Effects of Velocity Slip and Temperature Jump on...

JOURNAL OF MECHANICAL ENGINEERING ISSN 2165 - 8145 VOL. 1, NO. 3 NOVEMBER 2012

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Abstract—The present study concerns the first and second law

analyses of an electrically conducting fluid past a porous rotating

disk in the presence of velocity slip and temperature jump

conditions. A semi numerical-analytical technique, combination

of differential transforms method (DTM) and padé approximant,

named DTM-Padé is employed to solve system of ordinary

differential equations that is convert form of the partial

differential equations governing the heat and flow motion.

Entropy generation equation is derived as a function of velocity

and temperature gradients and non-dimensionalized using

geometrical and flow physical field-dependent parameters. The

velocity profiles in radial, tangential and axial directions,

temperature distribution and averaged entropy generation

number are obtained. The effects of flow physical parameters

such as magnetic interaction parameter, suction parameter,

Reynolds number, Knudsen number, Prandtl number, and

Brinkman number on the all fluid velocity components,

temperature distribution, and averaged entropy generation

number are checked and discussed and the path for minimizing

the entropy is also proposed.

Index Terms—velocity slip, temperature jump, entropy

generation, MHD flow, rotating disk, DTM-Padé

I. INTRODUCTION

N most of the investigations no-slip boundary condition (the

assumption that a liquid adheres to a solid boundary) is

established and 0Kn , but in some situations such as

emulsions, suspensions, foams and polymer solution [1], the

no-slip condition is not adequate. For the range of

0.01 0.1Kn (slip flow) the standard Navier–Stokes and

energy equations can still be used by taking into account

velocity slip and temperature jump. In recent years, the slip-

flow regime has been widely studied and researchers have

been concentrating on the analysis of micro-scale in micro-

electro-mechanical systems (MEMS) associated with the

Manuscript received November 13, 2012. Mohammad Mehdi Rashidi is with the Mechanical Engineering

Department, Engineering Faculty of Bu-Ali Sina University, Hamedan, Iran,

(corresponding author to provide phone: +98 811 8257409; fax: +98 811 8257400; e-mail: [email protected], [email protected]).

Navid Freidooni Mehr is with the Mechanical Engineering Department,

Engineering Faculty of Bu-Ali Sina University, Hamedan, Iran (e-mail: [email protected]).

embodiment of velocity slip and temperature jump. Because of

the micro-scale dimensions, the slip flow greatly differs from

the traditional no-slip flow [2]–[4]. Sparrow et al [5]

considered the fluid flow due to the rotation of a porous

surfaced disk and employed a set of linear slip flow

conditions. A substantial reduction in torque occurred as a

result of surface slip. Turkyilmazoglu and Senel [6]

investigated the effects of roughness on the heat and mass

transfer for the flow over a rotating disk subjected to a wall

suction or injection. Arikoglu et al [7] presented the effect of

slip on entropy generation over a rotating disk in MHD flow

by semi-numerical analytical solution technique. Sahoo [8]

studied the effects of partial slip, viscous dissipation, and

Joule heating on the flow and heat transfer of an electrically

conducting non-Newtonian fluid due to a rotating disk.

Entropy generation minimization should be taken into

account in thermal systems in some different situations, for

example when we have thermodynamics irreversibility, heat

transfer through finite temperature gradient, convective heat

transfer characteristics and, viscous effects. The equipment

performance degrades because of irreversibility and it should

be noted that the entropy generation is a scale of process

irreversibility [9], [10]. In order to increase the efficiency in

all types of manufacturing systems and to minimize the

entropy generation, the researchers have been focusing on the

second law of thermodynamics in design of thermal

engineering systems. The system efficiency calculations using

the second law of thermodynamics based on the entropy

generation is much more reliable and suitable than the

calculations by the first law of thermodynamics. Accurate

calculation of the entropy generation plays an important role

in the development of thermal system components and the

entropy generation is a criterion of work demolition in the

systems. Optimal system design will result in reduction of

entropy generation [11], [12]. Many researchers have been

motivated to conduct the second law analysis applications in

the design of thermal engineering systems, in recent years.

