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Nonlinear radiation effects on squeezing flow of a Casson fluid between parallel disks
Accepted Manuscript Nonlinear radiation effects on squeezing flow of a Casson Fluid between parallel disks
Syed Tauseef Mohyud-Din, Sheikh Irfanullah Khan
PII: DOI: Reference:
S1270-9638(15)00328-4 http://dx.doi.org/10.1016/j.ast.2015.10.019 AESCTE 3463
To appear in:
Aerospace Science and Technology
Received date: Revised date: Accepted date:
6 June 2015 10 October 2015 18 October 2015
Please cite this article in press as: S.T. Mohyud-Din, S.I. Khan, Nonlinear radiation effects on squeezing flow of a Casson Fluid between parallel disks, Aerosp. Sci. Technol. (2015), http://dx.doi.org/10.1016/j.ast.2015.10.019
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Nonlinear Radiation Effects on Squeezing Flow of a Casson Fluid between Parallel Disks *Syed Tauseef Mohyud-Din, Sheikh Irfanullah Khan, Department of Mathematics, Faculty of Sciences, HITEC University, TaxilaCantt, Pakistan *
Corresponding Author ([email protected])
Abstract: In this paper, we investigate the nonlinear radiation effects in a timedependent two-dimensional flow of a Casson fluid squeezed between two parallel disks when upper disk is taken to be impermeable and lower one is porous. Suitable similarity transforms are employed to convert governing partial differential equations into system of ordinary differential equations. Well known Homotopy Analysis Method (HAM) is employed to obtain the expressions for velocity and temperature profiles. Effects of different physical parameters such as squeeze number S, Eckert number Ec and the dimensionless length on the flow when keeping ܲ ݎൌ are also discussed with the help of graphs for velocity and temperature coupled with a comprehensive discussions. Mathematica Package BVPh2.0 is utilized to formulate the total error of the system for both the suction and injection cases. The skin friction coefficient and local Nusselt number are presented with graphical aids for emerging parameters.
Key words:Homotopy analysis method (HAM); Casson fluid; nonlinear radiation effect; squeezing flow; parallel disks.
1.Introduction Squeezing flow between parallel disks is an important area of interest because of its application in bio fluid mechanics like pumping of heart, flow through certain arteries, polymer industry process, injection modeling, compression, liquid-metal lubrication, and the squeezed films in power transmission. Most frequently, squeezing flows were illustrated in modeling of metal and plastic sheets, thin fiber and paper sheets formations etc. After the pioneer work done by Stefan (1874) [1], several attempts are reported that extended the traditional problem to heat transfer case. Different studies are available in literature that used various solution schemes to get analytical and numerical solutions for the said problem. Siddiqui et al. [2] examined two-dimensional MHD squeezing flow 1
between parallel plates. For parallel disk similar problem has been discussed by Domairry and Aziz [3]. Both used Homotopy perturbation method (HPM) to determine the solution. Joneidi et al. [4] studied the mass transfer effect on squeezing flow between parallel disks using Homotopy analysis method (HAM). Most recently influence of heat transfer in the MHD squeezing Flow between parallel disks has been investigated by T. Hayat et al. [5]. They used HAM to solve the resulting nonlinear system of ordinary differential equations. Since then squeezing flow has been of great interests for various researchers [6-8]. Investigations with heat transport phenomenon are important in many industrial applications such as wire coating, hot rolling, metal spinning, glass blowing, paper manufacturing, glass fiber, glass sheet productions, the drawing of plastic films, continuous casting and the aerodynamic extrusion of plastic sheets. Radiative heat transfer plays an important role in controlling where convective heat transfer coefficients are small. In polymer industry, the radiative heat transfer is an essential phenomenon for the design of reliable equipment’s, nuclear plants, gas turbines, etc. The process of radiative heat transfer is also important in free and forced convection flows. Keeping all these applications in mind, many authors studied various research problems for the case of radiative heat transfer. Some of these can be seen in [9-12] and references there in. High nonlinearity of governing equations in various flow problems means unlikeliness of exact solution. To cope up with these problems, many approximation techniques have been developed. Highly nonlinear problems such as the ones discussed above are therefore solved by using these techniques. These include, Adomian's Decomposition Method (ADM), Variational Iteration Method, Variation of Parameters Method, Homotopy Perturbation Method, Homotopy Analysis Method, etc. [13-20]. Mathematical models describing the realistic flow problems mostly involve the NonNewtonian fluids. One of these fluids is known as Casson fluid and its formulation is provided in [21-22] is found to suitable for blood flow problems up to large extent. Casson fluid flow between squeezing disks under radiative heat transfer has not been considered yet. To fill out this gap, heat transfer analysis for squeezing flow of Casson fluid between parallel disks under the effects of thermal radiation is presented. Well 2
known Homotopy Analysis Method (HAM) [23-36] is employed to solve the problem. Graphs are plotted to analyze the effects of different emerging parameters on velocity and temperature profiles.
2.Governing Equations We have investigated parallel infinite disks h distance apart with magnetic field practiced
vertically and is proportional to ሺ െ ࢇ࢚ሻ . Incompressible Casson fluid is used in between the disks. Magnetic field is negligible for low Reynold numbers.
