This document discusses a numerical study of the effect of thermal radiation on free convection boundary layer flow over a vertical wavy cone. The governing equations for steady, laminar, two-dimensional flow are presented and non-dimensionalized. These equations are then solved using the Mathematica technique. Graphs of the dimensionless temperature, velocity, skin friction coefficient, and Nusselt number are generated for various values of the Prandtl number, radiation parameter, surface wave amplitude, and cone half-angle. The results are discussed to analyze the impact of thermal radiation on the flow and heat transfer characteristics.
Boundary Layer Flow in the Vicinity of the Forward Stagnation Point of the Sp...iosrjce
Exact solutions are important not only in its own right as solution of particular flows, but also serve as accuracy check for numerical solution. Exact solution of the Navier-Strokes equation are, for example, those
of steady and unsteady flows near a stagnation point, Stagnation point flows can either be viscous or inviscid,
steady or unsteady, two dimensional or three dimensional, normal or oblique and forward or reverse. The classic problems of two dimensional and three dimensional stagnation point flow are associated with the names of Hiemenz and Homan A novel radial stagnation point flow impinging axi symmetrically on a circular cylinder was reported by Wang. The present paper deals with the laminar boundary layer flow and heat transfer in the
stagnation region of a rotating and translating sphere with uniform magnetic fields. The governing equations of
flow are derived for ξ = 0 (t*=0) and ξ=1 (t*→∞) and solutions in the closed form are obtained. The
temperature and velocity fields for ξ = 0 are numerically computed. This shows that the thermal boundary layer
thickness decreases as Prandtl number Princreases.The surface heat transfer (28) increases with the Prandtl
number Pr. The surface heat transfer (28) at the starting of motion is found to be strangely dependent on the
Prandtl number Pr. But it is dependent of magnetic field, buoyancy force Bp and Rotation Parameter Ro.
The effect of magnetic field on the boundary layer flow over a stretching she...IAEME Publication
The boundary layer flow created due to a linearly stretching sheet in a nanofluid is studied numerically. The boundary value problem consisting of nonlinear partial differential equations are converted into nonlinear ordinary differential equations, using similarity transformation and are solved numerically using Runge-Kutta Fourth order method, with shooting technique. The transport equations include the effects of Brownian motion and thermophoresis.
Naiver strokes equations :- These balance equations arise from applying Issac Newton’s second law to fluid motion , together with assumption that the stress in the fluid is the sum of a diffusing viscous and a pressure term- hence describing viscous flow.
Boundary Layer Flow in the Vicinity of the Forward Stagnation Point of the Sp...iosrjce
Exact solutions are important not only in its own right as solution of particular flows, but also serve as accuracy check for numerical solution. Exact solution of the Navier-Strokes equation are, for example, those
of steady and unsteady flows near a stagnation point, Stagnation point flows can either be viscous or inviscid,
steady or unsteady, two dimensional or three dimensional, normal or oblique and forward or reverse. The classic problems of two dimensional and three dimensional stagnation point flow are associated with the names of Hiemenz and Homan A novel radial stagnation point flow impinging axi symmetrically on a circular cylinder was reported by Wang. The present paper deals with the laminar boundary layer flow and heat transfer in the
stagnation region of a rotating and translating sphere with uniform magnetic fields. The governing equations of
flow are derived for ξ = 0 (t*=0) and ξ=1 (t*→∞) and solutions in the closed form are obtained. The
temperature and velocity fields for ξ = 0 are numerically computed. This shows that the thermal boundary layer
thickness decreases as Prandtl number Princreases.The surface heat transfer (28) increases with the Prandtl
number Pr. The surface heat transfer (28) at the starting of motion is found to be strangely dependent on the
Prandtl number Pr. But it is dependent of magnetic field, buoyancy force Bp and Rotation Parameter Ro.
The effect of magnetic field on the boundary layer flow over a stretching she...IAEME Publication
The boundary layer flow created due to a linearly stretching sheet in a nanofluid is studied numerically. The boundary value problem consisting of nonlinear partial differential equations are converted into nonlinear ordinary differential equations, using similarity transformation and are solved numerically using Runge-Kutta Fourth order method, with shooting technique. The transport equations include the effects of Brownian motion and thermophoresis.
Naiver strokes equations :- These balance equations arise from applying Issac Newton’s second law to fluid motion , together with assumption that the stress in the fluid is the sum of a diffusing viscous and a pressure term- hence describing viscous flow.
HOW TO PREDICT HEAT AND MASS TRANSFER FROM FLUID FRICTIONbalupost
In this paper, the „Generalized Lévêque Equation (GLE)“, which allows to calculate heat or mass transfer coefficients – or the corresponding Nusselt and Sherwood numbers – from frictional pressure drop or friction forces in place of the flow rates or Reynolds numbers is used in external flow situations, such as a single sphere or a single cylinder in cross flow.