Aïboud and Saouli [13] displayed the application of the

second law analysis of thermodynamics to viscoelastic MHD

flow over a stretching surface. The effect of slip and joule

dissipation on entropy generation in MHD flow over a single

rotating disk was studied by Arikoglu et al [7] via semi-

Effects of Velocity Slip and Temperature Jump

on the Entropy Generation in

Magnetohydrodynamic Flow over a Porous

Rotating Disk

M.M. Rashidi, and N. Freidooni Mehr

I


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numerical analytical solution method, named differential

transform method. Their work is most similar work to the

present study. Mahian et al [14] analyzed the first and second

laws of thermodynamics to demonstrate the effects of MHD

flow on the distributions of velocity, temperature and entropy

generation between two concentric rotating cylinders. San et

al [15] presented the entropy generation analysis for combined

forced convection heat and mass transfer in a two dimensional

channel.

The use of an external magnetic field is a very important

issue in many industrial applications, especially as a

mechanism to control the material construction [16]. In

addition, study the heat transfer and flow in a closed cavity or

a channel in the presence of a magnetic field due to many

engineering applications such as nuclear cooling reactors,

MHD marine propulsion, MHD micro pumps, electronic

packages and microelectronic devices are very important. The

first studies on the effects of MHD were carried out in 1907.

In that year, Northrup built an MHD pump [17]. Studying the

effects of rotation and magnetic field factors on the fluid flow

was one of the most important research topics for many

researchers in recent years.

Some of strongly nonlinear equations used to describe

physical systems in the form of mathematical modeling, do

not have exact solutions. The numerical or analytical methods

can be applied to solve these nonlinear equations. Despite all

the benefits, there are many disadvantages for the numerical

methods in comparison with the analytical methods.

Therefore, we use one of the semi numerical-analytic

techniques named differential transform method (DTM) [18],

[19] to solve the system of nonlinear differential equations. In

some especial cases with high order of nonlinearity, applying

DTM cannot satisfy the infinity boundary conditions. To

overcome this problem, the padé approximant is applied to the

DTM results to enlarge convergence radius of them. In recent

years, DTM and DTM-Padé were employed to solve for many

kind of high nonlinear problems [20]–[23].

The current article is mainly motivated by the need to

understand the entropy generation in the MHD flow over a

porous rotating disk in the presence of the velocity slip and

temperature jump conditions. We examine the entropy

generation analysis in this perusal, because of more reliability

of second law of thermodynamics analysis than the first one.

The combination of the DTM and Padé approximant, DTM-

Padé, is employed to investigate the effect of flow physical

parameters such as magnetic interaction parameter, suction

parameter, Reynolds number, Knudsen number, and Brinkman

number on the all fluid velocity components, temperature

distribution, and averaged entropy generation number. The

obtained results of present study can be applied to design

thermal systems with reduced sources of irreversibility’s.

II. GOVERNING EQUATIONS AND MATHEMATICAL

FORMULATION

Consider the axially symmetric laminar flow of an

incompressible Newtonian fluid past a porous rotating disk

that has a constant angular velocity, in the presence of

externally applied uniform vertical magnetic field. The

coordinate system and geometry of the problem are shown in

Fig. 1. The three dimensional governing equations for the

continuity, momentum and energy in laminar MHD

incompressible boundary layer flow in cylindrical coordinates

can be presented, respectively, as follows

1

0,w

rur r z

(1)

2

22 2

0

2 2 2

1

1,

u u v pu w

r z r r

Bu u u uu

r rr z r

(2)

22 2

0

2 2 2

1,

v v uvu w

r z r

Bv v v vv

r rr z r

(3)

2 2

2 2

1 1,

w w p w w wu w

r z z r rr z

(4)

2 2

2 2

1,

p

T T k T T Tu w

r z C r rr z

(5)

where is the fluid density, p is the fluid pressure, is the

kinematic viscosity, is the electrical conductivity, k is the

thermal conductivity and pC is the specific heat at constant

pressure. The flow velocity components are in the directions

of increasing cylindrical polar coordinates , ,r z . An

external uniform magnetic field B is applied normal to the

disk surface that has a constant magnetic flux density 0B ,

which is assumed constant by taking small magnetic Reynolds

number much smaller than the fluid Reynolds number. The

surface of the rotating disk is maintained at a uniform

temperature wT , while the temperature and pressure of the

ambient fluid are T and p , respectively. Assuming the

effect of velocity slip is very important and should be included

in the modeling of flow field for the more accurate prediction.

