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(ISSN: 2992-4421 )                                                                                                   Dhananjaiah D. S1* 

https://ijojournals.com/                                                       Volume 07 Issue 03 || March, 2024 || 

A Mathematical analysis of oscillatory free convection in a vertical wavy channel with Chemical reaction 
 

 

 

A Mathematical analysis of oscillatory free convection in a 
vertical wavy channel with Chemical reaction 

 
Dhananjaiah D. S1, Prof. K. Shivashankara2, Venuprasad K. K3, Prakasha.P4 

1Department of Mathematics, Government First Grade College K.R.Nagar, Mysuru, India 
E-mail id: dhanu2614@gmail.com 

2Department of Mathematics, Yuvaraja’s College, University of Mysore, Mysore, India 
E-mail id: drksshankara@gmail.com 

3Department of Mathematics, Government First Grade College K.R.Pete, Mandya, India 
E-mail id: kkvpmaths@gmail.com 

4Department of Mathematics, Government First Grade College, Madagi, Ramnagar, India 
E-mail id: profprakasha@gmail.com 

 

Abstract 
The effect of chemical reaction on unsteady combined heat and mass transfer flow of a 

viscous electrically conducting fluid in a vertical wavy channel with oscillatory flux. The non-
linear governing equations are solved by employing a regular perturbation technique with the 
slope  of the wavy wall as a perturbation parameter. The velocity, the temperature and the 
concentration are analyzed for different variations of the governing parameters. The rate heat and 
mass transfer are evaluated for different variations. 
 
Keywords:  Heat Transfer, Mass Transfer, Chemical reaction, Wavy channel,  

1. Introduction 
 
Due to growing significances, the application of non-Newtonian liquid is mandatory in 

the engineering and industry. It is outstanding to those plentiful applications in more than a few 
regions, they are, the plastic manufacturing, performance of lubricant, food processing, and/ or 
movement of biological liquids. The second graded fluid preserve many fluids these are diluted 
polymer solution, slurry flow, as well as industrial oil, in addition to a lot of flow problems by a 
choice of geometry as well as dissimilar mechanical and/or thermal boundary cir- cumstances 
have been deliberated. Tan and Masuoka [1] found the Stokes first problems for the second 
graded fluids and Rashidi et al. [2] discussed by the unsteady compressible flows of the second 
order fluids. Hayat et al. [3] explored by the unsteady stagnation point flow of second grade 
fluids with changeable free stream.Due to complicated relation between stress and strain in non-
Newtonian fluids and their technological application, their study in fluid dynamics is more 
valuable than Newtonian fluids. Viscous fluids flow has attracted the attention of scientists and 
engineers because of its important applications notably in the flow of oil through porous rocks, 
the extraction of energy from geothermal regions, the filtration of solids from liquids and drug 
penetration through human skin. Second grade fluid is a subclass of non-Newtonian fluid in 
which velocity field has up to two derivatives in stress strain tensor relationship where as in New- 
tonian fluid it has derivatives up to first order. Flow of second grade fluid gains attention of the 
researchers in many boundary layer flows and have been successfully studied in various kinds of 
flows. Study of heat transfer in non-Newtonian fluids is much interesting for researchers now-a-
days. 

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mailto:dhanu2614@gmail.com
mailto:drksshankara@gmail.com
mailto:kkvpmaths@gmail.com
mailto:profprakasha@gmail.com


IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Dhananjaiah D. S1* 

https://ijojournals.com/                                                       Volume 07 Issue 03 || March, 2024 || 

A Mathematical analysis of oscillatory free convection in a vertical wavy channel with Chemical reaction 
 

 

 

