Effect of Variable Properties on Heat and Mass Transfer Flow of … · 2018-12-01 · Effect of Variable Properties on Heat and Mass Transfer Flow of Nanofluid over a Vertical Cone

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 22 (2018) pp. 15611-

© Research India Publications. http://www.ripublication.com

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Effect of Variable Properties on Heat and Mass Transfer Flow of Nanofluid

over a Vertical Cone Saturated by Porous Medium under Enhanced

Boundary Conditions

B. Kalavathamma 1* and C. Venkata Lakshmi 2

1, 2Department of Mathematics, Sri Padmavati Mahila Visvavidyalayam (Women’s University) Tirupati,

Andhra Pradesh-517502, India

*Corresponding Author

ABSTRACT

We have investigated the impact of variable viscosity and

thermal conductivity on MHD boundary layer flow, heat and

mass transfer of nanofluid over a vertical cone saturated by

porous medium with thermal radiation and chemical reaction.

Further, the viscosity and thermal conductivity are considered

as the function of nanoparticle volume fraction (𝜙). By using

suitable similarity variables the governing equations

represents the velocity, temperature and volume fraction of

nanoparticles are transformed into the set of ordinary

differential equations. These equations together with

associated boundary conditions are solved numerically by

using an optimized, extensively validated, variational Finite

element method. Effects of different parameters such as

variable viscosity, Buoyancy, magnetic, radiation, variable

thermal conductivity, thermophoresis, Brownian motion,

Lewis number and chemical reaction parameters on velocity,

temperature and concentration profiles are examined and the

results are presented in graphical from. Furthermore, the skin-

friction coefficient, Nusselt number and Sherwood number are

also investigated and are shown in tabular form.

Keywords: Nanofluid, Vertical cone, Variable viscosity,

Variable thermal conductivity, Chemical reaction, Thermal

radiation.

NOMENCLATURE

km Thermal conductivity Nux Nusselt number

𝜙 Nanoparticle volume fraction 𝜙w Nanoparticle volume fraction on the plate

𝜙∞ Ambient nanoparticle volume fraction (x, y) Cartesian coordinates

Tw Temperature at the plate T∞ Ambient temperature attained

T Temperature on the plate 𝑅𝑎𝑥 Rayleigh number

𝑞𝑤 Wall heat flux 𝐽𝑤 Wall mass flux

DB Brownian diffusion DT Thermophoretic diffusion coefficient

𝛽0 Strength of magnetic field g Gravitational acceleration vector

Nt Thermophoresis parameter Le Lewis number

P Pressure Nb Brownian motion parameter

𝑞𝑟 Thermal radiation M Magnetic parameter

𝑆ℎ𝑥 Local Sherwood number Nv Variable viscosity parameter

𝐾𝑟 Rate of chemical Reaction

Nc Variable thermal conductivity parameter Cr Chemical reaction parameter

Nr Buoyancy ratio K Permeability of the porous medium

Greek symbols:

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μ viscosity ε porosity

γ proportionality constant 𝛼𝑚 Thermal diffusivity .

𝜌𝑓 Fluid density 𝜌𝑝 Nanoparticle mass density

ψ Stream function V Kinematic viscosity of the fluid

τ Parameter defined by ε (𝜌𝑐)𝑝

(𝜌𝑐)𝑓 (ρc)f Heat capacity of the fluid

∅ (η) Dimensionless nanoparticle volume fraction η Similarity variable

θ (η) Dimensionless temperature (ρc)p Effective heat capacity of the nanoparticle

α Acute angle of the plate to the vertical β Volumetric expansion coefficient

Subscripts:

w Condition on the plate ∞ Condition far away from the plate

ɳ Similarity variable f Base fluid

1. INTRODUCTION

In recent years the field of science and technology,

nanotechnology has become more popular because of its

specific application to the arenas of electronics, fuel cells,

space, fuels, better air quality, batteries, solar cells, medicine,

cleaner water, chemical sensors and sporting goods.

Nanoparticles are the particles which are of 1-100 nm in size.

The convectional heat transfer fluids like water, oil, kerosene

and ethylene glycol have poor heat transfer capabilities due to

their low thermal conductivity. To improve the thermal

conductivity of these fluids nano/micro-sized materials are

suspended in liquids. Due to the nanofluid thermal

enhancement, performance, applications and benefits in

several important arenas, the nanofluid has contributed

significantly well in the field of microfluidics, manufacturing,

microelectronics, advanced nuclear systems, polymer

technology, transportation, medical, saving in energy.

