The Upshots of Dufour and Soret in Stretching Porous Flow of Convective Maxwell Nanofluid with Nonlinear Thermal Emission

Michael Williams; Isah Bala Yabo

doi:doi:10.11648/j.ijtam.20241003.12

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The Upshots of Dufour and Soret in Stretching Porous Flow of Convective Maxwell Nanofluid with Nonlinear Thermal Emission

Michael Williams^*

, Isah Bala Yabo

Published in International Journal of Theoretical and Applied Mathematics (Volume 10, Issue 3)

Received: 15 August 2024 Accepted: 6 September 2024 Published: 18 October 2024

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Abstract

In this paper, the combined upshot of Soret and Dufoue of a convective Maxwell nanofluid on a porous perpendicular surface with nonlinear thermal emission was investigated. In the present work, the impact of permeable stretching sheet, nonlinear thermal emission, heat sour sink, Dufour and Soret effect, chemical reaction, Brownian motion and thermophoresis in a convective Maxwell nanofluid flow is widely discussed. The governing equations derived for the problem are highly nonlinear coupled partial differential equations. The governing equations were transformed into ordinary differential equations using Lie symmetry group alterations. The BVP4C MATLAB solver was employed to solve the ordinary differential equations numerically after validating the convergence of the method with existing results in the literature. The numerical results were established and discussed using tables and graphs. It was found that variations in porosity parameter (K), Dufour (Du) and Soret (Sr) improves velocity, temperature and concentration profiles respectively and the present of nonlinear thermal radiation and heat source emit more heat for the flow. Also, it is exciting to report that both porosity (K) and Dufour (Du) parameters has a strong impact on the flow of skin frictions, Nusselt number and Sherwood number. However, the current results may present applications in the areas of petroleum reservoir, heat exchangers, steel industries, cooling applications, nuclear waste disposal and so on.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 10, Issue 3)
DOI	10.11648/j.ijtam.20241003.12
Page(s)	38-50
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Soret, Dufour, Heat Source, and Nonlinear Thermal Radiation

1. Introduction

The attentions of several researchers on the area of combine effect of Soret and Dufour have been drowned due to the recent awakening engineering and scientific applications. It is applicable in the areas of petroleum reservoir, heat exchangers, steel industries, cooling applications, nuclear waste disposal and so on. Soret and Dufour played a vital role in the double-diffusive area. This is because their upshots orders of magnitude are smaller. The product of mass flux situated in mass equation caused by temperature slope is called Soret effect while the product of energy flux located in energy equation instigated by mass gradient is called Dufour effect. Ismail et al.

[1]

established the upshots of thermo-diffusion and diffusion-thermo on a parallel porous plate. It was solved analytically and finite difference method was used to validate their results numerically. However, they noted different parallel flow behaviors due to buoyance ratio- diffusion thermo. Safae et al.

[2]

dissected the upshot of heat source and thermo diffusion carried out numerically within the stretching sheet. They noted that Soret upshot is increased with the presence of R on the mean concentration and reduced with absent of R. Some analysis of Soret and Dufour effects can be seen in the studies of Bidemi and Ahmad

[3]

, Michael and Isah

[4]

The modern exploration in nanotechnology to develop the effectiveness of refined thermal systems was established as a result of novel energy sources. The size of the particles in nanofluid is nanometer which is made by suppressing the particles in a fundamental liquid. Nanofluids have been form by several forms of base fluid and nanoparticles as a means of heat transfer for diverse procedures. The most common base fluids in nanofluids are ethylene glycol, water, and motor engine oil. While water is a renewable resource, it is not ideal due to its low thermal conductivity. On the other hand, motor engine oil and ethylene glycol have high viscosity but are environmentally toxic. Additionally, mixtures of water or ethylene glycol with nanoparticles are commonly used as car coolants to enhance engine performance. Computers of high performance make used of cooling electronic technology to reach an extreme power of 100,300 W/cm² in a microprocessor circuit. Rana et al.

