- Email: [email protected]
Boundary layer analysis of micropolar dusty fluid with TiO2 nanoparticles in a porous medium under the effect of magnetic field and thermal radiation over a stretching sheet
Accepted Manuscript Boundary layer analysis of micropolar dusty fluid with TiO2 nanoparticles in a porous medium under the effect of magnetic field and thermal radiation over a stretching sheet
S.S. Ghadikolaei, Kh. Hosseinzadeh, M. Yassari, H. Sadeghi, D.D. Ganji PII: DOI: Reference:
S0167-7322(17)33149-5 doi: 10.1016/j.molliq.2017.08.111 MOLLIQ 7819
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
14 July 2017 27 August 2017 29 August 2017
Please cite this article as: S.S. Ghadikolaei, Kh. Hosseinzadeh, M. Yassari, H. Sadeghi, D.D. Ganji , Boundary layer analysis of micropolar dusty fluid with TiO2 nanoparticles in a porous medium under the effect of magnetic field and thermal radiation over a stretching sheet, Journal of Molecular Liquids (2017), doi: 10.1016/j.molliq.2017.08.111
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Boundary layer analysis of Micropolar dusty fluid with TiO2 nanoparticles in a porous medium under the effect of magnetic field and thermal radiation over a stretching sheet
S.S. Ghadikolaeia, Kh. Hosseinzadehb, M.Yassaria, H. Sadeghia, D.D. Ganjib* a
PT
Department of Mechanical Engineering, Mazandaran university science and Technology , Babol, Iran
b
Corresponding author,
E-mail:[email protected]
SC
*
RI
Department of Mechanical Engineering, Babol Noushirvani University of Technology, Babol, Iran
NU
Abstract
This paper analyzed the boundary layer flow and heat transfer of an incompressible TiO2-
MA
water nanoparticle on micropolar fluid with homogeneously suspended dust particles in the presence of thermal radiation. Since the nanoparticles have high thermal conductivity
D
coefficient compared to base fluids, so their distribution leads to an increase in the thermal
PT E
conductivity of the fluids and they are considered to be the main parameters of heat transfer. In the following, nonlinear equations that can describe this problem are presented in this
CE
article. These nonlinear equations are numerically analyzed using the Runge-Kutta-Fehlberg method in MAPLE software. The main goal of this paper is to study and analyze the behavior
AC
of the velocity profile in two phases of liquid and dust for different values of parameters suction/injection (f0), mass concentration (L), Material (R), Penetrable (Kp), Magnetic (M), Solid volume fraction (𝜑), fluid particle interaction (β) and also temperature profile effectiveness from radiation parameter (Nr) changes, Eckert number (Ec), Prandtl number (Pr) in two PST and PHF cases for both fluid and dust phases, in which dual behavior of the velocity profile compared to the β changes in the fluid and dust phases and also Lorentz force generated by the magnetic field is also mentioned. Finally, the effect of changes of Kp and M 1
ACCEPTED MANUSCRIPT in the presence of β on the coefficient of surface friction, the effect of Ec changes in the presence of Nr and Pr changes in the presence of β on the Nusselt number in two PST and PHF cases and also R changes in the presence of f0, β and M on couple stress has been investigated and analyzed.
PT
Key words: boundary layer flow, micropolar fluid, dust nanoparticles, thermal radiation,
RI
magnetic field.
k .....permeability of the porous medium
Uw .....stretching sheet velocity
k * .....mean absorption coefficient
Vw .....suction velocity
r .....mass of the dust particle
Greek Symbols
B0 .....uniform magnetic field strength
MA
S .....number density of the dust particles
R .....material parameter
.....fluid particle interaction parameter
.....solid volume fraction parameter
.....ratio of specific heat
.....dimension less temperature
PT E
M .....magnetic parameter
D
N ..... angular velocity K .....Stokes drag constant
NU
SC
Nomenclature
.....dynamic viscosity of nanofluid
.....kinematic viscosity
Kp ..... permeability parameter
.....density
Ec .....Eckert number
.....dimensionless variable
AC
CE
Nr ..... radiation parameter
Pr .....Prandtl number
* ..... Stefan-Boltzman constant
C ..... specific heat
v .....relaxation time of the particle phase
T ..... temperature
T .....thermal equilibrium time
T .....free stream temperature
Subscripts
j ..... refrence length
p .....dust particle
l .....mass concentration parameter
f .....fluid
k .....thermal conductivity
nf .....nanofluid 2
ACCEPTED MANUSCRIPT 1. Introduction From last decades till now, the study of viscous fluids on a stretching sheet has attracted many researchers attention. This is due to the fact that many processes like casting, hot rolling, drawing wires of copper, glass blowing, and extrusion sheets of plastic and also
PT
polymer industries have benefited from it. First, Sakiadis [1] investigated boundary layer flow of an incompressible viscous fluid
RI
over moving continuous solid surfaces. Then to develop this research, Crane [2] studied two-
SC
dimensional flow past a linearly stretching sheet. His experiment played a significant role in
NU
the polymer industry. A little later, same work was done by Wang [3] for three dimensional. Since then, many researches have focused on stretching sheet issues. Problems which the
MA
conditions and effects of different parameters such as MHD, thermal radiation, suction/injection, slip effects, permeable surface and etc. are considered. For an example
D
viscoelastic fluid flow past a stretching surface was investigated by Rajagopal et al. [4].
PT E
Chakrabarti and Gupta [5] discussed about MHD flow and heat transfer on a stretching surface. Kelson and Desseaux [6] studied the effects of effective surface heat transfer in a
CE
micropolar fluid on a permeable stretching sheet. The study on heat and mass transfer effect in boundary layer MHD flow of a viscous fluid in the presence of thermal radiation and
AC
exponentially stretching sheet through porous medium in order to investigate the effects of temperature and velocity field and also Nusselt number were conducted by Swain et al. [7]. Hayat et al. [8] analyzed the effects of melting heat in flow on variable thicked stretching surface. In their paper, effects of various physical parameters on velocity distribution, temperature, and fluid concentration had been investigated. Fathy and Salem [9] investigated radiation effects on forced free movement of the laminar flow in the vicinity of vertical stretching sheet considering the magnetic field and the unsteady viscosity. In their study, velocity and temperature equations of sheet are written using power law form. MHD flow 3
ACCEPTED MANUSCRIPT and heat transfer on a stretching sheet with variable thermal conductivity has been analyzed by Nandeppanavarnet al. [10] considering the effect of partial slip and using similar conversions for converting partial equations to ordinary equations. Most mentioned studies have been related to Newtonian fluids. In recent years, the study
PT
on non-Newtonian fluids has also become extended, due to wide application of these fluids in the oil industry, the polymer industry, and other industries. Non-Newtonian fluids such as
RI
color, blood, shampoo, palm oil, mayonnaise, etc., are such that their mathematical modeling
SC
is much more complicated than conventional Newtonian fluids and cannot be modeled with the Navier-Stokes equations related to Newtonian fluids. These fluids are divided into several
NU
categories, in which micropolar fluids are an example of them. Micropolar fluids are nonNewtonian fluids with microstructure that consists of rigid particles suspended in a viscous
MA
medium in which particle deformation is ignored. These fine particles can be rotated in fluid, so in modeling these type of fluids, the motion of fluid is described by two velocity vectors,
PT E
D
classical velocity vector and micro rotation (spin) vector. A micro fluid has many applications in various fields. Among them, colloids and polymeric suspensions, liquid crystal and lubrication problem can be pointed out. The first time micropolar fluids theory
CE
was raised by Eringen [11] but Ariman et al. [12] revealed the many practical applications of
AC
this type of fluid in their article. Other studies were done in this field with different condition, too. The finite element method was used to analyze the micropolar fluid flow in a porous medium under uniform suction by Bhargava et al. [13]. Pal and Mandal [14] analyzed electrically conduction micropolar nanofluid flow over a stretching surface that was under thermal radiation and a uniform heat source by Runge-Kutta numerical method. Madani et al. [15] investigated magneto-micropolar fluid past a continuously moving plate considering radiation effects. There were blowing and suction in this research. The study of suction or injection effects on flow characterization which past on continuous moving plate in the 4
ACCEPTED MANUSCRIPT presence of radiation in micropolar fluid and temperature analysis inside the boundary layer was done by EL-Arabawy [16]. Yacob et al. [17] investigated a steady stagnation point flow of a micropolar fluid which past on a linearly shrinking/stretching sheet. Micropolar fluid flow on a permeable stretching sheet considering suction was investigated by Rasoli et al. [18]. At the end, Nering and Rup [19] study can be mentioned among new studies about fluid
PT
types. They experimented the effect of adding nanoparticles on a micropolar fluid in
RI
increasing the amount of heat transfer during an unsteady natural convection.
SC
Another topic that has been found over the last few years in the scientific activities of researchers is the subject of nanofluids. These fluids come from adding very fine particles to
NU
nano dimensions in common fluids such as water or oil to improve their thermal conductivity. Nanofluids have different applications in industry and engineering that for example treatment
MA
of cancer infected tissues, electronic and space cooling can be pointed out. In 1995, the first time nanofluid word was raised by Choi [20]. Since then, scientists have made many articles
PT E
D
on this subject. The study and investigation of melting process heat transfer of nano-enhanced phase-change material consisting of nanoparticle and hybrid nanoparticles was performed on a horizontal heated cylinder located in the middle of a square hole by Chamkha et al. [21].
CE
They investigated the effects of the volume fraction of nanoparticles, the turbulent
AC
conductivity parameter, the viscosity parameter and the Fourier number on melting and heat transfer after numerical solving of nonlinear governing differential equations the problem with the Galerikin finite element method. They concluded that the highest melting rate for Fourier numbers was in the range (0-0.5), however, for the values greater than 0.5, the melting rate decreases. Simulation of heat transfer and entropy generation of a nanofluid flow inside a hot squared cavity using water as a base fluid and four types of metallic and nonmetallic nanofluids of Al2O3, Cu, Ag, and TiO2 using Lattice Boltzman method and also calculating the thermal conductivity and the effective viscosity using Brinkman and 5
ACCEPTED MANUSCRIPT Maxwell-Garnett (MG) model were carried out by Sheikholeslami et al. [22]. Ghalambaz et al. [23] in an article examine the heat transfer of melting process of a nano-enhanced phasechange material using nanoparticles and hybrid nanoparticles separately in a hot cavity in isothermal condition with isolated left and right walls, floor and roof at hot and cold temperatures, respectively. They also studied of the effects of parameters and dimensionless
PT
numbers on melting process and heat transfer. They concluded significant changes in solid-
RI
liquid interface for large values of Fourier numbers due to the increase of the nanoparticles
SC
volume fraction, conductivity parameter, and viscosity parameter. Sheikholeslami et al. [24] studied and investigated the effects of magnetic field dependent viscosity on free convection
NU
heat transfer of a single-phase nanofluid considering its Brownian motion inside a chamber and simulated the problem of control volume based finite element method. They also
MA
examined the effects of various parameters on hydrothermal behavior, and concluded that the reduction of Nusselt number due to magnetic field dependent viscosity for low Hartman
D
number and high Raylight number is very sensetive. Studying free convection heat transfer of
PT E
a two-phase nanofluid and achieving inverse relation between Nusselt number and Lewis number, and increasing Nusselt number by increasing Buoyancy ratio number were done by
CE
Sheikholeslami et al. [25].There are also very useful and applicable articles in this field which
AC
can be study in Ref. [26-30].
As was told in the previous paragraph about combining nano particles in pure fluids such as water and oil, the presence of particles in milli and/or micro in base fluids cause improve their thermal conductivity. As we know, it is rarely possible to find an impurity-free fluid such as dust particles in nature. Therefore, the presence of these micro or millimeter sized particles (dust particles) in the base fluids cause to improve the properties of the heat transfer of the fluids. Considered name to these fluids is dusty fluids. Research and study on these fluids has been useful in many practical applications such as combustion, petroleum 6
ACCEPTED MANUSCRIPT transport, lunar ash flows, power plant piping and many other engineering branches. In 1970, Marble [31] examined the effects of dusty gases on fluid dynamics. In recent years, many papers are written in this field. Analysis of dusty fluid flow past a stretching surface in the presence of radiation effects was done by Ramesh and Gireesha [32]. They used Runge-Kutta numerical method in their analysis. Gireesha et al. [33] studied the effects of radiation and
PT
space-dependent heat source in convective of a dusty fluid along a vertical stretching surface.
