 Open Access
 Authors : T. Rajeshwari, S. Hari Singh Naik, K. Ramesh
 Paper ID : IJERTV12IS070106
 Volume & Issue : Volume 12, Issue 07 (July 2023)
 Published (First Online): 07082023
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Numerical Analysis of Nano fluid on Stagnation Flow Past a stretching sheet in the Presence of Magneto hydrodynamics (MHD), Convective Heating in the presence of porous media
1T.Rajeshwari
DepartmentofMathematicalSciences,N.T.R Govt.degreecollegeforwomen, Telangana,India
2S. Hari Singh Naik Department of Mathematics, UCS, Osmania
University, Hyderabad, Telangana, 500007, India.
3K.Ramesh
Department of Mathematics, Sreenidhi Institute of Science and Technology, Yamnampet, Ghatkesar, Madchal, T.S, India
Abstract – In the present content an analysis is carried out to examine the effect of convective heat transfer on stagnationpoint flow of Nano fluid towards a stretched sheet through porous medium in the presence of a magnetic field. Using suitable similarity transformations, the governing boundarylayer equations corresponding to the momentum, energy and concentration are reduced to nonlinear ordinary differential equation. The transformed equations are then solved numerically using Keller Box method. The alterations in velocity, temperature and concentration for different values of physical parameters are plotted through graphs. An evaluation is made with previous results and we found a good similarity. Tables are used to analyse skin friction and the Nusselt and Sherwood values. The main findings of investigations are velocity increases as the increasing in velocity ratio parameter, porosity and decreases as increasing in magnetic parameter. The temperature profile increases with the enhancement in the values of Prandtle number and Bi finally concentration increases with the increasing values of Nt and Le and decreases with the increasing values of Brownian motion parameter.
Keywords Nanofluid, Porous media, Stagnation Flow, MHD, Convective Heating.
INTRODUCTION
The thermal characteristics of base fluids could enhance by introduce the nanoparticles, the resultant fluids are called nanofluids. The nanofluids are homogeneous mixture of very small particles of size 109 m. Incorporation the
nanoparticles of metals (Al, Cu, Ag, etc.) or metal oxides (CuO, TiO2, Al2O3, etc.) or nitride ceramics (AlN, S iN) or carbide ceramics (SiC, TiC), etc. to the based fluids like water, ethylene glycol, oil etc., the thermal conductivity is enhanced by an order of magnitude. The said nanofluids offer improved heat absorption, and also heat transfer rate as the specific heat of metal, its based particles is very low as compared to water or other liquids. Nanofluids have potential applications in engineering and biomedical, etc. like in heat exchanger, refrigerators, microelectronics, engine cooling / vehicles thermal managements, hybridpowered engines, coolants of nuclear reactor, cooling agent in airplanes, micro machines in micro reactors, automobiles, etc. Choi [1] introduced first time by usage of nanoparticles in base fluids and the results of experiments witnessed the thermal properties of fluids are enhanced through nanoparticles. Buongiorno group [2] developed a mathematical model, which exhibited the characteristic of thermophoresis and Brownian motion of nanoparticles. Later, many researchers used the nanofluids in various directions particularly for stretching sheets [36] cylinders [710] etc.
, Magnetonanofluids have specific applications in biomedicine, optical modulators, magnetic cell separation, magnetooptical wavelength filters, silk float separation,
nonlinear optical materials, hyperthermia, optical switches,
past over a linear stretching surface in the direction of x axes
drug delivery, optical gratings etc. A magnetonanofluid has both the liquid and magnetic properties. The used magnetic
with the velocity
uw (x) ax
where a is a constant, and y
field influences the suspended particles and reorganizes their concentration in the fluid regime which powerfully influences the heat transfer analysis of the flow. Magnetonanofluids are useful to guide the particles up the blood stream to a tumor with magnets. This is because the magnetic nanoparticles are regarded more adhesive to tumor cells than nonmalignant cells. Such particles absorb more power than microparticles in alternating current magnetic fields tolerable in humans i.e., for
cancer therapy. Several authors[11, 12, 13, 14] have discussed
axis is normal to the sheet. A hot fluid with temperature Tf is
applied to heat up or cool down the surface of the sheet at lower level by convective heat transfer mode, which provides the heat transfer coefficient hf . A uniform magnetic field B0
is applied in the transverse direction y normal to the surface
and the induced magnetic field is assumed to be small compared to the applied magnetic field and so neglected. The ambient fluid temperature and Nano particles fraction have
the MHD boundary layer flow, heat and mass transfer
constant value T
and C , respectively. Under the above
characteristics of nanofluids.
Motivating by above research work, the aim of this research work is to extend the research work of Ibrahim and Haq [15] and Ch.Janaiah and G.Upender Reddy [16] by excluding the effects Soret (Thermal diffusion) and Dufour (Diffusion thermo) on magnetohydrodynamic nanofluid flow towards a nonlinear stretching sheet in presence of porous media, Thermophoresis, Brownian motion and Convective heating effects using numerical solutions.
The present research work is aims to study the effect of convective heat transfer on stagnationpoint flow of nanofluid via a stretched sheet and porous medium in the presence of a magnetic field. Using suitable similarity transformations, the governing boundarylayer equations corresponding to the momentum, energy and concentration are reduced to nonlinear ordinary differential equation. The transformed equations are then solved numerically by use of KellerBox method. The alterations in velocity, temperature and concentration for different values of physical parameters are plotted. A comparison is made with the previous results and the results are found a good similarity. The evaluated skinfriction coefficients, Nusselt and Sherwood numbers are tabulated.
FORMULATION OF THE PROBLEM:
Consider a steady, laminar, incompressible electrically conducting, two dimensional flow of a nanofluid
conditions the physical coordinate system is shown in figure 1.
Figure1: Geometry representation of the fluid
The governing equations [16] of the conservation of mass, momentum, energy and nanoparticle fraction is
(1)
(2)
(3)
(14)
The effect of the terms , in equation (3)
and (4) are very negligible even though in the absence of these terms results obtained are in good agreement with the results obtained by Ch. Janaiah et al. [16] .
(5)
Where the dimensionless parameters are
Velocity Ratio Parameter A= , Nt
= ,
, Magnetic field M=
METHODOLOGY:
Stage 1: In the early stage, all ODEs must be converted to firstorder ODEs. (11) – (13).
f ' p , p' q , ' t ,' g
Momentum Equation
(15)
(16)
Equation of Thermal Energy
Equation of species Concentration
The Boundary Conditions for Nano Fluid flow are
(6)
(7)
(8)
17)
In order to solve the system of ordinary differential equations (11)(13).The Following steps are used.