In the base of slip flow theory, one can declare that the fluid

velocity at the surface is different from the wall velocity

compared to the local velocity gradient in normal direction.

According to Karniadakis and Beskok [24], the following

form for the slip flow equation has been proposed:

22 3 1

,2

vs w

v

u Kn Re TU U

z Ec r

(6)

where sU is the velocity of the fluid near to the disk surface,

wU is the wall velocity, and v is the tangential momentum

accommodation coefficient, which is usually determined


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empirically [25] and depends on fluid and surface finish and

is the ratio of specific heats. Thermal creep effect is shown

in the second term of right-hide side of (6). This effect induces

velocity slip along the wall due to the temperature gradient

near to the wall along the surface. It can be illustrated that this

term is of second-order in terms of Knudsen number. Thus, it

is negligible comparing to the first term that is of first-order in

the slip-flow regime [25]–[27]. For thermal boundary

condition, we have considered temperature jump effect that

was derived by von Smoluchowski [24]. The temperature

jump condition can be given by [24]:

2 2,

1

t ss w

t

TT T

Pr z

(7)

where sT is the temperature of the fluid near to the disk

surface, wT is the disk wall temperature, and t is the energy

accommodation coefficient that is usually determined

experimentally and depends on the surface finish, the fluid

temperature, and the pressure. Thus, following above

description about boundary conditions, the appropriate

boundary conditions, subjected to uniform suction 0w through

the disk, become:

0

2,

2,

at 0,

,

2 2,

1

0, 0, , at .

v

v

v

v

t sw

t

uu

z

vv r

z z

w w

TT T

Pr z

u v T T z

(8)

The non-dimensional forms of the mean flow velocities and

temperature distributions of (1)-(5) are given by Von

Karman’s exact self-similar solution of the Navier-Stokes

equations:

1 2

1 2

,

, , ,

, .w

z

u r F v r G w H

p p P T T T T

(9)

Substituting (9) in (1)-(5), we obtain the following system

of ordinary differential equations:

F HF F G MF (10)

G HG FG MG (11)

H F (12)

,Pr H (13)

where , , F G H and are the non-dimensionless functions

of modified dimensionless vertical coordinate ,

2

0M B is the magnetic interaction parameter,

pPr C k is the Prandtl number and primes denote

differentiation with respect to . The transformed boundary

conditions can be written as follows:

0 0 , 0 1 0 ,

0 , 0 1 0 ,

0, 0, 0, as ,

s

F F G G

H W

F G

(14)

where 2 2v v v v Kn Re

is

the slip factor,

1 2

0sW w is the suction parameter that

0sW shows a uniform suction at the disk surface,

2 2 1t t Kn Pr Re is the temperature

jump factor, and Re is the rotational Reynolds number. The

values of tangential momentum accommodation number,

energy accommodation coefficient and the specific heat ratio

for air are considered as 0.9, 0.9, and 1.4, respectively [24].

III. ENTROPY GENERATION ANALYSIS

Generally, the local volumetric entropy generation rate, in

the presence of axial symmetry and magnetic field, can be

expressed as (for more details, see [7], [10])

2

2

1. ,

gen

ww

w

kS T

TT

J QV E V BT

(15)

where

2

,T T T

r r z

(16)

,

u v w v wu

r z z rr

w u u vr

r z r r r

(17)

,J E V B (18)

where is viscous dissipation, J is electric current, Q is

electric charge density, V is the velocity vector and E is

electric field. It is assumed that the electric force per unit

charge, in compared with V B in (15) is negligible and we

also consider that the electric current is much greater than

.QV Thus, the entropy generation rate reduces in this case to:


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2gen

w

Thermal irreversibility

w

Fluid friction irreversibility

2

0

k TS

zT

u wu

r zr

T v u vr

z z r r

B

T

.w

Joule dissipation irreversibility

u v

(19)