The combined heat and mass transport problems through the chemical reaction are of 
significance in a lot of processes and have obtained an extensive value of concentration in current 
years. In developments such as drying, disappearance at the external of a fluid body, energy 
transportation in a drenched cooling increase and the flow in a desert cooler, heat and mass 
transport happen simultaneously. Possible applications of that category of flow can be established 
in numerous industries. Some examples, in the power industries, between the techniques of 
generating electric energy is solitary in this electrical energy are extracted directly exciting from 
a conducting fluid. It is predominantly attracted in cases of diffusion and chemical reaction 
occurs at approximately the identical speediness. Once diffusion is to a great extent faster than 
chemical reaction, then merely chemical reaction influences the rate of chemical reaction; when 
diffusion is not much quicker than chemical reaction, the diffusion as well as kinetics interacts to 
construct very dissimilar consequences. The investigation of heat generation or absorption 
consequences in moving fluids is significant in sight of quite a few substantial problems, they 
are, and flu- ids undergo exothermic or else endothermic chemical reaction. Outstanding to the 
quick development of electronic technology, effectual freezing of electronic apparatus has 
become certified and freezing of electronic apparatus ranges from own transistors to foremost 
structure computers and from energy providers to telephone switch panels and thermal diffusion 
impacts has been exploited for isotopes separation in the combination among gases with 
extremely low molecular weight (H2 and He) and average molecular weight. 
Bestman [4] investigated the free convection boundary layer flow with simultaneous heat and 
mass transfer in a porous medium when the boundary walls move in its own plane with suction. 
Abdus Sattar and Hamid Kalim [5] studied the unsteady free convection interaction with thermal 
radiation in a boundary layer flow past a vertical porous plate. Makinde [6] explored the 
combined free convection boundary layer flow with thermal radiation and mass transfer past a 
permeable vertical plate. Makinde et al. [7] investigated the problem of unsteady convection with 
chemical reaction and radiative heat transfer past a flat porous plate moving through a binary 
mixture in an optically thin environment. Muthu- cumaraswamy and Ganesan [8] explored the 
impact of the chem- ical reaction and injection on flow characteristics in an unsteady upward 
motion of an isothermal plate.  
 In many chemical engineering processes, there does occur the chemical reaction between 
a foreign mass and the fluid in which the plate is moving. These processes take place in 
numerous industrial applications viz., polymer production, manufacturing of ceramics or 
glassware and food processing. Das et al[9] have studied the effects of mass transfer on flow past 
an impulsively started infinite vertical plate with constant heat flux and chemical reaction. 
Muthukumaraswamy[10] has studied the effects of reaction on a long surface with suction.  
Radiation and mass transfer on an unsteady two-dimensional laminar convective boundary layer 
flow of a viscous incompressible chemically reacting fluid along a semi-infinite vertical plate 
with suction by taking into account the effects of viscous dissipation. 

Kandaswamy et al[11] have discussed the Effects of chemical reaction, heat and mass 
transfer on boundary layer flow over a porous wedge with heat radiation in the presence of 
suction or injection.  

The study of heat transfers and mixed convection flow in enclosures of various shapes has 
received attention [12] due to its practical applications. Interest in these convection flow and heat 
transfer in porous medium has been motivated by a broad range of applications to geothermal 
systems, crude oil production, storage of nuclear waste materials, ground water pollution, fiber 
and granular insulations solidification of castings. In a wide range of such problems, the physical 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Dhananjaiah D. S1* 

https://ijojournals.com/                                                       Volume 07 Issue 03 || March, 2024 || 

A Mathematical analysis of oscillatory free convection in a vertical wavy channel with Chemical reaction 
 

 

 

system can be modeled as a two-dimensional rectangular enclosure with vertical walls held at 
different temperatures and the connecting adiabatic horizontal walls. Convective heat transfers in 
a rectangular porous duct whose vertical walls are maintained at two different temperatures and 
horizontal walls insulated received attention by several investigators [13]. Furthermore, in 
references [14 and 15] numerical results are being presented.  

Coupled heat and mass transfer phenomenon in porous media is gaining attention due to 
its interesting applications. The flow phenomenon is relatively complex rather than that of the 
pure thermal convection process. Underground spreading chemical wastes and other pollutants, 
grain storage, evaporation cooling and solidification are the few other application areas where the 
combined thermo-solutal natural convection in porous media are observed. Combined heat and 
mass transfer by free convection under boundary layer approximations has been studied by Bejan 
and Khair[16],Lai and Kulacki[17].The free convection heat and mass transfer in a porous 
enclosure has been studied recently by Angirasa et al[18]. The combined effects of thermal and 
mass diffusion in channel flows has been studied in recent times by a few authors, notably, 
Nelson and Wood [19]. 

  Theoretical and experimental investigations of natural convection MHD flow over a 
vertical porous plate in presence of chemical reaction plays an important role in Agriculture, 
geophysics and astrophysics. To study the underground water resources, filtration and water 
purification process in chemical engineering one must know the concepts of the fluid flow 
through porous medium. The porous medium is like a non homogeneous medium but for the 
sake of analysis, it may be possible to replace it with a homogeneous fluid. Oscillatory flows 
are associated with high rates of heat and mass transfer. Many studies have been done to 
understand its characteristics in different systems such as pulse combustors and reciprocating 
engines. Many investigators reported oscillatory flows by involving various physical situations.  