Convert the system of ordinary differential equations into a set of equations of the first order.

To solve ordinary differential equations, write the difference equatios using the central differences.
For solving equations (5) to (8) we introduced new transformations
, ,
(9)
Where represents the Stream function and is defined by

Using the Newton method, linearize the algebraic equations, and then write them down in matrix form.

Use the block tridiagonal elimination method to solve the linear system. Substitute the above values in equations (15) –

and write the first order ODEs into finite differences by
( f j f j1 ) ( pj pj1 )
using
transformation and
hj 2
Using (9) the flow Eqs.(5) – (8) become
(10)
linearize the difference Equations. numerical procedure
(11)
(13)
Boundary Conditions
FIGB: FLOW DIAGRAM OF THE KELLER BOX METHOD
The study investigated the momentum, energy, and concentration equationgoverned nonNewtonian Nano fluid on Stagnation Flow Past a stretching sheet model, which is impacted by the Velocity ratio parameter, magnetic field, thermal radiation, , and chemical reaction. To solve the system
of highly nonlinear ordinary differential equations, we use a
4
3.5
A=0.2, 0.4,
0.6, 0.8, 1.0, 1.2,
1.4
3
2.5
f'()
2
1.5
1
0.5
0
Keller Box Method. When fixing the values of , , , ,,.
Fig.3 illustrates the variation in velocity with velocity ratio parameter A. Velocity ratio parameter is represents the relation of freestream velocity to stretching sheet velocity. An increase in A indicates the freestream velocity is higher than the stretching sheet velocity as a result impeding force acting on the fluid will decrease, and velocity increases. Fig.4 explains the effect of magnetic field on velocity. When the magnetic field's intensity is increased, a resistive Lorentz force is generated. The decrease in velocity profile is caused by this resistive force. Fig.5 shows how the Concentration profile is affected by the Brownian motion parameter Nb. The fluid
molecules move randomly as the Brownian motion factor Nb
1
0.9
0.8
0.7
f'()
0.6
0.5
0.4
0.3
0.2
0 1 2 3 4 5 6 7
Figure2: Effect of on velocity
M = 0.5, 0.8, 1, 1.2,
1.4
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
rises, which causes an increase in the rate of fluid conversion and a decrease in concentration. Fig.6 examines the influence of Biot number Bi on temperature. The Biot number is a measure of how much thermal resistance there is for conduction inside a body compared to convection at the surface. As Biot number increases the resistance to conduction transmission is also increases, which causes the temperature at surface increase and thickness of thermal boundary layer also enhance.
Figure3: Effect of on velocity
I.
0.1
0
Nb=0.1, 0.2, 0.3,
0.4
0.1
()
0.2
0.3
0.4
0.5
0 5 10 15
Figure4: Effect of on concentration
1
0.9
0.8
0.7
0.6
()
0.5
0.4
0.3
0.2
0.1
0
Bi = 5, 7, 9,
11
0 1 2 3 4 5 6 7
Figure5: Effect of on temperature
1
0.9
0.8
0.7
0.6
0.5
Pr = 0.025,
0.71. 3, 7, 11
0.4
0.3
0.2
0.1
Figure6: Effect of on concentration
Fig .7 demonstrates the influence of Lewis number on concentration profile. Ratio of species diffusivity to thermal
diffusivity is denoted by the Lewis number. Increase values of species diffusivity larger than thermal diffusivity as