Keeping above applications, Choi et al. [1] have reported in

his experimental study that there is 150% enhancement in

thermal conductivity when carbon nanotubes are added to the

ethylene glycol or oil. Copper nanoparticles dispersed in

ethylene glycol have higher intrinsic thermal conductivity

than the nanofluids consisting oxide particles and found that

40% increase in the thermal conductivity of nanofluids

consisting ethylene glycol with 0.3 vol % copper

nanoparticles of 10 nm diameter [2]. Buongiorno [3]

explained the thermal properties of nanofluid and noticed

Brownian diffusion and thermophoresis are main slip effects

in heat transfer process. Kuznetsov et al. [4] studied

analytically the convective flow of a nanofluid through

vertical plate using Buongiorno’s mathematical model.

Sudarsana Reddy et al.[5] have studied heat transfer process

over stretching sheet and reported that the rate of heat transfer

is rises with increasing values of nano particle volume fraction

parameter. Some captivating investigations on nanofluid flow

in permeable media can be found in [6, 7]. Chamkha et al. [8]

observed the impact of heat generation/absorption on entropy

generation of Cu-water based nanofluid flow through porous

medium with magnetic field effect and noticed that increase in

the volume fraction detracts the nusselt number and entropy

generation. Sulochana et al. [9] discussed the MHD nanofluid

flow over vertical revolving surface with Soret and radiation

effects. Sabour et al.[10] have analyzed the augmentation in

heat transfer of nanofluids in a square cavity. Thirupathi

Thumma et al. [11] investigated the heat transfer phenomena

of two types of nanofluids Cu – water and Al2O3 – water over

inclined porous plate. The results indicated that the thickness

of thermal and concentration boundary layer of Cu-water is

more as compared with Al2O3-water nanofluid. In addition to

this, the heat flow along the inclined porous plate using hybrid

approach has discussed in [12]. Javed et al. [13] used FEM to

investigate the MHD laminar flow of Cu-water nanofluid

inside a triangular cavity while left wall is heated

uniformly/non-uniformly. Recently, Dulal Pal et al. [15]

studied the Hall current effect on the magneto hydrodynamic

thermal characteristics of nanofluids over stretching as well as

shrinking surface. They found that Hall parameter reduces

temperature profiles over stretching sheet, whereas, revers

trend found in shrinking sheet. Mansour et al. [16] discussed

the effect of viscous dissipation on entropy generation due to

MHD convection flow of nanofluid in porous square

enclosure with active parts and observed that the increase in

length of the heated and cold parts leads to the reduction in

the local total entropy generation. Sudasana Reddy et al. [17]

have explored numerically the heat and mass transfer

boundary layer flow of Cu-water and Ag-water based

nanofluids (with volume fraction 10% and 30%) over vertical

surface utilizing FEM method. The heat and mass flow

phenomena of Non Newtonian fluid through vertical surface

with convective conditions using Buongiornio’s model

studied by Subba Rao [18]. Recently, The bio convection heat

transfer in porous truncated cone was carried out by Mahdy

[19]. Sudhagar et al. [20] investigated the effects of different

parameters on laminar, mixed convection boundary flow over

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vertical cone filled with a nanofluid for the application of loud

speaker cooling ,in which Nur decreases with increase of M

ultimately it leads to the decrease of the voice coil

temperature in the loud speaker. Based on previous works,

Prabhavathi et al. [21] explained the flow of two different

nanofluids Al2O3-water and Cu-water in the presence of heat

generation/absorption. Siddiqa et al. [22] investigated the

presence of dust particles in nanofluid along the vertical wavy

frustum of a cone. Some interesting works on convection flow

can be found in [23, 24]. Since the bio convection has its

applications in bio-microsystems, the numerical solution to

the bio convection flow of nanofluid containing gyrotactic

microorganisms through vertical wavy cone was found by

Siddiqa et al. [25] and conclude that the amplitude of the

wavy surface of the cone and half cone angle has dominated

effect on heat and mass transfer coefficients as well as density

number of the microorganisms. Chandra Shekar Balla et al.

