Journal of Indian Acad. Math.

ISSN: 0970-5120

Vol. 47, No. 1 (2025) pp. 231-249

Hemangini Shukla¹

INVARIANT ANALYSIS OF HEAT

GENERATION AND THERMAL

Shivanshi Dave²

Kapil K Dave³

A. K. Rathod ⁴

N. D. Patel⁵

and

RADIATION EFFECTS ON MHD

NON-NEWTONIAN POWER-LAW

NANOFLUID OVER LINEARLY

STRETCHING SURFACE WITH

CONVECTIVE BOUNDARY CONDITIONS

J. A. Prajapati⁶

Abstract: This study examines the effects of thermal radiation and heat

generation along a stretching surface. The power-law non-Newtonian model

under the influence of Brownian motion and thermophoresis for nanofluids

is analysed for determining their effects on various parameters of nanofluid

like temperature, velocity etc. The uniform magnetic field and boundary

conditions for convective mode are also considered for nanofluid flow. The

objective of similarity invariants is to convert non-linear partial differential

equations into ordinary differential equations invariantly. The numerical

results of the investigation for the impacts of various parameters on skin

friction coefficients, Nusselt-Sherwood numbers are determined. The

behaviour of different physical factors on skin friction coefficients in

y

x

and

directions, on the local Nusselt number, and on the Sherwood number is

analysed. An increment in the power-law index increases the Nusselt

number. The results of the experiment indicates that an increase in the heat

generation parameter will result in a drop in the Nusselt number and an

increase in the Sherwood number. Sherwood number will decrease and

Nusselt number will increase with an increase in thermal radiation

parameter.

Keywords: MHD Nanofluid; Non-Newtonian Power-law Model; Heat

Generation and Radiation; Similarity Invariants; Convective

Boundary Condition.

Mathematics Subject Classification No.: 35Q35, 76M60, 58J70, 35G60.

H. SHUKLA, S. DAVE, K.K. DAVE, A.K. RATHOD, N.D.PATEL AND J.A. PRAJAPATI

232

1. Introduction

The dispersion of nanoparticles in a base fluid, such as water, ethylene

glycol, or oil, is known as nanofluid. It was introduced and studied by Choi(1995). In

his experimental research, he also noticed that heat transfer was enhanced in

nanofluid compared to regular fluids. There are many attractive applications of

nanofluid like coolants, brake fluid, gear lubrication in automobile industries. It is

useful in solar devices, as delivery of cancer drugs in the medical field, and coolants

in electronic devices. So, it is an essential to study the influence of different physical

factors and various physical situations on nanofluid flow.

Tesfaye et al. (2020) analysed the erratic flow of Williamson nanofluid over

a stretched sheet under the influence of a magnetic field, heat radiation, and chemical

reaction. Kalidas et al. (2018) examined heat generation/absorption effects for

Oldroyd-B type nanofluid, two-dimensional flow over a permeable stretching surface

under the effect of magnetic field and slip velocity. Umadevi and Nithyadevi (2016)

investigated two-dimensional nanofluid flows under uniform heat generation or

absorption with a uniform magnetic field for different thermal boundaries. Bilal et al.

(2018) examined the impact of the various physical factors for three-dimensional

Maxwell nanofluid MHD flow passing through a bidirectional stretching surface

under nonlinear thermal radiation. Hayat et al. (2017) addressed three-dimensional

Maxwell MHD nanofluid flow under the influence of heat generation-absorption and

thermal radiation on a stretching surface. Burger’s nano-liquid flow over a stretching

sheet was studied by Ganesh et al. (2018) with the impact of non-linear radiation and

non-uniform heat generation and absorption. The thermal radiation effects on the

MHD stagnation point, the two-dimensional flow of a non-Newtonian Williamson

fluid, over a stretching plate, were examined by Hasmawani et al. (2019) by applying

similarity transformations. The two-dimensional flow of Maxwell nanofluid on a

linearly stretching surface under heat generation and absorption impacts was

investigated by Awais et al. (2015). The two-dimensional flow passing over an

exponentially stretching sheet of MHD Casson fluid was studied with internal heat

generation by Animasaun et al. (2016).

Waqas et al. (2017) modelled and analysed Oldroyd-B nano-liquid two-

dimensional flow over a moving sheet with heat generation and absorption effects

using the Homotopy analysis method. The MHD nanofluid three- dimensional flow

over a shrinking sheet under viscous dissipation and heat generation and absorption

with entropy generation was examined by Hiranmoy et al. (2019). The solution for

unsteady, two-dimensional nanofluid flow over a stretching surface was studied

numerically by utilising the fourth-fifth order RKF technique under the influence of

radiation, thermophoresis, and heat generation and absorption by Pandey and Manoj

(2018). Ahmed et al. (2019) examined MHD Maxwell nanofluids flow over a

INVARIANT ANALYSIS OF HEAT GENERATION

233

stretching surface under the influence of heat generation-absorption and non-linear

thermal radiation in the porous medium by applying similarity variables and the

shooting technique. Kalpna and Sumit (2017) investigated two- dimensional (MHD)

Jeffrey

nanofluid

flow

in

the

presence

of

thermal

radiation,

heat

generation/absorption, and viscous dissipation over an impermeable surface by

assuming similarity transformations and applying the Homotopy analysis method.

