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(ISSN: 2992-4421 )                                                                                                                       *1 Boniface Inalu Obi*  

https://ijojournals.com/                                                           Volume 08 || Issue 09 || September, 2025 || 

“Numerical Computation of Reactive Flow of Third Grade Fluid With Heat Generation" 

 

 

Numerical Computation of Reactive Flow of Third Grade Fluid With Heat Generation 

*1 Boniface Inalu Obi,2 Edwin Esekhaigbe, 3Uchenna Awucha Uka 
 

1Department of Mathematics, Imo State University, Owerri, Nigeria 
2Mathematics and Computer Science Department, University of Africa, Bayelsa State, Nigeria 

3Basic Science Department, School of Science and Technology, Babcock University, Ogun State, 
Nigeria 

Corresponding author *1
 Boniface Inalu Obi, 

Abstract 

Computation of reactive flow of third grade fluid in cylindrical pipe with heat generation is 
considered. The resulting governing equations of motion are highly nonlinear and are solved 
using collocation method in verifying the impacts of some material variables involved. Results 
indicate that increase in the non-Newtonian variable increases the flow velocity  and decreases 
the temperature of the walls of the cylindrical pipe. It is observed that the critical Frank-
Kamenetskii variable exist for which the solution fails to be distinct. It is further observed that 
for c  , a steady state solution does not exist suggesting a thermal runaway which could be 

avoided by setting 1.8879565  .  

Keywords: Numerical, third grade, heat generation, reactive, computation.  

1. Introduction 

Third grade fluid is in the class of non- Newtonian fluids of the differential type. It is a class of 

non- Newtonian fluid where the stress tensor is the addition of all the tensors that can be 

developed from the velocity field with up to three derivatives. Example of this class of fluid 

include ketchup, paints, blood etc. chemical reactions involved in fluid which is capable of 

altering the fluids properties and composition in flow process is referred to as reactive flow. 

There are some studies now available in literature. Some of the earliest work on third grade 

fluid are Fosdick and Rajagopal [3] analyzed the thermodynamic third grade fluid and showed 

the restrictions on the stress constitutive model. Rajagopal [10] examined the stability 

properties of third grade fluids. Szeri and Rajagopal [13] investigated the flow of third grade 

fluids between heated parallel plates. Ellahi et al [1] examined the impacts of slip on the 

nonlinear flows of a third grade fluid. In the investigation, the results of no-slip condition were 

inferred as a restricting case when the slip variable is equal to zero. 

https://doi.org/10.5281/zenodo.17510312 IJO JOURNALS

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Boniface Inalu Obi*  

https://ijojournals.com/                                                           Volume 08 || Issue 09 || September, 2025 || 

“Numerical Computation of Reactive Flow of Third Grade Fluid With Heat Generation" 

 

 

Makinde [4] investigated a steady flow of a reactive variable viscosity fluid in a cylindrical pipe 

with an isothermal wall. Makinde [5] studied the thermal criticality for reactive gravity-driven 

thin film flow of a third grade fluid with adiabatic free surface down an inclined plane. 

Salawuland Fatumbi [11] examination the inherent irreversibility of hydromagnetic third grade 

reactive poiseuille flow with variable viscosity. The employed the weighted residual method for 

the solution. The study shows that the heat dissipation of reactive exothermic chemical in a 

uniform magnetic field moved past fluid in a porous medium in an irreversible mode. The study 

on steady flow of a reactive viscous fluid in porous cylindrical pipe was carried out byFarayola 

[2]. The investigation was done using regular perturbation for the solution of the nonlinear 

equation. Okedayo et al [9] numerically investigated the reactive MHD flow of thrd grade fluid 

in pipe.The study involved the weighted residual collocation method which was use as a 

computational means of solution and were able to display the influence of various thermo-

physical parameter. It was observed that the critical value of the Frank-Kamenetskii and third 

grade parameters exists for which the solution sizes to be unique. Obi et al [7] analyzed the 

incompressible flow of third grade fluid in an inclined rotating cylindrical pipe with isothermal 

wall and Joule heating. They solved the nonlinear equations by perturbation method and the 

effect of some parameter on the flow presented graphically. Yurusoyand Pakdemirli [14] 

investigated the fluid flow of third grade fluid in pipe with heat transfer with a case of constant 

viscosity. Reynold’sand Vogel’s models were introduced to account for temperature-dependent 

viscosity. They analytically examined the flow and compared the result with the finite 

difference procedure earlier given by Massaudi and Christie [6]. Okedayo et al [8] focuses on 

Gaterkinweighted residual method for magnetohydrodynamic mixed convection flow in a 

vertical channel filled with porous media. Results obtained were analyzed using tables and 

graphs. Siddiqui et al [12] analyzed the thin film flow of third grade fluid down an inclined plane 

using the combined traditional perturbation as well as the homotopy perturbation methods 

and comparison made of the two results obtained from the two techniques. 

This research seeks to examine the  consequences of the Frank-Kamenetskii parameter on the 

reactive flow of third grade fluid in cylindrical pipe.  