[5]

applied Buongiorno model to recommend a non-homogeneous type which identified seven components that influenced the upgrading of warmness modification to Nanofluid, thermophoresis and Brownian motion had been determined to be the extreme contributing components. Madhukesh et al.

[6]

investigated the consequences of the states of the higher temperature variation, thermal emission and viscous heat. In addition, they improved Buongiorno mathematical model with Brownian motion, thermophoresis and gyrotactic microorganisms. Therefore, the quest for the relevant application of nanofluid drives the efforts of many authors in conducting research on the related topic in recent years which include Puneeth

[7]

, Khan

[8]

, Hayat

[9]

, Gireesha

[10]

, Hussain

[11]

and Isah

[12]

Heat and mass transfer over an extended surface is another attractive area of research with applications in many industrial and engineering activities. The prediction of heat and mass transport behaviors in non-Newtonian fluids encompasses applications in thermal engineering, metal spinning, and sphere bed processes, glass fiber, hot rolling, and so on. Zaid et al

[13]

investigated the free convective flow of a viscous material concerning heat and mass transfer in a permeable channel using cubic B-spline collocation outline. Hayat et al.

[14]

studied the flow of Eyring-Powell fluid of heat and mass transfer on a stretching sheet. Mahanthesh et al.

[15]

addressed the heat and mass flow of chemical responses over motionless and moving pattern. Selimefendigil et al.

[16]

investigated the configurations of heat and mass transfer transportation in a fluid flow through the channel pattern. Ahmed et al.

[17]

analyzed the microstructural implications of the finite difference scheme to evaluate the numerical simulations of heat and mass transfer in a viscous liquid.

The nonlinear thermal radiation has attracted the attention of many researchers due to several engineering applications. Though, processing industries make use of nonlinear thermal radiation to produce heat to get a good finished product. The plants for electricity generation, rocket propulsion, turbine for gas and so on are produced from the finished products. Khan et al.

[18]

inspected the significance of nonlinear Rosseland approximation in a Walter-B nanoliquid by utilizing Brownian motion, thermophoresis, and convective boundary conditions. Their results show that an increasing Schmidt number reduces the concentration of the nanofluid. Ilias et al.

[19]

explored the magnetohydrodynamic heat transfer on an extending sheet. Their findings revealed how heat transfer and fluid flow impact the leading surface. Jha and Samaila

[20]

supported the significance of nonlinear Rosseland approximation and thermophoresis on heat and mass transfer in mixed convective vertical channels. Increasing the radiative heat flux enhances the fluid temperature, which, in turn, causes the velocity and concentration near the porous surface to rise.

The Lie symmetry group approach has been applied by several researchers to investigate countless convective boundary problems under several flow patterns in aerodynamics, plasma physics, meteorology, fluid mechanics, chemical engineering, and various other branches of engineering. It is a scaling symmetry group used to reduce two independent variables of the model to a single variable. Several papers on Lie symmetry group approach have been published which include Ahmad et al.

[21]

, Das et al.

[22]

, Rashidi et al.

[23]

, Kandasamy et al.

[24]

and so on.

Here, our primary objective is to build upon the work of Ahmad et al.

[21]

by incorporating nonlinear thermal radiation into the energy equation, considering the effects of Soret and Dufour. This will allow us to investigate the mass and energy flux, as well as the convective Maxwell nanofluid flow, on a stretching porous sheet.

2. Mathematical Analyses

The characteristics of Soret and Dufour effects in convective Maxwell nanofluid flow on a stretching porous parallel plate with nonlinear thermal radiation are examined. The nanofluid flow, driven by thermophoresis and Brownian motion, is influenced by a magnetic field

. According to Figure 1, The Maxwell nanofluid is presumed to flow along the x-direction. The Boussinesq approximation is considered, and the flow is fully developed and laminar.

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Figure 1. The physical channel of the parallel porous plate.

The governing expressions that drive the flow are as Ahmad et al.

[21]

and Li et al.