Fluid Dynamics describes the physics of fluids at level of Undergraduate in science (physics, math, engineering). For comments or improvements please contact solo.hermelin@gmail.com. Thanks.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Heat Transfer in the flow of a Non-Newtonian second-order fluid over an enclo...IJMERJOURNAL
ABSTRACT : The problem of the heat transfer in the flow of an incompressible non-Newtonian second-order fluid over an enclosed torsionally oscillating discs in the presence of the magnetic field has been discussed. The obtained differential equations are highly non-linear and contain upto fifth order derivatives of the flow and energy functions. Hence exact or numerical solutions of the differential equations are not possible subject to the given natural boundary conditions; therefore the regular perturbation technique is applied. The flow functions 퐻, 퐺, 퐿 and 푀 are expanded in the powers of the amplitude (taken small) of the oscillations. The behaviour of the temperature distribution at different values of Reynolds number, phase difference, magnetic field and second-order parameters has been studied and shown graphically. The results obtained are compared with those for the infinite torsionally oscillating discs by taking the Reynolds number of out-flow 푅푚 and circulatory flow 푅퐿 equals to zero. Nusselt number at oscillating and stator disc has also been calculated and its behaviour is represented graphically.
Numerical Study of Forced Convection in a Rectangular Channel
Original Research Article
Journal of Chemistry and Materials Research Vol. 1 (1), 2014, 7–11
Salim Gareh
m - projective curvature tensor on a Lorentzian para – Sasakian manifoldsIOSR Journals
In this paper we studied m-projectively flat, m-projectively conservative, 𝜑-m-projectively flat LP-Sasakian manifold. It has also been proved that quasi m- projectively flat LP-Sasakian manifold is locally isometric to the unit sphere 𝑆𝑛(1) if and only if 𝑀𝑛 is m-projectively flat.
HOW TO PREDICT HEAT AND MASS TRANSFER FROM FLUID FRICTIONbalupost
In this paper, the „Generalized Lévêque Equation (GLE)“, which allows to calculate heat or mass transfer coefficients – or the corresponding Nusselt and Sherwood numbers – from frictional pressure drop or friction forces in place of the flow rates or Reynolds numbers is used in external flow situations, such as a single sphere or a single cylinder in cross flow.
Fluid Dynamics describes the physics of fluids at level of Undergraduate in science (physics, math, engineering). For comments or improvements please contact solo.hermelin@gmail.com. Thanks.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Heat Transfer in the flow of a Non-Newtonian second-order fluid over an enclo...IJMERJOURNAL
ABSTRACT : The problem of the heat transfer in the flow of an incompressible non-Newtonian second-order fluid over an enclosed torsionally oscillating discs in the presence of the magnetic field has been discussed. The obtained differential equations are highly non-linear and contain upto fifth order derivatives of the flow and energy functions. Hence exact or numerical solutions of the differential equations are not possible subject to the given natural boundary conditions; therefore the regular perturbation technique is applied. The flow functions 퐻, 퐺, 퐿 and 푀 are expanded in the powers of the amplitude (taken small) of the oscillations. The behaviour of the temperature distribution at different values of Reynolds number, phase difference, magnetic field and second-order parameters has been studied and shown graphically. The results obtained are compared with those for the infinite torsionally oscillating discs by taking the Reynolds number of out-flow 푅푚 and circulatory flow 푅퐿 equals to zero. Nusselt number at oscillating and stator disc has also been calculated and its behaviour is represented graphically.
Numerical Study of Forced Convection in a Rectangular Channel
Original Research Article
Journal of Chemistry and Materials Research Vol. 1 (1), 2014, 7–11
Salim Gareh
m - projective curvature tensor on a Lorentzian para – Sasakian manifoldsIOSR Journals
In this paper we studied m-projectively flat, m-projectively conservative, 𝜑-m-projectively flat LP-Sasakian manifold. It has also been proved that quasi m- projectively flat LP-Sasakian manifold is locally isometric to the unit sphere 𝑆𝑛(1) if and only if 𝑀𝑛 is m-projectively flat.
To study the factors effecting sales of leading tractor brands in Haryana (In...IOSR Journals
Every aspect of the economic life in India is influenced by the agriculture. Agriculture contributes nearly 32% of the national income of India and it offers live hood nearly 70% of the total population and the agriculture is influenced by the tractors industry. Tractor industry plays an important role on the development of agriculture. Indian tractor market is very complex so marketer must care in analysing consumer behaviour. Green Revolution in India had its origin in northern India where Haryana is situated. Thus Haryana’s Contribution to Green Revolution in India is the maximum, In 1966-67 production of food grains in Haryana was 2090 thousand tones. In 1970-71 it increased to 3939 thousand tones and in 1994-2000 it further rose to 131 lakh tones, all this due to the development of tractor manufacturing industries like FARMTRAC, HMT, EICHER, TAFE etc. Present work covers studying sales of different tractor brands in Haryana (India) and how various brands have become the choice of agriculturist on the basis of getting experienced by others. The best brand so for is found to be FARMTRAC by agriculturist by the recommendation of relatives who have experinecd the same. It was depicted from the studies that farmers purchasing tractors by recommendations of relatives are not much educated.