Makinde (2011) introduced similarity variables and used the fourth-order Runge-

Kutta method and the shooting method to examine the impacts of internal heat

generation on two-dimensional boundary layer flow on a vertical plate with a

convective surface boundary condition. Lalrinpuia and Surender (2019) used the

homotopy analysis approach to assess MHD nanofluid flow in a saturated porous

medium, in an inclined channel with a heat source/sink, accounting for hydrodynamic

slip and convection at the boundary. Khan et al. (2014) analysed the impacts of heat

generation/ absorption on the 3-D flow of an Oldroyd-B nanofluid over a sheet

stretching in both x and y directions. They applied similarity transformations.

The influence of heat generation, radiation, and viscous dissipation on the

flow of MHD nanofluid over a sheet stretched exponentially in a porous medium was

studied by Thiagarajan and Dinesh Kumar (2019). The MHD-Carreau nanofluid flow

over a radially stretched sheet under the influence of chemical reaction, nonlinear

thermal radiation, and heat generation/absorption was examined by Dianchen et al.

(2018). The second grade Cattaneo-Christov two-dimensional fluid flow caused by a

linear stretched Riga plate was studied under the impact of heat generation/absorption

by Aisha et al. (2018). Abdul Khan et al. (2018) analysed Williamson nanofluid flow

in three dimensions across a linear porous stretching surface for the impact of thermal

radiation. Sulochana et al. (2016) investigated Newtonian and non-Newtonian,

3-D magnetohydrodynamic fluid flow across a stretched sheet. Chuo-Jeng and Kuo-

Ann (2021) examined the effects of zero nanoparticle flux, internal heat generation,

nonlinear radiation, and changing viscosity on free convection on a non-Newtonian

power-law nanofluid flowing via a vertical truncated cone embedded in a fluid-

saturated

porous

medium.

Considering

thermal

radiation

and

heat

absorption/generation, Mabood et al. (2020) investigated MHD Oldroyd-B two-

dimensional, thermal stratified flow across an inclined linearly stretched sheet.

Recently, Newtonian and various non-Newtonian fluid models like Sisko, Powell-

Eyring, Power-Law Model, Prandtl-Eyring were analysed using invariant analysis via

the group- theoretic technique by deriving dependent and independent invariants.

(Patel et al. 2015, Shukla et al. 2017, 2018, 2020). Impact of heat

generation/absorption in the context of nonlinear thermal radiation on

magnetohydrodynamic stagnation-point two-dimensional Newtonian nanofluid flow

across a convective stretching surface were examined by Feroz et al. (2018). Shukla

et al. (2020) analysed flow over linearly stretching surface for 3-D Power-low

nanofluid.

H. SHUKLA, S. DAVE, K.K. DAVE, A.K. RATHOD, N.D.PATEL AND J.A. PRAJAPATI

234

Due to the significance role of heat generation and thermal radiation on

nanofluid flow, we have extended the work done by Shukla et al. (2020) and

considered the influence of heat generation and thermal radiation. In this paper, we

have studied a power-law fluid flow in three dimensions on a linearly stretched sheet.

A survey of the literature shows that most studies have focused on flows in two

directions, X and Y. The scenario is more real in three dimensions, X, Y, and Z. We

have also examined the effects of thermophoresis, magnetic field, and Brownian

motion on heat generation and thermal radiation. The convective boundary conditions

have been considered for the analysis of the present non-Newtonian fluid flow model.

Various parameters like Nusselt number, skin friction coefficients, and Sherwood

number have been considered for analysing the flow. Similarity-dependent and

independent invariants have been used with the aim of transforming the nonlinear

PDEs into ODEs invariantly.

2. Governing Equation of the Boundary Value Problem

Here, we have considered the three-dimensional power-law nano non-

Newtonian fluid model. The flow is incompressible, steady, laminar over a linearly

stretching sheet with the velocity u_w ax and v_w by in X and Y-direction

respectively. Here, the stretched sheet is exposed to a homogeneous magnetic field B

that is directed in the surface's normal direction. The conditions of convective

boundaries are considered for the flow analysis. Heat generation/absorption impacts,

as well as the impact of thermal radiation, are also considered in the heat transfer

study.