 
2.Mathematical model 
 

We consider the motion of viscous, incompressible fluid through a porous medium in a 
vertical channel bounded by flat walls. The thermal buoyancy in the flow field is created by a 
traveling thermal wave imposed on the boundary wall at y =L while the boundary at y = -L is 
maintained at constant temperature T1

 while both the walls are maintained at uniform 
concentration. The Boussinesq approximation is used so that the density variation will be 
considered only in the buoyancy force. We choose a rectangular Cartesian system 0 (x, y) with x-
axis in the vertical direction and y-axis normal to the walls. The walls of the channel  are  at 
y=L. 

The equations governing the unsteady flow, heat and mass transfer in terms of stream 
function . 

     







2
0

0
4222

)()(

)(])()()[(







k
CCg

TTg

y

yxyyxt

        (2.1) 

     )()()( 1
2

oope CCQTTQ
yxxyt

C 
























        (2.2) 

 )()( 1
2

oCCkD
yxxyt


























                                          (2.3) 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Dhananjaiah D. S1* 

https://ijojournals.com/                                                       Volume 07 Issue 03 || March, 2024 || 

A Mathematical analysis of oscillatory free convection in a vertical wavy channel with Chemical reaction 
 

 

 

The boundary conditions for the velocity and temperature fields are  

             11      ,     ,0    ,0 CCTT
xy









 
   on y = -L  

             22    ),(       ,0    ,0 CCntmxSinTTT
xy

e 







 
 on  y = L     (2.4) 

 
Introducing the non-dimensional variables as   

21

2

21

22 ,,/,,/,
cC

CC

TT

TT
mttLyymxx









          (2.5)      

the governing equations in the non-dimensional form ( after dropping the dashes ) are  

     
2

2
22

1
14

1

2
12

1 ))(()
),(

),(
)((

y
MDN

R

G

yx
R yyt









  




          (2.6) 

    


 2
2
1)( Q

yxxyt
P 






















                              (2.7) 




 




















 2
1)(

yxxyt
Sc       (2.8) 

where 

 


UL
R          (Reynolds number)    

2

3



 LTg
G e
 (Grashof number) 

1k

c p
 ( Prandtl number),     

k

L
D

2
1  (Darcy parameter), 

1D
Sc


 (Schmidt number)   

2

222
2



 LH
M oe ( Hartmann Number) 




2QL
 (Heat source parameter)  

(Radiation   absorption parameter) 

1

2
1

1
D

LK
   (Chemical reaction parameter)  Lm (Aspect ratio) 

2m

n


    (non-dimensional thermal wave velocity) 

 
The corresponding boundary conditions are  
           1)1()1(      

 10,0 








yat

yx


         (2.9) 

2

2

2

2
22

1
yx 







 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Dhananjaiah D. S1* 

https://ijojournals.com/                                                       Volume 07 Issue 03 || March, 2024 || 

A Mathematical analysis of oscillatory free convection in a vertical wavy channel with Chemical reaction 
 

 

 

2.5 

 

 
2.0 

 

 
1.5 

 

 
1.0 

 

 
0.5 

---- u 

R=5 w 

R=6 

R=7 

 
 
 
 

R=8 

R=9 

 0.2 0.4 0.6 0.8 1.0 

2.5 
 

 
2.0 

 

 
1.5 

 

 
1.0 

 

 
0.5 

   ----- u 
Gm= 10 

Gm= 8 w 

Gm= 6 

 
Gm= 4 

 

 
Gm= 2 

 0.2 0.4 0.6 0.8 1.0 

 
10,)(),(

11,1),(





yonCtxSinyx

yonCyx




 1,)(),(  CtxSinyx     

            00,0 








yat

y

C

y


               (2.10) 

The value of  on the boundary assumes the constant volumetric flow in consistent with the 
hypothesis. Also the wall temperature varies in the axial direction in accordance with the 
prescribed arbitrary function t. 
 
3. Nusselt number and Sherwood number 
 

Knowing the temperature & concentration the local rate of heat and mass transfer on the 
walls have been calculated using the formula  

                        1)(
1







 y

wm y
Nu




 

where   



1

1

5.0 dym    and           1)(
1







 y

wm y

C

CC
Sh  

where   



1

1

5.0 dyCCm  

where 1421 .......,..........,.......... ddd  are constants. 

 
4. Discussion of the numerical results 
 

In this analysis we investigate the effect of Chemical reaction on convective Heat and 
mass transfer flow of a viscous fluid in a vertical wavy channel.  