RI
Hazarika and Konch [34] analyzed MHD free convection of a dusty fluid over a vertical
SC
permeable surface by a numerical method. It should be noted that in that problem, thermal conductivity effects and variable viscosity on convection was intended. Investigation of
NU
unsteady flow and heat transfer of a dusty fluid between two parallel plates with variable physical properties is conducted by Makinde and Chingoka [35]. The effects of viscosity,
MA
uniform magnetic field, wall slip parameter and electric conductivity on temperature field and fluid velocity and dust particles were investigated in their research. Good articles in this area
PT E
D
could be study in Ref. [36-40].
According to mentioned content of this section, the aim of this study is to analyze the MHD boundary layer flow and heat transfer of a micropolar dusty fluid with TiO2
CE
nanoparticles over a permeable stretching surface in the presence of thermal radiation. An
AC
analytical method used to solve governing differential equations this problem is solved by fourth-fifth Rung-Kutta Fehlberg method that is done with the help of the Maple software.
2. Mathematical formulation of the problem A steady two-dimensional laminar flow and heat transfer of a dusty micropolar fluid suspended with TiO2 nanoparticles over a permeable stretching sheet is studied here. The surface is along x-axis and moves with a linear velocity Uw (x) = bx, where b is constant. The fluid occupies the space 𝑦 > 0 as it is shown in Fig.1. In addition a uniform magnetic field B0
7
ACCEPTED MANUSCRIPT is applied along the positive y-direction. It is assumed that the particles of dust are of the same size and the number density of them remains constant throughout the flow. Also the effects of thermal radiation are considered in this problem.
The boundary layer governing equations the behavior of the mentioned fluid can be
PT
written as follows:
u u k 2u KS k N B02 v ( nf ) 2 (u p u ) u nf u , x y nf y nf nf y nf k
(4)
D
u p u v p p ) KS (u u p ), x y
(3)
MA
u p v p 0, x y
(5)
PT E
p (u p
(2)
NU
N N 2 N k u u v (2N ), 2 x y nf j y nf j y
SC
u
(1)
RI
u v 0, x y
Here (??, ??) and (𝑢𝑝 ,𝑣𝑝 ) represent the velocity components of nanofluid and dust phases
CE
in the ?? and ?? directions respectively. Furthermore, Stokes drag constant (𝐾), the density of dust particles (𝜌𝑝 ), the effective density of the nanofluid (𝜌𝑛𝑓 ) and spin-gradient viscosity (𝛾)
AC
are defined as:
K 6f a , p rS , nf (1 )f s , (nf k ) j nf (1 R ), (6) 2
2
Where the dynamic viscosity of the nanofluid (𝜇𝑛𝑓 ), the material parameter (𝑅) and reference length (𝑗) are given by:
nf
f k ,R , j nf , 2.5 nf a (1 )
(7)
8
ACCEPTED MANUSCRIPT The related boundary conditions are:
u Uw (x ),v Vw (x ), N n
u y
at y 0,
u v 0,u p 0,v p v , N 0 as y .
(8)
By introducing the following similarity transformations [41]:
PT
Uw (x ) 12 U (x ) 12 U (x ) 12 ) y, ( w ) y , N Uw (x)( w ) h (), f x f x f x
(
RI
Where, Uw (x ) bx , Vw (x ) f 0 bf .
(9)
SC
And the continuity Equations (1) and (4) are identically satisfied by introducing a stream
, v , y x
MA
u
NU
function 𝜓 as follows:
Similarly, it can be done for 𝑢𝑝 and 𝑣𝑝 .
PT E
ordinary differential equations:
s ](ff (f )2 ) (1 )2.5[l (F f ) Mf ] f
CE
(1 R )f (1 )2.5[(1 ) Rh k p f 0
D
Equations (2), (3) and (5) along with boundary condition (8) are reduced to the following
(10)
(11)
FF (F )2 (f F ) 0,
(12)
AC
R (1 )h (1 )2.5[(1 ) s ](fh f h ) R (2h f ) 0, 2 f
Here prime indicates the differentiation with respect to 𝜂. In addition, the parameters in the above equations are defined as follows:
9
ACCEPTED MANUSCRIPT rS
r , the fluid f K B 02 1 particle interaction parameter for velocity , the magnetic parameter M , the bv f b The mass concentration l
permeability parameter K p
, the relaxation time of the particle phase v
f
bk
.
PT
The boundary condition (9) becomes:
f () 1, f () f 0 , h() m at 0,
(13)
SC
RI
f () 0, F () 0, F () 1,h() 0 as .
NU
2.1. Heat transfer Analysis
MA
The governing boundary layer heat transport equations for dusty nanofluid are:
p T T 2T pc pf u v ] K nf ( T T ) (u p u )2 nf ( )2 p 2 x y T v y y 1 qr , (c p )nf y
(14)
(pcmf )[u p
PT E
D
(c p )nf [u
T p T c v p p ] p pf (T p T ), x y T
(15)
CE
Where the thermal conductivity (𝑘𝑛𝑓 ) and the heat capacitance of the nanofluid
AC
(𝜌𝐶𝑝 )𝑛𝑓 are introduced by the following formulas:
(c p )nf (1 )(c p )f ( cp )s ,
k 2k f 2(k f k s ) k nf s , kf k s 2k f 2(k f k s )
(16)
By employing the Rosseland approximation for radiation, the radiative heat flux (𝑞𝑟 ) is given by:
qr
4 *T 4 , 3k *y
(17)
10
ACCEPTED MANUSCRIPT In addition it is assumed that the temperature differences within the flow are sufficiently small, thus by expanding 𝑇 4 in a truncated Taylor series about 𝑇∞ and ignoring the higherorder terms, it can be written as a linear function of the temperature. Then we obtain:
T 4 4TT3 3T3 ,
(18)
16 *T 3 2T c T T v ] K nf ( * ) 2 p pf (T p T ) P (u p u )2 x y T v 3k k nf y
RI
(c p )nf [u
PT
By substituting equations (4) and (5) into equation (1), it becomes:
(19)
SC
u nf ( )2 , y
It is a fact that the solution of the equations (16) and (19) depends on the nature of the
NU
prescribed boundary conditions. So, here two types of heating process are applied.
MA
The prescribed boundary conditions are introduced as follows:
T x x qw D ( )2 (PHF case) at y 0, T Tw T A ( )2 (PST case), K y l l
D
(20)
PT E
T T ,T p T as y .
Where the characteristic length 𝑙 is defined by l
f b
.
T T T T ,p () p , Tw T Tw T
(21)
AC
()
CE
The non-dimensional nanofluid phase temperature ( ) and dust phase temperature p ( ) can be given by:
x l
Where, T T A ( )2 () (PST case), Tw T
D x 2 f (PHF case). ( ) t l b
By substituting (9), (16) and (21) into (15) and (19), we get:
(c p )s k nf 4 (1 Nr ) Pr[(1 ) ](f 2f ) Pr l T ( p ) kf 3 (c p )f
(22)
Pr lEc (F f ) Pr Ecf 0, 2
2
Fp 2F p T ( p ) 0,
(23) 11
ACCEPTED MANUSCRIPT The dimensionless parameters defined in the problem are: The Prandtl number Pr
(c p )f b 2l 2 , the Eckert number Ec Ac pf kf
(PST case) and
b 2l 2t b 4 *T 3 (PHF case), the radiation parameter Nr * , fluid particle interaction Dc pf f k k nf c pf 1 parameter for temperature T , the ratio of specific heat . bT cmf
PT
Ec
() 1 (PST mood), () 1 (PHF mood) at 0,
(24)
NU
3. Numerical method for solution
SC
() 0,p () 0 as .
RI
The subjected thermal boundary conditions become:
MA
In this section, the non-linear ordinary differential equations (10-12) and (22, 23) with respect to the boundary condition (13) and (24) are solved numerically by employing Runge
D
-Kutta Fehlberg 45 method with the help of Maple software. In this technique the boundary
PT E
value problem must be transformed into the initial value problem. In addition, it is crucial to choose a finite values of 𝜂∞ .
25 1408 2197 1 k0 k2 k 3 k 4 ), 216 2565 4109 5
CE
y m 1 = y m h(
AC
16 6656 28561 9 2 y m 1 y m h( k 0 k2 k 3 k 4 k 5 ), 135 12825 56430 50 55
(25) (26)
Where (25) and (26) are the approximations of fourth order and fifth order to the solution respectively, and also;
12
ACCEPTED MANUSCRIPT k 0 f (x m y m ), k f (x h , y hk 0 ), m 1 4 m 4 k 2 f (x m 3 h, y m ( 3 k 0 9 k 1 )h ), 8 32 32 12 1932 7200 7296 k0 k1 k 2 )h ), k 3 f (x m h , y m ( 13 2197 2197 2197 439 3860 845 k 4 f (x m h, y m ( 216 k 0 8k 1 513 k 2 4104 k 3 )h ), k f (x h , y ( 8 k 2k 3544 k 1859 k 11 k )h ), m 1 5 2 m 27 0 2565 2 4104 3 40 4
RI
PT
(27)
SC
Now, in order to solve the problem of this paper by applying the mentioned method, first the non-linear boundary value problem must be converted to the system of first order
NU
differential equations. So, new set of variables are given by:
𝑓 = 𝑞1 , 𝑓 ′ = 𝑞2 , 𝑓 ′′ = 𝑞3 , ℎ = 𝑞4 , ℎ′ = 𝑞5 , 𝐹 = 𝑞6 , 𝐹′ = 𝑞7 , 𝜃 = 𝑞8 , 𝜃′ = 𝑞9 ,
MA
𝜃𝑝 = 𝑞10
(28)
After applying the above equations into the equations (10–12) and (22, 23), they are
D
reduced to the following system of ordinary differential equations:
s ](q1q3 q22 ) (1 )2.5[l (q7 q2 ) Mq2 ] f
PT E
(1 R )q3 (1 )2.5[(1 ) Rq5 Kpq2 0
(29)
(30)
q6q7 q72 (q2 q7 ) 0
(31)
(c p )s k nf 4 (1 Nr )q9 Pr[(1 ) ](q q 2q2q8 ) Pr l T (q10 q8 ) kf 3 (c p )f 1 9
(32)
AC
CE
R (1 )q5 (1 )2.5[(1 ) s ](q1q5 q1q4 ) R (2q4 q3 ) 0 2 f
Pr Ec (q7 q2 )2 Pr Ecq32 0
q6q10 2q7q10 T (q8 q10 ) 0
(33)
The boundary conditions (13) and (24) become: 𝑞1 (0) = 𝑓0 , 𝑞2 (0) = 1 , 𝑞3 (0) = 𝑠1 , 𝑞4 (0) = −𝑚 , 𝑞5 (0) = 𝑠2 , 𝑞6 (0) = 𝑠3 , 𝑞7 (0) = 𝑠4 ,
(34) 13
ACCEPTED MANUSCRIPT q8 (0) 1, q9 (0) s5 , q10 (0) s 6 ,(PSTcase ) q8 (0) s5 , q9 (0) 1, q10 (0) s 6 ,(PHFcase).
(35)
Where unknown initial conditions 𝑠1 , 𝑠2 , 𝑠3 , 𝑠4 , 𝑠5 and 𝑠6 are calculated using iterative method called shooting method until boundary the conditions 𝑓 ′ (𝜂) → 0 , ℎ(𝜂) → 0 , 𝐹(𝜂) → 1, 𝐹 ′ (𝜂) → 0 , 𝜃(𝜂) → 0 , 𝜃𝑝 (𝜂) → 0 as 𝜂 → ∞ are satisfied. The above procedure will be
PT
repeated since the nonlinear solution converges with a convergence criterion of10−6. In
RI
addition, the step size is chosen as 𝛥𝜂 = 0.001.
SC
A comparison has been done between the the current obtained results of velocity profile with those of the previously published study of Ghalambaz et al [42] for the special cases in figure
NU
2. A very good agreement can be observed between them.
MA
4. Results and discussion
First, the behavior of velocity and temperature within the boundary layer is shown in the
D
case of different suction/ injection parameter (f0) in Figs. 3-7. Observations show that the
PT E
additive values of f0 reduce the velocity and temperature within the boundary layer. Therefore, the suction/injection parameter can also be referred to as a flow velocity reducer.