The electronic band parameters calculated by the Triangular potential model f...IOSR Journals
This work reports on theoretical investigation of superlattices based on Cd1-xZnxS quantum dots
embedded in an insulating material. This system, assumed to a series of flattened cylindrical quantum dots with
a finite barrier at the boundary, is studied using the triangular potential. The electronic states and the effective
mass of 1 Γ miniband have been computed as a function of inter-quantum dot separation for different zinc
compositions. Calculations have been made for electrons, heavy holes and light holes. Results are discussed and
compared with those of the Kronig-Penney and sinusoidal potentials
Spectrophotometric determination of a few commercial drugs using NBS and Rhod...IOSR Journals
Simple, sensitive and selective methods are developed for the spectrophotometric determination of drugs, viz., Montelukast sodium, Prasugrel, Ondensetron, Rosuvastatin calcium, Amlodepine besylate based on their reactivity towards N- bromosuccinimide (NBS). The method involves the addition of excess NBS of known concentration in the presence of 1M HCl, reactants are allowed to react and the unreacted NBS is estimated by the measurement in the decrease in the absorbance of the Rhodamine-B dye (λmax 557nm). This method has been applied for the determination of drugs in their pure form as well as in tablet formulations
Design and Implementation of Encoder for (15, k) Binary BCH Code Using VHDL a...IOSR Journals
Abstract: In this paper we have designed and implemented(15, k) a BCH Encoder on FPGA using VHDL for reliable data transfers in AWGN channel with multiple error correction control. The digital logic implementation of binary encoding of multiple error correcting BCH code (15, k) of length n=15 over GF (24) with irreducible primitive polynomial x4+x+1 is organized into shift register circuits. Using the cyclic codes, the reminder b(x) can be obtained in a linear (15-k) stage shift register with feedback connections corresponding to the coefficients of the generated polynomial. Three encoder are designed using VHDL to encode the single, double and triple error correcting BCH code (15, k) corresponding to the coefficient of generated polynomial. Information bit is transmitted in unchanged form up to k clock cycles and during this period parity bits are calculated in the LFSR then the parity bits are transmitted from k+1 to 15 clock cycles. Total 15-k numbers of parity bits with k information bits are transmitted in 15 code word. Here we have implemented (15, 5, 3), (15, 7, 2) and (15, 11, 1) BCH code encoder on Xilinx Spartan 3 FPGA using VHDL and the simulation & synthesis are done using Xilinx ISE 13.3. BCH encoders are conventionally implemented by linear feedback shift register architecture. Encoders of long BCH codes may suffer from the effect of large fan out, which may reduce the achievable clock speed. The data rate requirement of optical applications require parallel implementations of the BCH encoders. Also a comparative performance based on synthesis & simulation on FPGA is presented. Keywords: BCH, BCH Encoder, FPGA, VHDL, Error Correction, AWGN, LFSR cyclic redundancy checking, fan out .
A Novel Approach of Area-Efficient FIR Filter Design Using Distributed Arithm...IOSR Journals
Abstract: In this paper, a highly area-efficient multiplier-less FIR filter is presented. Distributed Arithmetic (DA) has been used to implement a bit-serial scheme of a general asymmetric version of an FIR filter, taking optimal advantage of the 3-input LUT-based structure of FPGAs. The implementation of FIR filters on FPGA based on traditional arithmetic method costs considerable hardware resources, which goes against the decrease of circuit scale and the increase of system speed. This paper presents the realization of area efficient architectures using Distributed Arithmetic (DA) for implementation of Finite Impulse Response (FIR) filter. The performance of the bit-serial and bit parallel DA along with pipelining architecture with different quantized versions are analyzed for FIR filter Design. Distributed Arithmetic structure is used to increase the resource usage while pipeline structure is also used to increase the system speed. In addition, the divided LUT method is also used to decrease the required memory units. However, according to Distributed Arithmetic, we can make a Look-Up-Table (LUT) to conserve the MAC values and callout the values according to the input data if necessary. Therefore, LUT can be created to take the place of MAC units so as to save the hardware resources. The simulation results indicate that FIR filters using Distributed Arithmetic can work stable with high speed and can save almost 50 percent hardware resources to decrease the circuit scale, and can be applied to a variety of areas for its great flexibility and high reliability. This method not only reduces the LUT size, but also modifies the structure of the filter to achieve high speed performance. Keywords: DSP, Digital Filters, FIR , FPGA, MAC, Distributed Arithmetic(DA),Divided LUT, pipeline
Non-NewtonianFluid Flow and Heat Transfer over a Non- Linearly Stretching Sur...IJERA Editor
This paper investigates the MHD flow and heat transfer of an electrically conducting non-newtonian power-law
fluid over a non-linearly stretching surface along with porous plate in porous medium. The governing equations
are reduced to non-linear ordinary differential equations by means of similarity transformations. These
equations are then solved numerically with the help ofRunge – Kutta shooting method. The effect of various
flow parameters in the form of dimensionless quantities on the flow field are discussed and presented
graphically.