Fig.1: Schematics diagram of the problem
Consequently, Casson fluid flow equation is delineated as [31-34]
W ij
½ ª § p y ·º ¸¸» eij , S ! S c ° °2« P B ¨¨ ° ° ¬« © 2S ¹¼» ¾ ® § p y ·º ° ° ª °2« P B ¨¨ 2S ¸¸» eij , S c ! S ° © c ¹»¼ ¿ ¯ «¬
where௬ is yield stress of fluid,ߤ the plastic-dynamic viscosity of the fluid andߨis the self-product of component of deformation rate with itself and S c is the critical value of the said self-product.ܶ௪ andܶ are constant temperatures lower d upper diskrespectively. The viscous dissipation effects in the energy equation are retained. We have chosen the cylindrical coordinates system ሺݎǡ ߶ǡ ݖሻǡwhere the upper disk is moving with
ష ࢇࡴሺିࢇ࢚ሻ
velocity
towards or away from the stationary lower disk.Thus, the constitutive
equations for two-dimensional flow and heat transfer of a viscous fluid under the effects 3
of thermal radiation can be written as:
wu u ww wr r wz
wu wu · § wu w U¨ u ¸ wr wz ¹ © wt
0
(1)
§ 1 E · § w 2 u 1 wu u w 2 u · V 2 wpˆ ¸ B t u , (2) ¸¸ ¨¨ 2 P ¨¨ wr r wr r 2 wz 2 ¸¹ U © E ¹ © wr
ww ww · § ww u w ¸ wr wz ¹ © wt
§ 1 E · § w 2 w w 2 w 1 ww · wpˆ ¸ , (3) ¸¸ ¨¨ 2 2 P ¨¨ wz r wr ¸¹ wz © E ¹ © wr
U¨
wT wT · § wT Cp¨ u w ¸ wr wz ¹ © wt
k § w 2T w 2T 1 wT · 1 w ¨ ¸ qr U ¨© wr 2 wz 2 r wr ¸¹ U wz 2 2 § § wu · 2 · ¨ 2¨ ¸ 2§¨ ww ·¸ 2 u ¸ (4) ¸ r2 § 1 ·¨ © wr ¹ © wr ¹ Q ¨¨1 ¸¸¨ ¸. 2 2 © E ¹¨ § wu · § ww · § wu ·§ ww · ¸ ¨ ¨ wz ¸ 2¨ wz ¸ 2¨ wz ¸¨ wr ¸ ¸ © ¹ © ¹© ¹ ¹ © © ¹
Supporting conditions are
^u
0, w
w0 ` z
®u ¯
0, w
dh ½ ¾ dt ¿ z
T T
Tw z Th z
0
(5) h t
0
(6)
h t
In the above equations u and w are the velocity components in the r- and z - directions respectively, while
U
is the density, viscosity P , pˆ the pressure, specific heat C p , T
the temperature, v kinematic viscosity, k thermal conductivity, k coefficient, Stefan Boltzmann constant
V , and w0
mean absorption
is suction/injection velocity.
Substituting the following transformations [8]
ar aH F c K , w F K , 2 1 at 1 at B0 T Th z B t , K , T K , Tw Th H 1 at 1 at
u
(7)
in Eqs. (2) - (4) and removing the pressure gradient from the subsequent equations, we finally obtain 4
§ 1· ~ ¨¨1 ¸¸ F iv S KF ccc 3F cc 2 F F ccc M 2 F cc © E¹
1 Rd (1 (T
w
(8)
0,
c § 1· 3 1)T T ' S Pr 2 FT c KT c ¨¨1 ¸¸ Pr Ec F cc2 12G 2 F c2 © E¹
0, (9)
with associated condition F 0 F 1
A, F c 0 0, T 0 1, 1 F c 1 0, T 1 0. 2,
(10)
Where A denotes the suction/injection, Eckert number Ec, Hartman number M, squeeze number S, Prandtl number Pr, radiation parameter Rd , T w is the temperature difference parameter, and
G the dimensionless length defined as S Ec
aB02 H 2
aH 2 , M2 2Q
Q
, Pr
PC p K0
16V *Th3 , 3k * k
H 2 1 at . r2
2
§ ar · ¨¨ ¸¸ , G 2 C p Tw Th © 2 1 at ¹ 1
, Rd
(11)
The skin friction coefficient and Nusselt number are defined as
W rz
z h t
C fr
U 2 1aH at
W rz
P¨
2
, Nu
H qw qr , K 0 Tw Th
(12)
where
§ wu ww · ¸ © wz wr ¹ z
h t
, qw
§ wT · k 0 ¨ ¸ © wz ¹ z
.
(13)
h t
In view of (7), Eq. (12) can be written as
C fr
§ 1 · F cc 1 r 2 ¨¨1 ¸¸ 2 , Nu © E ¹ H Re r
1 Rd T c 1 , 1 at
raH 1 at . 2Q
(14)
12
Re r
(15)
3.Solution Procedure for HAM Tracking the methodology proposed by Liao [23-24], we can initiate initial guesses as F0 K
A
3 1 2 A K 2 1 2 A K 3 , 2
5
(16)
T 0 K K.
(17)
Linear operator can be chosen as d 4F and LT dK 4
LF
d 2T dK 2
(18)
Above operators satisfies the following linearity, M
S
Pr
Ec
G
Total Error
ߠ௪
E
LF C1 C2K C3K 2 C4K 3
Rd
=1
=2
0,
LT C5 C6K 0,
of System
(19) (20)
where Ci i 1 6 are constants. 3.1
Convergence of Solution The convergence of the series solutions is highly depends on the auxiliary
parameters = F and = 4 . We define total error of the system for the functions F , 4 to determine the optimal values of these parameters as 2
ª § m ·º ¨ ¸» . N F i x ' « F ¨¦ j ¦ ¸ i 0 ¬ « ©j 0 ¹¼» K
Em,1 = F
1 K
Em, 2 = 4
ª § m ·º 1 ¨ ¸» . N i x 4 ' « ¦ 4 ¦ j ¸ K i 0 «¬ ¨© j 0 ¹»¼
Where 'x
2
K
1 K . For a given order of approximation m, the optimal values of = F , = 4
can be determine by minimizing the total error of the system given in the equations by using the HAM package BVPh2.0. The optimal values of these auxiliary parameters are given in table 1 and 2, corresponding to different values of the parameters.