2. Formulation of the Problem 

https://doi.org/10.5281/zenodo.17510312 IJO JOURNALS

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Boniface Inalu Obi*  

https://ijojournals.com/                                                           Volume 08 || Issue 09 || September, 2025 || 

“Numerical Computation of Reactive Flow of Third Grade Fluid With Heat Generation" 

 

 

The fundamental models of an incompressible viscous flow are the mass and momentum 

conservation laws which in the absence of heat transfer process, such laws are defined through 

continuity and momentum equations.  

In vector form, 

 . 0                                                                                                                               1

f+div -                                              

u

Du u
p

Dt k


  

 

                                                  2
 

where 

 =density, u =velocity, p =pressure,  =stress tensor, f = body force and

 =material derivative
D

Dt

 

 

Stress tensor defining a third grade fluid is given by 

 

 

3

1

1 1

2
2 1 1 2 1

2
2 1 2 2 1 1

                                                                                                                            3

where

tr                          

i
i

A

A A

A A A





 

 



 

 

  

  



                                                                            4
 

 is the coefficient of dynamic viscosity, 1 1 1 1, , ,     are constants. 

 The Rivlin-Ericksen tensors nA  are defined by  0 1A  , being the identity tensor. 

     1
1 1,  1                                                                         5

tn
n n n

DA
A A u u A n

Dt


        

Assuming the surface tension to be negligible, we have the velocity field in t he 

    , 0,0                                                                                                                   6U u r  

https://doi.org/10.5281/zenodo.17510312 IJO JOURNALS

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Boniface Inalu Obi*  

https://ijojournals.com/                                                           Volume 08 || Issue 09 || September, 2025 || 

“Numerical Computation of Reactive Flow of Third Grade Fluid With Heat Generation" 

 

 

Substituting the values of .  and v  in equation (2), yields 

 2
0

1
                                                                                    7

d du dp
r u B u

r dr dr k dz


 
 

   
 

 

 
2

2
0 0

1
exp 0                                                  8

d dT du E
r B u QC A

r dr dr k dr RT




     
         

     
 

         00 0 0, 0,                                                                              9
du dT

u a T a T
dr dr

     

where 

u is fluid velocity, T  is absolute temperature,  is dynamic viscosity,  ius 

electricalconductivity, 0T  is reference temperature, a is radius of the pipe, r is 

radial distance, E is the activation energy, R  is the universal gas constant, 0C is 

concentration,  Q  is heat activationand A  is the rate constant. Introducing the following  

dimensionless parameters: 
 

   
02

0 0
0 0 2

0 0

, , , ,                                        10

E

RTRT Ea QC AE
r ar u u u T T

RT E RT k
 



     


 
 
Using eqn (10) in eqns (7-9), yields 
 

 
1

1                                                                                           11
d du
r u Mu

r dr dr


 
    

   

 1

2

21
0                                                                      12r

d d du
r B Mu e

r dr dr dr







    

      
   

 

         0 0 0, 1 0, 0 0                                                                             13
du d

u
dr dr


   

 

3. Method of Solution 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Boniface Inalu Obi*  

https://ijojournals.com/                                                           Volume 08 || Issue 09 || September, 2025 || 

“Numerical Computation of Reactive Flow of Third Grade Fluid With Heat Generation" 

 

 

In order to solve the nonlinear momentum and energy equation of (11) and (12) with the 

condition (13), we employ collocation method for the solution. This technique is a function 

approximation method which reduces the nonlinear ordinary differential equations to  

algebraic equations which can be solved byany iterative fixed point technique. 

The approximate solution is of the form 

     
0

                                                                                                             14
n

j j
j

u r a r


  

Equation (14) is the trial function over the region which must satisfy the given boundary 

conditions. In this technique, the one, two and three term coefficients are employed and the 

maximum velocity and temperature ascertained.  

     3 3 2 3
0 0 1 0 1(1 ), (1 ) ( )                                                             15u r a r u r a r a r r       

Similarly, 

     3 3 2 3
0 2 1 2 3(1 ), (1 ) ( )                                                             16r a r r a r a r r      

 

 
max

max
max 1 0.000875

0.875
4.500000000 0.05321635176                                                       17

e 


 


  

 

 0 0.2222222222                                                                                                            18a   

The thermalcritical property is determined bythe relationship between maximum temperature 

and the Frank-Kamenetskii variable and achieved from equations (17)and (18) with other 

thermo-solutal parameters. Plotting equation (17) results in figure 5.    

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Boniface Inalu Obi*  

https://ijojournals.com/                                                           Volume 08 || Issue 09 || September, 2025 || 

“Numerical Computation of Reactive Flow of Third Grade Fluid With Heat Generation" 

 

 

 

 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Boniface Inalu Obi*  

https://ijojournals.com/                                                           Volume 08 || Issue 09 || September, 2025 || 

“Numerical Computation of Reactive Flow of Third Grade Fluid With Heat Generation" 

 

 

 

 

 

4. Results and Discussions 

In this section, the impacts of some variables that are of great importance to the study are 

discussed. These effects come from the graphs presented in earlier section. In figures 1 and 2, 

the influence of non-Newtonian and magnetic field variables are respectively shown. It is seen  

from the results that increase in both parameters enhance the flow velocity. Figure 3 shows the 

temperature profiles for  values of the non-Newtonian variable. Result shows that increase in 

the parameter , decreases the temperature of the cylindrical pipe at the walls. Figure 4 is the 

temperature profiles for various values of the magnetic field parameter. Results indicate that 

increase in the magnetic field parameter increases the temperature of the system. This is 

because magnetic field can generate heat through electrical resistance, thereby leads to a rise 

in temperature. Figure 5 shows the critical value of the Frank-Kamenetskii parameter

1.8879565c  .  

