Theoretical analysis of slip flow on a rotating cone with viscous dissipation effects

616

2015,27(4):616-623

DOI: 10.1016/S1001-6058(15)60523-6

Theoretical analysis of slip flow on a rotating cone with viscous dissipation effects* SALEEM S.1, NADEEM S.2 1. Department of Mathematics, COMSATS Institute of Information Technology, Attock 43600, Pakistan, E-mail: [email protected] 2. Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan (Received August 10, 2014, Revised September 25, 2014) Abstract: This paper is concerned with the mutual effects of viscous dissipation and slip effects on a rotating vertical cone in a viscous fluid. Similarity solutions for rotating cone with wall temperature boundary conditions provides a system of nonlinear ordinary differential equations which have been treated by optimal homotopy analysis method (OHAM). The obtained analytical results in comparison with the numerical ones show a noteworthy accuracy for a special case. Effects for the velocities and temperature are revealed graphically and the tabulated values of the surface shear stresses and the heat transfer rate are entered in tables. From the study it is seen that the slip parameter γ enhances the primary velocity while the secondary velocity reduces. Further it is observed that the heat transfer rate NuRex−1/2 increases with Eckert number Ec and Prandtl number Pr . key words: mixed convection, incompressible flow, differential equations, slip effects, viscous dissipation

Introduction A study which involves the equivalent participation of both forced and natural convection is termed as mixed convection. It plays a key role in atmospheric boundary layer flows, heat exchangers, solar collectors, nuclear reactors and in electronic equipment’s. Such processes occur when the effects of buoyancy forces in forced convection or the effects of forced flow in natural convection become much more remarkable. The interaction of both convections is mostly noticeable in physical situations where the forced convection flow has low velocity or moderate and large temperature differences. In the concerned analysis, a rotating cone is placed in a Newtonian fluid with the axis of the cone being in line with the external flow is inspected. The mixed convective heat transfer problems with cones are generally used by automobile and chemical industries. Some important applications are 0F

* Biography: SALEEM S. (1986-), Male, Ph. D., Assistant Professor

design of canisters for nuclear waste disposal, nuclear reactor cooling system, etc.. Practically, the unsteady mixed convective flows do not give similarity solutions and for the last few years, various problems have been deliberated, where the non-similarities are taken into account. The unsteadiness and non-similarity in such type of flows is due to the free stream velocity, the body curvature, the surface mass transfer or even possibly due to all these effects. The crucial mathematical difficulties elaborate in finding non-similar solutions for such studies have bounded several researchers to confine their studies either to the steady nonsimilar flows or to the unsteady semi-similar or selfsimilar flows. A solution is recognized as self-similar if a system of partial differential equations can be reduced to a system of ordinary differential equations. If the similarity transformations are able to reduce the number of independent variables only, then the reduced equations are named semi-similar and the corresponding solutions are the semi-similar solutions. Hering and Grosh[1] studied steady mixed convection boundary layer flow from a vertical cone in an ambient fluid for the Prandtl number of air. Himasekhar et al.[2] carried out the similarity solution of the mixed convection boundary layer flow over a vertical rota-