6
M
S
Pr
Ec
G
E
ߠ௪
Rd
0.1
0.2
0.71
0.2
0.1
1
1.1
0.2
=1
=2
-0.502
-0.819
Total Error of System 7.91ൈ ͳͲି଼
0.4
-0.513
-0.817
1.31ൈ ͳͲି
0.7
-0.511
-0.817
1.50ൈ ͳͲି
0.1
-0.527
-0.815
1.87ൈ ͳͲି
0.3
-0.503
-0.821
3.97ൈ ͳͲି
0.5
-0.504
-0.824
4.28ൈ ͳͲିହ
0.1
-0.501
-0.811
1.63ൈ ͳͲି଼
0.3
-0.501
-0.813
7.23ൈ ͳͲି଼
0.5
-0.501
-0.816
3.41ൈ ͳͲି
0.1
-0.504
-0.819
8.04ൈ ͳͲି଼
0.3
-0.521
-0.817
6.18ൈ ͳͲି
0.5
-0.535
-0.818
4.53ൈ ͳͲିହ
0.2
-0.514
-0.818
1.41ൈ ͳͲି
0.4
-0.516
-0.819
2.47ൈ ͳͲି
0.6
-0.519
-0.820
7.22ൈ ͳͲି
0.5
-0.349
-0.819
5.25ൈ ͳͲି
1.5
-0.603
-0.818
8.22ൈ ͳͲି
2.5
-0.718
-0.818
6.61ൈ ͳͲି
1.05
-0.511
-0.827
5.64ൈ ͳͲି
1.1
-0.514
-0.817
1.25ൈ ͳͲି
1.15
-0.519
-0.805
8.23ൈ ͳͲି
0.1
-0.512
-0.903
8.05ൈ ͳͲି
0.4
-0.520
-0.687
1.08ൈ ͳͲିହ
0.7
-0.528
-0.555
5.38ൈ ͳͲିହ
0.2
0.2
0.71
0.2
0.1
1
1.1
Table 1: Optimal Values for A> 0.
Table 2: Optimal values for A< 0.
7
0.1
-0.493
-0.796
6.42ൈ ͳͲିଷ
0.4
-0.495
-0.796
6.38ൈ ͳͲିଷ
0.7
-0.497
-0.795
6.31ൈ ͳͲିଷ
0.1
-0.498
-0.802
3.59ൈ ͳͲିଷ
0.3
-0.490
-0.790
1.02ൈ ͳͲିଶ
0.5
-0. 486
-0.778
2.19ൈ ͳͲିଶ
0.1
-0.495
-0.806
2.13ൈ ͳͲି
0.3
-0.494
-0.803
1.56ൈ ͳͲିସ
0.5
-0.494
-0.799
1.36ൈ ͳͲିଷ
0.1
-0.494
-0.795
4.51ൈ ͳͲିସ
0.3
-0.493
-0.797
3.19ൈ ͳͲିଶ
0.5
-0.492
-0.797
2.50ൈ ͳͲିଵ
0.2
-0.493
-0.796
7.68ൈ ͳͲିଷ
0.4
-0.492
-0.795
1.55ൈ ͳͲିଶ
0.6
-0.488
-0.796
4.53ൈ ͳͲିଶ
0.5
-0.330
-0.797
2.75ൈ ͳͲିଶ
1.5
-0.590
-0.795
3.45ൈ ͳͲିଷ
2.5
-0.701
-0.796
1.95ൈ ͳͲିଷ
1.05
-0.490
-0.811
8.37ൈ ͳͲିସ
1.1
-0.493
-0.796
6.41ൈ ͳͲିଷ
1.15
-0.497
-0.779
2.50ൈ ͳͲିଶ
0.1
-0.492
-0.882
1.66ൈ ͳͲିଷ
0.4
-0.495
-0.667
2.18ൈ ͳͲିଶ
0.7
-0.496
-0.537
4.61ൈ ͳͲିଶ
0.2
0.2
0.2
0.71
0.71
0.2
0.2
0.1
0.1
1
1
1.1
1.1
0.2
4. Results and discussions For convenience, we divide our discussions in to two parts one dedicated to investigate the upshots on varying physical parameter for the case suction ( A ! 0 ) and the other one 8
describes the same effects for the case of blowing ( A 0 ). After ensuring the convergence of series solution our concern now is to see the influences of suction/blowing parameter, squeeze number S andCasson fluid parameter
E on
velocity and temperature profile is examined. 4.1. Suction Case To analyze the impact of emerging parameters on velocity and temperature profile for suction, effects of squeezing parameter S on the velocity and temperature profile are discussed in Figs. 2a & 2b. In Fig. 2a velocity profile decreases in magnitude near the porous walls for 0. 4, and the velocity profile increases in magnitude for increasing value of squeeze numbers, while in Fig. 2b temperature profile decreases with increase in squeeze number S. Effect of Casson fluid parameterߚfor velocity and temperature profile is discussed in Fig. 3a & 3b. In Fig. 3a, increase in magnitude for velocity near the porous wall for Casson fluid parameter and decreases for increasing value of ߚ, while, in Fig. 3b temperature decreases for increase in Casson fluid parameter.