We have taken the following parameters for the flow analysis.

T_- Temperature at Infinite distance from the sheet's surface

C_- Concentration at Infinite distance from the sheet's surface

h_f- Heat transfer coefficient

h_s- Convective mass transfer coefficient

Convective heat transfer mode is used to heat or cool the sheet's surface by

maintaining a hot fluid temperature T_fand a convective concentration of fluid C_f

.

We have used the following boundary value flow governing equations.

u

v

w



 0

(1)

x

y

z

INVARIANT ANALYSIS OF HEAT GENERATION

235

u

   u ⁿ

B²

u

 v

 w

 



u

(2)









x

y

z

 z

z



v

 



u ^{n 1}v

B²









 v

 w



v

(3)





x

y

z

 z

z

















²



T

²T

z²

 T C 

D_T T 



u

 v

 w

 

  D_B



















x

y

z

z z

T_

z









Q₀

1

q_r



(T  T_) 

(4)

(5)

c_p

c_pz

2













C

 C

D_T²T

u

 v

 w

 D_B



z²

x

y

z

T_

Boundary values for convective mode are given by

C

T

u  u_w ax

,

v  v_w by

,

w  0

,

, D_B

 h_f(T T)

 h_s(C_f C)

(6)

k

f

z

at z  0, u  0, v  0, w  0,T  T_,C  C_at z  

Where,

u

-Velocity in the

the direction

x

direction,

v

-Velocity in the

y

direction,

w

-Velocity in

z

T

- Fluid temperature,

capacitance ratio

C

-Fluid concentration, ꢀ-Fluid density, ꢁ-Heat

D_T- Thermophoresis diffusion coefficient, D_B-Brownian diffusion coefficient,

: flow index

n

ꢂ (> 0) - Rheological constant, ꢃ-electrical conductivity of the fluid, ꢄ-thermal

diffusivity

Q₀: coefficient of internal heat generation, q_r-radiative heat flux.

H. SHUKLA, S. DAVE, K.K. DAVE, A.K. RATHOD, N.D.PATEL AND J.A. PRAJAPATI

236

3

4^T_

T ⁴

q_ris defined as q_r



3k^

z

Where

^

- the Stefan-Boltzmann constant,

k

^- absorption coefficient.

Now, expanding T ⁴about T_and neglecting higher terms, we get following

expression:

4

3

T ⁴ T_ 4T_T  4T_T_



3









4 T_

q_r



T ⁴



3k^









z

3

16^T_

q_r

²T

z²

3k^

z

(7)

q_r

By putting

in equation (4), we get the following equation.

z



²



T

²T

z²

 T C 

 D_TT 



u

 v

 w

 

  DB



















x

y

z

z z

T

z











3

16^T_

Q₀

1

²T

z²



(T  T_) 

(8)

3k^

c_p

3. Invariance Analysis by Generalized Group Theoretic Method

We have used the following dependent and independent absolute invariants

to convert governing partial differential equations into ordinary differential equations

invariantly. (Shukla et al. 2020)

INVARIANT ANALYSIS OF HEAT GENERATION

237

1n



  d₁z(x)^1n



u

H₁() 

d x

2

v

H₂() 

d y



3

W





H₃() 

n 1

1n

d (x)

4

T T



H₄()   

T

T

f

C C

H₅()  φ 

C

C





f

We have assumed the following values for the coefficients and parameters.

1

n 1

1

n 1

















a^2n

a

n 2



d¹



, d₂ a d₃ b, d₄ a























(u_w)^2n_xn

B²

c_pu_wx

2

pr 

(Re) _{n 1}, Re 

, M

k





3

h_f

(C_f C_)

16T_

Q₀

1

x(Re) _{n 1}, N_b D_B

Bi₁

,

R_d

,

₁

3k^

k



c_p

(10)

Where Re -local Reynolds number, pr -generalised Prandtl number, Le -the

Lewis number, N_b-Brownian motion parameter, N_t-thermophoresis parameter Bi₁

and Bi₂generalised Biot number. We have taken skin-friction coefficients C_fxand

C_fyalong the - and -axes, the Nusselt number, and the Sherwood number for

-

x

y

analysing the fluid flow. We have used the following equations for above parameters:

1

(Re)^{n 1}C_fx (H₁^'(0))ⁿ

(11)

(12)

1

av_w

(Re)^{n 1}C_fy 

(H₁^'(0))^{n 1}H₂^'(0)

bu_w

H. SHUKLA, S. DAVE, K.K. DAVE, A.K. RATHOD, N.D.PATEL AND J.A. PRAJAPATI

238

1

(Re)^{n 1}Nu_x (1  R_d)H^'₄(0)

(13)

1

(Re)^{n 1}Sh_x H^'₅(0)

(14)

Differentiating absolute invariants of equation (9) with respect to similarity

independent variable



and applying on governing equations (1 to 5, 8), we obtain

following equations.