Numerical study of mhd boundary layer stagnation point flow and heat transfer...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
MHD convection flow of viscous incompressible fluid over a stretched vertical...IJERA Editor
The effect of thermal radiation, viscous dissipation and hall current of the MHD convection flow of the viscous incompressible fluid over a stretched vertical flat plate has been discussed by using regular perturbation and homotophy perturbation technique with similarity solutions. The influence of various physical parameters on velocity, cross flow velocity and temperature of fluid has been obtained numerically and through graphs.
Unsteady Free Convection MHD Flow of an Incompressible Electrically Conductin...IJERA Editor
In this paper we investigate unsteady free convection MHD flow of an incompressible viscous electrically
conducting fluid through porous medium under the influence of uniform transverse magnetic field between two
heated vertical plate with one plate is adiabatic. The governing equations of velocity and temperature fields with
appropriate boundary conditions are solved by the Integral Transform Technique. The obtained results of
velocity and temperature distributions are shown graphically and are discussed on the basis of it. The effects of
Hartmann number, Darcy parameter, Prandtl number and the decay factor, and effects of adiabatic plate on the
velocity and temperature fields are discussed.
MHD Free Convection from an Isothermal Truncated Cone with Variable Viscosity...IJERA Editor
This paper presents a study of MHD free convection flow of an electrically conducting incompressible fluid with
variable viscosity about an isothermal truncated cone in the presence of heat generation or absorption. The fluid
viscosity is assumed to vary as a inverse linear function of temperature. The non-linear coupled partial
differential equations governing the flow and heat transfer have been solved numerically by using an implicit
finite - difference scheme along with quasilinearization technique. The non-similar solutions have been obtained
for the problem, overcoming numerical difficulties near the leading edge and in the downstream regime. Results
indicate that skin friction and heat transfer are strongly affected by, both, viscosity-variation parameter and
magnetic field. In fact, the transverse magnetic field influences the momentum and thermal fields, considerably.
Further, skin friction is found to decrease and heat transfer increases near the leading edge. Also, it is found that
the direction of heat transfer gets reversed during heat generation.
MHD Free Convection from an Isothermal Truncated Cone with Variable Viscosity...IJERA Editor
This paper presents a study of MHD free convection flow of an electrically conducting incompressible fluid with
variable viscosity about an isothermal truncated cone in the presence of heat generation or absorption. The fluid
viscosity is assumed to vary as a inverse linear function of temperature. The non-linear coupled partial
differential equations governing the flow and heat transfer have been solved numerically by using an implicit
finite - difference scheme along with quasilinearization technique. The non-similar solutions have been obtained
for the problem, overcoming numerical difficulties near the leading edge and in the downstream regime. Results
indicate that skin friction and heat transfer are strongly affected by, both, viscosity-variation parameter and
magnetic field. In fact, the transverse magnetic field influences the momentum and thermal fields, considerably.
Further, skin friction is found to decrease and heat transfer increases near the leading edge. Also, it is found that
the direction of heat transfer gets reversed during heat generation.
International Journal of Engineering and Science Invention (IJESI)inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Analysis of mhd non darcian boundary layer flow and heat transfer over an exp...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
MHD Natural Convection Flow of an incompressible electrically conducting visc...IJERA Editor
We consider a two-dimensional MHD natural convection flow of an incompressible viscous and electrically
conducting fluid through porous medium past a vertical impermeable flat plate is considered in presence of a
uniform transverse magnetic field. The governing equations of velocity and temperature fields with appropriate
boundary conditions are solved by the ordinary differential equations by introducing appropriate coordinate
transformations. We solve that ordinary differential equations and find the velocity profiles, temperature profile,
the skin friction and nusselt number. The effects of Grashof number (Gr), Hartmann number (M) and Prandtl
number (Pr), Darcy parameter (D-1) on velocity profiles and temperature profiles are shown graphically.