Fig. 2: (a) Influence of S on ࡲᇱ ሺࣁሻ
Fig. 2: (b) Influence of S onࣂሺࣁሻ
9
Fig. 3:(a) Influence ofࢼ onࡲᇱ ሺࣁሻ
Fig. 3: (b) Influence ofࢼ on ࣂሺࣁሻ
The graphical representations of key parameters in case of suction for temperature profile are displayed in Figs. 4-7. Fig. 4 depicts the influence of radiation parameter for suction case, increase in radiation parameter ܴ݀ results in decreased temperature. Physically, increase in radiation parameter can give us higher values of temperature that can be used in many industrial appliances.
Fig. 4:Influence ofࡾࢊ on ࣂሺࣁሻ
Fig. 5:Influence of Ec on ࣂሺࣁሻ
10
Fig. 6:Influence of ࣂ࢝ onࣂሺࣁሻ
Fig. 7:Influence of ࢾ onࣂሺࣁሻ
Fig. 5 depicts the behavior of Ec on the temperature distribution, where rapid rise in temperature is seen with higher Ec. However, for the higher Ec number boundary layer thickness decreases. This increase in temperature is due to the presence of viscous dissipation term in the energy equation. Fig. 6 depicts the behavior of temperature for increasing values of ߠ ௪ . A slight fall in temperature distribution is seen for the higher values of ߠ௪ .Here, one can easily that the parameter ߠ௪ don’t affect the temperature near upper disk. The effect of dimensionless length ߜ is almost like Eckert number that can be seen from Fig. 5. 4.2. Blowing Case To analyze the behavior of all emerging parameter on velocity and temperature profile for blowing case Figs. 8-13 are displayed. In Fig. 8a, the behavior of squeeze number on velocity profile is opposite in blowing case as compared to suction case.However, for the temperature distribution all the physical parameters have same behavior in blowing case as compared to those in suctions case. Only difference is the effects of Casson parameterࢼthat decrease the velocity profile near the porous walls for ߟ ͲǤͷ, and then starts increasing for higher value of Casson parameter ߚǤ
11
Fig. 8: (a) Influence of S onࡲԢሺࣁሻ
Fig. 8: (b) Influence of S onࣂሺࣁሻ
Fig. 9: (a) Influence ofࢼ on ࡲᇱ ሺࣁሻ
Fig. 9: (b) Influence of ࢼ on ࣂሺࣁሻ
Fig. 10:Influence ofࡾࢊon ࣂሺࣁሻ
Fig. 11:Influence of Ec onࣂሺࣁሻ
12
Fig. 12:Influence of ࣂ࢝ onࣂሺࣁሻ
Fig. 13:Influence of ࢾ onࣂሺࣁሻ
Fig. 14: Influence of S and M of ࢌ࢘
Fig. 25: Influence of S and ࢼ of ࢌ࢘
The effect of skin friction coefficient on velocity profile are discussed in Figs. 14-15. In Fig. 14, it is clear that, for increasing value of squeeze number S along with the magnetic parameter M, the skin friction coefficient increases for the suction case (A > 0) and a slight fall is observed in the blowing case (A < 0). However, in Fig 15, one can easily see that the behavior is opposite for increasing value of Casson parameter ߚ for both the suction and injection cases.
13
Fig. 16: Influence of ࡾࢊ and ࢼ for ࡺ࢛
Fig. 17: Influence of ࡾࢊ and ࡱࢉ for ࡺ࢛
The combined effects of local Nusselt number are portrayed in Figs. 17-18. Fig. 16 depicts the behavior of local Nusselt number following the changes in radiation ܴ݀ and Casson parameter, it is clear that for higher valuesof Casson parameterߚ, the rate of heat transfer at the upper disk decreases with increasing value of ܴ݀.
Fig. 18: Influence of ࡾࢊ and ࣂ࢝ for ࡺ࢛
In Fig. 17. A rapid rise is seen in rate of heat transfer at upper disk for increasing values of ܴ݀ along with smaller variation in Eckert number for both suction and injection cases. While in Fig. 18, a slight increase in the rate of heat transfer is observed for increasing value of ߠ௪ . Comparison of linear and nonlinear radiation effects are presented in Table 3. Table 3: Comparison of Linear and Nonlinear radiation
Linear Radiation
Nonlinear Radiation
14
ߟ
ܨሺߟሻ
ߠሺߟሻ
ܨሺߟሻ
ߠሺߟሻ
0.
1.5
1
1.5
1
0.1
1.471160
0.994200
1.471160
0.997129
0.2
1.393540
0.935382
1.393540
0.941030
0.3
1.280080
0.849048
1.280080
0.856950
0.4
1.143300
0.751859
1.143300
0.761416
0.5
0.995343
0.652861
0.995343
0.663498
0.6
0.848116
0.554388
0.848116
0.565575
0.7
0.713334
0.452615
0.713334
0.463724
0.8
0.602627
0.337797
0.602627
0.347799
0.9
0.527620
0.194229
0.527620
0.201156
1.