[25]

(1)

(2)

(3)

(4)

The above modals boundary conditions are as Ahmed et al.

[21]

(5)

Below is the introduction of non- dimensional variables to dimensionless equations (1) to (5)

(6)

Rosseland approximation is used to simplify the radiative heat flux as follows:

(7)

Expand

into Taylor’s series expansion In order to linearize equation (7) gives

(8)

By putting equations (6) to (8) into equations (1) to (5) the dimensionless equations below emerged

(9)

(10)

(11)

(12)

The fresh boundary conditions are:

(13)

Introducing the stream function defined by

(14)

The following are obtained by putting equation (14) into equations (9) to (13). Equation (9) is satisfied identically.

(15)

(16)

(17)

With boundary conditions as:

(18)

2.1. Lie Group Alteration

The scaling symmetry group techniques are introduced below as Ahmed et al.

[21]

(19)

By applying equation (19) into equations (15) to (18) where

is the real number and

is the parameter of the group we got:

(20)

(21)

(22)

By equating the powers of

the invariant of the technique established for equations (20) - (22)

(23)

From equation (18), the invariance of the boundary conditions becomes:

(24)

Center on the outcomes of equation (23) put on equation (24) the group alterations (19) becomes:

(25)

Furthermore, subject to Taylor series expansions we have:

(26)

Since

, altering equation (26), the following developed:

(27)

Currently, in terms of differentials equation (27) yields

(28)

(29)

The similarity below is attained after solving equations (28) – (29)

(30)

Where

is the similarity variable

By applying equation (30) into equations (15) to (18) yielded the following:

(31)

(32)

(33)

While the boundary conditions reads:

(34)

The concerned engineering physical quantities of practical interest are

, and

as defined below:

(35)

(36)

(37)

2.2. Method of Computation

The following transformation was applied to convert equations (31) – (34) into first-order initial value problems.

(38)

(\begin{matrix} h_{1} \\ h_{2} \\ h_{3} \\ h_{4} \\ h_{5} \\ h_{6} \\ h_{7} \end{matrix}) = (\binom{y_{2}}{\begin{matrix} y_{3} \\ [\frac{1}{1 - β y_{1}^{2}}] [- 2 β y_{2} y_{1} y_{3} - [y_{1} y_{3} - y_{2}^{2}] + [(M + \frac{1}{K}) (y_{2} - β y_{1} y_{3})] + Ra [y_{4} - Nr y_{6}]] \\ y_{5} \\ \frac{1}{[1 + \frac{4}{3} N {(C_{T} + y_{4})}^{3}]} [- 4 N {(Ct + y_{4})}^{2} y_{5}^{2} - \Pr (y_{1} y_{5} + Q y_{4} + Nb y_{7} y_{5} + Nt y_{5}^{2} Du y_{7}^{,})] \\ y_{7} \\ - \frac{Nt}{Nb} {Sry}_{5}^{’} - Le [y_{1} y_{7} - γ y_{6}] \end{matrix}})

(39)

The boundary conditions

(40)

3. Validation of Results

The analysis to validate the results was conducted in Table 1, showing excellent agreement with those of Hayat et al.

[26]

, Turkyilmazoglu

[27]

and Ahmad et al.

[21]

Table 1. Evaluation of results for

when

and

, with previous published works.

Ahmad et al.

[21]

Hayat et al.

[26]

Turkyilmazoglu et al.

[27]

Present results

0.0

-1.00000

-1.000000

-1.00000000

0.5

-1.22474

-1.224744

-1.22474487

1.0

-1.41421

-1.414213

-1.41421356

4. Results and Discussion

The result of the combine upshot of Soret and Dufore of a convective Maxwell nanofluid on a porous stretching sheet with nonlinear thermal emission is ascertained. The impact of the appropriate nondimensional flow parameters has been considered using tables and graphs. In this study, the defaulted values are:

(41)

Unless otherwise detailed.