EFFECT OF SLIP PARAMETER OF A BOUNDARY-LAYER FLOW FOR NANOFLUID OVER A VERTIC...IAEME Publication
In this paper we analyze the effect of momentum slip, thermal slip and solutal slip on stagnation point flow of MHD nanofluid towards stretching sheet .The governing partial differential equation of flow, heat and mass transfer on considered flow are converted into the ordinary differential equations by means of similarity trans formations .The resulting equations are solved by the Runge-Kutta fourth order method with efficient shooting technique. Effects of various governing parameters on flow, heat and mass transfer are studied through the plots. The various numerical tables which are calculated and tabulated. A comparison of our present results with a previous study has been done and we found that an excellent agreement is there with the earlier results and of ours.
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Chemical Reaction on Heat and Mass TransferFlow through an Infinite Inclined ...iosrjce
The numerical studies are performed to examine the mass transfer flow with thermal diffusion and
diffusion thermo effect past an infinite, inclined vertical plate in a porous medium in the presence of chemical
reaction. First of all, the governing equations are transformed to a system of dimensionless coupled partial
equations. Explicit finite difference method has been used to solve these dimensionless equations for momentum,
concentration and energy equations. During the course of discussion, it is found that various parameters related
to the problem influence the calculated result. Finally, the profiles of velocity, concentration and temperature
are analyzed and illustrated with graphs.
1. IOSR Journal of Applied Physics (IOSR-JAP)
e-ISSN: 2278-4861.Volume 4, Issue 5 (Sep. - Oct. 2013), PP 72-78
www.iosrjournals.org
www.iosrjournals.org 72 | Page
Effect of thermal radiation on free convection boundary layer
flow over a vertical wavy cone
a
E. M. A. Elbashbeshy, b
T.G. Emam, c
E. A. Sayed
a
Mathematics Department, Faculty of science, Ain Shams University, Abbassia, Cairo, Egypt,
b
Mathematics Department, The German University in Cairo-GUC, New Cairo City, Cairo, Egypt,
c
Department of Physics& Engineering Mathematics, Faculty of Engineering-Mattaria, Helwan University,
Cairo, Egypt,
Abstract: Effect of thermal radiation on a steady two-dimensional free convection laminar boundary layer flow
of a viscous incompressible optically thick fluid over a vertical wavy cone has been investigated. Using
appropriate transformations, the basic governing equations are transformed into non-dimensional boundary-
layer equations. These equations are then solved numerically by using Mathematica technique. The effect of the
radiation parameter on velocity, temperature, skin friction and local Nusselt number has been discussed with
graphical representation.
Keywords: Laminar boundary layer, free convection, Vertical wavy cone, radiation effect.
I. Introduction
Convection problems associated with irregular surfaces have received less attention than the cases
with regular surfaces. Surfaces are sometimes roughened to disturb the flow and alter the rate of heat transfer on
such surfaces. Thus, it is clear that convection problems associated with wavy surfaces occur frequently in
practice.
Many articles have been published [1-12] dealing the problem of natural convection over a vertical
wavy cone, frustum of a wavy cone and wavy vertical surface.
Convective heat transfer with thermal radiation is great importance in processes involving high
temperatures such as in gas turbines, nuclear power plants, and thermal energy storage among others. The
thermal radiation of gray fluid which is emitting and absorbing in a nonscattering medium has been examined
by Hossain et al. [13&14] , Raptis et al. [15] , and Cortell [16-19].
In this article, the free convection boundary layer flow over a vertical wavy cone including thermal
radiation effect has been investigated.
II. Mathematical formulation
The boundary layer analysis outlined below allows the shape of the wavy surface, )ˆ(ˆ x to be arbitrary,
but our detailed numerical work will assume that the surface exhibits sinusoidal deformations. Thus the wavy
surface of the cone is described by
)/ˆsin(ˆ)ˆ(ˆˆ lxaxy (1)
where l is the fundamental wavelength associated with wavy surface and aˆ is the amplitude of the surface
waves.
The physical model of the problem and the two dimensional coordinate system are shown in Figure 1,
where is the half angle of the flat surface of the cone and )ˆ(ˆ xr is the local radius of the flat surface of the
cone which is defined by
)sin(ˆˆ xr (2)
2. Effect of thermal radiation on free convection boundary layer flow over a vertical wavy cone
www.iosrjournals.org 73 | Page
Fig. 1: Physical model and coordinates system
The conservation equations for the flow characterized with steady, laminar and two-dimensional boundary
layer; under the usual Boussinesq approximation, the continuity, momentum and energy equations can be
written as:
0
ˆ
)ˆˆ(
ˆ
)ˆˆ(
y
vr
x
ur
(3)
)(cosˆˆ
ˆ
ˆ1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ 2
TTgu
x
p
y
u
v
x
u
u
(4)
)(sinˆˆ
ˆ
ˆ1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ 2
TTgv
y
p
y
u
v
x
u
u
(5)
y
q
C
T
C
k
y
T
v
x
T
u r
pp
ˆ
1ˆ
ˆ
ˆ
ˆ
ˆ 2
(6)
where ( yx ˆ,ˆ ) are the dimensional coordinates in the vertical and horizontal directions, )ˆ,ˆ( vu are the velocity
components parallel to ( yx ˆ,ˆ ), g is the acceleration due to gravity, pˆ is the pressure of the fluid, is the
density, is the kinematic viscosity, is the coefficient of volume expansion, T is the temperature of the
fluid, T is the ambient temperature, k is the thermal conductivity of the fluid, pC is the specific heat at
constant pressure and rq is the radiative heat flux in the yˆ direction.