The above equation reveals that the entropy generation is

due to three effects, the first effect, a conductive effect, is the

local entropy generation due to heat transfer irreversibility

HTIN , which contains the entropy generation by heat

transfer due to axial conduction from the rotating disk, the

second one, a viscous effect, is due to fluid friction

irreversibility FFIN and the last effect denotes to the

magnetic effects in the form of joule dissipation irreversibility

JDIN that is caused by the movement of electrically

conducting fluid under the magnetic field inducing electric

currents that circulate in the fluid [7]. The entropy generation

number, dimensionless form of entropy generation rate,

represents the ratio between the actual entropy generation rate

genS and characteristic entropy generation rate 0 .S The

similarity transformation parameters of (9) are employed to

non-dimensionalized the local entropy generation given in

(19), thus the entropy generation number becomes:

2

,G

F HRe Re

N Br F Gr

M F G

(20)

where wT T is the dimensionless temperature

difference, Br R k T is the rotational Brinkman

number, r r R is the dimensionless radial coordinate and

G gen wN S k T T is the dimensionless entropy

generation rate.

The averaged entropy generation number that is an

important measure of total global entropy generation can be

evaluated using the following formula

1

,0 0

12 ,

m

G av GN r N dr d

(21)

where is the considered volume. In order to consider both

the velocity and thermal boundary layers, we calculate the

volumetric entropy generation in a large finite domain. Thus,

integration in (21) is obtained in the domain 0 1r and

0 ,m where m is a sufficiently large number.

IV. ANALYTICAL APPROXIMATIONS BY MEANS OF

DTM-PADÉ

Taking differential transform of (12)‒(13) (for more details

of DTM theory, see [21], [28]), one can obtain

0

1 2 2

1 1

0,

k

r

k k F k

k r H r F k r

F r F k r G r G k r

M F k

(22)

0

1 2 2

1 1

2

0,

k

r

k k G k

k r H r G k r

F r G k r

M G k

(23)

1 1 2 0,k H k F k

(24)

0

1 2 2

1 1 0,k

r

k k k

Pr k r H r k r

(25)

where , , ,F k G k H k and k are the transformed

functions of , , ,F G H and , respectively

and are given by:

,k

k =0

F = F k

(26)

,k

k =0

G = G k

(27)

,k

k =0

H = H k

(28)

.k

k =0

= k

(29)

The transformed boundary conditions become

0 1 , 1 ,

0 1 1 , 1 ,

0 1 1 , 1 ,

0 ,s

F F F a

G G G b

c

H W

(30)

where ,a b and c are constants. Substituting (30) into (22)‒

(25) and with a recursive method, one can calculate the values

of , , ,F k G k H k and .k Therefore, substituting


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all , , ,F k G k H k and k into (26)‒(29), we have

the series solutions as follows

2

22 2

3

22 2

1

2 1

2 1

1,

61

s

s

s

aW a MF a a

a b

a M b b

a W a MW

a b

(31)

2

3

111

2 2 1

2 11

,16

2 1

s

s

s

b W M bG b b

a b

b M a b

b W M bW

a b

(32)

2

23 2 2

2

11 ,

3

s

s

H W a a

a W a M a b

(33)

2

3 2 2

11

2

12 .

6

s

s

c c c PrW

c Pr W ac Pr

(34)

The Padé approximant is applied to enlarge the convergence

radius of the truncated series solution. In this way, the

polynomial approximation converts into a ratio of two

polynomials. As it is shown in Fig. 2-5, without using the

Padé approximant, the DTM solution, cannot satisfy boundary

conditions at infinity. Therefore, it is important to combine the

DTM with the Padé approximant series solution to provide an

effective tool to handle infinite boundary value problems.

Thus, we apply the Padé approximation to (31)‒(34) and using

the boundary conditions of (14) at , one can obtain ,a b

and c . The number of required terms is characterized by the

convergence of the numerical values to one’s desired

accuracy.

V. RESULTS AND DISCUSSION

For the present investigation, we have considered the value

of the Prandtl number is equal to 1. The default values of other

flow physical parameters are referred in each of the graphs.