CE
Figs. 8-10 show the effect of change in the mass concentration parameter (L) on the
AC
velocity and temperature profiles. Carried out analysis on these graphs shows inverse relation between the values of L the velocity and temperature profiles. That is, velocity and temperature both decrease by increasing L value. The effect of upside changes of material parameter values on the velocity and temperature profiles are shown in Figs. 11-15. Fig.11 show an increase in the velocity profiles in terms of the increase in the value of R, but in contrast, Figs. 12-15 can be seen that with increasing R, the temperature profiles in both PST and PHF cases decrease for both the liquid and dust phases. 14
ACCEPTED MANUSCRIPT Figs. 16-20 show the results of effects of increasing value of the penetrable parameter (Kp) on velocity and temperature profiles, respectively. Obviously, porosity in the environment is restrictive of fluid flow and resistance to its movement. The same behavior is true for increasing values of KP. That is, increasing in Kp is the agent of creation resistance to the motion of the fluid. As we know, air resistance always reduces the velocity and produces
PT
heat. Therefore, increasing the value of Kp causes decrease in fluid velocity and,
RI
consequently, increasing the temperature in the thermal boundary layer.
SC
The effect of increasing magnetic parameter (M) values on the velocity and temperature profiles is shown in Figs. 21-25. The existence of an inverse magnetic field, proportional to
NU
the direction of flow that causes resistance drag force, such as Larentz force in the opposite direction of flow, has led to the results of Fig. 21 for increasing the values of M, the velocity
MA
reduces in both the liquid phase and the dust. Figs. 22-25, show that with increasing M values , the velocity gradient increases in both PST and PHF cases, respectively, because the
PT E
D
thickness of the momentum boundary layer decreases with increasing M. The behavior of the velocity and temperature profiles for a certain variation range of nano
CE
particle volume fraction parameter (𝜙) is shown in Figs. 26-28. Observations show that by increasing the 𝜙 value in the specific range, the velocity and temperature profiles have two
AC
completely different behaviors. The reaction of the velocity profile to increasing in 𝜙, is positive and increasing in both liquid and dust phases, but in contrast, increasing in 𝜙 in both liquid and dust phases cause temperature profile reduction in both PST and PHF cases. Figs. 29-31 show the effects of increasing fluid particle interaction parameter (β) on velocity and temperature distribution. From the observations, it seems that the effect of β variations on the velocity profile, in contrast to the effect of the variation in other studied parameters in two phases of liquid and dust, that is, the process of increasing the values of β
15
ACCEPTED MANUSCRIPT decreases the velocity of fluid phase, on the other hand, it has increased the velocity of the dust phase. Therefore, it can be concluded that the β variation has an inverse relation with the velocity of the fluid phase and is directly related to the velocity of the dust phase. But the results from Figs. 30 and 31 show the uniformity of the temperature profile in terms of changes in the positive beta direction. Therefore, increasing beta increases the temperature of
PT
both fluid and dust phases both PST and PHF cases.
RI
In Figs. 32-35, the behavior of the temperature profile is shown in the exposure to
SC
different values of the radiation parameter (Nr) in PST and PHF cases, respectively. Observations show that by increasing the value of Nr, the temperature rise is also achieved,
NU
because in higher value of Nr, the heat transfer through the conduction reduces the thermal boundary layer thickness and floating force a more than heat transfer through the radiation.
3 4𝜎∗ 𝑇∞
𝐾∗ 𝑘𝑛𝑓
, that is, the direct relations between Nr changes and temperature.
D
of 𝑁𝑟 =
MA
However, it should be noted that the results from Figs. 32-35 are a solid proof of the relation
PT E
The speed of temperature distribution effectiveness in two PST and PHF cases in relation to the changes in Eckert number (Ec) is shown in Figs. 36 and 37. Observations prove that
CE
the speed of temperature distribution increases by increasing Ec in both liquid and dust phases. However, the results from Figs. 36 and 37 are clear because Ec has direct relation
AC
kinetic energy and kinetic energy is directly related to the changes in temperature and thickness of the thermal boundary layer, so it is natural that the kinetic energy also increases with increasing value of Ec, and as a result, increasing temperature and thickness of the thermal boundary layer are followed. The temperature profile reaction in PST and PHF cases in terms of Prandtl number (Pr) is shown in Figs.38-41. Observations show that increasing Pr has inverse relation with temperature and is temperature reduction agent in both liquid and dust phases. 16
ACCEPTED MANUSCRIPT Figs. 42 and 43 show surface friction coefficients (-f "(0)) in terms of β, for incremental changes in the values of Kp and M, respectively. Observations prove an increase in surface friction coefficients as well as increase in Kp and M values in the presence of β. Figs. 44-47, show heat transfer properties in two PST and PHF cases for Ec variations in terms of Nr and Pr in the presence of β. Thus, it can be concluded that, by increasing the Ec
PT
in the presence of Nr, the values of the Nusselt number(𝜃 ′ (0)) for the PST case will decrease
RI
and increase for the PHF case, but for the incremental values of Pr in the presence of β, the
SC
Nusselt number (𝜃(0)) shows a different behavior than the Ec changes. That is, with the increase of Pr in the presence of β, the Nusselt number increases in the PST case and
NU
decreases in PHF case.
MA
Finally, Fig. 48, show the outcomes of incremental changes of R in the presence of β on the couple stress (ℎ′ (0)). The graph show that, with increasing R in the presence ??, couple
D
stress is increased.
PT E
5. Conclusion
The goal of this article is to discuss about the impact of radiation in the flow of a dusty
CE
fluid suspended with TiO2 nanoparticles over a permeable stretching sheet. The partial non-
AC
linear differential equations defining this issue were converted to ordinary forms by means of similarity transformations, so they could be solved easily by an analytical method. Here Runge-kutta Fehlberg fourth-fifth order scheme was employed as a numerical method with the help of Maple software. After solving those equations, the results were displayed in the tables and figures where the influences of different parameters in both PST and PHF cases on skin friction coefficient, couple stress, heat transfer characteristics, temperature and velocity profiles were indicated. Some of important findings of these profiles are listed below.
17
ACCEPTED MANUSCRIPT
The velocity profile 𝑓′(𝜂) decreases by enhancing 𝑓0 ,𝐾𝑝 , 𝑙 and 𝑀 in additions it increases by enhancing 𝜙, 𝑅.
The temperature 𝜃(𝜂) has direct relationships with 𝐾𝑝 ,??, 𝐸𝑐, 𝑀 and 𝑁𝑟 and also it has inverse correlations with 𝑅 , 𝑙 , 𝜙 ,Pr and 𝑓0 for both types of heating process( PHF and PST cases). Increasing in the value of radiation parameter 𝑁𝑟 make the thermal boundary layer
PT
RI
thicker. The couple stress value ℎ′(0) augments with a rise in 𝑅.
For increasing the values of Ec the Nusselt number increases for PHF case and
SC
Skin friction coefficient −𝑓′′(0) has direct relationship with 𝐾𝑝 , 𝑀.
MA
NU
decreases for PST case but Pr has invers impact on nusselt number.
D
References:
PT E
[1] B.C. Sakiadis, Boundary layer behavior on continuous solid surface: I. Boundary-layer equations for two dimensional and axisymmetric flow, AIChEJ, 7 (1961) 26–28. [2] L.J. Crane, Flow past a stretching plane, Z. Angew. Math Phys. 21 (1970) 645–647.
CE
[3] C.Y. Wang, The 3-dimensional flow due to a stretching flat surface, Phys. Fluids, 27
AC
(1984) 1915-1917.
[4] K. R. Rajagopal, T. Y. Na, A. S. Gupta, Flow of a viscoelastic fluid over a stretching sheet, Rheol. Acta, 23, (1984) 213–215. [5] A.Chakrabarti and A.S.Gupta, Hydromagnetic flow and heat transfer over a stretching sheet, Quart.Appl. Math., 37 (1979) 73-78. [6] N.A. Kelson, A. Desseaux, Effect of surface conditions on flow of a micropolar fluid driven by a porous stretching sheet, International Journal of Engineering Science, 39 (2001) 1881-1897.
18
ACCEPTED MANUSCRIPT [7] I. Swain, S. R. Mishra, H. B. Pattanayak, Flow over Exponentially Stretching Sheet through Porous Medium with Heat Source/Sink, Journal of Engineering, 2015 (2015) 7 pages. [8] Tasawar Hayat, Muhammad Ijaz Khan, Ahmad Alsaedi, Muhammad Imran Khan, Homogeneous-heterogeneous reactions and melting heat transfer effects in the MHD flow by a stretching surface with variable thickness, Journal of Molecular Liquids, 223 (2016) 960-
PT
968. [9] A. M. Salem, R. Fathy, Effect of Thermal Radiation on MHD Mixed Convective Heat
RI
Transfer Adjacent to a Vertical Continuously Stretching Sheet in the Presence of Variable
SC
Viscosity, Journal of the Korean Physical Society , 57 (2010) 1401-1407. [10] Mahantesh M. Nandeppanavar, K. Vajravelu, M. Subhas Abel, M. N. Siddalingappa,
NU
MHD flow and heat transfer over a stretching surface with variable thermal conductivity and partial slip, Meccanica, 48 (2013) 1451–1464.
MA
[11] A.C. Eringen, Theory of micropolar fluids, Journal of Mathematics and Mechanics, 16 (1966) 1-18.
D
[12] T. Ariman, M.A. Turk, N.D. Sylvester, Microcontinuum fluid mechanics—A review,
PT E
International Journal of Engineering Science,11 (1973) 905-930. [13] R. Bhargava, L. Kumar, H.S. Takhar, Finite element solution of mixed convection micropolar flow driven by a porous stretching sheet, International Journal of Engineering
CE
Science, 41 (2003) 2161-2178.
[14] Dulal Pal, Gopinath Mandal, Thermal radiation and MHD effects on boundary layer
AC
flow of micropolar nanofluid past a stretching sheet with non-uniform heat source/sink, International Journal of Mechanical Sciences, 126 (2017) 308-318. [15] M. Madani, Y. Khan, M. Fathizadeh, A .Yildirim, Application of homotopy perturbation and numerical methods to the magneto-micropolar fluidflow in the presence of radiation, Engineering computations, 29 ( 2012) 277-294. [16] Hassan A.M. El-Arabawy, Effect of suction/injection on the flow of a micropolar fluid past a continuously moving plate in the presence of radiation, International Journal of Heat and Mass Transfer, 46 (2003) 1471-1477.
19
ACCEPTED MANUSCRIPT [17]
Nor Azizah Yacob, Anuar Ishak, Ioan Pop, Melting heat transfer in boundary layer
stagnation-point flow towards a stretching/shrinking sheet in a micropolar fluid, Computers & Fluids, 47 (2011) 16-21. [18] Haliza Rosali, Anuar Ishak, Ioan Pop, Micropolar fluid flow towards a stretching/shrinking sheet in a porous medium with suction, International Communications in Heat and Mass Transfer, 39 (2012) 826-829.
RI
fluid, Superlattices and Microstructures, 98 (2016) 283-294.
PT
[19] Konrad Nering, Kazimierz Rup, The effect of nanoparticles added to heated micropolar
[20] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, The
SC
Proceedings of the 1995, ASME International Mechanical Engineering Congress and Exposition, San Francisco, USA, ASME 1995, pp. 99–105 FED 231/MD 66. E.Izadpanahi, M.Ghalambaz, Phase-change heat
NU
[21] A.J.Chamkha, A.Doostanidezfuli,
transfer of single/hybrid nanoparticles-enhanced phase-change materials over a heated
MA
horizontal cylinder confined in a square cavity, Advanced Powder Technology, 28 (2017) 385-397.
[22] M. Sheikholeslami, H.R. Ashorynejad, P. Rana, Lattice Boltzmann simulation of
PT E
214 (2016) 86–95.
D
nanofluid heat transfer enhancement and entropy generation, Journal of Molecular Liquids,
[23] M.Ghalambaz, A.Doostani, E.Izadpanahi, A.J.Chamkha, Phase-change heat transfer in a cavity heated from below: The effect of utilizing single or hybrid nanoparticles as additives,
CE
Journal of the Taiwan Institute of Chemical Engineers, 72 (2017) 104-115. [24] M. Sheikholeslami, M.M. Rashidi, T.Hayat, D.D. Ganji, Free convection of magnetic
399.