The boundary conditions are given by
yasTTu
xyyTTvu ww
ˆ,0ˆ
)ˆ(ˆat,0ˆ,0ˆ
(7)
where wT is the surface temperature
Using the Rosseland approximation for radiation [15], radiative heat flux is simplified as
,
ˆ3
4 4
*
*
y
T
qr
(8)
where
*
and
*
are the Stefan-Boltzmann constant and the Rosseland coefficient.
We assume that the temperature differences within the flow are such that the term
4
T may be expressed as a
linear function of temperature. Hence, expanding
4
T in a Taylor series about T and neglecting higher-order
terms we get
.34 434
TTTT (9)
Using equations (8) and (9), the energy equation (6) becomes
3. Effect of thermal radiation on free convection boundary layer flow over a vertical wavy cone
www.iosrjournals.org 74 | Page
2
2
*
3*
2
ˆ3
16ˆ
y
T
c
T
T
y
T
v
x
T
u
p
(10)
Now introduce the following non-dimensional variables
xd
d
l
x
x
υ
lTTgβ
Gr
TT
TT
θuvGr
l
vuGr
l
u
l
a
apGr
l
p
l
r
r(Gr)
l
xy
y
l
x
x
x
w
w
x
/
ˆ
ˆ
,
)ˆ(ˆ
)(,
)(cos
,),ˆˆ(,ˆ
,
ˆ
,ˆ,
ˆ
,
)ˆ(ˆˆ
,
ˆ
2
3
4/12/1
1
2
2
41
(11)
where is the dimensionless temperature, is the dynamic viscosity and Gr is the Grashof number.
Substituting the transformations given in (11) into (3)-(6) and ignoring terms of small order in Gr we obtain
the following boundary layer equations:
0
)()(
y
rv
x
ru
(12)
2
2
24/1
)1(
y
u
y
p
Gr
x
p
y
u
v
x
u
u xx (13)
tan)1()( 2
2
24/12
y
u
y
p
Gru
y
u
v
x
u
u xxxxx (14)
2
2
2
)1(
Pr
1
y
N
y
v
x
u Rx
(15)
where
*3
3/16 kTNR is the radiation parameter and
k
Cp
Pr is the Prandtl number.
The corresponding boundary conditions to be satisfied are:
yasu
yvu
0,0
0at1,0,0
(16)
Equation (13) indicates that the pressure gradient along the y-direction is of )( 4/1
GrO , which implies that
lowest order pressure gradient along x -direction can be determined from the inviscid flow solution. However,
this pressure gradient is zero, since there is no externally induced free stream. On the other hand, equation (14)
shows that ypGr
/4/1
is of )1(O and is determined by the left-hand side of this equation. Thus, the
elimination of
y
p
from equations (13) and (14) leads to
tan)
1
tan1
()1(
1
22
2
22
2
x
x
x
x
xxx
y
u
u
y
u
v
x
u
u
(17)
If we introduce the non similar variables:
sin),,(
),,( 4/14/3
xrx
yxxrfx
(18)
where is the pseudo- similarity variable, ),( yxf is dimensionless stream function and is the stream
function which is defined according to yu / and xv / . Equations (15) and (17) are now
transformed to
4. Effect of thermal radiation on free convection boundary layer flow over a vertical wavy cone
www.iosrjournals.org 75 | Page
)()
1
tan1
()
12
1
()
4
7
()1( 2
2
2
2
x
f
f
x
f
fxf
x
fff
x
x
x
xxx
x
(19) )(
4
7
)1(
Pr
1 2
x
f
x
fxfNRx
(20)
The boundary conditions (16) now take the form
(21)
as0,0
0at1,0
f
ff
Here the primes denote the differentiation with respect to . The equations (19) and (20) subjected to the
boundary conditions (21) are solved by using Mathematica program.
The quantities of physical interest are shearing stress and rate of heat transfer in terms of the skin friction
coefficient fC and the Nusselt number Nu , respectively and can be written as
)(
ˆˆ
and2
TTk
xq
Nu
U
C
w
ww
f
.
Where 0ˆ00ˆ )(and yrywyw nqTnkqun , and n is the unit normal to the wavy
surface of the cone.