Figs. 6-9 display the effect of magnetic interaction parameter

on the all velocity components and temperature distribution. A

drag-like force that named Lorentz force is created by the

infliction of the vertical magnetic field to the electrically

conducting fluid. This force has the tendency to slow down the

flow around the disk at the expense of increasing its

temperature. Indeed, the magnetic field as a body force

increases in effect of friction forces and the great resistances

on the fluid particles, which cause to generate heat in the fluid,

apply as the vertical magnetic field increases. Due to the

above reasons, the velocity profiles in radial, tangential and

magnitude of axial directions reduce and the thermal boundary

layer thickness increases with the increase in magnetic

interaction parameter. It must be pointed that the radial and

axial velocity component distributions decay rapidly along

with increasing the magnetic interaction parameter and also

the magnetic interaction parameter variations has the less

effect on the temperature distribution and the radial velocity

profile experiences a maximum value close to the disk surface

due to the existence of the centrifugal forces.

The effect of suction parameter on the all velocity

component distributions and temperature field is shown in

Figs. 10-13. When the suction applies at the disk surface, the

radial and tangential velocity profiles reduce and the

magnitude of axial velocity increases. In addition, the radial

velocity component becomes very small, for strong values of

the suction parameter. The usual decay of temperature occurs

for the higher values of suction.

The effects of Reynolds number and Knudsen number on

the radial, tangential and axial velocity components and

temperature distribution are demonstrated in Figs. 14-21,

respectively. The effects of these two parameters have been

studied together; because of their similar effects on the slip

boundary condition. The maximum of radial velocity

component decreases, and its location moves toward the disk

in the presence of the slip condition. In addition, the radial

velocity profile caused by the centrifugal forces starts from

zero only in the case of no slip condition 0Kn . The fluid

velocity components in all directions and temperature

distribution decrease with increasing the values of the

Reynolds and Knudsen numbers. In other word, less amount

of flow is drawn and pushed away in the velocity directions,

as the slips get stronger and decreasing in the heat generation

in the vicinity of the disk occurs in the slip condition.

Fig. 22 shows the effect of Prandtl number on the

temperature distribution. As we know, the thermal boundary

layer thickness is inversely proportional to the square root of

Prandtl number. Hence, the thermal boundary layer

thicknesses get decreased with increasing the value of the

Prandtl number. In other word, the flow with large Prandtl

number prevents spreading of heat in the fluid.

Figs. 23-27 depict the results of averaged entropy

generation number as a function of suction parameter,

Reynolds number, Knudsen number, Prandtl number, and

Brinkman number for a wide range of the magnetic interaction

parameter. With applying magnetic effect, the radial and

tangential velocity components decrease and axial velocity

and temperature distribution increase, as it is explained before.

Therefore, from (19), some of the entropy generation

irreversibility mechanisms produce greater rate of entropy

with increasing magnetic interaction parameter whiles the

others generate decreasing rate. Fig. 23 shows the effect of

suction parameter on the averaged entropy generation number.

By applying the suction at the disk surface, the averaged

entropy generation number decreases. The effects of Knudsen


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and Reynolds number on the averaged entropy generation

number are presented in Figs. 24 and 25. Both the temperature

and the velocity gradients decrease, as the Knudsen and

Reynolds number increases. This results in a decrease in the

averaged entropy generation. Increasing the velocity slip and

temperature jump at the wall cause to decreasing heat transfer

and momentum transfer from the wall to the fluid and this also

brings about the reduction in the averaged entropy generation.

Fig. 26 represents the effect of Prandtl number on the

averaged entropy generation number. As the Prandtl number

increases, the value of averaged entropy generation increases.

Fig. 27 demonstrates the effect of Brinkman number on the

averaged entropy generation number versus magnetic

interaction parameter. An increase in the entropy generation

produced by fluid friction and joule dissipation brings with

increasing the value of the Brinkman number.

VI. CONCLUSIONS

In the current perusal, the mathematical formulation has

been derived for the entropy generation analysis in MHD flow

over a porous rotating disk in the presence of the velocity slip

and temperature jump. DTM-Padé is applied to solve the

system of ordinary differential equations. An excellent

agreement can be observed between the results of this study

and the numerical results obtained by shooting method, in an

especial case. The effects of physical flow parameters such as

magnetic interaction parameter, suction parameter, Reynolds

number, Knudsen number, and Brinkman number on the fluid

velocity in radial, tangential, and axial directions, temperature

distribution, and averaged entropy generation are illustrated.