AC
nanofluid considering MFD viscosity effect, Journal of Molecular Liquids, 218 (2016) 393–
[25] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Soheil Soleimani, Thermal management for free convection of nanofluid using two phase model, Journal of Molecular Liquids, 194 (2014) 179-187. [26] K.V. Prasad, K. Vajravelu, Hanumesh Vaidya, Robert A. Van Gorder, MHD flow and heat transfer in a nanofluid over a slender elastic sheet with variable thickness, Results in Physics, 7 (2017) 1462-1474.
20
ACCEPTED MANUSCRIPT [27] M. Sheikholeslami, D.D. Ganji, Nanofluid hydrothermal behavior in existence of Lorentz forces considering Joule heating effect, Journal of Molecular Liquids. 224 (2016) 526-537. [28] Srinivas Reddy C, Kishan Naikoti, Mohammad Mehdi Rashidi, MHD flow and heat transfer characteristics of Williamson nanofluid over a stretching sheet with variable thickness and variable thermal conductivity, Transactions of A. Razmadze Mathematical
PT
Institute, 171 (2017) 195-211. [29] M. Sheikholeslami, Soheil Soleimani, D.D. Ganji, Effect of electric field on
RI
hydrothermal behavior of nanofluid in a complex geometry, Journal of Molecular Liquids.
SC
213 (2016) 153-161.
[30] M. Sheikholeslami, D.D. Ganji, Free convection of Fe3O4-water nanofluid under the
NU
influence of an external magnetic source, Journal of Molecular Liquids. 229 (2017) 530-540. [31] F.E. Marble, Dynamics of dusty gases, Ann. Rev. Fluid Mech, 2 (1970) 397- 446.
MA
[32] G. K. Ramesh; B. J. Gireesha, Flow Over a Stretching Sheet in a Dusty Fluid with Radiation Effect, J. Heat Transfer. 135 (2013) 6 pages.
D
[33] B.J. Gireesha, A.J. Chamkha, S. Manjunatha, C.S. Bagewadi, Mixed convective flow of
PT E
a dusty fluid over a vertical stretching sheet with non‐uniform heat source/sink and radiation, International Journal of Numerical Methods for Heat & Fluid Flow, 23 (2013) 598-612. [34] G.C. Hazarika, J. Konch, Effects of variable viscosity and thermal conductivity on
CE
magnetohydrodynamic free convection dusty fluid along a vertical porous plate with heat generation, Turk J Phys, 40 (2016) 52 – 68.
AC
[35] O.D. Makinde, T. Chinyoka, MHD transient flows and heat transfer of dusty fluid in a channel with variable physical properties and Navier slip condition, Computers & Mathematics with Applications, 60 (2010) 660-669. [36] M.A. Ezzat, A.A. El-Bary, M.M. Morsey, Space approach to the hydro-magnetic flow of a dusty fluid through a porous medium, Computers & Mathematics with Applications, 59 (2010) 2868-2879. [37] N.C. Ghosh, B.C. Ghosh, L. Debnath, The hydromagnetic flow of a dusty viscoelastic fluid between two infinite parallel plates, Computers & Mathematics with Applications, 39 (2000) 103-116. 21
ACCEPTED MANUSCRIPT [38] N. Sandeep, C. Sulochana, B. Rushi Kumar, Unsteady MHD radiative flow and heat transfer of a dusty nanofluid over an exponentially stretching surface, Engineering Science and Technology, an International Journal, 19 (2016) 227-240. [39] R. Muthuraj, K. Nirmala, S. Srinivas, Influences of chemical reaction and wall properties on MHD Peristaltic transport of a Dusty fluid with Heat and Mass transfer, Alexandria Engineering Journal, 55 (2016) 597-611.
PT
[40] M.M. Bhatti, A. Zeeshan, N. Ijaz, O. Anwar Bég, A. Kadir, Mathematical modelling of nonlinear thermal radiation effects on EMHD peristaltic pumping of viscoelastic dusty fluid
RI
through a porous medium duct, Engineering Science and Technology, an International
SC
Journal, 20 (2017) 1129-1139.
[41] Aminreza Noghrehabadins, Ehsan Izadpanahi, Mohammad Ghalambaz, Analyze of fluid
NU
flow and heat transfer of nanofluids over a stretching sheet near the extrusion slit, Computers & Fluids, 100 (2014) 227-236
MA
[42] M. Ghalambaz, E. Izadpanahi, A. Noghrehabadi, A. Chamkha, Study of the boundary layer heat transfer of nanofluids over a stretching sheet: Passive control of nanoparticles at
PT E
D
the surface, Canadian Journal of Physics , 93 (2015) 725-733.
Table.1. Density, specific heat and thermal conductivity of base fluids and nanoparticles.
𝑷𝒓𝒐𝒑𝒆𝒓𝒕𝒊𝒆𝒔\𝒄𝒐𝒏𝒔𝒕𝒊𝒕𝒖𝒆𝒏𝒕𝒔
Tio2
997.1
4250
𝑱 𝑺𝒑𝒆𝒄𝒊𝒇𝒊𝒄 𝒉𝒆𝒂𝒕, 𝑪𝒑 ( ⁄𝑲𝒈 𝑲)
4179
686.2
𝑻𝒉𝒆𝒓𝒎𝒂𝒍 𝒄𝒐𝒏𝒅𝒖𝒄𝒕𝒊𝒗𝒊𝒕𝒚, 𝒌 (𝑾⁄𝒎 𝑲)
0.613
8.9538
CE
H2 O
𝑲𝒈 ⁄ 𝟑) 𝒎
AC
𝑫𝒆𝒏𝒔𝒊𝒕𝒚, 𝝆 (
List of figures: 22
ACCEPTED MANUSCRIPT Fig. 1. Geometry of the problem . Fig. 2. Comparison between the results of velocity profile and the results of Ref. [42] Fig. 3. Influence of f0 on velocity profile when
1,Kp 0.4,M 0.4,R 0,m 0.6, 6.1, l 0.5, 0.2, Nr 0.4,Ec 0.3,Pr 6.2.
Fig. 4 & 5. Influence of f0 on temperature profile for PST case. Fig. 6 & 7. Influence of f0 on temperature profile for PHF case. Fig. 8. Influence of l on velocity profile when
RI
Fig. 10. Influence of l on temperature profile for PHF case.
PT
1,Kp 0.4,M 0.4,R 0,m 0.6, 6.1,f0 0.01, 0.2, Nr 0.4,Ec 0.3,Pr 6.2. Fig. 9. Influence of l on temperature profile for PST case. Fig. 11. Influence of R on velocity profile when
SC
1,Kp 0.4,M 0.4,f0 0.01,m 0.6, 6.1, l 0.5, 0.2, Nr 0.4,Ec 0.3,Pr 6.2.
Fig. 12 & 13. Influence of R on temperature profile for PST case.
NU
Fig. 14 & 15. Influence of R on temperature profile for PHF case. Fig.16. Influence of Kp on velocity profile when
MA
1,M 0.4,R 0,f0 0.01,m 0.6, 6.1, l 0.5, 0.2, Nr 0.4,Ec 0.3,Pr 6.2. Fig. 17 & 18. Influence of Kp on temperature profile for PST case. Fig. 19 & 20. Influence of R on temperature profile for PHF case.
D
Fig. 21. Influence of M on velocity profile when
PT E
1,Kp 0.4,R 0,f0 0.01,m 0.6, 6.1, l 0.5, 0.2, Nr 0.4,Ec 0.3,Pr 6.2. Fig. 22 & 23. Influence of M on temperature profile for PST case. Fig. 24 & 25. Influence of M on temperature profile for PHF case.
CE
Fig. 26. Influence of ?? on velocity profile when
AC
1,Kp 0.4,M 0.4,R 0,f0 0.01,m 0.6, 6.1, l 0.5, Nr 0.4,Ec 0.3,Pr 6.2. Fig. 27. Influence of ?? on temperature profile for PST case. Fig. 28. Influence of ?? on temperature profile for PHF case. Fig. 29. Influence of ?? on velocity profile when
Kp 0.4,M 0.4,R 0,f0 0.01,m 0.6, 6.1, l 0.5, 0.2, Nr 0.4,Ec 0.3,Pr 6.2. Fig. 30. Influence of ?? on temperature profile for PST case. Fig. 31. Influence of ?? on temperature profile for PHF case. Fig. 32 & 33. Influence of Nr on temperature profile for PST case when
1,Kp 0.4,M 0.4,R 0,f0 0.01,m 0.6, 6.1, l 0.5, 0.2,Ec 0.3,Pr 6.2. 23
ACCEPTED MANUSCRIPT Fig. 34 & 35. Influence of Nr on temperature profile for PHF case. Fig. 36. Influence of Ec on temperature profile for PST case when
1,Kp 0.4,M 0.4,R 0,f0 0.01,m 0.6, 6.1, l 0.5, 0.2, Nr 0.4,Pr 6.2. Fig. 37. Influence of Ec on temperature profile for PHF case. Fig. 38 & 39. Influence of Pr on temperature profile for PST case when
1,Kp 0.4,M 0.4,R 0,f0 0.01,m 0.6, 6.1, l 0.5, 0.2, Nr 0.4,Ec 0.3.
PT
Fig. 40 & 41. Influence of Pr on temperature profile for PHF case. Fig. 42 & 43. Influence of Kp & M on skin friction coefficient in the presence of β.
RI
Fig. 44 & 45. Influence of Ec on heat transfer properties in the presence of Nr for PST and PHF cases.
SC
Fig. 46 & 47. Influence of Pr on heat transfer properties in the presence of β for PST and PHF cases.
AC
CE
PT E
D
MA
NU
Fig. 48. Influence of Pr on couple stress in the presence of β.
Fig.1. Geometry of the problem . 24
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
AC
CE
PT E
D
MA
Fig.2.
Fig.3.
25
RI
PT
ACCEPTED MANUSCRIPT
PT E
D
MA
NU
SC
Fig.4.
AC
CE
Fig.5.
Fig.6. 26
RI
PT
ACCEPTED MANUSCRIPT
Fig.8.
AC
CE
PT E
D
MA
NU
SC
Fig.7.
Fig.9. 27
Fig.10.
PT E
D
MA
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
AC
CE
Fig.11.
Fig.12. 28
RI
PT
ACCEPTED MANUSCRIPT
Fig.14.
AC
CE
PT E
D
MA
NU
SC
Fig.13.
Fig.15. 29
RI
PT
ACCEPTED MANUSCRIPT
PT E
D
MA
NU
SC
Fig.16.
AC
CE
Fig.17.
Fig.18.
30
Fig.20.
AC
CE
PT E
D
MA
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
Fig.21.
31
Fig.19.
RI
PT
ACCEPTED MANUSCRIPT
PT E
D
MA
NU
SC
Fig.22.
AC
CE
Fig.23.
Fig.24.
32
RI
PT
ACCEPTED MANUSCRIPT
Fig.26.
AC
CE
PT E
D
MA
NU
SC
Fig.25.
Fig.27. 33
Fig.28.
PT E
D
MA
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
AC
CE
Fig.29.
Fig.30. 34
RI
PT
ACCEPTED MANUSCRIPT
Fig.32.
AC
CE
PT E
D
MA
NU
SC
Fig.31.
35
RI
PT
ACCEPTED MANUSCRIPT
PT E
D
MA
NU
SC
Fig.33.
AC
CE
Fig.34.
Fig.35.
36
PT E
D
MA
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
AC
CE
Fig.37.
Fig.38.
37
Fig.36.
PT E
SC
D
MA
NU
Fig.39.
RI
PT
ACCEPTED MANUSCRIPT
AC
CE
Fig.40.
Fig.41. 38
PT E
D
MA
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
AC
CE
Fig.43.
Fig.44. 39
Fig.42.
PT E
D
MA
NU
SC
Fig.45.
RI
PT
ACCEPTED MANUSCRIPT
AC
CE
Fig.46.
Fig.47. 40
AC
CE
PT E
D
MA
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
41
Fig.48.