1  n

aH₁ bH₂ aH¹

aH₁^' 0

(15)

(16)

3

1  n

1  n

a(H₁)² aH₁^'H₃

aH₁^'H₁ na(H₁^')^{n 1}H₁^'' MH₁ 0

1  n

1  n

b(H₁)² aH₂^'H₃

aH₂^'H₁ a(n  1)(H₁^')^{n 2}H₂^'H₁^'''

1  n

a(H₁^')^{n 1}H₂^'' MH₂ 0

(17)

1  n

prH^'₄H₃ pr

H₁H^'₄ H^'₄^' R_dH^'₄^' ₁H₄ N_bH^'₄H^'₅ N_t(H^'₄)² 0

1  n

(18)

(19)

N_t

1  n

H^'₅^'



H^'₄^' pr LeH₅^'H^'₃ pr Le

H₁H^'₅ 0

N_b

1  n

Similarly, we have obtained the following equations of boundary conditions

from equation (6).

At  0

,

H₁ 1

,

H₂ 1

,

H₃ 0

,

H^'₄ Bi₁(1  H₄)

,

H₅^' Bi₂(1  H₅)

,

At  

,

H₁ H₂ H₃ H₄ H₅ 0

.

(20)

The following equations are obtained from equations (15-19)

H₁ ^'₁, H₂ ^'₂,H₃



₁

₂

^'₁

(21)

2n

1n

b

a

1n

1n

INVARIANT ANALYSIS OF HEAT GENERATION

239

2n

1n

a(^'₁)² b^'₁^'₂

a^'₁^'₁ na(^'₁^')^{n 1}^'₁^'' M^'₁ 0,

(22)

2n

1n

b(^'₂)² b^'₂^'₂

a^'₂^'₁ a(n  1)(^'₁^')^{n 2}^'₂^'^'₁^''

a(₁^'')^{n 1}^'₂^'' M₂ 0,

(23)

b

2n

H^'₄^' N_bH^'₄H₅^' N_t(H^'₄)²

prH^'₄₂

pr ₁()H^'₄ R_aH^'₄ ₁H₄(24)

a

1  n

N_t

N_b

b

2n

H₅^''



H^'₄^'



Le pr H₅^'₂

pr Le₁H₅^' 0

(25)

a

1  n

4. Numerical Solution

We have transformed the aforementioned system of equations into a system

of first order differential equations in order to use Bvp4c - MATLAB software.

By replacing functions ₁, ^'₁, ₁^'', ₂, ^'₂, ₂^'', H₄, H^'₄, H₅, H₅^'by y_i, for

i  1, 2, , 10 respectively, we get the following equations.

y'₁ y₂

(26)

(27)

y'₂ y₃

(a(y₂)² by₃y₄

ay₁y₃ My₂)

2n

1n

y'₃

(28)

na(y₃)^{n 1}

y'₄ y₅

(29)

(30)

y'₅ y₆

b(y₅)² by₄y₆

ay₁y₆ a(n  1)(y₃)^{n 2}y'₃y₆ My₅

2n

1n

y'₆

(31)

(32)

a(y₃)^{n 1}

y'₇ y₈

H. SHUKLA, S. DAVE, K.K. DAVE, A.K. RATHOD, N.D.PATEL AND J.A. PRAJAPATI

240

N_by₈y₁₀ N_t(y₈)²

pr y₄y₈

pry₁y₈ y₇

b

a

2n

1n

y'₈



(33)

1  R_d

y'₉ y₁₀

(34)

(35)

N_t

b

2n

y'₁₀ 

y'₈

Lepr y₄y₁₀



prLey₁y₁₀

N_b

a

1  n

  0  y₁ y₄ 0, y₂ y₅ 1

y₈ Bi₁(1  y₇(0)), y₁₀ Bi₂(1  y₉(0))

    y₁ 0, y₄ 0, y₇ 0, y₉ 0

(36)

(37)

We have obtained the following equations from equations (11-14)

1

(Re)^{n 1}C_fx  (H₁^'(0))ⁿ  (₁^''(0))ⁿ  (y₃(0))n

1

av_w

bu_w

av_w

bu_w

(Re)^{n 1}C_fy 

(H₁^'(0))^{n 1}H₂^'(0)  

(₁^''(0))^{n 1}₂^''(0)

1

u_w

a

(Re)^{n 1}C_fy  (y₃(0))^{n 1}y₆(0)

(38)

(39)

v_w

b

1

(Re)^^{n 1}Nu_x (1  R_d)H^'₄(0)  (1  R_d)y₈(0)

1

(Re)^^{n 1}Sh_x H^'₅(0)   y₁₀(0) (40)

6. Results and Discussion

We have used MATLAB bvp4c solver for analysing fluid flow problem.