 

  

Fig.1 Velocity Profile for various values of Chemical Reaction   Fig.2 Velocity Profile for various values of Grashof Number  

increase of �2 (rotation parameter) increase the Primary velocity but the reverse process exists for 
the secondary velocity. At the same time in certain stage after that reverse processes exists for the 
secondary velocity in the fluid flow.The increasing of permeability parameter K., Grashof 
number for heat transfer Gr(Figs.1&2) 
 

  

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Dhananjaiah D. S1* 

https://ijojournals.com/                                                       Volume 07 Issue 03 || March, 2024 || 

A Mathematical analysis of oscillatory free convection in a vertical wavy channel with Chemical reaction 
 

 

 

1.0 

 

 
0.8 

 

 
0.6 

 

 
0.4 

 

 
0.2 

 
 

R=1.25 

R=2.25 

R=3.25 

R=4.25 

R=5.25 

 0.2 0.4 0.6 0.8 1.0 

1.0 0.8 0.6 0.4 0.2 

 
 

Pe=1.25 

Pe=2.25 

Pe=3.25 

pe=4.25 

Pe=5.25 

1.0 

 

0.8 

 

0.6 

 

0.4 

 

0.2 

R=1.0 

0.18 5 10 15 20 2 4 
Pe=5 

6 8 10 12 14 

R=1.25 3.0 

0.16 Pe=4 

0.14 
2.5 

Pe=3 
0.12 

R=1.85 

0.10 2.0 
R=2.00 

Pe=2 

0.08 

1.5 
0.06 

R=2.25 Pe=1 

  

Fig.3 Temperature Profile for various values of Chemical Reaction    Fig.4 Concentration Profile for various values of Peclet Number 
 

 Fig.3 shows that,The increase effects of temperature exists,  the reverse processes exists if 
increase of chemical reaction parameter.Concentration profile shows the decrease effects 
while increasing of Peclet number(Fig.4). 

 

Fig.5 Mass flux for various value of Chemical Reaction Fig.6 Heat flux for various value of Peclet Number 

 

Mass flux shows decrease effects while increasing of chemical reaction (Fig.5)..Heat flux 
shows increasing effects while increase of Peclet Number(Fig.6). 

The Tables 1–3 symbolize the skin friction, Nusselt number and Sherwood number for 
dissimilar deviations in the per- tinent parameters. When the magnetic field is large, then the 
Hall current will be developed in the flow field. 

 

 

 

 

 

 

 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Dhananjaiah D. S1* 

https://ijojournals.com/                                                       Volume 07 Issue 03 || March, 2024 || 

A Mathematical analysis of oscillatory free convection in a vertical wavy channel with Chemical reaction 
 

 

 

 

 

Table.1 The shear stresses 
 

M k R Pr Gr Gm Sc Kc H Q1 be bi S 
2 1 1 0.7 5 3 0.2 1 1 1 1 0 2.532394 

3                       1.884042 

4                       1.618128 

  1                     2.913229 

  2                     3.049695 

    2                   2.464944 

    3                   2.451027 

      3                 1.313291 

      7                 1.280852 

        10               4.461883 

        15               6.441776 

          6             3.706505 

          9             4.90861 

            0.3           3.081731 

            0.6           13.56434 

              2         3.501852 

              3         5.160271 

                2       1.271224 

                3       1.219895 

                  2     6.037402 

                  3     9.631637 

                    2   2.960489 

                    3   3.181527 

                      0 2.57465 

                      1 2.624382 

 
 
 
 
 
 
 
 
 
 
 
 
 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Dhananjaiah D. S1* 

https://ijojournals.com/                                                       Volume 07 Issue 03 || March, 2024 || 

A Mathematical analysis of oscillatory free convection in a vertical wavy channel with Chemical reaction 
 

 

 

 
 
 

Table.2 The Nusselt number 
 

Kc Ql Sc H Pr n t Nu 

1 1 0.22 1 0.71 0.5 0.5 0.727227 

2             0.850651 

3             0.924433 

  2           -0.14204 

  3           -0.0117 

    0.3         0.813039 

    0.6         1.004183 

      2       1.249786 

      3       1.60923 

        3     3.39786 

        7     7.54639 

          1   0.72799 

          1.5   0.728996 

            1 0.727683 

            1.5 0.72827 

 
 
 