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ting cone in an ambient fluid for a wide range of Prandtl numbers. A few years back, Anilkumar and Roy[3] obtained the self-similar solutions of unsteady mixed convection flow from a rotating cone in a rotating fluid. Unsteady heat and mass transfer from a rotating vertical cone with a magnetic field and heat generation or absorption effects were examined by Chamkha and Al-Mudhaf[4].The non-similar solution to study the effects of mass transfer (suction/injection) on the steady mixed convection boundary layer flow over a vertical permeable cone were presented by Ravindran et al.[5]. Also Nadeem and Saleem[6] explore the analytical study of mixed convection flow of non-Newtonian fluid on a rotating cone. Hall effects on unsteady flow due to noncoaxially rotating disk and a fluid at infinity were presented by Hayat et. al.[7]. Fluids revealing slip are significant in technological applications such as in the polishing of artificial heart valves and internal cavities. Slip also occurs on hydrophobic surfaces, particularly in micro- and nano-fluidics. Makinde and Osalsui[8] studies MHD steady flow in a channel with slip at the permeable boundaries. Ellahi et. al.[9] examined the study of generalized Couette flow of a third grade fluid with slip: the exact solutions. Some relevant studies on this phenomenon are given in Refs.[10-15]. The influence of variable viscosity and viscous dissipation on the nonNewtonian flow was explored by Hayat et. al.[16]. In general it is challenging to handle nonlinear problems, especially in an analytical way. Perturbation techniques like variation of iteration method (VIM) and homotopy perturbation method (HPM)[17,18] were frequently used to get solutions of such mathematical investigation. These techniques are dependent on the small/large constraints, the supposed perturbation quantity. Unfortunately, many nonlinear physical situations in real life do not always have such nature of perturbation parameters. Additional, both of the perturbation techniques themselves cannot give a modest approach in order to adjust or control the region and rate of convergence series. Liao[19] presented an influential analytic technique to solve the nonlinear problems, explicitly the homotopy analysis method (HAM). It offers a suitable approach to control and regulate the convergence region and rate of approximation series, once required. Encouraged by all above findings, the main emphasis of the present paper is to examine the effects of slip on boundary layer flow over a rotating cone in a viscous fluid with viscous dissipation. The concerned nonlinear partial differential for rotating cone are transformed to system of nonlinear ordinary differential equations with proper similarity transformations and then solved by optimal homotopy analysis method (OHAM)[19-29]. Also the effects of related physical parameters on velocities, surface stress tensors, temperature and heat transfer rate are reported and discussed

through graphs and tables. 1. Analysis of the problem Consider the unsteady, axi-symmetric, incompressible viscous fluid flow of over a rotating cone in a Newtonian fluid. It is assumed that only the cone is in rotation with angular velocity which is a function of time. This develops unsteadiness in the flow field. Rectangular curvilinear coordinate system is taken to be fixed. Here u , v and w be the components of velocity in x , y and z - directions, respectively. The temperature as well as concentration variations in the flow fluid are responsible for the existence of the buoyancy forces. The gravity g acts downward in the direction of axis of the cone. Moreover, the wall temperature Tw and wall concentration Cw are linear functions of x , while the temperature T∞ and concentration C∞ far away from the cone surface are taken to be constant. The physical model and coordinate system is shown in Fig.1.

Fig.1 Physical model and coordinate system

By using Boussinesq approximation and boundary layer theory, the governing momentum and energy equations are deliberated as: ∂ ( xu ) ∂ ( xw) + =0 ∂x ∂z

(1)

∂u ∂u ∂u v2 ∂ 2u +u +w − = ν 2 + g β cos α * (T − T∞ ) (2) ∂t ∂x ∂z x ∂z ∂v ∂v ∂ v uv ∂2v +u +w + =ν 2 ∂t ∂x ∂z x ∂z ∂T ∂T ∂T ∂ 2T µ +u +w =α 2 + ∂t ∂x ∂z ∂z ρC p

(3)  ∂ u  2  ∂ v  2    +     ∂z   ∂z   (4)

618

where ν is the kinematic viscosity, ρ is the density,

γ = N ρn 1/ 2 (Ω sin α * )1/ 2 (1 − st * )−1/ 2

(6)

∗

g is the gravity, α is the semi-vertical angle of the cone, β is the volumetric coefficient of expansion for

temperature, α is the thermal diffusivity and C p specific heat of the fluid. The boundary conditions appropriate to the viscous flow problem are stated below u ( x, 0, t ) = N µ Nµ

The Eq.(3) is trivially satisfied and Eqs.(4) and (7) takes the form f ′′′ − ff ′′ +

1 2 1   f ′ − 2 g 2 − 2λθ − s  f ′ + η f ′′  = 0 (7) 2 2  

1   g ′′ − f g ′ + g f ′ − s  g + η g ′  = 0 2  

∂u v ( x, 0, t ) = Ω x sin α * (1 − st * ) −1 + ∂z

(8)

 θ 1   θ ′′ − Pr  f ′ − f θ ′ + s  2θ + ηθ ′  −

∂v , ∂z



w( x, 0, t ) = 0 , T ( x, 0, t ) = Tw , u ( x, ∞, t ) = 0 , v ( x, ∞, t ) = 0 , T ( x, ∞, t ) = T∞

(5)

here Ω is the dimensionless angular velocity of the cone, T∞ is the temperature far away from fluid, N is the velocity slip factor and t ∗ is the dimensionless time. It is suitable to reduce system of partial differential equations in to nonlinear ordinary differential equations with the help of following similarity transformation[3].