[26] explained the Finite element analysis of unsteady

convection flow along a vertical cone by considering both

magnetic field and thermal radiation and conclude that

velocity, temperature profiles decreases with magnetic

parameter. On the other hand, the temperature profile

depreciates with increase in radiation parameter. Hady et al.

[27] presented Non-Darcy convective flow along a vertical

cone filled with porous medium. Ramana et al. [28] studied

the behavior of MHD dirty nanofluids flow. They

recognized the enhancement in heat transfer in Cu-water dusty

nanofluid than Al2O3-water dusty nanofluid. The effects of

nano particle volume fraction and the nano particles diameter

on thermal conductivity of Al2O3-water nanofluid utilizing

the drift-flux model by Ghalambaz [29] and reported that

decrease in the size of nano particles or increase in the volume

fraction of nanoparticles reduces the rate of heat transfer from

the cone when the cone surface is hot. Khan et al. [30]

investigated for the solution of natural-convective flow along

both vertical cone and plate embedded in porous media filled

with power law nanofluids by considering convective

boundary conditions. In addition to this, different aspects

related to flow and heat transfer have been discussed in

[31,32&34-36]. Carbon nanotubes are the best heat

conducting materials and are the best choice in

nanotechnology. Because of this, Sreedevi et al. [33] have

analyzed the MHD flow of single and multi-walled carbon

nanotubes about vertical cone with convective boundary

conditions.

2. MATHEMATICAL ANALYSIS OF THE PROBLEM:

We assume a steady two-dimensional viscous incompressible

natural convection nanofluid flow over a vertical cone

embedded by saturated porous medium in the presence of

variable viscosity and thermal conductivity with the

coordinate system given in Fig.1. The fluid is assumed an

electrically conducted through a non-uniform magnetic field

of strength 𝐵0 is applied in the direction normal to the surface

of the cone. It is assumed that 𝑇𝑤, and 𝜙𝑤 are the temperature

and nanoparticle volume fraction at the surface of the cone

(𝑦 = 0) and 𝑇∞ and 𝜙∞ are the temperature and nanoparticle

volume fraction of the ambient fluid, respectively. By

employing the Oberbeck - Boussinesq approximation the

governing equations describing the steady-state conservation

of mass, momentum, energy as well as conservation of

nanoparticles for nanofluids take the following form:

Fig.1. Physical model and coordinate system

𝜕(𝑟𝑢)

𝜕𝑥+

𝜕(𝑟𝑣)

𝜕𝑦 = 0 (1)

𝜕𝑝

𝜕𝑦= 0 (2)

𝜕𝑝

𝜕𝑥 = −

𝜇(𝜙)

𝐾𝑢 + 𝑔 [(1 − 𝜙∞)𝜌𝑓∞𝛽(𝑇 − 𝑇∞) − (𝜌𝑝 −

𝜌𝑓∞)(𝜙 − 𝜙∞)] 𝑐𝑜𝑠 𝛾 −𝜎𝛽𝑜

2

𝜌𝑓𝑢 (3)

(𝑢𝜕𝑇

𝜕𝑥+ 𝑣

𝜕𝑇

𝜕𝑦) =

1

(𝜌𝑐𝑝)𝑛𝑓

𝜕

𝜕𝑦(𝑘𝑚(𝜙)

𝜕𝑇

𝜕𝑦) +

(𝜌𝑐)𝑝

(𝜌𝑐𝑝)𝑛𝑓

[𝐷𝐵𝜕𝜙

𝜕𝑦 .

𝜕𝑇

𝜕𝑦+

(𝐷𝑇

𝑇∞) (

𝜕𝑇

𝜕𝑦)

2

] − 1

(𝜌𝑐𝑝)𝑛𝑓

𝜕

𝜕𝑦(𝑞𝑟) (4)

1

𝜀(𝑢

𝜕𝜙

𝜕𝑥+ 𝑣

𝜕𝜙

𝜕𝑦) = 𝐷𝐵

𝜕2𝜙

𝜕𝑦2 + (𝐷𝑇

𝑇∞)

𝜕2𝑇

𝜕𝑦2 − 𝐾𝑟(𝜙 − 𝜙∞) (5)

The boundary conditions based on the problem description are

as follows:

𝑢 = 0, 𝑇 = 𝑇𝑤 , 𝐷𝐵𝜕𝜙

𝜕𝑦+ (

𝐷𝑇

𝑇∞)