Tables 1 and 2 show the values for Skin friction coefficients, Nusselt number, and

Sherwood number.

INVARIANT ANALYSIS OF HEAT GENERATION

241

Table 1: Skin friction coefficient values for various parameters in the

directions

x

and

y

pr

Cf_y

N_tN_b

R_d

0

ꢂ

0

Cf_x

n

1

a

b

M

Le

1 1 1 0.1 0.1 0.5

1 1 1 0.2 0.1 0.5

1 1 1 0.3 0.1 0.5

1 1 1 0.1 0.1 0.5

1 1 1 0.1 0.2 0.5

1 1 1 0.1 0.3 0.5

1 1 1 0.1 0.1 0.5

1 1 1 0.1 0.1 1.2

1 1 1 0.1 0.1 1.5

1 1 1 0.1 0.1 0.5

0.2

1.7538893508

1.7538897120

1.7538902961

1.7538893508

1.7538895038

1.7538896175

1.7538893508

1.9058126194

1.9704591416

1.7538900002

1.7538901651

1.7538907395

1.7538899995

1.7538899984

1.7538900002

1.5511878626

2.8042554030

3.1138892553

1.7538893508

1.7538897120

1.7538902961

1.7538893508

1.7538895038

1.7538896175

1.7538893508

1.9058126194

1.9704591416

1.7538900002

1.7538901651

1.7538907395

1.7538899995

1.7538899984

1.7538900002

1.8971685570

1.3992670114

1.1463275550

0.2 0.2

1 1 1 0.1 0.1 0.5 0.1 0.2 0.2

1 1 1 0.1 0.1 0.5 0.2 0.2 0.2

1 1 1 0.1 0.1 0.5

0

0.2

0.1 0.2

0.2 0.2

1 10 2 2 0.1 0.1 0.5 0.1 0.2 0.2

1 10 4 2 0.1 0.1 0.5 0.1 0.2 0.2

1 10 6 2 0.1 0.1 0.5 0.1 0.2 0.2

From Table 1, it is observed that the value of skin friction coefficient in

X

and

Y

direction both enhances with rising values of the thermophoresis parameter

N_tas well as thermal radiation R_d. The reason behind it is that if the thermophoresis

parameter is increasing the temperature and concentration, differences between the

surface of the semi-infinite vertical plate and the ambient fluid are increasing and

hence accelerates the heat transfer rate. Table 1 shows the skin friction coefficient

for various values of the Brownian motion parameter N_b

.

H. SHUKLA, S. DAVE, K.K. DAVE, A.K. RATHOD, N.D.PATEL AND J.A. PRAJAPATI

242

The skin friction coefficient in both directions is seen to grow with

increasing values of the Brownian motion parameter N_band opposite behaviour

observed for Lewis number Le . The skin friction coefficient increases as the

magnetic field parameter

M

increases because it reflects an increase in surface

velocity gradients. A similar phenomenon is noticed in Table 1. Effect of stretching

ratio parameter significantly affects skin friction coefficient. An increase in

parameter

b

, the skin friction coefficient in the

X

direction rises, whereas the

Y

direction exhibits the opposite behaviour.