2

2





1  Ec  ( f ′′) 2 + ( g ′) 2   = 0 4 

(9)

here λ is the mixed convection parameter, s is the unsteady parameter and the flow is accelerated for s > 0 and retarded for s < 0 , Pr is the Prandtl number, Ec is the Eckert number and γ is the slip parameter. The boundary conditions in non-dimensional form for the concerned flow problem are given as: f (0) = 0 , f ′(0) = γ f ′′(0) , g (0) = 1 + g g ′(0) ,

u = − 2−1 Ω x sin α * (1 − st * )−1 f ′(η ) ,

θ (0) = 1 , f ′(∞) = 0 , g (∞) = 0 , θ (∞) = 0

v = Ω x sin α * (1 − st * ) −1 g (η ) ,

The surface stress tensors in primary and seconddary directions for the present analysis are:

w = (v Ω sin α * )1/ 2 (1 − st * )−1/ 2 f (η ) , Cfx =

T = T∞ + (Tw − T∞ )θ (η ) ,

x Tw − T∞ = (T0 − T∞ )   (1 − st * )−2 , L

Cf y =

1/ 2

 Ω sin α *  t * = (Ω sin α * )t , η =   v  

ReL = Ω sin α *

(1 − st * ) −1/ 2 z ,

L2 ν , Pr = , v k

Gr = g β cos α * (T0 − T∞ ) xLκ (Ω sin α * )5 / 2 , Ec = cρ q0n 1/ 2

Gr L3 , λ= 2 , 2 ReL v

(10)

t xz z = 0 ∂u   = − Rex−1/ 2  2 µ  , * * −1 2 ∂z  z =0 ρ [Ω x sin α (1 − st ) ]  t yz

∂v   = − Rex−1/ 2  2 µ  ∂z  z =0 ρ [Ω x sin α (1 − st ) ]  *

z =0

* −1 2

or in dimensionless form C f x Re1/x 2 = [− f ′′]η =0 , 0.5C f y Re1/x 2 = [− g ′]η =0

(11)

The heat transfer coefficient in dimensionless form is stated as Nu Rex−1/ 2 = −θ ′(0)

where Rex = Ω x 2 sin α * (1 − st * )−1 /n Reynolds number.

(12) is the local

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2. Optimal homotopy analysis procedure The solutions of the coupled nonlinear parabolic ordinary differential equations given in Eqs.(10)-(13) are carried out analytically by optimal homotopy analysis method (OHAM) which was established by Liao[12]. The following initial guesses and linear operators for velocity components and temperature fields are used f 0 , g 0 and θ 0 respectively is f 0 (η ) = 0 g 0 (η ) =

(13)

1 exp(−η ) 1+ g

(14)

optimal values of non-zero auxiliary parameters c0f , c0g and c0θ it is used here the so-called average residual error specified by[19]. 2

m m  1 j   m ˆ  ˆ ε = η η θˆ(η )  N f g ( ), ( ),   dy ∑ ∑ ∑ f ∑ j +1 i =0   n =0 n =0 n =0  y =i d y  (19) f m

2

m  1 j   m ˆ  ε =  N g  ∑ f (η ), ∑ gˆ (η )   dy ∑ j +1 i =0   n =0 n =0  y =i d y  g m

(20)

2

θ 0 (η ) = exp(−η )

(15)

L f = f ′′′ − f ′

(16)

Lg = g ′′ − 1

(17)

Lθ = θ ′′ − 1

(18)

The standard procedure of homotopy analysis method can be follow as[19,29]. Table 1 Local optimal convergence control paraments and total averaged squared residual errors using BVPh2.0 M