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Figure 2. Influence of

Table 2. Impact of

, and

. Impact of

, and

. Impact of

, and

K	C_f	Nur	Shr
0.2	-2.77175	0.06222	0.47366
0.4	-2.17873	0.06672	0.47621
0.6	-1.94287	0.06871	0.47769

Table 3. Impact of

, and Shr.

Ct	C_f	Nur	Shr
0.2	-2.04041	0.06787	0.47704
0.7	--2.04040	0.06772	0.47723
0.9	-2.04039	0.06760	0.47739

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Figure 3. Influence of

. Influence of

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Figure 4. Influence of <i></i> on

However, Figure 2 examines the raising effects of

outlined. This developed the momentum boundary layer as a result of

which gives interior heat to the flow and the permeable layer becomes wide. Physically, by enhancing

the permeable hole becomes large which leads to the fast flow of fluid. Figure 3 depicts the influence of

outcome. From the picture, it is clear that the

outcome increase with rising values of

. Actually,

is the proportion of Buoyancy to the product of heat diffusion and viscous. The upshot

outline is exemplified by Figure 4. By boosting the values of

consequently reduced the results of

profile.

is used to check the fluid material and characters. Although,

is quite inconsequential if the fluid material materializes. Figure 5 explains the upshot of

profile with growing values of

which in turn diminishes the

profile. The retarding impact of

outline is indicated in Figure 6. This physically shows that as

raises it magnifies the magnetic strength and it additionally upsurge the fluid particles which in turn diminishes the

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Figure 5. Influence of

Download: Download full-size image

Figure 6. Influence of

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Figure 7. Influence of

Table 4. Impact of

on C_f,

, and

Sr	C_f	Nur	Shr
0.2	-2.04041	0.06766	0.48656
0.6	-2.04040	0.06794	0.47387
0.8	-2.04040	0.06808	0.46754

Table 5. Impact of Du on C_f,

, and

Du	C_f	Nur	Shr
0.3	-2.03952	0.05364	0.47274
0.6	-2.03823	0.03288	0.46652
0.9	-2.03700	0.01278	0.46067

Figure 7 gives the growing impact of

outline. Consequently,

upsurge the buoyance force which improves the fluid velocity. Figure 8 brings foreword the action of

profile. Obviously the raising of

improves the

outline. The improvement

is as results of excessive concentration gradient originate from

. Also, energy transport and mass diffusion take place in higher rate in the particles which in turn boost the thickness of

. The features of

for

are described in Figure 9 which exhibits raising behavior when

is augmented. This is because

measured the relatively amount of interior energy within the body. Figure 10 Shows

variations for

. The upsurge of

improves

. Basically,

occurrence improved greatly as a result of the presence of

. Figure 11 demonstrates the impact of

. The upsurge of

boost

. Thermal gradient has a great effect on

. Actually, greater

produces greater thermal gradient which in turn upsurged

. Figure 12 which exhibits decaling behavior when

is augmented. Actually,

= thermal diffusivity/ mass diffusivity. Figure 13 illustrates the effect of

. The upsurge of

decreases

Table 6. Impact of

, and

. Impact of

, and

. Impact of

, and

Nb	C_f	Nur	Shr
0.5	-2.04024	0.06478	0.48772
1.0	-2.03990	0.05928	0.49139
2.0	-2.03910	0.04626	0.49336

Table 7. Impact of γ on C_f, Nur, and Shr.