Table.3. The Sherwood number 
 

Kc Sc N t Sh 

1 0.22 0.5 0.5 0.715581 

2    0.90641 

3    1.054072 

 
0.3   0.841856 

 
0.6   1.25509 

1  0.715965 

1.5  0.716476 

1 0.715813 

   
1.5 0.716109 

 
The skin friction magnitudes are described in Table 1. An Increases in the Hartmann number 
precede decreases in skin friction. because the frictional drag was decreased by the Lorentz effect 
on a viscous fluid. An increase in the rotation parameter, Prandtl number, and heat source 
parameter is used to examine the comparable behavior. Furthermore, an increase in the 
permeability parameter K leads to increased skin friction in significant ways on the surface 
boundary. Similarly, increases in the radiation-absorption parameter, Schmidt number, chemical 
reaction parameter, thermal Grashof number, mass Grashof number, Hall, and ion slip parameters 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Dhananjaiah D. S1* 

https://ijojournals.com/                                                       Volume 07 Issue 03 || March, 2024 || 

A Mathematical analysis of oscillatory free convection in a vertical wavy channel with Chemical reaction 
 

 

 

near the surface boundary are examined for the same effect. Table 2 indicates that an increase in 
the chemical reaction parameter, Schmidt number, Prandtl number, heat source parameter, 
oscillation frequency, and time all contribute to an increase in the Nusselt number. It decreases as 
the radiation-absorption parameter increases. According to Table 3, a stronger Sherwood number 
is preceded by an increase in the Schmidt number, chemical reaction parameter, oscillation 
frequency, or time. 
 
 
5.References. 
[1] W. Tan, T. Masuoka, Stokes’ first problem for a second grade fluid in a porous half- space          

      with heated boundary, Int. J. Non- Linear Mech. 40 (2005) 515–522. 

[2] M.M. Rashidi, S.A. Majid, A. Mostafa, Application of homotopy analysis method to the    
      unsteady squeezing flow of a second-grade fluid between circular plates, Math. Probl. Eng.   
     18 (2010), 706840. 
[3] T. Hayat, M. Qasim, S.A. Shehzad, A. Alsaedi, Unsteady stagnation point flow of  
          second grade fluid with variable free stream, Alexandria Eng. J. 53 (2014) 455–461. 
[4] A.R. Bestman, Natural convection boundary layer with suction and  mass transfer in a porous  
       medium, Int. J. Energy Res. 14 (1990) 389–396. 
[5] M.D. Abdus Sattar, M.D. Hamid Kalim, Unsteady  free-convection  interaction with thermal  
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[6] O.D. Makinde, Free convection flow with thermal radiation and mass transfer past a moving  
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[7] O.D. Makinde, P.O. Olanrewaju, W.M. Charles, Unsteady convection with chemical reaction and  
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[8]  R. Muthucumaraswamy, P. Ganesan, Effect of the chemical reaction and injection on flow          
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(2001) 665–671. 
[9]   U.N. Das, R. Deka, V.M. Soundalgekar, Effects of mass transfer on flow past an impulsively  
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[10]  R. Muthukumaraswamy, Effects of a chemical reaction on a moving isothermal surface with suction., 

Acta Mechnica,V.155,p.65, 2002 

[11]  P. Kandaswamy, Wahid Abd, B.Md.Raj, B. Azme Khamis, Effects of chemical reaction, heat and 

mass transfer on boundary layer flow over a porous wedge with heat radiation in the presence of 

suction or injection, Theoret. Appl. Mech., V.33. No.2, pp.123-148, 2006 

[12]  Hiroxhi Iwai, Kazuyoshi nakabe, Kenjiro Suzuki: Flow and Heat transfer characteristics of 

backward-facing step laminar flow in a rectangular duct., Int.J.Heat and Mass transfer,V.43, pp.457-

471(2000) 

[13]      Teoman Ayhan, Hayati Olgum : Betul Ayhan : Heat transfer and flow structure in a Rectangualr 

channel withwing -1, type vortex Generator. Tr. J. of Engineering and Environmental Science, pp, 

85-195, 22 (1998). 

[14]  Cheng K.S. and J.R. Hi.: Steady, Two-dimensional, natural convection in rectangular enclosures with 

differently heated walls transaction of the ASME, v. 109, p, 400, (1987). 

[15]  Chan, B.K.C, Ivey, U.M and Barry, J.M: Natural convection in enclosed porous medium with       

rectangular boundaries ASME journal of heat transfer, v. 92, pp, 21-27 (1970). 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                   Dhananjaiah D. S1* 

https://ijojournals.com/                                                       Volume 07 Issue 03 || March, 2024 || 

A Mathematical analysis of oscillatory free convection in a vertical wavy channel with Chemical reaction 
 

 

 

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	A Mathematical analysis of oscillatory free convection in a vertical wavy channel with Chemical reaction
	Dhananjaiah D. S1, Prof. K. Shivashankara2, Venuprasad K. K3, Prakasha.P4