ε mf

ε mg

ε mθ

CPU time/s

4

4.18×10−5

1.85×10−5

1.57×10−4

4.50

8

5.22×10−6

6.62×10−7

6.35×10−6

21.25

12

5.21×10−7

5.18×10−8

6.03×10−7

56.00

16

−8

−9

−8

127.0

7.44×10

7.31×10

2.71×10

m  1 j   m ˆ  εm =  Nθ  ∑ f (η ), ∑ θˆ(η )   dy ∑ j +1 i =0   n =0 n =0  y =i d y 

(21)

ε mt = ε mf + ε mg + ε mθ

(22)

θ

here ε mt is the total squared residual error, δ y = 0.5 and j = 20 . Tables 1 and 2 displays the values for several optimal convergence control parameter. These tables show that the averaged squared residual errors and total averaged squared residual errors are going smaller and smaller with the order of approximation increases, which assures that the solution is convergent at higher order approximations. The results will be similar if we choose the values of the optimal convergence parameters from any higher order approximation. We choose the 10th iteration set of optimal values to plot figures and draw tables in the coming sections. Hence, optimal homotopy analysis method provides us a sensible way to choose any set of local convergence control parameters to attain the convergent solutions.

Table 2 Individual averaged squared residual errors using optimal values at m = 10 from Table 1 M

c0f

c0g

c0θ

ε mt

2

−1.16

−0.35

−1.28

5.49×10−4

−1.30

1.37×10

−4

25.70

−5

150.28 881.06

4

−1.21

−0.29

CPU time/s

6

−0.99

−0.28

−1.33

2.27×10

8

−0.86

−0.28

−1.50

5.62×10−6

−0.85

−6

10

−1.03

−0.30

1.83×10

3.37

2 527.84

Generally homotopy analysis solutions involve the non-zero auxiliary parameters c0f , c0g and c0θ hich are helpful in finding the convergence-region and rate of the homotopy series solutions. In order to attain the

Fig.2 Variation of − f ′(η ) for λ

3. Results and discussion This portion of study involves the graphical and numerical results of various significant parameters on

620

Fig.3 Variation of − f ′(η ) for γ

Fig.7 Variation of θ (η ) for Pr

Fig.4 Variation of g (η ) for λ

Fig.8 Variation of −θ ′(0) for λ

Fig.5 Variation of g (η ) for γ

Fig.6 Variation of θ (η ) for Ec

velocities, temperature, surface stress coefficients and heat transfer coefficient. Such variations have been observed in Figs.2-7. Figure 2 is sketched to display the behavior of primary velocity − f ′(η ) for mixed convection parameter λ in the presence of slip and no slip parameters. The positive buoyancy parameter acts like a favorable pressure gradient, with property to accelerate the fluid. It is expected from Fig.2 that − f ′(η ) and boundary layer thickness increases with increasing values of λ , further it is noticed that the primary velocity − f ′(η ) has greater magnitude for γ = 0.5 (i.e., in the presence of slip parameter). The influence of slip parameter γ on primary velocity − f ′(η ) is shown in Fig.3. It is devoted from the figure that − f ′(η ) enhances its magnitude with an increase in γ . The influence of mixed convection parameter λ and slip parameter γ is to reduce the secondary velocity g (η ) respectively (See Fig.4 and 5). Moreover it is seen that the secondary velocity g (η ) has least magnitude for γ = 0.5 (i.e., in the presence of slip parameter). Figure 6 is devoted to show the influence of Eckert number Ec on temperature θ (η ) . The figure shows that the temperature θ (η ) is an

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Table 3 Comparison of values of Skin friction coefficients and heat transfer for s = γ = Ec = 0

Pr

λ

0 0.7

1 10 0

10

1 10

Numerical results[3]

Present analytical results

− f ′′(0)

− g ′(0)

−θ ′(0)

1.0255

0.6154

0.4299

2.2010 8.5042 1.0255 1.5630 5.0820

0.8493 1.3992 0.6158 0.6835 0.9845

0.6121

azimuthal surface stress tensor 0.5C f y Rex−1/ 2 . Mixed convection parameter λ and the unsteady parameter

