /s)
Figure 38 illustrates the impact of magnetic field at the Stream function, (m2 /s)
surface (Ms ) on species concentration whilst the effects on Fluid viscosity (kg/m/s)
bulk fluid (Mb ) is illustrated in Figure 39. It can be seen T The thermal coefficients (1/K)
that Ms has a positive impact on the concentration bound- C Concentration expansion coefficients (1/kgm3)
ary layer with Mb showing an adverse effect on the species
concentration in the fluid. The heat generation or absorp- Recommendations
tion parameter Q shows minimal effects on the species The incorporation of magnetized surfaces in cooling or
concentration. heating systems is recommended. The heat generation can
J. Nanofluids, 11, 629–645, 2022 643Heat Transfer on a Chemically Reacting Non-Newtonian Casson Fluid Over a Vertically Stretched Magnetized Surface Aloliga et al.
be controlled through modulation of the magnetization 31. C. Ram-Reddy and P. Naveen, Heat Transf. 48, 2122 (2019).
strength at the bulk surface. 32. S. Hashmi, Comprehensive Materials Processing, Newnes,
Amsterdam (2014).
33. J. Sun, J. Hensel, T. Nitschke-Pagel, and K. Dilger, Materials 12,
Conflict of Interest 2700 (2019).
There is no conflict of interest in this study. 34. A. Ali, F. Iqbal, D. N. M. Khan, S. Asghar, and M. Awais, Heat
Transfer Res. 49, 614 (2019).
35. M. Das, G. Mahanta, S. Shaw, and S. B. Pareda, Heat Transfer Asian
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