Using the transformation (9) and (16), then fC and Nu take the following form:
)0,()1(1)(
)0,(1)(
/24/1
24/1
xNNu
x
Gr
xfC
x
Gr
Rx
xf
III. Results and discussion
The set of non-linear partial differential equations (19) and (20) satisfying the boundary conditions (21) have
been solved numerically using the Mathematica method for several values of the involved parameters, namely
Prandtl number Pr , radiation parameter RN , wavy surface amplitude parameter a and cone half angle
parameter .
1 2 3 4 5 6 7 8 9 10
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9 10
0.2
0.4
f
Fig. 2: The temperature profiles )( for Fig. 3: The velocity profiles )(f for various Pr
various Pr at 2.0a , 0
30 , 1x and 2rN at 2.0a , 0
30 , 1x and 2rN
2
1
30
2.0
0
RN
x
a
––– 72.0Pr
...... 3Pr
----- 7Pr
––– 72.0Pr
...... 3Pr
----- 7Pr
2
1
30
2.0
0
RN
x
a)( )(f
5. Effect of thermal radiation on free convection boundary layer flow over a vertical wavy cone
www.iosrjournals.org 76 | Page
2 4 6 8 10
0.2
0.4
0.6
0.8
1
2 4 6 8 10
0.2
0.4
0.6
f
Fig. 5: The temperature profiles )( for Fig. 4: The velocity profiles )(f for various RN
various RN at 2.0a , 0
30 , 1x and at 2.0a , 0
30 , 1x and 72.0Pr
72.0Pr
0 1 2 3 4
x
0.2
0.4
0.6
0.8
1.0
f
x , 0
0 1 2 3 4
x
0.2
0.4
0.6
0.8
1.0
x , 0
Fig.6: The skin- friction coefficient for various Pr Fig.7: The Nusselt number for various Pr
at 2.0a , 0
30 ,and 2RN at 2.0a , 0
30 ,and 2RN
0 1 2 3 4
x
0.2
0.4
0.6
0.8
1.0
1.2
f
x , 0
0 1 2 3 4
x
0.2
0.4
0.6
0.8
1.0
x , 0
Fig.8: The skin- friction coefficient for various RN Fig.9: The Nusselt number for various RN
at 2.0a , 0
30 and 72.0Pr at 2.0a , 0
30 and 72.0Pr
)(f )(
x
4
1
)(
x
Gr
C f
4
1
)(
x
Gr
Nu
––– 0RN
...... 1RN
----- 10RN
72.0Pr
1
30
2.0
0
x
a
72.0Pr
1
30
2.0
0
x
a
––– 72.0Pr
...... 5Pr
----- 10Pr
4
1
)(
x
Gr
Nu
––– 72.0Pr
...... 5Pr
----- 10Pr
2
30
2.0
0
RN
a
72.0Pr
30
2.0
0
a
4
1
)(
x
Gr
C f
2
30
2.0
0
RN
a
x
––– 0RN
...... 1RN
----- 10RN
––– 0RN
...... 1RN
----- 10RN
72.0Pr
30
2.0
0
a
––– 0RN
...... 1RN
----- 10RN
6. Effect of thermal radiation on free convection boundary layer flow over a vertical wavy cone
www.iosrjournals.org 77 | Page
The effects of vary Prandtl Pr number and radiation parameter RN , on the dimensionless velocity profiles
)(/
f and temperature profiles )( are shown in Figures 2-5. Figures 2&3 show the effects of vary Prandtl
number Pr on the dimensionless velocity and dimensionless temperature against for wavy surface amplitude
parameter 2.0a and cone half angle parameter
0
30 , radiation parameter 2RN and the
dimensionless distance 1x . It can be observed from these figures, the velocity and temperature profiles are
decreases for increasing values of Prandtl number Pr . In figures 4&5 represent the influence of the radiation
parameter RN , on the velocity and temperature profiles in the boundary layer for 2.0a , 72.0Pr
and
0
30 , while 1x . These figures display the effect of radiation parameter, RN on both the velocity
and temperature profiles in the boundary layer regime. As radiation parameter, RN increases, both velocity and
temperature increase within the boundary layer regime. In figure 4, for all values of RN , there exists a local
maximum of the velocity within the boundary layer. As radiation parameter, RN increases, the maximum
values of velocity increases. In Figs. 6and 7 found the skin friction coefficient 4
1
)(
x
Gr
C f and the rate of heat
transfer coefficient 4
1
)(
x
Gr
Nu for different values of Prandtl number Pr for 2.0a , 2RN
and
0
30 , while 1x . The influence of the parameters RN and a for different values of Prandtl number
Pr, the decreasing skin friction coefficient becomes slower in the downstream region along x direction. On the
other hand, increasing the values of Prandtl number Pr the temperature gradient as well as the rate of heat
transfers consequently increases. In Figs. 8 & 9, represent the influence of the radiation parameter, RN on the
local skin friction coefficient 4
1
)(
x
Gr
C f and the rate of heat transfer 4
1
)(
x
Gr
Nu for 2.0a , 72.0Pr
and
0
30 , while 1x . From Figure 8, for increasing RN , the skin friction coefficient
4
1
)(
x
Gr
C f increases. Increasing values of RN leads to increasing the velocity of the fluid (see figure 4) within
boundary layer. For enhanced velocity, the velocity gradient of the fluid increases. From figure 9 Increasing
values of RN leads to an decrease in the heat absorption intensity of the fluid, consequently the local rate of
heat transfer decrease.