Table 2: Sherwood number and Nusselt number Values for different parameters

pr

Rd

0

ꢂ

n

b

n

M

Le

N_t

N_b

Sh_x

Nu_x

1

0.1 0.1 0.5 1 0.2

0.2 0.1 0.5 1 0.2

0.3 0.1 0.5 1 0.2

0.1 0.1 0.5 1 0.2

0.1 0.2 0.5 1 0.2

0.1 0.3 0.5 1 0.2

0.1 0.1 0.5 1 0.2

0.1 0.1 1.2 1 0.2

0.1 0.1 1.5 1 0.2

0.1 0.1 0.5 0.7 0.2

0.1 0.1 0.5 1.2 0.2

0.1 0.1 0.5 1.7 0.2

0

0.1055262378

0.3824750511

1

0

-0.0451858252 0.3801468924

-0.1918276471 0.3777829817

0

0.1055262378

0.1838693036

0.2099947839

0.1055262378

0.1066120068

0.1070520017

0.3824750511

0.3801079022

0.3777045682

0.3824750511

0.3811253420

0.3803950922

0

0.6808097648 -0.0591201311

0

0.4517559332

0.1075016163

0.0640340920

0.0926736299

0.1055262378

0.1210982576

0.0767238041

0.4341594390

0.3645373365

0.3644023170

0.3824750511

0.3642872395

0.4960100946

0.4966261688

0.4968748668

0.4970963117

0

0.1 0.1 0.5 1

0

0.2

0.1 0.1 0.5 1 0.1 0.2

0.1 0.1 0.5 1 0.2

0.1 0.1 0.5 1 0.2 0.2

0

0.1 0.1 0.5 2 0.2 0.2 0.1 0.1154854564

1 1.2 0.1 0.1 0.5 2 0.2 0.2 0.1 0.1151724519

1 1.3 0.1 0.1 0.5 2 0.2 0.2 0.1 0.1150959844

1 1.4 0.1 0.1 0.5 2 0.2 0.2 0.1 0.1150477775

INVARIANT ANALYSIS OF HEAT GENERATION

243

Table 2 indicates the effect of various parameters on the Sherwood number

and Nusselt number. Growing thermophoresis parameter values are accompanied by

decreasing Sharwood and Nusselt numbers. Table 2 demonstrates that when the

Brownian motion parameter increases, the rate of heat transmission slows down,

resulting in a fall in the Nusselt number and an observed increase in the Sherwood

number. It is observed that the Nusselt number decreases and the Sherwood number

increases with an acceleration of the magnetic parameter .

The Lorentz force is increased when the magnetic parameter increases,

slowing down fluid motion and lowering the rate of heat flux in the process. By

increasing the value of the Lewis number, nanoparticle volume fraction distribution

decreases, because of reduction in mass diffusion. This, in turn, increases the

Sherwood number, with the opposite effect being seen on the Nusselt number. Based

on the table's numerical values, it can be determined that as the radiation parameter is

raised, the Sherwood number falls and the Nusselt number rises. An analogous result

was noted with the Prandtl number. The Sherwood number rises, the heat generation

parameter lambda increases, and the Nusselt number decreases. The Sherwood

number decreases as

increases.

n

(the power-law index) increases, but the Nusselt number

Figures 1 and 2 depict, how the Lewis number changes the Nusselt and

Sherwood numbers in response to thermophoresis and thermal radiation, respectively.

The Nusselt number decreases as the thermophoresis parameter and Lewis number

grow, while inverse patterns are seen as the thermal radiation parameter increases. As

thermophoresis and Lewis numbers rise, Sherwood number tends to increase;

conversely, as the thermal radiation parameter increases, it tends to decrease.

Figures 3 and 4 show the impact of the heat source/sink parameter under the

influence of thermal radiation and thermophoresis parameter on the Nusselt number

and Sherwood number. Figures 5 and 6 demonstrate the influence of the Brownian

motion parameter, the thermophoresis parameter, and the thermal radiation parameter

on the Sherwood number and Nusselt number respectively. Sherwood number

decreases as thermal radiation parameter value increases. Nusselt number increasing

as a result of the thermal radiation parameter increasing.

H. SHUKLA, S. DAVE, K.K. DAVE, A.K. RATHOD, N.D.PATEL AND J.A. PRAJAPATI

244

INVARIANT ANALYSIS OF HEAT GENERATION

245

5. Conclusion

We have used both similarity dependent and independent invariants to get a

similarity solution for the boundary value problem associated with power-law

nanofluid flow. The power-law nanofluid problem's governing equations have been

converted into ordinary differential equations with the help of invariants. The

numerical solutions of derived ordinary differential equations are utilized by using

MATLAB bvp4c software to find the effects of various parameters like Nusselt

number, Sharwood number and Skin friction coefficients on fluid flow. The

following are the findings of the analysis of the fluid flow using invariants.

H. SHUKLA, S. DAVE, K.K. DAVE, A.K. RATHOD, N.D.PATEL AND J.A. PRAJAPATI

246



The findings indicate that an increase in the Lewis number Le results in a

drop in the coefficient of skin friction in the

x

and

y

directions, an increase

in the Sharwood number, and a decrease in the Nusselt number.



An increase in the magnetic parameter

to increase in both the and directions, the Nusselt number decreases,

and the Sharwood number increases.

M

causes the skin friction coefficient

x

y



A rise in the power-law index , a fall in the Sherwood number, and an

increase in the Nusselt number.

The Sherwood and Nusselt numbers decrease with an increase in the

thermophoresis parameter.

As the radiation parameter increases, the Nusselt number rises while the

Sherwood number reduces.

Conflict of Interest Statement

We (authors) do not have any conflict of interest (financial or academic) for

this work.

REFERENCES

Abdul, S., Yufeng N., Zahir, S., Abdullah, D., Waris, K. and Saeed, I. (2018): Three-

Dimensional Nanofluid Flow with Heat and Mass Transfer Analysis over a Linear

Stretching Surface with Convective Boundary Conditions, Applied Sciences, Vol. 8,

pp. 1-18.