AC
CE
PT E
D
MA
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
Graphical abstract 42
ACCEPTED MANUSCRIPT Highlights
AC
CE
PT E
D
MA
NU
SC
RI
PT
TiO2 nanoparticle homogeneously suspended in Micropolar fluid Magnetic field and thermal radiation effect on Micropolar dusty fluid are considered The Runge-Kutta-Fehlberg numerical method is employed to solving the nonlinear equations Increasing in the value of radiation parameter Nr make the thermal boundary layer thicker For increasing the values of Ec the Nusselt number increases for PHF case and decreases for PST
43
S.S. Ghadikolaei, Kh. Hosseinzadeh, M. Yassari, H. Sadeghi, D.D. Ganji PII: DOI: Reference:
S0167-7322(17)33149-5 doi: 10.1016/j.molliq.2017.08.111 MOLLIQ 7819
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
14 July 2017 27 August 2017 29 August 2017
Please cite this article as: S.S. Ghadikolaei, Kh. Hosseinzadeh, M. Yassari, H. Sadeghi, D.D. Ganji , Boundary layer analysis of micropolar dusty fluid with TiO2 nanoparticles in a porous medium under the effect of magnetic field and thermal radiation over a stretching sheet, Journal of Molecular Liquids (2017), doi: 10.1016/j.molliq.2017.08.111
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Boundary layer analysis of Micropolar dusty fluid with TiO2 nanoparticles in a porous medium under the effect of magnetic field and thermal radiation over a stretching sheet
S.S. Ghadikolaeia, Kh. Hosseinzadehb, M.Yassaria, H. Sadeghia, D.D. Ganjib* a
PT
Department of Mechanical Engineering, Mazandaran university science and Technology , Babol, Iran
b
Corresponding author,
E-mail:[email protected]
SC
*
RI
Department of Mechanical Engineering, Babol Noushirvani University of Technology, Babol, Iran
NU
Abstract
This paper analyzed the boundary layer flow and heat transfer of an incompressible TiO2-
MA
water nanoparticle on micropolar fluid with homogeneously suspended dust particles in the presence of thermal radiation. Since the nanoparticles have high thermal conductivity
D
coefficient compared to base fluids, so their distribution leads to an increase in the thermal
PT E
conductivity of the fluids and they are considered to be the main parameters of heat transfer. In the following, nonlinear equations that can describe this problem are presented in this
CE
article. These nonlinear equations are numerically analyzed using the Runge-Kutta-Fehlberg method in MAPLE software. The main goal of this paper is to study and analyze the behavior
AC
of the velocity profile in two phases of liquid and dust for different values of parameters suction/injection (f0), mass concentration (L), Material (R), Penetrable (Kp), Magnetic (M), Solid volume fraction (𝜑), fluid particle interaction (β) and also temperature profile effectiveness from radiation parameter (Nr) changes, Eckert number (Ec), Prandtl number (Pr) in two PST and PHF cases for both fluid and dust phases, in which dual behavior of the velocity profile compared to the β changes in the fluid and dust phases and also Lorentz force generated by the magnetic field is also mentioned. Finally, the effect of changes of Kp and M 1
ACCEPTED MANUSCRIPT in the presence of β on the coefficient of surface friction, the effect of Ec changes in the presence of Nr and Pr changes in the presence of β on the Nusselt number in two PST and PHF cases and also R changes in the presence of f0, β and M on couple stress has been investigated and analyzed.
PT
Key words: boundary layer flow, micropolar fluid, dust nanoparticles, thermal radiation,
RI
magnetic field.
k .....permeability of the porous medium
Uw .....stretching sheet velocity
k * .....mean absorption coefficient
Vw .....suction velocity
r .....mass of the dust particle
Greek Symbols
B0 .....uniform magnetic field strength
MA
S .....number density of the dust particles
R .....material parameter
.....fluid particle interaction parameter
.....solid volume fraction parameter
.....ratio of specific heat
.....dimension less temperature
PT E
M .....magnetic parameter
D
N ..... angular velocity K .....Stokes drag constant
NU
SC
Nomenclature
.....dynamic viscosity of nanofluid
.....kinematic viscosity
Kp ..... permeability parameter
.....density
Ec .....Eckert number
.....dimensionless variable
AC
CE
Nr ..... radiation parameter
Pr .....Prandtl number
* ..... Stefan-Boltzman constant
C ..... specific heat
v .....relaxation time of the particle phase
T ..... temperature
T .....thermal equilibrium time
T .....free stream temperature
Subscripts
j ..... refrence length
p .....dust particle
l .....mass concentration parameter
f .....fluid
k .....thermal conductivity
nf .....nanofluid 2
ACCEPTED MANUSCRIPT 1. Introduction From last decades till now, the study of viscous fluids on a stretching sheet has attracted many researchers attention. This is due to the fact that many processes like casting, hot rolling, drawing wires of copper, glass blowing, and extrusion sheets of plastic and also
PT
polymer industries have benefited from it. First, Sakiadis [1] investigated boundary layer flow of an incompressible viscous fluid
RI
over moving continuous solid surfaces. Then to develop this research, Crane [2] studied two-
SC
dimensional flow past a linearly stretching sheet. His experiment played a significant role in
NU
the polymer industry. A little later, same work was done by Wang [3] for three dimensional. Since then, many researches have focused on stretching sheet issues. Problems which the
MA
conditions and effects of different parameters such as MHD, thermal radiation, suction/injection, slip effects, permeable surface and etc. are considered. For an example
D
viscoelastic fluid flow past a stretching surface was investigated by Rajagopal et al. [4].
PT E
Chakrabarti and Gupta [5] discussed about MHD flow and heat transfer on a stretching surface. Kelson and Desseaux [6] studied the effects of effective surface heat transfer in a
CE
micropolar fluid on a permeable stretching sheet. The study on heat and mass transfer effect in boundary layer MHD flow of a viscous fluid in the presence of thermal radiation and
AC
exponentially stretching sheet through porous medium in order to investigate the effects of temperature and velocity field and also Nusselt number were conducted by Swain et al. [7]. Hayat et al. [8] analyzed the effects of melting heat in flow on variable thicked stretching surface. In their paper, effects of various physical parameters on velocity distribution, temperature, and fluid concentration had been investigated. Fathy and Salem [9] investigated radiation effects on forced free movement of the laminar flow in the vicinity of vertical stretching sheet considering the magnetic field and the unsteady viscosity. In their study, velocity and temperature equations of sheet are written using power law form. MHD flow 3
ACCEPTED MANUSCRIPT and heat transfer on a stretching sheet with variable thermal conductivity has been analyzed by Nandeppanavarnet al. [10] considering the effect of partial slip and using similar conversions for converting partial equations to ordinary equations. Most mentioned studies have been related to Newtonian fluids. In recent years, the study
PT
on non-Newtonian fluids has also become extended, due to wide application of these fluids in the oil industry, the polymer industry, and other industries. Non-Newtonian fluids such as
RI
color, blood, shampoo, palm oil, mayonnaise, etc., are such that their mathematical modeling
SC
is much more complicated than conventional Newtonian fluids and cannot be modeled with the Navier-Stokes equations related to Newtonian fluids. These fluids are divided into several
NU
categories, in which micropolar fluids are an example of them. Micropolar fluids are nonNewtonian fluids with microstructure that consists of rigid particles suspended in a viscous
MA
medium in which particle deformation is ignored. These fine particles can be rotated in fluid, so in modeling these type of fluids, the motion of fluid is described by two velocity vectors,
PT E
D
classical velocity vector and micro rotation (spin) vector. A micro fluid has many applications in various fields. Among them, colloids and polymeric suspensions, liquid crystal and lubrication problem can be pointed out. The first time micropolar fluids theory
CE
was raised by Eringen [11] but Ariman et al. [12] revealed the many practical applications of
AC
this type of fluid in their article. Other studies were done in this field with different condition, too. The finite element method was used to analyze the micropolar fluid flow in a porous medium under uniform suction by Bhargava et al. [13]. Pal and Mandal [14] analyzed electrically conduction micropolar nanofluid flow over a stretching surface that was under thermal radiation and a uniform heat source by Runge-Kutta numerical method. Madani et al. [15] investigated magneto-micropolar fluid past a continuously moving plate considering radiation effects. There were blowing and suction in this research. The study of suction or injection effects on flow characterization which past on continuous moving plate in the 4
ACCEPTED MANUSCRIPT presence of radiation in micropolar fluid and temperature analysis inside the boundary layer was done by EL-Arabawy [16]. Yacob et al. [17] investigated a steady stagnation point flow of a micropolar fluid which past on a linearly shrinking/stretching sheet. Micropolar fluid flow on a permeable stretching sheet considering suction was investigated by Rasoli et al. [18]. At the end, Nering and Rup [19] study can be mentioned among new studies about fluid
PT
types. They experimented the effect of adding nanoparticles on a micropolar fluid in
RI
increasing the amount of heat transfer during an unsteady natural convection.
SC
Another topic that has been found over the last few years in the scientific activities of researchers is the subject of nanofluids. These fluids come from adding very fine particles to
NU
nano dimensions in common fluids such as water or oil to improve their thermal conductivity. Nanofluids have different applications in industry and engineering that for example treatment
MA
of cancer infected tissues, electronic and space cooling can be pointed out. In 1995, the first time nanofluid word was raised by Choi [20]. Since then, scientists have made many articles
PT E
D
on this subject. The study and investigation of melting process heat transfer of nano-enhanced phase-change material consisting of nanoparticle and hybrid nanoparticles was performed on a horizontal heated cylinder located in the middle of a square hole by Chamkha et al. [21].
CE
They investigated the effects of the volume fraction of nanoparticles, the turbulent
AC
conductivity parameter, the viscosity parameter and the Fourier number on melting and heat transfer after numerical solving of nonlinear governing differential equations the problem with the Galerikin finite element method. They concluded that the highest melting rate for Fourier numbers was in the range (0-0.5), however, for the values greater than 0.5, the melting rate decreases. Simulation of heat transfer and entropy generation of a nanofluid flow inside a hot squared cavity using water as a base fluid and four types of metallic and nonmetallic nanofluids of Al2O3, Cu, Ag, and TiO2 using Lattice Boltzman method and also calculating the thermal conductivity and the effective viscosity using Brinkman and 5
ACCEPTED MANUSCRIPT Maxwell-Garnett (MG) model were carried out by Sheikholeslami et al. [22]. Ghalambaz et al. [23] in an article examine the heat transfer of melting process of a nano-enhanced phasechange material using nanoparticles and hybrid nanoparticles separately in a hot cavity in isothermal condition with isolated left and right walls, floor and roof at hot and cold temperatures, respectively. They also studied of the effects of parameters and dimensionless
PT
numbers on melting process and heat transfer. They concluded significant changes in solid-
RI
liquid interface for large values of Fourier numbers due to the increase of the nanoparticles
SC
volume fraction, conductivity parameter, and viscosity parameter. Sheikholeslami et al. [24] studied and investigated the effects of magnetic field dependent viscosity on free convection
NU
heat transfer of a single-phase nanofluid considering its Brownian motion inside a chamber and simulated the problem of control volume based finite element method. They also
MA
examined the effects of various parameters on hydrothermal behavior, and concluded that the reduction of Nusselt number due to magnetic field dependent viscosity for low Hartman
D
number and high Raylight number is very sensetive. Studying free convection heat transfer of
PT E
a two-phase nanofluid and achieving inverse relation between Nusselt number and Lewis number, and increasing Nusselt number by increasing Buoyancy ratio number were done by
CE
Sheikholeslami et al. [25].There are also very useful and applicable articles in this field which
AC
can be study in Ref. [26-30].
As was told in the previous paragraph about combining nano particles in pure fluids such as water and oil, the presence of particles in milli and/or micro in base fluids cause improve their thermal conductivity. As we know, it is rarely possible to find an impurity-free fluid such as dust particles in nature. Therefore, the presence of these micro or millimeter sized particles (dust particles) in the base fluids cause to improve the properties of the heat transfer of the fluids. Considered name to these fluids is dusty fluids. Research and study on these fluids has been useful in many practical applications such as combustion, petroleum 6
ACCEPTED MANUSCRIPT transport, lunar ash flows, power plant piping and many other engineering branches. In 1970, Marble [31] examined the effects of dusty gases on fluid dynamics. In recent years, many papers are written in this field. Analysis of dusty fluid flow past a stretching surface in the presence of radiation effects was done by Ramesh and Gireesha [32]. They used Runge-Kutta numerical method in their analysis. Gireesha et al. [33] studied the effects of radiation and
PT
space-dependent heat source in convective of a dusty fluid along a vertical stretching surface.