0.5
0
0.5
0
5. Conclusions This article studies the squeezing flow of a Casson fluid between parallel disks under the effects of nonlinear radiation. Governing equations transformed to set of ordinary differential equations are solved using a well-known method, i.e., Homotopy Analysis Method (HAM). Effects of different emerging parameters on velocity and temperature profiles are discussed with the help of graphs. Numerical solution is also carried out to confirm the results obtained by HAM. It can be concluded that for the case of suction, velocity profile behaves quite oppositely as compared to the case of injection for all varying parameters. While the temperature profile has almost similar behavior for both the cases. Acknowledgement: Authors are thankful to the anonymous reviewers for the valuable comments that really helped to improve the quality of the presented work. References [1] M. J. Stefan, Versuch¨ Uber die scheinbareadhesion,Sitzungsberichte der Akademie der Wissenschaften in Wien. Mathematik-Naturwissen, 69, (1874) 713--721. 15
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19
Syed Tauseef Mohyud-Din, Sheikh Irfanullah Khan
PII: DOI: Reference:
S1270-9638(15)00328-4 http://dx.doi.org/10.1016/j.ast.2015.10.019 AESCTE 3463
To appear in:
Aerospace Science and Technology
Received date: Revised date: Accepted date:
6 June 2015 10 October 2015 18 October 2015
Please cite this article in press as: S.T. Mohyud-Din, S.I. Khan, Nonlinear radiation effects on squeezing flow of a Casson Fluid between parallel disks, Aerosp. Sci. Technol. (2015), http://dx.doi.org/10.1016/j.ast.2015.10.019
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Nonlinear Radiation Effects on Squeezing Flow of a Casson Fluid between Parallel Disks *Syed Tauseef Mohyud-Din, Sheikh Irfanullah Khan, Department of Mathematics, Faculty of Sciences, HITEC University, TaxilaCantt, Pakistan *
Corresponding Author ([email protected])
Abstract: In this paper, we investigate the nonlinear radiation effects in a timedependent two-dimensional flow of a Casson fluid squeezed between two parallel disks when upper disk is taken to be impermeable and lower one is porous. Suitable similarity transforms are employed to convert governing partial differential equations into system of ordinary differential equations. Well known Homotopy Analysis Method (HAM) is employed to obtain the expressions for velocity and temperature profiles. Effects of different physical parameters such as squeeze number S, Eckert number Ec and the dimensionless length on the flow when keeping ܲ ݎൌ are also discussed with the help of graphs for velocity and temperature coupled with a comprehensive discussions. Mathematica Package BVPh2.0 is utilized to formulate the total error of the system for both the suction and injection cases. The skin friction coefficient and local Nusselt number are presented with graphical aids for emerging parameters.
Key words:Homotopy analysis method (HAM); Casson fluid; nonlinear radiation effect; squeezing flow; parallel disks.
1.Introduction Squeezing flow between parallel disks is an important area of interest because of its application in bio fluid mechanics like pumping of heart, flow through certain arteries, polymer industry process, injection modeling, compression, liquid-metal lubrication, and the squeezed films in power transmission. Most frequently, squeezing flows were illustrated in modeling of metal and plastic sheets, thin fiber and paper sheets formations etc. After the pioneer work done by Stefan (1874) [1], several attempts are reported that extended the traditional problem to heat transfer case. Different studies are available in literature that used various solution schemes to get analytical and numerical solutions for the said problem. Siddiqui et al. [2] examined two-dimensional MHD squeezing flow 1
between parallel plates. For parallel disk similar problem has been discussed by Domairry and Aziz [3]. Both used Homotopy perturbation method (HPM) to determine the solution. Joneidi et al. [4] studied the mass transfer effect on squeezing flow between parallel disks using Homotopy analysis method (HAM). Most recently influence of heat transfer in the MHD squeezing Flow between parallel disks has been investigated by T. Hayat et al. [5]. They used HAM to solve the resulting nonlinear system of ordinary differential equations. Since then squeezing flow has been of great interests for various researchers [6-8]. Investigations with heat transport phenomenon are important in many industrial applications such as wire coating, hot rolling, metal spinning, glass blowing, paper manufacturing, glass fiber, glass sheet productions, the drawing of plastic films, continuous casting and the aerodynamic extrusion of plastic sheets. Radiative heat transfer plays an important role in controlling where convective heat transfer coefficients are small. In polymer industry, the radiative heat transfer is an essential phenomenon for the design of reliable equipment’s, nuclear plants, gas turbines, etc. The process of radiative heat transfer is also important in free and forced convection flows. Keeping all these applications in mind, many authors studied various research problems for the case of radiative heat transfer. Some of these can be seen in [9-12] and references there in. High nonlinearity of governing equations in various flow problems means unlikeliness of exact solution. To cope up with these problems, many approximation techniques have been developed. Highly nonlinear problems such as the ones discussed above are therefore solved by using these techniques. These include, Adomian's Decomposition Method (ADM), Variational Iteration Method, Variation of Parameters Method, Homotopy Perturbation Method, Homotopy Analysis Method, etc. [13-20]. Mathematical models describing the realistic flow problems mostly involve the NonNewtonian fluids. One of these fluids is known as Casson fluid and its formulation is provided in [21-22] is found to suitable for blood flow problems up to large extent. Casson fluid flow between squeezing disks under radiative heat transfer has not been considered yet. To fill out this gap, heat transfer analysis for squeezing flow of Casson fluid between parallel disks under the effects of thermal radiation is presented. Well 2
known Homotopy Analysis Method (HAM) [23-36] is employed to solve the problem. Graphs are plotted to analyze the effects of different emerging parameters on velocity and temperature profiles.
2.Governing Equations We have investigated parallel infinite disks h distance apart with magnetic field practiced
vertically and is proportional to ሺ െ ࢇ࢚ሻ . Incompressible Casson fluid is used in between the disks. Magnetic field is negligible for low Reynold numbers.