Ahmed, S. E., Ramadan, A. M., Abd Elraheem, M. A. and Mahmoud, S. S. (2019):

Magnetohydrodynamic Maxwell Nanofluids Flow over a Stretching Surface through

a Porous Medium: Effects of Non-Linear Thermal Radiation, Convective Boundary

Conditions and Heat Generation/Absorption, International Journal of Aerospace and

Mechanical Engineering (World Academy of Science, Engineering and

Technology), Vol. 13, pp. 436-443.

Aisha, A., Mir, N. A., Farooq, M., Javed, M., Ahmad, S., Malik, M. Y. and Alshomrani, A. S,

(2018): Physical aspects of heat generation/absorption in the second-grade fluid flow

due to Riga plate: Application of Cattaneo-Christov approach, Results in Physics,

Vol. 9, pp. 955-960.

Animasaun, I. L., Adebile, E. A. and Fagbade, A. I. (2016): Casson fluid flow with variable

thermo-physical property along exponentially stretching sheet with suction and

exponentially decaying internal heat generation using the homotopy analysis method,

Journal of the Nigerian Mathematical Society, Vol. 35, pp. 1-17.

INVARIANT ANALYSIS OF HEAT GENERATION

247

Awais, M., Hayat, T., Irum S. and Alsaedi, A., (2015): Heat Generation/Absorption Effects in

a Boundary Layer Stretched Flow of Maxwell Nanofluid: Analytic and Numeric

Solutions, PLOS ONE, Vol. 10, No. 6. Bilal, M., Sagheer, M. and Hussain, S.

(September 2018), Maxwell nanofluid flow with nonlinear radiative heat flux,

Alexandria Engineering Journal, Vol. 57(3), pp. 1917-1925.

Chuo-Jeng, H. and Kuo-Ann, Y. (2021): Nonlinear Radiation and Variable Viscosity Effects

on Free Convection of a Power-Law Nanofluid Over a Truncated Cone in Porous

Media with Zero Nanoparticles Flux and Internal Heat Generation, Journal of

Thermal Science and Engineering Applications, Vol. 13.

Dianchen, L., Ramzan, M., Huda, N., Jae, D. C., and Umer, F. (2018): Nonlinear radiation

efect on MHD Carreau nanofuid fow over a radially stretching surface with zero

mass fux at the surface, Scientific Reports, Vol. 8, p. 3709.

Feroz, A. S., Rizwan, U. H., Qasem, M. and Qiang, Z. (March 2018): Heat

generation/absorption and nonlinear radiation effects on stagnation point flow of

nanofluid along a moving surface, Results in Physics, Vol. 8, pp. 404-414.

Ganesh, K., Ramesh, G. K. and Gireesha, B. J. (2018): Thermal analysis of generalized

Burgers nanofluid over a stretching sheet with nonlinear radiation and non-uniform

heat source/sink, Archives of Thermodynamics, Vol. 39(2), pp. 97-122.

Kalpna, S. and Sumit, Gupta (2017): Viscous Dissipation and Thermal Radiation effects in

MHD flow of Jeffrey Nanofluid through Impermeable Surface with Heat

Generation/Absorption, Nonlinear Engineering, Vol. 6(2), pp. 153-166.

Hasmawani, H., Muhammad, K. A., Nazila I., Norhafizah, M. S., and Mohd, Z. S. (2019):

Thermal Radiation Effect on MHD Stagnation Point flow of Williamson Fluid over a

stretching Surface, Results in Physics, Conference Series 1366(2019) 012011, 2nd

International Conference on Applied & Industrial Mathematics and Statistics 23-25

July 2019, Kuantan, Pahang, Malaysia: IOP

Hayat, T., Qayyum, S., Shehzad, S. A. and Alsaedi, A. (2017): Simultaneous effects of heat

generation/absorption and thermal radiation in magnetohydrodynamics (MHD) flow

of Maxwell nanofluid towards a stretched surface, Results in Physics, Vol. 7,

pp. 562-573.

Hiranmoy, M., Mohammed, A. and Precious S., (2019): Dual solutions for three- dimensional

magnetohydrodynamic nanofluid flow with entropy generation, Journal of

Computational Design and Engineering, Vol. 6(4), pp. 657-665.

Kalidas, D., Tanmoy, C. and Prabir, K. K. (September 2018): Effect of magnetic field on

Oldroyd-B type nanofluid flow over a permeable stretching surface, Propulsion and

Power Research, Vol. 7(3), pp. 238-246.

H. SHUKLA, S. DAVE, K.K. DAVE, A.K. RATHOD, N.D.PATEL AND J.A. PRAJAPATI

248

Khan, W. A., Masood, K., and Malik, R. (2014): Three-Dimensional Flow of an Oldroyd-B

Nanofluid towards Stretching Surface with Heat Generation/Absorption, PLOS ONE,

Vol. 9(8).