𝜕𝑇

𝜕𝑦= 0 𝑎𝑡 𝑦 = 0 (6)

𝑢 → 0, 𝑇 ⟶ 𝑇∞, 𝜙 ⟶ 𝜙∞ 𝑎𝑡 𝑦 → ∞ (7)

In the present study, the viscosity and thermal conductivity

are taken as the reciprocal and a linear function of

nanoparticle volume fraction, respectively. Therefore, the

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viscosity in terms of nanoparticle volume fraction can be

written as follows:

1

𝜇=

1

𝜇∞[1 + 𝛾(𝜙 − 𝜙∞)] (8)

The above equation can be simplified as

1

𝜇= 𝑚𝜇(𝜙 − 𝜙𝑟) (9)

Where, 𝑚𝜇 and 𝜙𝑟 are can be defined as

𝑚𝜇 =𝛾

𝜇∞ and 𝜙𝑟 = 𝜙∞ −

1

𝛾 .

In the above equations (8) and (9), 𝜇∞, 𝑚𝜇, 𝜙∞, 𝜙𝑟, and γ are

constant values. The thermal conductivity as a function of

volume fraction of nanoparticle is defined as

𝑘𝑚(𝜙) = 𝑘𝑚,∞(1 + 𝑚𝑘(𝜙 − 𝜙∞)) (10)

Where, 𝑚𝑘 =𝑁𝑐

(𝜙𝑤−𝜙∞) and Nc is the variable thermal

conductivity parameter, 𝑘𝑚,∞ is the effective thermal

conductivity. By using Rosseland approximation for radiation

[34], the radiative heat flux 𝑞𝑟 is defined as

𝑞𝑟 = −4𝜎∗

3𝐾∗

𝜕𝑇4

𝜕𝑦 (11)

where 𝜎∗ is the Stephan-Boltzman constant, 𝐾∗ is the mean

absorption coefficient. We assume that the temperature

differences within the flow are such that the term 𝑇4 may be

expressed as a linear function of temperature. This is

accomplished by expanding 𝑇4 in a Taylor series about the

free stream temperature 𝑇∞ as follows:

𝑇4 = 𝑇∞4 + 4𝑇∞

3(𝑇 − 𝑇∞) + 6𝑇∞2(𝑇 − 𝑇∞)2 + ⋯ (12)

Neglecting higher-order terms in the above equation (12)

beyond the first degree in (𝑇 − 𝑇∞), we get

𝑇4 ≅ 4𝑇∞3𝑇 − 3𝑇∞

4 . (13)

Thus, substituting Eq. (13) into Eq. (11), we get

𝑞𝑟 = −16𝑇∞

3𝜎∗

3𝐾∗

𝜕𝑇

𝜕𝑦. (14)

The continuity equation (1) is satisfied by introducing a

stream function (𝜓) as

𝑢 =1

𝑟

𝜕ψ

𝜕𝑦, 𝑣 = −

1

𝑟

𝜕ψ

𝜕𝑥 (15)

The following similarity transformations are introduced to

simplify the mathematical analysis of the problem

η = 𝑦

𝑥𝑅𝑎𝑥

12⁄

, 𝑓(𝜂) =𝜓

𝛼𝑚.𝑟.𝑅𝑎𝑥

12⁄

, 𝜃(𝜂) = 𝑇−𝑇∞

𝑇𝑤−𝑇∞ , ∅(𝜂) =

𝜙−𝜙∞

𝜙𝑤−𝜙∞ , 𝑁𝑣 =

𝜙𝑟−𝜙∞

𝜙𝑤−𝜙∞ (16)

Where 𝑅𝑎𝑥 is the local Rayleigh number and is defined as

𝑅𝑎𝑥 =𝑔𝛽𝐾𝜌𝑓∞(1 − 𝜙∞)(𝑇𝑤−𝑇∞) 𝑥 𝐶𝑜𝑠𝛾

𝜇∞𝛼𝑚 (17)

and ‘ r ’ can be approximated by the local radius of the cone,

if the thermal boundary layer is thin, and is related to the x

coordinate by 𝑟 = 𝑥 𝑠𝑖𝑛𝛾.

Using the similarity variables (16) and making use of Eqn.