γ	C_f	Nur	Shr
1.5	-2.04041	0.06603	0.57719
2.0	-2.04042	0.06452	0.66166
4.0	-2.04039	0.05994	0.92270

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Figure 8. Influence of

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Figure 9. Influence of

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Figure 10. Influence of

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Figure 11. Influence of

Download: Download full-size image

Figure 12. Influence of

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Figure 13. Influence of

Tables 2 to 7 were constructed to portray the effects of

, on

, and

. The outcomes revealed that by considering variations in values of

and

, the magnitude

accelerate while it depicts reverse behavior for

. Also,

and

declined the magnitude of

and inclined the magnitude with

. Similarly, it is experimentally perceived that rise in

, and

upsurge the

and decelerate with

5. Conclusion

The combined upshot of Soret and Dufore of a convective Maxwell nanofluid on a porous stretching sheet with nonlinear thermal emission was carried out. The results of the study are in form of velocity, temperature, concentration, skin friction, Nusselt number and Sherwood number. The key highlights for the present work are as follow:

i. Variations of

and

outcome improves

and

respectively

ii. Both

and

exhibits raising behavior for

, and

iii. Both

and

retard the

outline

iv.

has an escalating effect on the

outline

Table 8. Nomenclature.

Description	Symbol	Description	Description	Symbol	Description
Reference temperature		Gravitational acceleration	Prandtl number		Hartmann number
Dimensionless velocity		Soret parameter	Rayleigh number		Lewis number
Velocity of the exterior stream		Transverse magnetic field	Buoyancy ratio		Brownian motion parameter
Condition far away from the plate		Constant	Thermophoresis parameter		Thermal radiation parameter
Chemical reaction parameter		Electrical conductivity	Magnetic field strength		Density of base fluid
Kinematic viscosity		Dufour parameter	Velocity components along x, y-axis	T	Temperature variable
Relaxation time		Similarity variable	Nanoparticles specific heat		Ambient liquid concentration
Stream function		Dimensionless temperature	Heat capacity ratio		Stefan-Boltzmann constant
Dimensionless concentration		Thermal diffusivity	Nanoparticles concentration		Non-uniform heat generation
Volumetric thermal expansion coefficient of the base fluid		The fluid viscosity	Mean absorption coefficient		Absorption coefficient
Fluid specific heat		Deborah number	Ambient liquid temperature		Heat source/sink
Temperature ratio	S	Mass transfer parameter	Fluid specific heat at constant pressure		Biot number
Thermal conductivity		Brownian diffusion	Termophretic diffusion		Specific heat at constant pressure
Porous material		Free stream velocity of the flow	Reference concenteration

Nur Nessult Number Shr Sherwood number

Conflicts of Interest

The authors declare no conflicts of interest.

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Williams, M., Yabo, I. B. (2024). The Upshots of Dufour and Soret in Stretching Porous Flow of Convective Maxwell Nanofluid with Nonlinear Thermal Emission. International Journal of Theoretical and Applied Mathematics, 10(3), 38-50. https://doi.org/10.11648/j.ijtam.20241003.12

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Williams, M.; Yabo, I. B. The Upshots of Dufour and Soret in Stretching Porous Flow of Convective Maxwell Nanofluid with Nonlinear Thermal Emission. Int. J. Theor. Appl. Math. 2024, 10(3), 38-50. doi: 10.11648/j.ijtam.20241003.12

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Williams M, Yabo IB. The Upshots of Dufour and Soret in Stretching Porous Flow of Convective Maxwell Nanofluid with Nonlinear Thermal Emission. Int J Theor Appl Math. 2024;10(3):38-50. doi: 10.11648/j.ijtam.20241003.12