Fig.1: Schematics diagram of the problem
Consequently, Casson fluid flow equation is delineated as [31-34]
W ij
½ ª § p y ·º ¸¸» eij , S ! S c ° °2« P B ¨¨ ° ° ¬« © 2S ¹¼» ¾ ® § p y ·º ° ° ª °2« P B ¨¨ 2S ¸¸» eij , S c ! S ° © c ¹»¼ ¿ ¯ «¬
where௬ is yield stress of fluid,ߤ the plastic-dynamic viscosity of the fluid andߨis the self-product of component of deformation rate with itself and S c is the critical value of the said self-product.ܶ௪ andܶ are constant temperatures lower d upper diskrespectively. The viscous dissipation effects in the energy equation are retained. We have chosen the cylindrical coordinates system ሺݎǡ ߶ǡ ݖሻǡwhere the upper disk is moving with
ష ࢇࡴሺିࢇ࢚ሻ
velocity
towards or away from the stationary lower disk.Thus, the constitutive
equations for two-dimensional flow and heat transfer of a viscous fluid under the effects 3
of thermal radiation can be written as:
wu u ww wr r wz
wu wu · § wu w U¨ u ¸ wr wz ¹ © wt
0
(1)
§ 1 E · § w 2 u 1 wu u w 2 u · V 2 wpˆ ¸ B t u , (2) ¸¸ ¨¨ 2 P ¨¨ wr r wr r 2 wz 2 ¸¹ U © E ¹ © wr
ww ww · § ww u w ¸ wr wz ¹ © wt
§ 1 E · § w 2 w w 2 w 1 ww · wpˆ ¸ , (3) ¸¸ ¨¨ 2 2 P ¨¨ wz r wr ¸¹ wz © E ¹ © wr
U¨
wT wT · § wT Cp¨ u w ¸ wr wz ¹ © wt
k § w 2T w 2T 1 wT · 1 w ¨ ¸ qr U ¨© wr 2 wz 2 r wr ¸¹ U wz 2 2 § § wu · 2 · ¨ 2¨ ¸ 2§¨ ww ·¸ 2 u ¸ (4) ¸ r2 § 1 ·¨ © wr ¹ © wr ¹ Q ¨¨1 ¸¸¨ ¸. 2 2 © E ¹¨ § wu · § ww · § wu ·§ ww · ¸ ¨ ¨ wz ¸ 2¨ wz ¸ 2¨ wz ¸¨ wr ¸ ¸ © ¹ © ¹© ¹ ¹ © © ¹
Supporting conditions are
^u
0, w
w0 ` z
®u ¯
0, w
dh ½ ¾ dt ¿ z
T T
Tw z Th z
0
(5) h t
0
(6)
h t
In the above equations u and w are the velocity components in the r- and z - directions respectively, while
U
is the density, viscosity P , pˆ the pressure, specific heat C p , T
the temperature, v kinematic viscosity, k thermal conductivity, k coefficient, Stefan Boltzmann constant
V , and w0
mean absorption
is suction/injection velocity.
Substituting the following transformations [8]
ar aH F c K , w F K , 2 1 at 1 at B0 T Th z B t , K , T K , Tw Th H 1 at 1 at
u
(7)
in Eqs. (2) - (4) and removing the pressure gradient from the subsequent equations, we finally obtain 4
§ 1· ~ ¨¨1 ¸¸ F iv S KF ccc 3F cc 2 F F ccc M 2 F cc © E¹
1 Rd (1 (T
w
(8)
0,
c § 1· 3 1)T T ' S Pr 2 FT c KT c ¨¨1 ¸¸ Pr Ec F cc2 12G 2 F c2 © E¹
0, (9)
with associated condition F 0 F 1
A, F c 0 0, T 0 1, 1 F c 1 0, T 1 0. 2,
(10)
Where A denotes the suction/injection, Eckert number Ec, Hartman number M, squeeze number S, Prandtl number Pr, radiation parameter Rd , T w is the temperature difference parameter, and
G the dimensionless length defined as S Ec
aB02 H 2
aH 2 , M2 2Q
Q
, Pr
PC p K0
16V *Th3 , 3k * k
H 2 1 at . r2
2
§ ar · ¨¨ ¸¸ , G 2 C p Tw Th © 2 1 at ¹ 1
, Rd
(11)
The skin friction coefficient and Nusselt number are defined as
W rz
z h t
C fr
U 2 1aH at
W rz
P¨
2
, Nu
H qw qr , K 0 Tw Th
(12)
where
§ wu ww · ¸ © wz wr ¹ z
h t
, qw
§ wT · k 0 ¨ ¸ © wz ¹ z
.
(13)
h t
In view of (7), Eq. (12) can be written as
C fr
§ 1 · F cc 1 r 2 ¨¨1 ¸¸ 2 , Nu © E ¹ H Re r
1 Rd T c 1 , 1 at
raH 1 at . 2Q
(14)
12
Re r
(15)
3.Solution Procedure for HAM Tracking the methodology proposed by Liao [23-24], we can initiate initial guesses as F0 K
A
3 1 2 A K 2 1 2 A K 3 , 2
5
(16)
T 0 K K.
(17)
Linear operator can be chosen as d 4F and LT dK 4
LF
d 2T dK 2
(18)
Above operators satisfies the following linearity, M
S
Pr
Ec
G
Total Error
ߠ௪
E
LF C1 C2K C3K 2 C4K 3
Rd
=1
=2
0,
LT C5 C6K 0,
of System
(19) (20)
where Ci i 1 6 are constants. 3.1
Convergence of Solution The convergence of the series solutions is highly depends on the auxiliary
parameters = F and = 4 . We define total error of the system for the functions F , 4 to determine the optimal values of these parameters as 2
ª § m ·º ¨ ¸» . N F i x ' « F ¨¦ j ¦ ¸ i 0 ¬ « ©j 0 ¹¼» K
Em,1 = F
1 K
Em, 2 = 4
ª § m ·º 1 ¨ ¸» . N i x 4 ' « ¦ 4 ¦ j ¸ K i 0 «¬ ¨© j 0 ¹»¼
Where 'x
2
K
1 K . For a given order of approximation m, the optimal values of = F , = 4
can be determine by minimizing the total error of the system given in the equations by using the HAM package BVPh2.0. The optimal values of these auxiliary parameters are given in table 1 and 2, corresponding to different values of the parameters.