5. Conclusion  

Computation of reactive flow of third grade fluid in cylindrical pipe with heat generation is 
considered. The resulting governing equations of motion are highly nonlinear and are solved 
using collocation method in verifying the impacts of some material variables involved. 
Results indicate that increase in the non-Newtonian variable increases the flow velocity  and 
decreases the temperature of walls of the cylindrical pipe. it is observed that the critical 

https://doi.org/10.5281/zenodo.17510312 IJO JOURNALS

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Boniface Inalu Obi*  

https://ijojournals.com/                                                           Volume 08 || Issue 09 || September, 2025 || 

“Numerical Computation of Reactive Flow of Third Grade Fluid With Heat Generation" 

 

 

Frank-Kamenetskii variable exist for which the solution fails to be distinct. It is further 

observed that for c  , a steady state solution does not exist suggesting a thermal 

runaway which could be avoided by setting 1.8879565  .  

 

Declarations  

1. Funding: Not applicable 

2. Informed Consent Statement: Not applicable 

3. Data Availability:  Not applicable 

4. Conflict of Interest Statement: No conflict of interest 

 

6. References 

 [1] Ellahi R., Hayat T., Mahomed F.M. and Asghar S. Effects of slip on the non-linear flow of a  
third grade fluid. Nonlinear Analysis: Real World Applications 11(2010),139-146.  
[2] Farayola P.I. On steady flow of a reactive viscous fluid in a porous cylindrical pipe. Open 
 Journal of Fluid Dynamics 7(2017),359-370  
[3] Fosdick R.L. And Rajagopal, K.R. Thermodynamics And Stability of Fluids ofThird Grade. 
Proc. R. Soc. Lond. 339(1980), 351-377. 
[4] Makinde O.D. On steady flow of a reactive variable viscosity fluid in a cylindrical pipe with 
an isothermal wall. International Journal of Numerical Methods for Heat and Fluid Flow, 
17(2007).187-194. 
[5] Makinde O.D. Thermal criticality for a reactive gravity driven thin film flow of a third grade 
fluid with Adiabatic free surface down an inclined plane.Applied Mathematics and 
Mechanics, 30(2009), 373-380. 
[6] Massoudi, M. And Christie, I. Effects of variable viscosity and viscous dissipation on 
theflow of a third –grade fluid in a pipe.    Int.  J. of Nonlinear Mech., 30(5)(1995) 
687-699. 
[7] Obi B.I.,Okedayo, G.T., Jiya, M. And Aiyesimi, Y.M. Analysis of flow of an 
incompressiblemhd third grade fluid in an inclined rotating cylindrical pipe with  
isothermal wall and Joule heating. International Journal For Research In Mathematics 
and Statistics. (2021); 7 (6). 
 [8] Okedayo G.T., Amumeji O.T. and Obi B.I. Galerkin weighted residual method for 
magnetohydrodynamic (MHD) mixed convectionflow in a vertical channel filled with 
porous media. International Journal for Research in Mathematics and Statistics, 4(5), (2018) 
 [9] Okedayo, G.T., Obi, B.I. &Olawuyi, O.M. A numerical study of reactive MHD flow of 
thirdgrade fluid. Journal of Mathematical Science and Computational Mathematics 
(JMSCM), 1(1), (2019). 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS 

(ISSN: 2992-4421 )                                                                                                                       *1 Boniface Inalu Obi*  

https://ijojournals.com/                                                           Volume 08 || Issue 09 || September, 2025 || 

“Numerical Computation of Reactive Flow of Third Grade Fluid With Heat Generation" 

 

 

  [10]Rajagopal K.R. On the stability of third grade fluids. Arch. Mech. 32(1980) 867-875. 
 [11] Salawul S.O. and Fatunmbi E.O. Inherent irreversibility of hydromagnetic third grade 
reactivepoiseuille flow of a variable viscosity in porous media with convective cooling. 
 Journal of the Serbian Society for Computational Mechanics 11(1)(2017), 46-58. 
  [12] Siddiqui A.M., Mahmood R, and Ghori Q.K. Homotopy perturbation method for thin film  
flow of a third grade fluid down an inclined plane. Chaos, Soliton and Fractals 35(2008) 
,140-147. 
[13] Szeri A.Z. Rajagopal K.R. Flow of a non-Newtonian fluid between heated parallel plates. 
Int. J. Non-linear Mech. 20(1985), 91-101. 
   [14] Yurusoy, M. and Pakdemirli, M.: Approximate analytical solutions for the flow of a third 
grade fluid in a pipe. International Journal of Non-Linear Mech. 37(2002),187-195. 
 

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