(14), the governing equations (3) - (5) together with boundary

conditions (6) and (7) reduce to

𝑁𝑣(𝑁𝑣 − ∅)𝑓 ′′ + 𝑁𝑣 𝑓 ′∅′ − (𝑁𝑣 − ∅)2(𝜃 ′ − 𝑁𝑟 ∅′) −

(𝑁𝑣 − ∅)2𝑀𝑓 ′ = 0 (18)

(1 + 𝑅)𝜃 ′′ + 𝑁𝑐 ∅ 𝜃 ′′ +3

2 𝑓 𝜃 ′ + 𝑁𝑐 𝜃 ′∅′ + 𝑁𝑡(𝜃 ′)2 +

𝑁𝑏 𝑓′𝜃′ = 0 (19)

∅′′ +3

2 𝐿𝑒 𝑓 ∅′ +

𝑁𝑡

𝑁𝑏 𝜃 ′′ − 𝐶𝑟 ∅ = 0 (20)

The transformed boundary conditions are

η = 0, 𝑓 = 0 , 𝜃 = 1 , Nb 𝑓 ′ + Nt 𝜃′ = 0.

η ⟶ ∞, 𝑓′ = 0 , 𝜃 = 0 , ∅ = 0. (21)

where prime denotes differentiation with respect to η, and the

key thermophysical parameters dictating the flow dynamics

are defined by

𝑁𝑟 =(𝜌𝑝 − 𝜌𝑓∞)(𝜙𝑤 − 𝜙∞)

𝜌𝑓∞𝛽(𝑇𝑤 − 𝑇∞)(1 − 𝜙∞), 𝑁𝑏 =

𝜀𝛽(𝜌𝑐)𝑝𝐷𝐵(𝜙𝑤 − 𝜙∞)

(𝜌𝑐)𝑓 𝛼𝑚,

𝑁𝑡 =𝜀(𝜌𝑐)𝑝𝐷𝑇(𝑇𝑤−𝑇∞)

(𝜌𝑐)𝑓 𝛼𝑚𝑇∞

, 𝐿𝑒 = 𝛼𝑚

𝜀𝐷𝐵 ,

𝑁𝑣 = −1

𝛾(𝜙𝑤−𝜙∞) , 𝑀 =

𝜎𝛽02𝑥

𝜌𝑅𝑎𝑥1/2 , 𝑅 =

16𝑇∞3𝜎∗

3𝐾∗𝑘 , 𝐶𝑟 =

𝐾𝑟𝑥2

𝐷𝐵𝑅𝑎𝑥.

Quantities of practical interest in this problem are local

Nusselt number Nux, and the local Sherwood number Shx, ,

which are defined as

𝑁𝑢𝑥 = 𝑥𝑞𝑤

𝑘𝑚 ,∞(𝑇𝑤−𝑇∞) , 𝑆ℎ𝑥 =

𝑥𝐽𝑤

𝐷𝑩(𝜙𝑤−𝜙∞) (22)

Here, 𝑞𝑤 is the wall heat flux and 𝐽𝑤 is the wall mass flux.

3. NUMERICAL METHOD OF SOLUTION:

The Finite-element method (FEM) is such a powerful method

for solving ordinary differential equations and partial

differential equations. The basic idea of this method is

dividing the whole domain into smaller elements of finite

dimensions called finite elements. This method is such a good

numerical method in modern engineering analysis, and it can

be applied for solving integral equations including heat

transfer, fluid mechanics, chemical processing, electrical

systems, and many other fields. Steps involved in finite-

element method [34, 35,36] are as follows.

(i) Finite-element discretization

(ii) Generation of the element equations

(iii) Assembly of element equations

(iv) Imposition of boundary conditions

(v) Solution of assembled equations

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4. RESULTS AND DISCUSSION

Numerical examination of the boundary value problem (18) –

(20) together with boundary conditions (21) has been

conducted at the eight influenced parameters to deliver the

physical incite of the flow problem and the results are shown

graphically from Figs. 2 – 21. The results obtained in the

present study are compared with the results of Noghrehabadi

et al. [31] and are shown in table 1. Thus it is very clear from

table 1 that the results of the present study are in close

agreement with the results published previously.

Table 1: Comparison of −𝜃′(0) with previously published

work with fixed values of 𝑁𝑡 = 10−6, 𝑁𝑏 = 10−5, 𝑁𝑟 =10−3, 𝑁𝑐 = 0.