RI
Hazarika and Konch [34] analyzed MHD free convection of a dusty fluid over a vertical
SC
permeable surface by a numerical method. It should be noted that in that problem, thermal conductivity effects and variable viscosity on convection was intended. Investigation of
NU
unsteady flow and heat transfer of a dusty fluid between two parallel plates with variable physical properties is conducted by Makinde and Chingoka [35]. The effects of viscosity,
MA
uniform magnetic field, wall slip parameter and electric conductivity on temperature field and fluid velocity and dust particles were investigated in their research. Good articles in this area
PT E
D
could be study in Ref. [36-40].
According to mentioned content of this section, the aim of this study is to analyze the MHD boundary layer flow and heat transfer of a micropolar dusty fluid with TiO2
CE
nanoparticles over a permeable stretching surface in the presence of thermal radiation. An
AC
analytical method used to solve governing differential equations this problem is solved by fourth-fifth Rung-Kutta Fehlberg method that is done with the help of the Maple software.
2. Mathematical formulation of the problem A steady two-dimensional laminar flow and heat transfer of a dusty micropolar fluid suspended with TiO2 nanoparticles over a permeable stretching sheet is studied here. The surface is along x-axis and moves with a linear velocity Uw (x) = bx, where b is constant. The fluid occupies the space 𝑦 > 0 as it is shown in Fig.1. In addition a uniform magnetic field B0
7
ACCEPTED MANUSCRIPT is applied along the positive y-direction. It is assumed that the particles of dust are of the same size and the number density of them remains constant throughout the flow. Also the effects of thermal radiation are considered in this problem.
The boundary layer governing equations the behavior of the mentioned fluid can be
PT
written as follows:
u u k 2u KS k N B02 v ( nf ) 2 (u p u ) u nf u , x y nf y nf nf y nf k
(4)
D
u p u v p p ) KS (u u p ), x y
(3)
MA
u p v p 0, x y
(5)
PT E
p (u p
(2)
NU
N N 2 N k u u v (2N ), 2 x y nf j y nf j y
SC
u
(1)
RI
u v 0, x y
Here (??, ??) and (𝑢𝑝 ,𝑣𝑝 ) represent the velocity components of nanofluid and dust phases
CE
in the ?? and ?? directions respectively. Furthermore, Stokes drag constant (𝐾), the density of dust particles (𝜌𝑝 ), the effective density of the nanofluid (𝜌𝑛𝑓 ) and spin-gradient viscosity (𝛾)
AC
are defined as:
K 6f a , p rS , nf (1 )f s , (nf k ) j nf (1 R ), (6) 2
2
Where the dynamic viscosity of the nanofluid (𝜇𝑛𝑓 ), the material parameter (𝑅) and reference length (𝑗) are given by:
nf
f k ,R , j nf , 2.5 nf a (1 )
(7)
8
ACCEPTED MANUSCRIPT The related boundary conditions are:
u Uw (x ),v Vw (x ), N n
u y
at y 0,
u v 0,u p 0,v p v , N 0 as y .
(8)
By introducing the following similarity transformations [41]:
PT
Uw (x ) 12 U (x ) 12 U (x ) 12 ) y, ( w ) y , N Uw (x)( w ) h (), f x f x f x
(
RI
Where, Uw (x ) bx , Vw (x ) f 0 bf .
(9)
SC
And the continuity Equations (1) and (4) are identically satisfied by introducing a stream
, v , y x
MA
u
NU
function 𝜓 as follows:
Similarly, it can be done for 𝑢𝑝 and 𝑣𝑝 .
PT E
ordinary differential equations:
s ](ff (f )2 ) (1 )2.5[l (F f ) Mf ] f
CE
(1 R )f (1 )2.5[(1 ) Rh k p f 0
D
Equations (2), (3) and (5) along with boundary condition (8) are reduced to the following
(10)
(11)
FF (F )2 (f F ) 0,
(12)
AC
R (1 )h (1 )2.5[(1 ) s ](fh f h ) R (2h f ) 0, 2 f
Here prime indicates the differentiation with respect to 𝜂. In addition, the parameters in the above equations are defined as follows:
9
ACCEPTED MANUSCRIPT rS
r , the fluid f K B 02 1 particle interaction parameter for velocity , the magnetic parameter M , the bv f b The mass concentration l
permeability parameter K p
, the relaxation time of the particle phase v
f
bk
.
PT
The boundary condition (9) becomes:
f () 1, f () f 0 , h() m at 0,
(13)
SC
RI
f () 0, F () 0, F () 1,h() 0 as .
NU
2.1. Heat transfer Analysis
MA
The governing boundary layer heat transport equations for dusty nanofluid are:
p T T 2T pc pf u v ] K nf ( T T ) (u p u )2 nf ( )2 p 2 x y T v y y 1 qr , (c p )nf y
(14)
(pcmf )[u p
PT E
D
(c p )nf [u
T p T c v p p ] p pf (T p T ), x y T
(15)
CE
Where the thermal conductivity (𝑘𝑛𝑓 ) and the heat capacitance of the nanofluid
AC
(𝜌𝐶𝑝 )𝑛𝑓 are introduced by the following formulas:
(c p )nf (1 )(c p )f ( cp )s ,
k 2k f 2(k f k s ) k nf s , kf k s 2k f 2(k f k s )
(16)
By employing the Rosseland approximation for radiation, the radiative heat flux (𝑞𝑟 ) is given by:
qr
4 *T 4 , 3k *y
(17)
10
ACCEPTED MANUSCRIPT In addition it is assumed that the temperature differences within the flow are sufficiently small, thus by expanding 𝑇 4 in a truncated Taylor series about 𝑇∞ and ignoring the higherorder terms, it can be written as a linear function of the temperature. Then we obtain:
T 4 4TT3 3T3 ,
(18)
16 *T 3 2T c T T v ] K nf ( * ) 2 p pf (T p T ) P (u p u )2 x y T v 3k k nf y
RI
(c p )nf [u
PT
By substituting equations (4) and (5) into equation (1), it becomes:
(19)
SC
u nf ( )2 , y
It is a fact that the solution of the equations (16) and (19) depends on the nature of the
NU
prescribed boundary conditions. So, here two types of heating process are applied.
MA
The prescribed boundary conditions are introduced as follows:
T x x qw D ( )2 (PHF case) at y 0, T Tw T A ( )2 (PST case), K y l l
D
(20)
PT E
T T ,T p T as y .
Where the characteristic length 𝑙 is defined by l
f b
.
T T T T ,p () p , Tw T Tw T
(21)
AC
()
CE
The non-dimensional nanofluid phase temperature ( ) and dust phase temperature p ( ) can be given by:
x l
Where, T T A ( )2 () (PST case), Tw T
D x 2 f (PHF case). ( ) t l b
By substituting (9), (16) and (21) into (15) and (19), we get:
(c p )s k nf 4 (1 Nr ) Pr[(1 ) ](f 2f ) Pr l T ( p ) kf 3 (c p )f
(22)
Pr lEc (F f ) Pr Ecf 0, 2
2
Fp 2F p T ( p ) 0,
(23) 11
ACCEPTED MANUSCRIPT The dimensionless parameters defined in the problem are: The Prandtl number Pr
(c p )f b 2l 2 , the Eckert number Ec Ac pf kf
(PST case) and
b 2l 2t b 4 *T 3 (PHF case), the radiation parameter Nr * , fluid particle interaction Dc pf f k k nf c pf 1 parameter for temperature T , the ratio of specific heat . bT cmf
PT
Ec
() 1 (PST mood), () 1 (PHF mood) at 0,
(24)
NU
3. Numerical method for solution
SC
() 0,p () 0 as .
RI
The subjected thermal boundary conditions become:
MA
In this section, the non-linear ordinary differential equations (10-12) and (22, 23) with respect to the boundary condition (13) and (24) are solved numerically by employing Runge
D
-Kutta Fehlberg 45 method with the help of Maple software. In this technique the boundary
PT E
value problem must be transformed into the initial value problem. In addition, it is crucial to choose a finite values of 𝜂∞ .
25 1408 2197 1 k0 k2 k 3 k 4 ), 216 2565 4109 5
CE
y m 1 = y m h(
AC
16 6656 28561 9 2 y m 1 y m h( k 0 k2 k 3 k 4 k 5 ), 135 12825 56430 50 55
(25) (26)
Where (25) and (26) are the approximations of fourth order and fifth order to the solution respectively, and also;
12
ACCEPTED MANUSCRIPT k 0 f (x m y m ), k f (x h , y hk 0 ), m 1 4 m 4 k 2 f (x m 3 h, y m ( 3 k 0 9 k 1 )h ), 8 32 32 12 1932 7200 7296 k0 k1 k 2 )h ), k 3 f (x m h , y m ( 13 2197 2197 2197 439 3860 845 k 4 f (x m h, y m ( 216 k 0 8k 1 513 k 2 4104 k 3 )h ), k f (x h , y ( 8 k 2k 3544 k 1859 k 11 k )h ), m 1 5 2 m 27 0 2565 2 4104 3 40 4
RI
PT
(27)
SC
Now, in order to solve the problem of this paper by applying the mentioned method, first the non-linear boundary value problem must be converted to the system of first order
NU
differential equations. So, new set of variables are given by:
𝑓 = 𝑞1 , 𝑓 ′ = 𝑞2 , 𝑓 ′′ = 𝑞3 , ℎ = 𝑞4 , ℎ′ = 𝑞5 , 𝐹 = 𝑞6 , 𝐹′ = 𝑞7 , 𝜃 = 𝑞8 , 𝜃′ = 𝑞9 ,
MA
𝜃𝑝 = 𝑞10
(28)
After applying the above equations into the equations (10–12) and (22, 23), they are
D
reduced to the following system of ordinary differential equations:
s ](q1q3 q22 ) (1 )2.5[l (q7 q2 ) Mq2 ] f
PT E
(1 R )q3 (1 )2.5[(1 ) Rq5 Kpq2 0
(29)
(30)
q6q7 q72 (q2 q7 ) 0
(31)
(c p )s k nf 4 (1 Nr )q9 Pr[(1 ) ](q q 2q2q8 ) Pr l T (q10 q8 ) kf 3 (c p )f 1 9
(32)
AC
CE
R (1 )q5 (1 )2.5[(1 ) s ](q1q5 q1q4 ) R (2q4 q3 ) 0 2 f
Pr Ec (q7 q2 )2 Pr Ecq32 0
q6q10 2q7q10 T (q8 q10 ) 0
(33)
The boundary conditions (13) and (24) become: 𝑞1 (0) = 𝑓0 , 𝑞2 (0) = 1 , 𝑞3 (0) = 𝑠1 , 𝑞4 (0) = −𝑚 , 𝑞5 (0) = 𝑠2 , 𝑞6 (0) = 𝑠3 , 𝑞7 (0) = 𝑠4 ,
(34) 13
ACCEPTED MANUSCRIPT q8 (0) 1, q9 (0) s5 , q10 (0) s 6 ,(PSTcase ) q8 (0) s5 , q9 (0) 1, q10 (0) s 6 ,(PHFcase).
(35)
Where unknown initial conditions 𝑠1 , 𝑠2 , 𝑠3 , 𝑠4 , 𝑠5 and 𝑠6 are calculated using iterative method called shooting method until boundary the conditions 𝑓 ′ (𝜂) → 0 , ℎ(𝜂) → 0 , 𝐹(𝜂) → 1, 𝐹 ′ (𝜂) → 0 , 𝜃(𝜂) → 0 , 𝜃𝑝 (𝜂) → 0 as 𝜂 → ∞ are satisfied. The above procedure will be
PT
repeated since the nonlinear solution converges with a convergence criterion of10−6. In
RI
addition, the step size is chosen as 𝛥𝜂 = 0.001.
SC
A comparison has been done between the the current obtained results of velocity profile with those of the previously published study of Ghalambaz et al [42] for the special cases in figure
NU
2. A very good agreement can be observed between them.
MA
4. Results and discussion
First, the behavior of velocity and temperature within the boundary layer is shown in the
D
case of different suction/ injection parameter (f0) in Figs. 3-7. Observations show that the
PT E
additive values of f0 reduce the velocity and temperature within the boundary layer. Therefore, the suction/injection parameter can also be referred to as a flow velocity reducer.