6
M
S
Pr
Ec
G
E
ߠ௪
Rd
0.1
0.2
0.71
0.2
0.1
1
1.1
0.2
=1
=2
-0.502
-0.819
Total Error of System 7.91ൈ ͳͲି଼
0.4
-0.513
-0.817
1.31ൈ ͳͲି
0.7
-0.511
-0.817
1.50ൈ ͳͲି
0.1
-0.527
-0.815
1.87ൈ ͳͲି
0.3
-0.503
-0.821
3.97ൈ ͳͲି
0.5
-0.504
-0.824
4.28ൈ ͳͲିହ
0.1
-0.501
-0.811
1.63ൈ ͳͲି଼
0.3
-0.501
-0.813
7.23ൈ ͳͲି଼
0.5
-0.501
-0.816
3.41ൈ ͳͲି
0.1
-0.504
-0.819
8.04ൈ ͳͲି଼
0.3
-0.521
-0.817
6.18ൈ ͳͲି
0.5
-0.535
-0.818
4.53ൈ ͳͲିହ
0.2
-0.514
-0.818
1.41ൈ ͳͲି
0.4
-0.516
-0.819
2.47ൈ ͳͲି
0.6
-0.519
-0.820
7.22ൈ ͳͲି
0.5
-0.349
-0.819
5.25ൈ ͳͲି
1.5
-0.603
-0.818
8.22ൈ ͳͲି
2.5
-0.718
-0.818
6.61ൈ ͳͲି
1.05
-0.511
-0.827
5.64ൈ ͳͲି
1.1
-0.514
-0.817
1.25ൈ ͳͲି
1.15
-0.519
-0.805
8.23ൈ ͳͲି
0.1
-0.512
-0.903
8.05ൈ ͳͲି
0.4
-0.520
-0.687
1.08ൈ ͳͲିହ
0.7
-0.528
-0.555
5.38ൈ ͳͲିହ
0.2
0.2
0.71
0.2
0.1
1
1.1
Table 1: Optimal Values for A> 0.
Table 2: Optimal values for A< 0.
7
0.1
-0.493
-0.796
6.42ൈ ͳͲିଷ
0.4
-0.495
-0.796
6.38ൈ ͳͲିଷ
0.7
-0.497
-0.795
6.31ൈ ͳͲିଷ
0.1
-0.498
-0.802
3.59ൈ ͳͲିଷ
0.3
-0.490
-0.790
1.02ൈ ͳͲିଶ
0.5
-0. 486
-0.778
2.19ൈ ͳͲିଶ
0.1
-0.495
-0.806
2.13ൈ ͳͲି
0.3
-0.494
-0.803
1.56ൈ ͳͲିସ
0.5
-0.494
-0.799
1.36ൈ ͳͲିଷ
0.1
-0.494
-0.795
4.51ൈ ͳͲିସ
0.3
-0.493
-0.797
3.19ൈ ͳͲିଶ
0.5
-0.492
-0.797
2.50ൈ ͳͲିଵ
0.2
-0.493
-0.796
7.68ൈ ͳͲିଷ
0.4
-0.492
-0.795
1.55ൈ ͳͲିଶ
0.6
-0.488
-0.796
4.53ൈ ͳͲିଶ
0.5
-0.330
-0.797
2.75ൈ ͳͲିଶ
1.5
-0.590
-0.795
3.45ൈ ͳͲିଷ
2.5
-0.701
-0.796
1.95ൈ ͳͲିଷ
1.05
-0.490
-0.811
8.37ൈ ͳͲିସ
1.1
-0.493
-0.796
6.41ൈ ͳͲିଷ
1.15
-0.497
-0.779
2.50ൈ ͳͲିଶ
0.1
-0.492
-0.882
1.66ൈ ͳͲିଷ
0.4
-0.495
-0.667
2.18ൈ ͳͲିଶ
0.7
-0.496
-0.537
4.61ൈ ͳͲିଶ
0.2
0.2
0.2
0.71
0.71
0.2
0.2
0.1
0.1
1
1
1.1
1.1
0.2
4. Results and discussions For convenience, we divide our discussions in to two parts one dedicated to investigate the upshots on varying physical parameter for the case suction ( A ! 0 ) and the other one 8
describes the same effects for the case of blowing ( A 0 ). After ensuring the convergence of series solution our concern now is to see the influences of suction/blowing parameter, squeeze number S andCasson fluid parameter
E on
velocity and temperature profile is examined. 4.1. Suction Case To analyze the impact of emerging parameters on velocity and temperature profile for suction, effects of squeezing parameter S on the velocity and temperature profile are discussed in Figs. 2a & 2b. In Fig. 2a velocity profile decreases in magnitude near the porous walls for 0. 4, and the velocity profile increases in magnitude for increasing value of squeeze numbers, while in Fig. 2b temperature profile decreases with increase in squeeze number S. Effect of Casson fluid parameterߚfor velocity and temperature profile is discussed in Fig. 3a & 3b. In Fig. 3a, increase in magnitude for velocity near the porous wall for Casson fluid parameter and decreases for increasing value of ߚ, while, in Fig. 3b temperature decreases for increase in Casson fluid parameter.