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@article{10.11648/j.ijtam.20241003.12,
  author = {Michael Williams and Isah Bala Yabo},
  title = {The Upshots of Dufour and Soret in Stretching Porous Flow of Convective Maxwell Nanofluid with Nonlinear Thermal Emission
},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {10},
  number = {3},
  pages = {38-50},
  doi = {10.11648/j.ijtam.20241003.12},
  url = {https://doi.org/10.11648/j.ijtam.20241003.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20241003.12},
  abstract = {In this paper, the combined upshot of Soret and Dufoue of a convective Maxwell nanofluid on a porous perpendicular surface with nonlinear thermal emission was investigated. In the present work, the impact of permeable stretching sheet, nonlinear thermal emission, heat sour sink, Dufour and Soret effect, chemical reaction, Brownian motion and thermophoresis in a convective Maxwell nanofluid flow is widely discussed. The governing equations derived for the problem are highly nonlinear coupled partial differential equations. The governing equations were transformed into ordinary differential equations using Lie symmetry group alterations. The BVP4C MATLAB solver was employed to solve the ordinary differential equations numerically after validating the convergence of the method with existing results in the literature. The numerical results were established and discussed using tables and graphs. It was found that variations in porosity parameter (K), Dufour (Du) and Soret (Sr) improves velocity, temperature and concentration profiles respectively and the present of nonlinear thermal radiation and heat source emit more heat for the flow. Also, it is exciting to report that both porosity (K) and Dufour (Du) parameters has a strong impact on the flow of skin frictions, Nusselt number and Sherwood number. However, the current results may present applications in the areas of petroleum reservoir, heat exchangers, steel industries, cooling applications, nuclear waste disposal and so on.
},
 year = {2024}
}

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TY  - JOUR
T1  - The Upshots of Dufour and Soret in Stretching Porous Flow of Convective Maxwell Nanofluid with Nonlinear Thermal Emission

AU  - Michael Williams
AU  - Isah Bala Yabo
Y1  - 2024/10/18
PY  - 2024
N1  - https://doi.org/10.11648/j.ijtam.20241003.12
DO  - 10.11648/j.ijtam.20241003.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 38
EP  - 50
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20241003.12
AB  - In this paper, the combined upshot of Soret and Dufoue of a convective Maxwell nanofluid on a porous perpendicular surface with nonlinear thermal emission was investigated. In the present work, the impact of permeable stretching sheet, nonlinear thermal emission, heat sour sink, Dufour and Soret effect, chemical reaction, Brownian motion and thermophoresis in a convective Maxwell nanofluid flow is widely discussed. The governing equations derived for the problem are highly nonlinear coupled partial differential equations. The governing equations were transformed into ordinary differential equations using Lie symmetry group alterations. The BVP4C MATLAB solver was employed to solve the ordinary differential equations numerically after validating the convergence of the method with existing results in the literature. The numerical results were established and discussed using tables and graphs. It was found that variations in porosity parameter (K), Dufour (Du) and Soret (Sr) improves velocity, temperature and concentration profiles respectively and the present of nonlinear thermal radiation and heat source emit more heat for the flow. Also, it is exciting to report that both porosity (K) and Dufour (Du) parameters has a strong impact on the flow of skin frictions, Nusselt number and Sherwood number. However, the current results may present applications in the areas of petroleum reservoir, heat exchangers, steel industries, cooling applications, nuclear waste disposal and so on.

VL  - 10
IS  - 3
ER  -

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Author Information

Michael Williams

Department of Mathematics, Usmanu Danfodiyo University, Sokoto, Nigeria

Contact Email

http://orcid.org/0009-0008-3803-873X
Isah Bala Yabo

Department of Mathematics, Usmanu Danfodiyo University, Sokoto, Nigeria

http://orcid.org/0000-0002-1308-6566

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Table 1

Table 1. Evaluation of results for when and, with previous published works.
Table 2

Table 2. Impact of on ,, and . Impact of on ,, and .
Table 3

Table 3. Impact of on , , and Shr.
Table 4

Table 4. Impact of on C_f, , and.
Table 5

Table 5. Impact of Du on C_f,, and.
Table 6

Table 6. Impact of on ,, and . Impact of on ,, and .
Table 7

Table 7. Impact of γ on C_f, Nur, and Shr.
Table 8

Table 8. Nomenclature.