CE
Figs. 8-10 show the effect of change in the mass concentration parameter (L) on the
AC
velocity and temperature profiles. Carried out analysis on these graphs shows inverse relation between the values of L the velocity and temperature profiles. That is, velocity and temperature both decrease by increasing L value. The effect of upside changes of material parameter values on the velocity and temperature profiles are shown in Figs. 11-15. Fig.11 show an increase in the velocity profiles in terms of the increase in the value of R, but in contrast, Figs. 12-15 can be seen that with increasing R, the temperature profiles in both PST and PHF cases decrease for both the liquid and dust phases. 14
ACCEPTED MANUSCRIPT Figs. 16-20 show the results of effects of increasing value of the penetrable parameter (Kp) on velocity and temperature profiles, respectively. Obviously, porosity in the environment is restrictive of fluid flow and resistance to its movement. The same behavior is true for increasing values of KP. That is, increasing in Kp is the agent of creation resistance to the motion of the fluid. As we know, air resistance always reduces the velocity and produces
PT
heat. Therefore, increasing the value of Kp causes decrease in fluid velocity and,
RI
consequently, increasing the temperature in the thermal boundary layer.
SC
The effect of increasing magnetic parameter (M) values on the velocity and temperature profiles is shown in Figs. 21-25. The existence of an inverse magnetic field, proportional to
NU
the direction of flow that causes resistance drag force, such as Larentz force in the opposite direction of flow, has led to the results of Fig. 21 for increasing the values of M, the velocity
MA
reduces in both the liquid phase and the dust. Figs. 22-25, show that with increasing M values , the velocity gradient increases in both PST and PHF cases, respectively, because the
PT E
D
thickness of the momentum boundary layer decreases with increasing M. The behavior of the velocity and temperature profiles for a certain variation range of nano
CE
particle volume fraction parameter (𝜙) is shown in Figs. 26-28. Observations show that by increasing the 𝜙 value in the specific range, the velocity and temperature profiles have two
AC
completely different behaviors. The reaction of the velocity profile to increasing in 𝜙, is positive and increasing in both liquid and dust phases, but in contrast, increasing in 𝜙 in both liquid and dust phases cause temperature profile reduction in both PST and PHF cases. Figs. 29-31 show the effects of increasing fluid particle interaction parameter (β) on velocity and temperature distribution. From the observations, it seems that the effect of β variations on the velocity profile, in contrast to the effect of the variation in other studied parameters in two phases of liquid and dust, that is, the process of increasing the values of β
15
ACCEPTED MANUSCRIPT decreases the velocity of fluid phase, on the other hand, it has increased the velocity of the dust phase. Therefore, it can be concluded that the β variation has an inverse relation with the velocity of the fluid phase and is directly related to the velocity of the dust phase. But the results from Figs. 30 and 31 show the uniformity of the temperature profile in terms of changes in the positive beta direction. Therefore, increasing beta increases the temperature of
PT
both fluid and dust phases both PST and PHF cases.
RI
In Figs. 32-35, the behavior of the temperature profile is shown in the exposure to
SC
different values of the radiation parameter (Nr) in PST and PHF cases, respectively. Observations show that by increasing the value of Nr, the temperature rise is also achieved,
NU
because in higher value of Nr, the heat transfer through the conduction reduces the thermal boundary layer thickness and floating force a more than heat transfer through the radiation.
3 4𝜎∗ 𝑇∞
𝐾∗ 𝑘𝑛𝑓
, that is, the direct relations between Nr changes and temperature.
D
of 𝑁𝑟 =
MA
However, it should be noted that the results from Figs. 32-35 are a solid proof of the relation
PT E
The speed of temperature distribution effectiveness in two PST and PHF cases in relation to the changes in Eckert number (Ec) is shown in Figs. 36 and 37. Observations prove that
CE
the speed of temperature distribution increases by increasing Ec in both liquid and dust phases. However, the results from Figs. 36 and 37 are clear because Ec has direct relation
AC
kinetic energy and kinetic energy is directly related to the changes in temperature and thickness of the thermal boundary layer, so it is natural that the kinetic energy also increases with increasing value of Ec, and as a result, increasing temperature and thickness of the thermal boundary layer are followed. The temperature profile reaction in PST and PHF cases in terms of Prandtl number (Pr) is shown in Figs.38-41. Observations show that increasing Pr has inverse relation with temperature and is temperature reduction agent in both liquid and dust phases. 16
ACCEPTED MANUSCRIPT Figs. 42 and 43 show surface friction coefficients (-f "(0)) in terms of β, for incremental changes in the values of Kp and M, respectively. Observations prove an increase in surface friction coefficients as well as increase in Kp and M values in the presence of β. Figs. 44-47, show heat transfer properties in two PST and PHF cases for Ec variations in terms of Nr and Pr in the presence of β. Thus, it can be concluded that, by increasing the Ec
PT
in the presence of Nr, the values of the Nusselt number(𝜃 ′ (0)) for the PST case will decrease
RI
and increase for the PHF case, but for the incremental values of Pr in the presence of β, the
SC
Nusselt number (𝜃(0)) shows a different behavior than the Ec changes. That is, with the increase of Pr in the presence of β, the Nusselt number increases in the PST case and
NU
decreases in PHF case.
MA
Finally, Fig. 48, show the outcomes of incremental changes of R in the presence of β on the couple stress (ℎ′ (0)). The graph show that, with increasing R in the presence ??, couple
D
stress is increased.
PT E
5. Conclusion
The goal of this article is to discuss about the impact of radiation in the flow of a dusty
CE
fluid suspended with TiO2 nanoparticles over a permeable stretching sheet. The partial non-
AC
linear differential equations defining this issue were converted to ordinary forms by means of similarity transformations, so they could be solved easily by an analytical method. Here Runge-kutta Fehlberg fourth-fifth order scheme was employed as a numerical method with the help of Maple software. After solving those equations, the results were displayed in the tables and figures where the influences of different parameters in both PST and PHF cases on skin friction coefficient, couple stress, heat transfer characteristics, temperature and velocity profiles were indicated. Some of important findings of these profiles are listed below.
17
ACCEPTED MANUSCRIPT
The velocity profile 𝑓′(𝜂) decreases by enhancing 𝑓0 ,𝐾𝑝 , 𝑙 and 𝑀 in additions it increases by enhancing 𝜙, 𝑅.
The temperature 𝜃(𝜂) has direct relationships with 𝐾𝑝 ,??, 𝐸𝑐, 𝑀 and 𝑁𝑟 and also it has inverse correlations with 𝑅 , 𝑙 , 𝜙 ,Pr and 𝑓0 for both types of heating process( PHF and PST cases). Increasing in the value of radiation parameter 𝑁𝑟 make the thermal boundary layer
PT
RI
thicker. The couple stress value ℎ′(0) augments with a rise in 𝑅.
For increasing the values of Ec the Nusselt number increases for PHF case and
SC
Skin friction coefficient −𝑓′′(0) has direct relationship with 𝐾𝑝 , 𝑀.
MA
NU
decreases for PST case but Pr has invers impact on nusselt number.
D
References:
PT E
[1] B.C. Sakiadis, Boundary layer behavior on continuous solid surface: I. Boundary-layer equations for two dimensional and axisymmetric flow, AIChEJ, 7 (1961) 26–28. [2] L.J. Crane, Flow past a stretching plane, Z. Angew. Math Phys. 21 (1970) 645–647.
CE
[3] C.Y. Wang, The 3-dimensional flow due to a stretching flat surface, Phys. Fluids, 27
AC
(1984) 1915-1917.
[4] K. R. Rajagopal, T. Y. Na, A. S. Gupta, Flow of a viscoelastic fluid over a stretching sheet, Rheol. Acta, 23, (1984) 213–215. [5] A.Chakrabarti and A.S.Gupta, Hydromagnetic flow and heat transfer over a stretching sheet, Quart.Appl. Math., 37 (1979) 73-78. [6] N.A. Kelson, A. Desseaux, Effect of surface conditions on flow of a micropolar fluid driven by a porous stretching sheet, International Journal of Engineering Science, 39 (2001) 1881-1897.
18
ACCEPTED MANUSCRIPT [7] I. Swain, S. R. Mishra, H. B. Pattanayak, Flow over Exponentially Stretching Sheet through Porous Medium with Heat Source/Sink, Journal of Engineering, 2015 (2015) 7 pages. [8] Tasawar Hayat, Muhammad Ijaz Khan, Ahmad Alsaedi, Muhammad Imran Khan, Homogeneous-heterogeneous reactions and melting heat transfer effects in the MHD flow by a stretching surface with variable thickness, Journal of Molecular Liquids, 223 (2016) 960-
PT
968. [9] A. M. Salem, R. Fathy, Effect of Thermal Radiation on MHD Mixed Convective Heat
RI
Transfer Adjacent to a Vertical Continuously Stretching Sheet in the Presence of Variable
SC
Viscosity, Journal of the Korean Physical Society , 57 (2010) 1401-1407. [10] Mahantesh M. Nandeppanavar, K. Vajravelu, M. Subhas Abel, M. N. Siddalingappa,
NU
MHD flow and heat transfer over a stretching surface with variable thermal conductivity and partial slip, Meccanica, 48 (2013) 1451–1464.
MA
[11] A.C. Eringen, Theory of micropolar fluids, Journal of Mathematics and Mechanics, 16 (1966) 1-18.
D
[12] T. Ariman, M.A. Turk, N.D. Sylvester, Microcontinuum fluid mechanics—A review,
PT E
International Journal of Engineering Science,11 (1973) 905-930. [13] R. Bhargava, L. Kumar, H.S. Takhar, Finite element solution of mixed convection micropolar flow driven by a porous stretching sheet, International Journal of Engineering
CE
Science, 41 (2003) 2161-2178.
[14] Dulal Pal, Gopinath Mandal, Thermal radiation and MHD effects on boundary layer
AC
flow of micropolar nanofluid past a stretching sheet with non-uniform heat source/sink, International Journal of Mechanical Sciences, 126 (2017) 308-318. [15] M. Madani, Y. Khan, M. Fathizadeh, A .Yildirim, Application of homotopy perturbation and numerical methods to the magneto-micropolar fluidflow in the presence of radiation, Engineering computations, 29 ( 2012) 277-294. [16] Hassan A.M. El-Arabawy, Effect of suction/injection on the flow of a micropolar fluid past a continuously moving plate in the presence of radiation, International Journal of Heat and Mass Transfer, 46 (2003) 1471-1477.
19
ACCEPTED MANUSCRIPT [17]
Nor Azizah Yacob, Anuar Ishak, Ioan Pop, Melting heat transfer in boundary layer
stagnation-point flow towards a stretching/shrinking sheet in a micropolar fluid, Computers & Fluids, 47 (2011) 16-21. [18] Haliza Rosali, Anuar Ishak, Ioan Pop, Micropolar fluid flow towards a stretching/shrinking sheet in a porous medium with suction, International Communications in Heat and Mass Transfer, 39 (2012) 826-829.
RI
fluid, Superlattices and Microstructures, 98 (2016) 283-294.
PT
[19] Konrad Nering, Kazimierz Rup, The effect of nanoparticles added to heated micropolar
[20] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, The
SC
Proceedings of the 1995, ASME International Mechanical Engineering Congress and Exposition, San Francisco, USA, ASME 1995, pp. 99–105 FED 231/MD 66. E.Izadpanahi, M.Ghalambaz, Phase-change heat
NU
[21] A.J.Chamkha, A.Doostanidezfuli,
transfer of single/hybrid nanoparticles-enhanced phase-change materials over a heated
MA
horizontal cylinder confined in a square cavity, Advanced Powder Technology, 28 (2017) 385-397.
[22] M. Sheikholeslami, H.R. Ashorynejad, P. Rana, Lattice Boltzmann simulation of
PT E
214 (2016) 86–95.
D
nanofluid heat transfer enhancement and entropy generation, Journal of Molecular Liquids,
[23] M.Ghalambaz, A.Doostani, E.Izadpanahi, A.J.Chamkha, Phase-change heat transfer in a cavity heated from below: The effect of utilizing single or hybrid nanoparticles as additives,
CE
Journal of the Taiwan Institute of Chemical Engineers, 72 (2017) 104-115. [24] M. Sheikholeslami, M.M. Rashidi, T.Hayat, D.D. Ganji, Free convection of magnetic
399.
AC
nanofluid considering MFD viscosity effect, Journal of Molecular Liquids, 218 (2016) 393–
[25] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Soheil Soleimani, Thermal management for free convection of nanofluid using two phase model, Journal of Molecular Liquids, 194 (2014) 179-187. [26] K.V. Prasad, K. Vajravelu, Hanumesh Vaidya, Robert A. Van Gorder, MHD flow and heat transfer in a nanofluid over a slender elastic sheet with variable thickness, Results in Physics, 7 (2017) 1462-1474.