Fig. 2: (a) Influence of S on ࡲᇱ ሺࣁሻ
Fig. 2: (b) Influence of S onࣂሺࣁሻ
9
Fig. 3:(a) Influence ofࢼ onࡲᇱ ሺࣁሻ
Fig. 3: (b) Influence ofࢼ on ࣂሺࣁሻ
The graphical representations of key parameters in case of suction for temperature profile are displayed in Figs. 4-7. Fig. 4 depicts the influence of radiation parameter for suction case, increase in radiation parameter ܴ݀ results in decreased temperature. Physically, increase in radiation parameter can give us higher values of temperature that can be used in many industrial appliances.
Fig. 4:Influence ofࡾࢊ on ࣂሺࣁሻ
Fig. 5:Influence of Ec on ࣂሺࣁሻ
10
Fig. 6:Influence of ࣂ࢝ onࣂሺࣁሻ
Fig. 7:Influence of ࢾ onࣂሺࣁሻ
Fig. 5 depicts the behavior of Ec on the temperature distribution, where rapid rise in temperature is seen with higher Ec. However, for the higher Ec number boundary layer thickness decreases. This increase in temperature is due to the presence of viscous dissipation term in the energy equation. Fig. 6 depicts the behavior of temperature for increasing values of ߠ ௪ . A slight fall in temperature distribution is seen for the higher values of ߠ௪ .Here, one can easily that the parameter ߠ௪ don’t affect the temperature near upper disk. The effect of dimensionless length ߜ is almost like Eckert number that can be seen from Fig. 5. 4.2. Blowing Case To analyze the behavior of all emerging parameter on velocity and temperature profile for blowing case Figs. 8-13 are displayed. In Fig. 8a, the behavior of squeeze number on velocity profile is opposite in blowing case as compared to suction case.However, for the temperature distribution all the physical parameters have same behavior in blowing case as compared to those in suctions case. Only difference is the effects of Casson parameterࢼthat decrease the velocity profile near the porous walls for ߟ ͲǤͷ, and then starts increasing for higher value of Casson parameter ߚǤ
11
Fig. 8: (a) Influence of S onࡲԢሺࣁሻ
Fig. 8: (b) Influence of S onࣂሺࣁሻ
Fig. 9: (a) Influence ofࢼ on ࡲᇱ ሺࣁሻ
Fig. 9: (b) Influence of ࢼ on ࣂሺࣁሻ
Fig. 10:Influence ofࡾࢊon ࣂሺࣁሻ
Fig. 11:Influence of Ec onࣂሺࣁሻ
12
Fig. 12:Influence of ࣂ࢝ onࣂሺࣁሻ
Fig. 13:Influence of ࢾ onࣂሺࣁሻ
Fig. 14: Influence of S and M of ࢌ࢘
Fig. 25: Influence of S and ࢼ of ࢌ࢘
The effect of skin friction coefficient on velocity profile are discussed in Figs. 14-15. In Fig. 14, it is clear that, for increasing value of squeeze number S along with the magnetic parameter M, the skin friction coefficient increases for the suction case (A > 0) and a slight fall is observed in the blowing case (A < 0). However, in Fig 15, one can easily see that the behavior is opposite for increasing value of Casson parameter ߚ for both the suction and injection cases.
13
Fig. 16: Influence of ࡾࢊ and ࢼ for ࡺ࢛
Fig. 17: Influence of ࡾࢊ and ࡱࢉ for ࡺ࢛
The combined effects of local Nusselt number are portrayed in Figs. 17-18. Fig. 16 depicts the behavior of local Nusselt number following the changes in radiation ܴ݀ and Casson parameter, it is clear that for higher valuesof Casson parameterߚ, the rate of heat transfer at the upper disk decreases with increasing value of ܴ݀.
Fig. 18: Influence of ࡾࢊ and ࣂ࢝ for ࡺ࢛
In Fig. 17. A rapid rise is seen in rate of heat transfer at upper disk for increasing values of ܴ݀ along with smaller variation in Eckert number for both suction and injection cases. While in Fig. 18, a slight increase in the rate of heat transfer is observed for increasing value of ߠ௪ . Comparison of linear and nonlinear radiation effects are presented in Table 3. Table 3: Comparison of Linear and Nonlinear radiation
Linear Radiation
Nonlinear Radiation
14
ߟ
ܨሺߟሻ
ߠሺߟሻ
ܨሺߟሻ
ߠሺߟሻ
0.
1.5
1
1.5
1
0.1
1.471160
0.994200
1.471160
0.997129
0.2
1.393540
0.935382
1.393540
0.941030
0.3
1.280080
0.849048
1.280080
0.856950
0.4
1.143300
0.751859
1.143300
0.761416
0.5
0.995343
0.652861
0.995343
0.663498
0.6
0.848116
0.554388
0.848116
0.565575
0.7
0.713334
0.452615
0.713334
0.463724
0.8
0.602627
0.337797
0.602627
0.347799
0.9
0.527620
0.194229
0.527620
0.201156
1.
0.5
0
0.5
0
5. Conclusions This article studies the squeezing flow of a Casson fluid between parallel disks under the effects of nonlinear radiation. Governing equations transformed to set of ordinary differential equations are solved using a well-known method, i.e., Homotopy Analysis Method (HAM). Effects of different emerging parameters on velocity and temperature profiles are discussed with the help of graphs. Numerical solution is also carried out to confirm the results obtained by HAM. It can be concluded that for the case of suction, velocity profile behaves quite oppositely as compared to the case of injection for all varying parameters. While the temperature profile has almost similar behavior for both the cases. Acknowledgement: Authors are thankful to the anonymous reviewers for the valuable comments that really helped to improve the quality of the presented work. References [1] M. J. Stefan, Versuch¨ Uber die scheinbareadhesion,Sitzungsberichte der Akademie der Wissenschaften in Wien. Mathematik-Naturwissen, 69, (1874) 713--721. 15
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