Plain Text BibTeX RIS

APA Style

Williams, M., Yabo, I. B. (2024). The Upshots of Dufour and Soret in Stretching Porous Flow of Convective Maxwell Nanofluid with Nonlinear Thermal Emission. International Journal of Theoretical and Applied Mathematics, 10(3), 38-50. https://doi.org/10.11648/j.ijtam.20241003.12

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ACS Style

Williams, M.; Yabo, I. B. The Upshots of Dufour and Soret in Stretching Porous Flow of Convective Maxwell Nanofluid with Nonlinear Thermal Emission. Int. J. Theor. Appl. Math. 2024, 10(3), 38-50. doi: 10.11648/j.ijtam.20241003.12

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AMA Style

Williams M, Yabo IB. The Upshots of Dufour and Soret in Stretching Porous Flow of Convective Maxwell Nanofluid with Nonlinear Thermal Emission. Int J Theor Appl Math. 2024;10(3):38-50. doi: 10.11648/j.ijtam.20241003.12

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@article{10.11648/j.ijtam.20241003.12,
  author = {Michael Williams and Isah Bala Yabo},
  title = {The Upshots of Dufour and Soret in Stretching Porous Flow of Convective Maxwell Nanofluid with Nonlinear Thermal Emission
},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {10},
  number = {3},
  pages = {38-50},
  doi = {10.11648/j.ijtam.20241003.12},
  url = {https://doi.org/10.11648/j.ijtam.20241003.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20241003.12},
  abstract = {In this paper, the combined upshot of Soret and Dufoue of a convective Maxwell nanofluid on a porous perpendicular surface with nonlinear thermal emission was investigated. In the present work, the impact of permeable stretching sheet, nonlinear thermal emission, heat sour sink, Dufour and Soret effect, chemical reaction, Brownian motion and thermophoresis in a convective Maxwell nanofluid flow is widely discussed. The governing equations derived for the problem are highly nonlinear coupled partial differential equations. The governing equations were transformed into ordinary differential equations using Lie symmetry group alterations. The BVP4C MATLAB solver was employed to solve the ordinary differential equations numerically after validating the convergence of the method with existing results in the literature. The numerical results were established and discussed using tables and graphs. It was found that variations in porosity parameter (K), Dufour (Du) and Soret (Sr) improves velocity, temperature and concentration profiles respectively and the present of nonlinear thermal radiation and heat source emit more heat for the flow. Also, it is exciting to report that both porosity (K) and Dufour (Du) parameters has a strong impact on the flow of skin frictions, Nusselt number and Sherwood number. However, the current results may present applications in the areas of petroleum reservoir, heat exchangers, steel industries, cooling applications, nuclear waste disposal and so on.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - The Upshots of Dufour and Soret in Stretching Porous Flow of Convective Maxwell Nanofluid with Nonlinear Thermal Emission

AU  - Michael Williams
AU  - Isah Bala Yabo
Y1  - 2024/10/18
PY  - 2024
N1  - https://doi.org/10.11648/j.ijtam.20241003.12
DO  - 10.11648/j.ijtam.20241003.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 38
EP  - 50
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20241003.12
AB  - In this paper, the combined upshot of Soret and Dufoue of a convective Maxwell nanofluid on a porous perpendicular surface with nonlinear thermal emission was investigated. In the present work, the impact of permeable stretching sheet, nonlinear thermal emission, heat sour sink, Dufour and Soret effect, chemical reaction, Brownian motion and thermophoresis in a convective Maxwell nanofluid flow is widely discussed. The governing equations derived for the problem are highly nonlinear coupled partial differential equations. The governing equations were transformed into ordinary differential equations using Lie symmetry group alterations. The BVP4C MATLAB solver was employed to solve the ordinary differential equations numerically after validating the convergence of the method with existing results in the literature. The numerical results were established and discussed using tables and graphs. It was found that variations in porosity parameter (K), Dufour (Du) and Soret (Sr) improves velocity, temperature and concentration profiles respectively and the present of nonlinear thermal radiation and heat source emit more heat for the flow. Also, it is exciting to report that both porosity (K) and Dufour (Du) parameters has a strong impact on the flow of skin frictions, Nusselt number and Sherwood number. However, the current results may present applications in the areas of petroleum reservoir, heat exchangers, steel industries, cooling applications, nuclear waste disposal and so on.

VL  - 10
IS  - 3
ER  -

Copy | Download