Parameter Noghrehabadi

et al. [31]

Present

Study

Nv Le −𝜃 ′(0) −𝜃 ′(0)

2.0

10.0

20.0

200.0

1000

1000

1000

1000

0.7584

0.7670

0.7680

0.7688

0.7591

0.7675

0.7686

0.7694

It is clearly noticed from Fig.2 that velocity profiles elevated

by the variable viscosity parameter (Nv) in the vicinity of the

cone surface. But, in the areas far away from the cone surface,

inside the boundary layer, the velocity profiles are poorly

affected by (Nv). The temperature distributions are

decelerated in the boundary layer region (Fig. 3) with the

rising values of (Nv). It is noted from Fig. 4 that nanoparticle

concentration distributions are elevated significantly from the

surface of the cone into the boundary layer as the values of

(Nv) increased.

It is observed from fig. 5 that the thickness of hydrodynamic

boundary layer is reduced with the enhancing values of

buoyancy ratio parameter (Nr). The temperature profiles of

the fluid rises with increasing values of (Nr) as shown in fig.

6. The concentration profiles enhances throughout the fluid

region with improving values of (Nr). This is because of the

fact that solutal boundary layer thickness elevates with

increasing values of Nr (Fig.7).

The impact of magnetic parameter (M) on velocity (𝑓′),

temperature (θ) and concentration of nanoparticle (∅) profiles

are depicted in Figs. 8 – 10. The thickness of hydrodynamic

boundary layer decelerates, whereas, the thickness of thermal

boundary layer heightens as the values of M rises. The

concentration of the fluid has changed its behavior at 𝜂 = 1.3,

fluids concentration diminishes when 𝜂 < 1.3, elevates when

𝜂 > 1.3 with enhance in the values of M (Fig 10).

With the higher values of radiation parameter (R) fluids

velocity and temperature rises in the boundary layer regime.

This is because of the fact that imposing thermal radiation into

the flow warmer the fluid, which causes an increment in the

thickness of hydrodynamic and thermal boundary layer in the

entire flow region (Fig.11 & 12). An increase in the values of

variable thermal conductivity parameter (Nc) the velocity,

temperature and concentration distributions deteriorates in the

fluid region as shown in figs. 13 – 15. This is because of the

fact that the thermal conductivity of the nanofluid decelerates

near the cone surface as the values of variable thermal

conductivity parameter (Nc) rises.

Figures 16 and 17 depict the temperature (θ) and

concentration (∅) distributions for various values of

thermophoretic parameter (Nt). Both the temperature and

concentration profiles elevate in the boundary layer region for

the higher values of thermophoretic parameter (Nt). Figures

18 – 19 shows the influence of Brownian motion parameter

(Nb) on thermal and solutal boundary layer thickness. It is

noticed that, with the increasing values of Brownian motion

parameter (Nb) the temperature of the fluid decelerates in the

boundary layer regime (Fig.18). Furthermore, the

concentration profiles are decelerated in the fluid regime as

the values of Nb rises (Fig. 19).

Fig.2: Effect of Nv on Velocity profiles.

Fig.3: Effect of Nv on Temperature profiles.

Nv 1.0, 3.0, 5.0, 7.0

R 0.1, Nt 0.5, Nr 0.5,

Le 5.0, Nb 0.5, M 0.5,

Cr 0.1, Nc 0.1.

1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

f

Nv 1.0, 3.0, 5.0, 7.0

R 0.1, Nt 0.5, Nr 0.5,

Le 5.0, Nb 0.5, M 0.5,

Cr 0.1, Nc 0.1.

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

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Fig.4: Effect of Nv on Concentration profiles.

Fig.5: Effect of Nr on Velocity profiles.

Fig.6: Effect of Nr on Temperature profiles.

Fig.7: Effect of Nr on Concentration profiles.

Fig.8: Effect of M on Velocity profiles.

Fig.9: Effect of M on Temperature profiles.

Nv 1.0, 3.0, 5.0, 7.0

R 0.1, Nt 0.5, Nr 0.5,

Le 5.0, Nb 0.5, M 0.5,

Cr 0.1, Nc 0.1.

1 2 3 4

0.10

0.05

0.05

0.10

Nr 0.1, 0.4, 0.7, 1.0

R 0.1, Nt 0.5, Nc 0.1,

Cr 0.1, Nb 0.5, M 0.5,

Nv 3.0, Le 5.0.