Lalrinpuia, T. and Surender, O. (2019): Entropy generation in MHD nanofuid fow with heat

source/sink, SN Applied Sciences, Vol. 1, No. 1672.

Mabood, F., Bognár, G. and Shafiq, A. (2020): Impact of heat generation/absorption of

magnetohydrodynamics Oldroyd-B fluid impinging on an inclined stretching sheet

with radiation, Sci Rep., Vol. 10.

Makinde (2011): Similarity Solution for Natural Convection from A Moving Vertical Plate

with Internal Heat Generation and A Convective Boundary Condition, Thermal

Science, Vol. 15(1), pp. 137-143.

Patel, M., Patel, J., and Timol, M. G., (2015): Laminar Boundary Layer Flow of Sisko Fluid,

Applications and Applied Mathematics, (AAM), Vol. 10(2), pp. 909-918.

Pandey, A. K. and Manoj, K. (2018): Effects Of Viscous Dissipation and Heat

Generation/Absorption on Nanofluid Flow Over An Unsteady Stretching Surface

With Thermal Radiation And Thermophoresis, Nanoscience and Technology An

International Journal, Vol. 9(4), pp. 325-341.

Sulochana, C., Ashwinkumar, G. P., and Sandeep, N. (2016): Similarity solution of 3D

Casson nanofluid flow over a stretching sheet with convective boundary conditions,

Journal of the Nigerian Mathematical Society, Vol. 35(1), pp. 128-141.

SUS., Choi. (1995): Enhancing thermal conductivity of fluids with nanoparticles,

developments and applications of non-Newtonian flows, ASME FED 231/MD

Vol. 66, pp. 99-105.

Tesfaye, K., Eshetu, H., Gurju, A. and Tadesse, W. (2020): Heat and Mass Transfer in

Unsteady Boundary Layer Flow of Williamson Nanofluids, Journal of Applied

Mathematics, Vol. 2020, 13 pages.

Thiagarajan, M. and Dinesh, M. (2019): Effects of thermal radiation and heat generation on

hydromagnetic flow of nanofluid over an exponentially stretching sheet in a porous

medium with viscous dissipation, World Scientific News, Vol. 128(2), pp. 130-147.

Umadevi, P. and Nithyadevi, N. (2016): Magneto-convection of water-based nanofluids inside

an enclosure having uniform heat generation and various thermal boundaries,

Journal of the Nigerian Mathematical Society, Vol. 35(1), pp. 82-92.

Waqas, M., Ijaz Khan, M., Hayat, T. and Alsaedi, A. (2017): Stratified flow of an Oldroyd-B

nanoliquid with heat generation, Results in Physics, Vol. 7, pp. 2489-2496.

INVARIANT ANALYSIS OF HEAT GENERATION

249

Shukla, H., Patel, J., Surati, H. C ., Patel, M. and Timol, M. G. (2017): Similarity solution of

forced convection flow of Powell-Eyring and Prandtl-Eyring fluids by group-

theoretic method, Mathematical Journal of Interdisciplinary Sciences, Vol. 5(2),

pp. 151- 165.

Shukla, H., Surati, H. C. and Timol, M. G. (2018): Similarity analysis of three dimensional

nanofluid flow by deductive group theoretic method, Applications and Applied

Mathematics-An International Journal, Vol. 13(2), pp. 1260-1272.

Shukla, H., Surati, H. C., and Timol, M. G. (2020): Similarity Analysis of MHD Three

Dimensional Nanofluid Flow for Non-Newtonian Power-Law Model over Linearly

Stretching Sheet with Convective Boundary Conditions, International Journal of

Heat and Technology, Vol. 38(1), pp. 203-212.

1. Department of Mathematics,

Government Engineering College,

Gandhinagar-382028, GTU, India

E-mail: hsshukla94@gmail.com

(Received, September 25, 2024)

(Revised, December 23, 2024)

2. B. Tech Student,

IIIT, Vadodara, Gandhinagar-382028, India

E-mail: shivanshidave10@gmail.com

3. Department of Instrumental and Control Engineering,

Government Engineering College,

Gandhinagar-382028, GTU, India;

E-mail: kcdave@gecg28.ac.in

4. Department of Mathematics,,

Government Engineering College,

Gandhinagar-382028, GTU, India

E-mail: ashwin.rathodmaths@gmail.com

5. Department of Mathematics,

Government Engineering College,

Gandhinagar-382028, GTU, India

E-mail: nirmaths@gmail.com

6. Department of Electrical, Engineering

Government Engineering College,

Gandhinagar-382028, GTU, India

E-mail: jignashaprajapati@gecg28.ac.in