result concentration profile increases. Fig. 8 illustrates the
influence of Prandtle number on temperature profile. The ratio of momentum diffusivity to thermal diffusivity is the
Prandtl number. An increase in the Prandtl number reduces the thickness of the thermal boundary layer.
Fig.9 depicted the impact of permeable parameter on
velocity curve. As size of the pores rises with growing values
permeable parameter as a result the velocity of the fluid is uplift. Fig.10 shows the influence of Thermophoresis
Figure7: Effect of on temperature
0
0 2 4 6 8
parameter Nt on concentration curve. It indicates the movement of the fluid particles through a fluid under the effect of temperature gradient, so that the fluid particles move form hot zone to cold zone as a result concentration of the fluid increases.
1
0.9
0.8
0.7
0.6
f'()
0.5
0.4
0.3
0.2
0.1
0
=0.35, 0.5, 0.7, 0.9
0 1 2 3 4 5 6 7
0.5
0
()
0.5
Nt = 0.1, 0.2, 0.3, 0.4
1
1.5
2
Figure8: Effect of on velocity
2.5
0 5 10 15
Figure9: Effect of on concentration
The effects of the most important factors on skinfriction, the rate of temperature transports, and the rate of mass transports are also shown in Table.1
Table I: Numerical values of skinfriction coefficient, rate of heat and mass transfer coefficient for variation of
, .
0.2
1.124723
0.437428
0.437428
0.4
0.927773
0.482356
0.482356
0.6
0.682591
0.520818
0.520818
0.5
1.124723
0.437428
0.437428
0.8
1.205664
0.431729
0.431729
1.0
1.257002
0.428215
0.428215
0.02
1.124723
0.083755
0.083755
0.71
1.114723
0.437428
0.437428
1
1.086423
0.936415
0.936415
0.1
1.098723
0.437428
0.437428
0.2
1.124563
0.436928
0.873855
0.3
1.144983
0.436427
1.309281
0.1
1.124723
0.437428
0.437428
0.2
1.124723
0.435824
0.532467
0.3
1.124723
0.421897
0.985232
0.1
1.026322
0.437428
0.437428
0.2
0.983022
0.436986
0.436986
0.3
0.966225
0.436593
0.436593
0
1.076823
0.445125
0.445125
0.3
1.214038
0.423321
0.423321
0.5
1.296371
0.410705
0.410705
5
1.083220
0.437428
0.437428
7
1.102447
0.448628
0.448628
9
1.111320
0.455102
0.455102
Ch.Janaiah[16]
Present Values
0.1
0.9588412405
1.202635
0.2
0.9052341045
1.124723
0.5
0.6533901124
0.810815
2.0
2.0095542375
2.049775
3.0
4.6822357889
4.832505
Table 2: Comparison of skinfriction coefficient results for different values of .
CONCLUSIONS:
In the present paper mainly concentrated on the influence of porous medium, velocity ratio and convective heating parameter on steady, incompressible nanofluid flow through a nonlinear stretching sheet. Using a numerical Keller – box technique, the main flow governing equations are resolved. The numerical outcomes for the concentration, temperature, and velocity profiles for the various parameters are plotted graphically and thoroughly discussed. The primary conclusions of this investigation are.

The velocity profiles are declines by the increasing magnetic parameter and raise by growing values of Velocity ratio and
Permeable parameter.

The Velocity ratio parameter, prandtle number, and Biotnuber constant all make the local Sherwood number go up.

The temperature distribution flattens out as the Prandtl number increases.

For increasing permeable parameter values, the magnitude of the skin friction parameter increases while the rates of heat and mass transfer decrease.

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