1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

1.2

f

Nr 0.1, 0.4, 0.7, 1.0

R 0.1, Nt 0.5, Nc 0.1,

Cr 0.1, Nb 0.5, M 0.5,

Nv 3.0, Le 5.0.

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

Nr 0.1, 0.4, 0.7, 1.0

R 0.1, Nt 0.5, Nc 0.1,

Cr 0.1, Nb 0.5, M 0.5,

Nv 3.0, Le 5.0.

1 2 3 4

0.10

0.05

0.05

M 0.5, 1.2, 1.9, 2.5

Nt 0.5, Nb 0.5, Nc 0.1,

Cr 0.1, Nr 0.5, R 0.1

Nv 3.0, Le 5.0.

1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

f

M 0.5, 1.2, 1.9, 2.5

Nt 0.5, Nb 0.5, Nc 0.1,

Cr 0.1, Nr 0.5, R 0.1

Nv 3.0, Le 5.0.

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 22 (2018) pp. 15611-

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15617

Fig.10: Effect of M on Concentration profiles.

Fig.11: Effect of R on Velocity profiles.

Fig.12: Effect of R on Temperature profiles.

Fig.13: Effect of Nc on Velocity profiles.

Fig.14: Effect of Nc on Temperature profiles.

Fig.15: Effect of Nc on Concentration profiles.

M 0.5, 1.2, 1.9, 2.5

Nt 0.5, Nb 0.5, Nc 0.1,

Cr 0.1, Nr 0.5, R 0.1

Nv 3.0, Le 5.0.

1 2 3 4

0.10

0.05

0.05

R 0.1, 0.3, 0.5, 0.7

Nt 0.5, Nb 0.5, Nc 0.1,

Cr 0.1, Nr 0.5, M 0.5,

Nv 3.0, Le 5.0.

1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

f

R 0.1, 0.3, 0.5, 0.7

Nt 0.5, Nb 0.5, Nc 0.1,

Cr 0.1, Nr 0.5, M 0.5,

Nv 3.0, Le 5.0.

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

Nc 0.5, 1.2, 1.9, 2.5

R 0.1, Nt 0.5, Nr 0.5,

Cr 0.1, Nb 0.5, M 0.5,

Nv 3.0, Le 5.0.

1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

1.2

f

Nc 0.5, 1.2, 1.9, 2.5

R 0.1, Nt 0.5, Nr 0.5,

Cr 0.1, Nb 0.5, M 0.5,

Nv 3.0, Le 5.0.

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

Nc 0.5, 1.2, 1.9, 2.5

R 0.1, Nt 0.5, Nr 0.5,

Cr 0.1, Nb 0.5, M 0.5,

Nv 3.0, Le 5.0.

1 2 3 4

0.10

0.05

0.05

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 22 (2018) pp. 15611-

© Research India Publications. http://www.ripublication.com

15618

Fig.16: Effect of Nt on Temperature profiles.

Fig.17: Effect of Nt on Concentration profiles.

Fig.18: Effect of Nb on Temperature profiles.

Fig.19: Effect of Nb on Concentration profiles.

Fig.20: Effect of Le on Concentration profiles.

Fig.21: Effect of Cr on Concentration profiles.

It is observed from fig. 20 that concentration distributions

decelerate with the increasing values of the Lewis number

(Le) in the entire boundary layer region. By definition, the

Lewis number represents the ratio of thermal diffusivity to the

mass diffusivity. Increasing the Lewis number means a

higher thermal diffusivity and a lower mass diffusivity, and

this produces thinner thermal and concentration boundary

layers. It is noticed from Fig.21 that concentration profiles are

Nt 0.1, 0.2, 0.3, 0.4

R 0.1, Nb 0.5, Nc 0.1,

Cr 0.1, Nr 0.5, M 0.5,

Nv 3.0, Le 5.0.

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

Nt 0.1, 0.2, 0.3, 0.4

R 0.1, Nb 0.5, Nc 0.1,

Cr 0.1, Nr 0.5, M 0.5,

Nv 3.0, Le 5.0.

1 2 3 4

0.04

0.02

0.02

0.04

0.06

Nb 0.3, 0.5, 0.7, 0.9

R 0.1, Nt 0.5, Nc 0.1,

Cr 0.1, Nr 0.5, M 0.5,

Nv 3.0, Le 5.0.