−θ ′(0)

1.0255*

0.6158*

0.4299*

1.0255

0.6158

0.4299

2.2014*

0.8497

*

0.6121*

2.2012

0.8496

0.6120

*

1.3992

*

1.0099*

8.5041

1.3995

1.0097

*

0.6158

*

1.4111*

1.0256

0.6158

1.4110

*

0.6838

*

1.5663*

1.5636

0.6837

1.5662

*

0.9841

*

2.3583*

5.0821

0.9840

2.3580

1.0255

1.4111

1.5638

1.5661

5.0825

2.3581

Values taken from Himasekhar et. al.[2].

slip parameter γ , but the variation is just opposite for

− g ′(0)

8.5045

1.0098

*

increasing function of Ec . The influence of the Prandtl number Pr on the temperature is drafted in Fig.7. It is clear from the respective figure that θ (η ) as well as the thermal boundary layer thickness decrease for Pr . Physically the fluid with higher Prandtl number has a lower thermal conductivity which effects in thinner thermal boundary layer and as a result heat transfer rate rises. For engineering phenomenon, the heat transfer rate must be small. This can be retained by keeping the low temperature difference between the surface and the free stream fluid, using a low Prandtl number fluid, keeping the surface at a constant temperature instead of at a constant heat flux, and by smearing the buoyancy force in the contrasting direction to that of forced flow. Figure 8 is sketched to observe the behavior of Nusselt number on mixed convection parameter λ . It is depicted that −θ ′(0) decreases with increasing λ . In order to get the authentication of accuracy of the analytical scheme, a comparison of the present results equivalent to the surface stress coefficients and heat transfer coefficient for γ = s = Ec = 0 with published literature of Chamkha et al.[4] and Himasekhar et al.[2] is presented and is found to be in remarkable agreement given in Table 3. Table 4 involves the numerical values of surface stress tensors for pertinent parameters. It is found from the table that the tangential surface stress tensor C f x Rex−1/ 2 increases for

− f ′′(0)

Table 4 Values for surface shear stresses when Pr = 1.0 and Ec = 0.5

γ

λ

S

C f x Rex−1/ 2

0.5C fy Rex−1/ 2

0

3.0614

0.9015

1.0

1.5074

0.4986

3.0

0.7507

0.2525

1.0

5.0993

1.0946

3.0

15.6067

1.7145

5.0

37.7863

2.3345

−0.5

5.0977

0.8149

0

5.0985

0.9602

0.5

5.0993

1.0949

Table 5 Values for reduced Nusselt number for interesting physical parameters

Ec

Pr

S

NuRex−1/ 2

0

0.3368

0.5

0.2798

1.0

0.2227 4.0

0.7494

7.0

0.7654

10.0

0.7976 −0.5

0.2823

0

0.2810

0.5

0.2798

622

S cause an increase in surface stress tensors in both directions (see Table 4). Table 5 depicts that as unsteady parameter S increases from −0.5 to 0.5, heat transfer rate Nu Rex−1/ 2 decreases. Similar behavior is observed for Eckert number. Moreover, it is seen that the Prandtl number Pr enhances the variation of heat transfer rate Nu Rex−1/ 2 .

4. Concluding remarks In this study we have deliberated the effects of slip on mixed convection flow of a fluid on a rotating cone in a viscous fluid with viscous dissipation. The non-linear partial differential equations are primarily reduced to a system of non-linear ordinary differential equation and then the solution is effectively carried out by optimal homotopy analysis method. The results shows that: (1) The primary velocity increases and secondary velocity decreases for both mixed convection parameter λ and slip parameter γ respectively. (2) Surface stress tensor in x - direction C f x Rex−1/ 2 enhances its magnitude for mixed convection parameter λ and unsteady parameter s , but possess opposite variation for slip parameter γ . (3) Temperature field is an increasing function of Eckert number Ec . (4) The heat transfer rate Nu Rex−1/ 2 has opposite variation Prandtl number Pr and Eckert number Ec . References [1] [2]

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