IV.Conclusions
In this paper, the problem of steady laminar free convection flow about a vertical wavy cone in the
presence of radiation effect is studied. From the present investigations, we may conclude the following:
1. The velocity and temperature distribution decreases with increasing the values of Pr and increases with
increasing the values of RN .
2. For increasing values of the Prandtl number Pr , the skin fiction coefficient decreases but local Nusselt
number increases.
3. An increasing in the values of radiation parameter RN leads to increases in the value of the skin friction
coefficient while the local heat transfer decrease.
V. References
[1] K. C. A. Alam, M. A. Hossain and D. A. S. Rees, Magnetohydrodynamic free convection along a vertical wavy surface, Appl.
Mech. Engineering, vol. 1, pp. 555–566, 1997.
[2] I. Pop and T. Y. Na, Natural convection from a wavy cone, Appl. Sci. Research, vol. 54, pp. 125–136, 1995.
[3] I. Pop and T. Y. Na, Natural convection over a vertical wavy frustum of a cone, Int. J. Non-Linear Mechanics, vol. 34, pp. 925–934,
1999.
[4] C. Y. Cheng, Natural convection heat and mass transfer near a vertical wavy cone with constant wall temperature and concentration
in a porous medium, Mech. Res. Communication, vol. 27, pp. 613–620, 2000.
[5] M. A. Hossain and D. A. S. Rees, Combined heat and mass transfer in natural convection flow from a vertical wavy surface, Acta
Mechanica, vol. 136, pp. 133 141, 1999.
7. Effect of thermal radiation on free convection boundary layer flow over a vertical wavy cone
www.iosrjournals.org 78 | Page
[6] M. A. Hossain, M. S. Munir and I. Pop, Natural convection flow of viscous fluid with viscosity inversely proportional to linear
function of temperature from a vertical cone, Int. J. Therm. Science, vol. 40, pp. 366–371, 2001.
[7] M. A. Hossain, S. Kabir and D. A. S. Rees, Natural convection of fluid with temperature dependent viscosity from heated vertical
wavy surface, ZAMP, vol. 53, pp. 48–52, 2002.
[8] T. Cebeci and P. Bradshaw, Physical and Computational Aspects of Convective heat Transfer, Springer, New York, 1984.
[9] C. C. Wang and C. K. Chen, Mixed convection boundary layer flow on inclined wavy plates including the magnetic field effect, Int.
J. Therm. Science, vol. 44, pp. 577–586, 2005.
[10] L. S. Yao, Natural convection along a vertical complex wavy surface, Int. J. Heat Mass Transfer, vol. 49, pp.281–286, 2006.
[11] M. M. Molla, M. A. Hossain and L. S. Yao, Natural convection flow along a vertical complex wavy surface with uniform heat flux,
ASME J. Heat Transfer, vol. 129 (no.10), pp. 1403-1407, 2007.
[12] Azad Rahman, Md M. Molla, M.M.A. Saker and A. Thohura, Effects of temperature dependent viscosity on natural convection
flow along a vertical wavy cone with heat flux, Int. J. of Energy and Technology, 3(8), pp.1-10, 2011.
[13] Hossain M.A., Alim M.A. and Rees D., The effect of radiation on free convection from a porous vertical plate. Int. J. Heat Mass
Transfer, 42, pp.181-191, 1999.
[14] Hossain M.A., Khanafer K. and Vafai K., The effect of radiation on free convection flow of fluid with variable viscosity from a
porous vertical plate. Int. J. Therm. Sci., 40, pp.115-124, 2001.
[15] Raptis A., Perdikis C. and Takhar H.S., Effect of thermal radiation on MHD flow, Appl. Math. Comp., 153, pp.645-649, 2004.
[16] Cortell R., Effects of heat source/sink, radiation and work done by deformation on flow and heat transfer of a viscoelastic fluid over
a stretching sheet. Comput. Math. Appl., 53pp. 305-316, 2007.
[17] Cortell R., Similarity solutions for boundary layer flow and heat transfer of a FENE-P fluid with thermal radiation. Phys. Lett. A.,
372, pp.2431-2439, 2008.
[18] Cortell R. ,Effects of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet. Phys. Lett.
A., 372, pp.631-636, 2008.
[19] Cortell R., Radiation effects in the Blasius flow, Appl. Math. Comp., 198, pp. 333-338, 2008.