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0 Nb 0.3, 0.5, 0.7, 0.9

R 0.1, Nt 0.5, Nc 0.1,

Cr 0.1, Nr 0.5, M 0.5,

Nv 3.0, Le 5.0.

1 2 3 4

0.05

0.05

0.10

0.15

Le 5.0, 7.0, 9.0, 11.0

R 0.1, Nt 0.5, Nr 0.5,

Cr 0.1, Nb 0.5, M 0.5,

Nv 3.0, Nc 0.1.

1 2 3 4

0.04

0.02

0.02

0.04

0.06

Cr 0.1, 0.3, 0.5, 0.7

R 0.1, Nt 0.5, Nr 0.5,

Le 5.0, Nb 0.5, M 0.5,

Nv 3.0, Nc 0.1.

1 2 3 4

0.05

0.05

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 22 (2018) pp. 15611-

© Research India Publications. http://www.ripublication.com

15619

highly influenced by the chemical reaction parameter and

declines in the fluid region.

It is noticed from table 2 that the magnitude of skin-friction

coefficient, Nusselt and Sherwood numbers are amplified in

the fluid regime as the values of variable viscosity parameter

(Nv) increases. The non-dimensional velocity and heat

transfer rates upsurges, whereas, mass transfer rate diminishes

with the higher values of (Nc). It is obvious from table 2 that

the dimensionless rates of velocity escalates, whereas, rates of

heat and mass transfer are both deteriorates with the higher

values of (Nt). It is also seen from this table that the values of

skin-friction coefficient and Nusselt number decelerates,

whereas, Sherwood number values enhances in the fluid

region as the values of (Nb) rises. Higher the values of

chemical reaction parameter (Cr) lower the values of non-

dimensional velocity, heat and mass transfer rates.

Table 2: Effect of various parameters on local skin-friction

co-efficient (𝐶𝑓), local Nusselt number ( 𝑁𝑢𝑥) and local

Sherwood number (𝑆ℎ𝑥).

Parameters −𝑓 ′′(0) −𝜃′(0) −𝜑′(0)

Nv Nc Nt Nb Cr

1.0

3.0

5.0

7.0

3.0

3.0

3.0

3.0

3.0

3.0

3.0

3.0

3.0

3.0

3.0

3.0

3.0

3.0

3.0

3.0

0.1

0.1

0.1

0.1

0.5

1.2

1.9

2.5

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.1

0.2

0.3

0.4

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.3

0.5

0.7

0.9

0.5

0.5

0.5

0.5

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.3

0.5

0.7

0.891364

1.117950

1.133760

1.120990

1.152860

1.163740

1.178620

1.198130

0.988353

1.029980

1.070410

1.109660

1.279170

1.147720

1.090000

1.057610

0.896364

0.893576

0.886547

0.876047

0.569568

0.662561

0.691567

0.691253

0.742135

0.897660

1.186630

1.815350

0.708372

0.701997

0.695130

0.687843

0.691365

0.680198

0.674447

0.670993

0.574568

0.572239

0.570990

0.568162

-0.502992

-0.815183

-0.937933

-0.992331

-0.484589

-0.603443

-1.891771

-1.998438

-0.141674

-0.280799

-0.417078

-0.550274

-1.152280

-0.680198

-0.481748

-0.372774

-0.299253

-0.311181

-0.466227

-0.562668

5. CONCLUSIONS

MHD natural convection boundary layer flow, heat and mass

transfer characteristics over a vertical cone embedded in a

porous medium saturated by a nanofluid under the impact of

variable viscosity, variable thermal conductivity, thermal

radiation and chemical reaction is investigated in this

research. The hydrodynamic, thermal and solutal boundary

layers thickness were analyzed for various values of the

pertinent parameters and the results are shown in figures.

Furthermore, the impact of these parameters on Nusselt

number and Sherwood number are also calculated. The

important findings of the present study are summarized as

follows.

(i) The hydrodynamic boundary layer thickness

heightens, whereas, the thickness of thermal

boundary layer decelerates as the values of (Nv)

rises.

(ii) With the increasing values of (Nc) the values of skin-

friction coefficient and Nusselt number elevates in

the fluid region.

(iii) Both the temperature and concentration profiles

elevates in the boundary layer regime as the values of

(Nt) increases.

(iv) With increase in the values (Nb) the temperature and

concentration distributions of the fluid, deteriorates.

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