20
ACCEPTED MANUSCRIPT [27] M. Sheikholeslami, D.D. Ganji, Nanofluid hydrothermal behavior in existence of Lorentz forces considering Joule heating effect, Journal of Molecular Liquids. 224 (2016) 526-537. [28] Srinivas Reddy C, Kishan Naikoti, Mohammad Mehdi Rashidi, MHD flow and heat transfer characteristics of Williamson nanofluid over a stretching sheet with variable thickness and variable thermal conductivity, Transactions of A. Razmadze Mathematical
PT
Institute, 171 (2017) 195-211. [29] M. Sheikholeslami, Soheil Soleimani, D.D. Ganji, Effect of electric field on
RI
hydrothermal behavior of nanofluid in a complex geometry, Journal of Molecular Liquids.
SC
213 (2016) 153-161.
[30] M. Sheikholeslami, D.D. Ganji, Free convection of Fe3O4-water nanofluid under the
NU
influence of an external magnetic source, Journal of Molecular Liquids. 229 (2017) 530-540. [31] F.E. Marble, Dynamics of dusty gases, Ann. Rev. Fluid Mech, 2 (1970) 397- 446.
MA
[32] G. K. Ramesh; B. J. Gireesha, Flow Over a Stretching Sheet in a Dusty Fluid with Radiation Effect, J. Heat Transfer. 135 (2013) 6 pages.
D
[33] B.J. Gireesha, A.J. Chamkha, S. Manjunatha, C.S. Bagewadi, Mixed convective flow of
PT E
a dusty fluid over a vertical stretching sheet with non‐uniform heat source/sink and radiation, International Journal of Numerical Methods for Heat & Fluid Flow, 23 (2013) 598-612. [34] G.C. Hazarika, J. Konch, Effects of variable viscosity and thermal conductivity on
CE
magnetohydrodynamic free convection dusty fluid along a vertical porous plate with heat generation, Turk J Phys, 40 (2016) 52 – 68.
AC
[35] O.D. Makinde, T. Chinyoka, MHD transient flows and heat transfer of dusty fluid in a channel with variable physical properties and Navier slip condition, Computers & Mathematics with Applications, 60 (2010) 660-669. [36] M.A. Ezzat, A.A. El-Bary, M.M. Morsey, Space approach to the hydro-magnetic flow of a dusty fluid through a porous medium, Computers & Mathematics with Applications, 59 (2010) 2868-2879. [37] N.C. Ghosh, B.C. Ghosh, L. Debnath, The hydromagnetic flow of a dusty viscoelastic fluid between two infinite parallel plates, Computers & Mathematics with Applications, 39 (2000) 103-116. 21
ACCEPTED MANUSCRIPT [38] N. Sandeep, C. Sulochana, B. Rushi Kumar, Unsteady MHD radiative flow and heat transfer of a dusty nanofluid over an exponentially stretching surface, Engineering Science and Technology, an International Journal, 19 (2016) 227-240. [39] R. Muthuraj, K. Nirmala, S. Srinivas, Influences of chemical reaction and wall properties on MHD Peristaltic transport of a Dusty fluid with Heat and Mass transfer, Alexandria Engineering Journal, 55 (2016) 597-611.
PT
[40] M.M. Bhatti, A. Zeeshan, N. Ijaz, O. Anwar Bég, A. Kadir, Mathematical modelling of nonlinear thermal radiation effects on EMHD peristaltic pumping of viscoelastic dusty fluid
RI
through a porous medium duct, Engineering Science and Technology, an International
SC
Journal, 20 (2017) 1129-1139.
[41] Aminreza Noghrehabadins, Ehsan Izadpanahi, Mohammad Ghalambaz, Analyze of fluid
NU
flow and heat transfer of nanofluids over a stretching sheet near the extrusion slit, Computers & Fluids, 100 (2014) 227-236
MA
[42] M. Ghalambaz, E. Izadpanahi, A. Noghrehabadi, A. Chamkha, Study of the boundary layer heat transfer of nanofluids over a stretching sheet: Passive control of nanoparticles at
PT E
D
the surface, Canadian Journal of Physics , 93 (2015) 725-733.
Table.1. Density, specific heat and thermal conductivity of base fluids and nanoparticles.
𝑷𝒓𝒐𝒑𝒆𝒓𝒕𝒊𝒆𝒔\𝒄𝒐𝒏𝒔𝒕𝒊𝒕𝒖𝒆𝒏𝒕𝒔
Tio2
997.1
4250
𝑱 𝑺𝒑𝒆𝒄𝒊𝒇𝒊𝒄 𝒉𝒆𝒂𝒕, 𝑪𝒑 ( ⁄𝑲𝒈 𝑲)
4179
686.2
𝑻𝒉𝒆𝒓𝒎𝒂𝒍 𝒄𝒐𝒏𝒅𝒖𝒄𝒕𝒊𝒗𝒊𝒕𝒚, 𝒌 (𝑾⁄𝒎 𝑲)
0.613
8.9538
CE
H2 O
𝑲𝒈 ⁄ 𝟑) 𝒎
AC
𝑫𝒆𝒏𝒔𝒊𝒕𝒚, 𝝆 (
List of figures: 22
ACCEPTED MANUSCRIPT Fig. 1. Geometry of the problem . Fig. 2. Comparison between the results of velocity profile and the results of Ref. [42] Fig. 3. Influence of f0 on velocity profile when
1,Kp 0.4,M 0.4,R 0,m 0.6, 6.1, l 0.5, 0.2, Nr 0.4,Ec 0.3,Pr 6.2.
Fig. 4 & 5. Influence of f0 on temperature profile for PST case. Fig. 6 & 7. Influence of f0 on temperature profile for PHF case. Fig. 8. Influence of l on velocity profile when
RI
Fig. 10. Influence of l on temperature profile for PHF case.
PT
1,Kp 0.4,M 0.4,R 0,m 0.6, 6.1,f0 0.01, 0.2, Nr 0.4,Ec 0.3,Pr 6.2. Fig. 9. Influence of l on temperature profile for PST case. Fig. 11. Influence of R on velocity profile when
SC
1,Kp 0.4,M 0.4,f0 0.01,m 0.6, 6.1, l 0.5, 0.2, Nr 0.4,Ec 0.3,Pr 6.2.
Fig. 12 & 13. Influence of R on temperature profile for PST case.
NU
Fig. 14 & 15. Influence of R on temperature profile for PHF case. Fig.16. Influence of Kp on velocity profile when
MA
1,M 0.4,R 0,f0 0.01,m 0.6, 6.1, l 0.5, 0.2, Nr 0.4,Ec 0.3,Pr 6.2. Fig. 17 & 18. Influence of Kp on temperature profile for PST case. Fig. 19 & 20. Influence of R on temperature profile for PHF case.
D
Fig. 21. Influence of M on velocity profile when
PT E
1,Kp 0.4,R 0,f0 0.01,m 0.6, 6.1, l 0.5, 0.2, Nr 0.4,Ec 0.3,Pr 6.2. Fig. 22 & 23. Influence of M on temperature profile for PST case. Fig. 24 & 25. Influence of M on temperature profile for PHF case.
CE
Fig. 26. Influence of ?? on velocity profile when
AC
1,Kp 0.4,M 0.4,R 0,f0 0.01,m 0.6, 6.1, l 0.5, Nr 0.4,Ec 0.3,Pr 6.2. Fig. 27. Influence of ?? on temperature profile for PST case. Fig. 28. Influence of ?? on temperature profile for PHF case. Fig. 29. Influence of ?? on velocity profile when
Kp 0.4,M 0.4,R 0,f0 0.01,m 0.6, 6.1, l 0.5, 0.2, Nr 0.4,Ec 0.3,Pr 6.2. Fig. 30. Influence of ?? on temperature profile for PST case. Fig. 31. Influence of ?? on temperature profile for PHF case. Fig. 32 & 33. Influence of Nr on temperature profile for PST case when
1,Kp 0.4,M 0.4,R 0,f0 0.01,m 0.6, 6.1, l 0.5, 0.2,Ec 0.3,Pr 6.2. 23
ACCEPTED MANUSCRIPT Fig. 34 & 35. Influence of Nr on temperature profile for PHF case. Fig. 36. Influence of Ec on temperature profile for PST case when
1,Kp 0.4,M 0.4,R 0,f0 0.01,m 0.6, 6.1, l 0.5, 0.2, Nr 0.4,Pr 6.2. Fig. 37. Influence of Ec on temperature profile for PHF case. Fig. 38 & 39. Influence of Pr on temperature profile for PST case when
1,Kp 0.4,M 0.4,R 0,f0 0.01,m 0.6, 6.1, l 0.5, 0.2, Nr 0.4,Ec 0.3.
PT
Fig. 40 & 41. Influence of Pr on temperature profile for PHF case. Fig. 42 & 43. Influence of Kp & M on skin friction coefficient in the presence of β.
RI
Fig. 44 & 45. Influence of Ec on heat transfer properties in the presence of Nr for PST and PHF cases.
SC
Fig. 46 & 47. Influence of Pr on heat transfer properties in the presence of β for PST and PHF cases.
AC
CE
PT E
D
MA
NU
Fig. 48. Influence of Pr on couple stress in the presence of β.
Fig.1. Geometry of the problem . 24
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
AC
CE
PT E
D
MA
Fig.2.
Fig.3.
25
RI
PT
ACCEPTED MANUSCRIPT
PT E
D
MA
NU
SC
Fig.4.
AC
CE
Fig.5.
Fig.6. 26
RI
PT
ACCEPTED MANUSCRIPT
Fig.8.
AC
CE
PT E
D
MA
NU
SC
Fig.7.
Fig.9. 27
Fig.10.
PT E
D
MA
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
AC
CE
Fig.11.
Fig.12. 28
RI
PT
ACCEPTED MANUSCRIPT
Fig.14.
AC
CE
PT E
D
MA
NU
SC
Fig.13.
Fig.15. 29
RI
PT
ACCEPTED MANUSCRIPT
PT E
D
MA
NU
SC
Fig.16.
AC
CE
Fig.17.
Fig.18.
30
Fig.20.
AC
CE
PT E
D
MA
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
Fig.21.
31
Fig.19.
RI
PT
ACCEPTED MANUSCRIPT
PT E
D
MA
NU
SC
Fig.22.
AC
CE
Fig.23.
Fig.24.
32
RI
PT
ACCEPTED MANUSCRIPT
Fig.26.
AC
CE
PT E
D
MA
NU
SC
Fig.25.
Fig.27. 33
Fig.28.
PT E
D
MA
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
AC
CE
Fig.29.
Fig.30. 34
RI
PT
ACCEPTED MANUSCRIPT
Fig.32.
AC
CE
PT E
D
MA
NU
SC
Fig.31.
35
RI
PT
ACCEPTED MANUSCRIPT
PT E
D
MA
NU
SC
Fig.33.
AC
CE
Fig.34.
Fig.35.
36
PT E
D
MA
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
AC
CE
Fig.37.
Fig.38.
37
Fig.36.
PT E
SC
D
MA
NU
Fig.39.
RI
PT
ACCEPTED MANUSCRIPT
AC
CE
Fig.40.
Fig.41. 38
PT E
D
MA
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
AC
CE
Fig.43.
Fig.44. 39
Fig.42.
PT E
D
MA
NU
SC
Fig.45.
RI
PT
ACCEPTED MANUSCRIPT
AC
CE
Fig.46.
Fig.47. 40
AC
CE
PT E
D
MA
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
41
Fig.48.
AC
CE
PT E
D
MA
NU
SC
RI
PT
ACCEPTED MANUSCRIPT
Graphical abstract 42
ACCEPTED MANUSCRIPT Highlights
AC
CE
PT E
D
MA
NU
SC
RI
PT
TiO2 nanoparticle homogeneously suspended in Micropolar fluid Magnetic field and thermal radiation effect on Micropolar dusty fluid are considered The Runge-Kutta-Fehlberg numerical method is employed to solving the nonlinear equations Increasing in the value of radiation parameter Nr make the thermal boundary layer thicker For increasing the values of Ec the Nusselt number increases for PHF case and decreases for PST
43