The results show that the main goal of second law of

thermodynamics that is minimizing entropy, are reached as the

magnetic interaction parameter, Prandtl number, and

Brinkman number decrease or suction parameter, Reynolds

number, and Knudsen number increase. It can be seen that the

disk surface acts as a strong source of irreversibility.

Fig. 1. Configuration of the flow and geometrical coordinates.

Fig. 2. The profile of F obtained by DTM for different value of n and

different order of DTM-Padé in comparison with the numerical solution when

1M , 1sW , 0.05Kn and Re 100 .

Fig. 3. The profile of G obtained by DTM for different value of n and


1M , 1sW , 0.05Kn and Re 100 .

Fig. 4. The profile of H obtained by DTM for different value of n and


1M , 1sW , 0.05Kn and Re 100 .


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Fig. 5. The profile of obtained by DTM for different value of n and


1M , 1sW , 0.05Kn and Re 100 .

Fig. 6. Effect of magnetic interaction parameter on radial velocity profile

when 1sW , 0.05Kn and Re 100 .

Fig. 7. Effect of magnetic interaction parameter on tangential velocity profile


Fig. 8. Effect of magnetic interaction parameter on axial velocity profile


Fig. 9. Effect of magnetic interaction parameter on temperature distribution


Fig. 10. Effect of suction parameter on radial velocity profile when 1M ,

0.05Kn and Re 100 .


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Fig. 11. Effect of suction parameter on tangential velocity profile when

1M , 0.05Kn and Re 100 .

Fig. 12. Effect of suction parameter on axial velocity profile when 1M ,

0.05Kn and Re 100 .

Fig. 13. Effect of suction parameter on temperature distribution when 1,M

0.05Kn and Re 100 .

Fig. 14. Effect of Reynolds number on radial velocity profile when 1M ,

1sW and 0.05Kn .

Fig. 15. Effect of Reynolds number on tangential velocity profile when

1M , 1sW and 0.05Kn .

Fig. 16. Effect of Reynolds number on axial velocity profile when 1M ,

1sW and 0.05Kn .


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Fig. 17. Effect of Reynolds number on temperature distribution when 1M ,

1sW and 0.05Kn .

Fig. 18. Effect of Knudsen number on radial velocity profile when 1M ,

1sW and Re 100 .

Fig. 19. Effect of Knudsen number on tangential velocity profile when

1M , 1sW and Re 100 .

Fig. 20. Effect of Knudsen number on axial velocity profile when 1M ,

1sW and Re 100 .

Fig. 21. Effect of Knudsen number on temperature distribution when 1M ,

1sW and Re 100 .

Fig. 22. Effect of Prandtl number on temperature distribution when 1M ,

1sW , 0.05Kn and Re 100 .


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Fig. 23. Change of ,G avN with respect to magnetic interaction parameter for

different values of suction parameter when 0.05Kn , Re 100 and

10Br .


different values of Reynolds number when 1sW , 0.05Kn and 5Br .


different values of Knudsen number when 1sW , Re 100 and 5Br .


different values of Prandtl number when 1sW , 0.05Kn , Re 100 and

10Br .


different values of Brinkman number when 1sW , 0.05Kn and

Re 100 .

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interior of an electric conductor” Physical Review (Series I), vol. 24, no. 6, pp. 474-497, June. 1907.

[18] A. Fatma, “Applications of differential transform method to differential-

algebraic equations” Applied Mathematics and Computation, vol. 152, no. 3, pp. 649-657, May. 2004.

[19] A. Fatma, “Solutions of the system of differential equations by

differential transform method” Applied Mathematics and Computation, vol. 147, no. 2, pp. 547-567, Jan. 2004.

[20] M.M. Rashidi and S.A. Mohimanian Pour, “A novel analytical solution

of heat transfer of a micropolar fluid through a porous medium with radiation by DTM-Padé” Heat Transfer—Asian Research, vol. 39, no.

8, pp. 575-589, Dec. 2010.

[21] M.M. Rashidi and E. Erfani, “The modified differential transform method for investigating nano boundary-layers over stretching surfaces”

International Journal of Numerical Methods for Heat & Fluid Flow, vol.

21, no. 7, pp. 864 - 883, Sep. 2011. [22] M.M. Rashidi and E. Erfani, “A new analytical study of MHD stagnation-

point flow in porous media with heat transfer” Computers and Fluids,

vol. 40, no. 1, pp. 172-178, Jan. 2011. [23] M.M. Rashidi and N. FreidooniMehr, “Series solutions for the flow in the

vicinity of the equator of an MHD boundary-layer over a porous rotating

sphere with heat transfer” Thermal Science, DOI: 10.2298/TSCI120301155R, 2012.

[24] G. Karniadakis, A. Beskok, and N. Aluru, Microflows: Fundamentals and

Simulation, Springer, 2001. [25] M. Renksizbulut, H. Niazmand, and G. Tercan, “Slip-flow and heat

transfer in rectangular microchannels with constant wall temperature”

International Journal of Thermal Sciences, vol. 45, no. 9, pp. 870-881, Sep. 2006.

[26] A. Arikoglu and I. Ozkol, “Analysis for Slip Flow Over a Single Free

Disk With Heat Transfer” Journal of Fluids Engineering, vol. 127, no. 3, pp. 624-627, May. 2005.

[27] A. Arikoglu, G. Komurgoz, I. Ozkol, and A.Y. Gunes, “Combined

Effects of Temperature and Velocity Jump on the Heat Transfer, Fluid Flow, and Entropy Generation Over a Single Rotating Disk” Journal of

Heat Transfer, vol. 132, no. 11, pp. 111703-111710, Nov. 2010.

[28] M.M. Rashidi, “The modified differential transform method for solving MHD boundary-layer equations” Computer Physics Communications,

vol. 180, no. 11, pp. 2210-2217, Nov. 2009.

Mohammad Mehdi Rashidi is born in Hamedan,

Iran in 1972. He received his B.Sc. degree in Bu-Ali Sina University, Hamedan, Iran in 1995. He also

received his M.Sc. and Ph.D degrees from Tarbiat

Modares University, Tehran, Iran in 1997 and 2002, respectively. His research (Areas of Specialization)

focuses on heat and mass transfer,

Thermodynamics, computational fluid dynamics (CFD), nonlinear analysis, engineering mathematics

and exergy and second law analysis.

He is an associate professor of Mechanical Engineering at the Bu-Ali Sina University, Hamedan, Iran. His works have

been published in the journal of Energy, Computers and Fluids,

Communications in Nonlinear Science and Numerical Simulation and several other international journals. He has published a book: Mathematical

Modelling of Nonlinear Flows of Micropolar Fluids (Germany, Lambert

Academic Press, 2011). He has published over 100 journal articles. Dr. Rashidi is a reviewer of several journals such as Applied

Mathematical Modelling, Computers and Fluids, Energy, Computers and

Mathematics with Applications, International Journal of Heat and Mass Transfer, International journal of thermal science, Mathematical and

Computer Modelling, etc. He was an invited professor in Génie Mécanique,

Université de Sherbrooke, Sherbrooke, QC, Canada J1K 2R (From Sep 2010-Feb 2012), Universite Paris Ouest, France (For Sep 2011) and University of

the Witwatersrand, Johannesburg, South Africa (For Aug 2012). He also is the

member of Islamic Educational, Scientific and Cultural Organization (ISESCO). He is the editor of the International Journal of Applied

Mathematical Research (IJAMR), Journal of Advanced Computer Science & Technology and Scientific Research and Essays, and Journal of Mechanical

Engineering.

Navid Freidooni Mehr received his B.Sc. degree in

Mechanical Engineering from Bu-Ali Sina University, Hamedan, Iran in 2010. He is currently

a M.Sc. student in Bu-Ali Sina University, Iran.

Born in Hamedan in 1988, his research interests include entropy generation analysis, heat and mass

transfer, nonlinear analysis and thermodynamics

cycle analysis.

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