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

effects

S. Nadeem1 and S. Saleem

1*

1Department of Mathematics, Quaid-i-Azam University 45320

Islamabad 44000, Pakistan * E-mail: salmansaleem_33@hotmail.com

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 enhances the primary velocity while the secondary

velocity reduces. Further it is observed that the heat transfer rate 12ReNu x

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 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

non-similar flows or to the unsteady semi-similar or self-

similar 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 rotating 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

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 non-Newtonian 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.

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.

Fig. 1 Physical model and coordinate system.

By using Boussinesq approximation and boundary layer

theory, the governing momentum and energy equations are

deliberated as

( ) ( )0

xu xw

x z

(1)

2 2

cos2

u u u v uu w g T T

t x z x z

(2)

2

2

v v v uv vu w

t x z x z

(3)

2 22

2

T T T T u vu w

t x z C z zz p

(4)

Where is the kinematic viscosity, is the density,

g is the gravity,

is the semi-vertical angle of the cone,

is the volumetric coefficient of expansion for

temperature, is the thermal diffusivity and Cp specific

heat of the fluid.

The boundary conditions appropriate to the viscous flow

problem are stated below

( ,0, )u

u x t Nz

1

( ,0, ) sin 1v

v x t x st Nz

( ,0, )w x t = 0, ( , 0, )T x t Tw

( , , ) 0,u x t ( , , ) 0,v x t = ( , , )T x t T (5)

here 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].

11

2 sin 1

11122sin 1 , ( sin ) 1

2

, 10

122 1

sin 2( sin ) , 1 ,Re sin , Pr

3

cos , , 0 2 2Re

u x st f

v x st g w v st f

xT T T T T T T T stw w

L

Lt t st z L

v v k

L GrGr g T T Ec

v L

5

2( sin )

1

20

1 1 122 2( sin ) 1 (6)

xL

c q

N st

The Eq. (3) is trivially satisfied and Eqs. (4) and (7) takes

the form

1 12 22 2 0

2 2( )f ff f g s f f

(7)

10

2( )g fg gf s g g

(8)

1 1 2 2Pr (2 ) 0

2 2 4[ { ( ) ( ) }]f f s Ec f g

(9)

here is the mixed convection parameter, s is the

unsteady parameter and the flow is accelerated for 0s

and retarded for 0,s Pr is the Prandtl number, Ec is the

Eckert number and is the slip parameter.

The boundary conditions in non-dimensional form for the

concerned flow problem are given as

' '' '(0) 0, (0) (0), (0) 1 (0), (0) 1

'( ) 0, ( ) 0, ( ) 0

f f f g g

f g

(10)

The surface stress tensors in primary and secondary

directions for the present analysis are

10 2Re 2

1 20[ sin 1 ]

uxz zC xfx

z zx st

10 2Re 2

1 20[ sin 1 ]

yz vzC xfy

z zx st

or in dimensionless form

12Re

0

120.5 Re

0

C ffx x

C gfy x

(11)

The heat transfer coefficient in dimensionless form is

stated as

1'2Re (0)Nu x

(12)

where

12sin 1Re

x st

x

is the local Reynolds

number.

OPTIMAL HOMOTOPY ANALYSIS

PROCEDURE

The solutions of the coupled nonlinear parabolic

ordinary differential equations given in Eqs. (7)- (10) are

carried out analytically by optimal homotopy analysis

method (OHAM) which was established by Liao [19]. The

following initial guesses and linear operators for velocity

components and temperature fields are used ,0f 0g and

0 respectively is

00f (13)

1

exp01

g

(14)

exp0 (15)

3

£3

d f df

df d (16)

2

£2

d gg

g d (17)

2

£2

d

d

(18)

The standard procedure of homotopy analysis method can

be follow as [19-29].

Optimal convergence-control parameters

Generally homotopy analysis solutions involve

the non-zero auxiliary parameters ,0f

c ,0g

c and

0c which are helpful in finding the convergence-region

and rate of the homotopy series solutions. In order to attain

the optimal values of non-zero auxiliary parameters

,0f

c ,0g

c and 0c it is used here the so-called average

residual error specified by [19].

2

1ˆ ˆˆ( ( ), ( ),

0 0 0 01

j m m mf

N f g dym fi n n nj y i y

(19)

21 ˆ ˆ( ( ), ( ),

1 0 0 0

j m mgN f g dym g

j i n n y i y

(20)

2

1ˆ ˆN ( ( ), )

0 0 01

j m m

f dymi n nj y i y

(21)

t f g

m m m m

22

here tm is the total squared residual error, 0.5y and

20.j 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 1: Local optimal convergence control parameters

and total averaged squared residual errors using BVPh2 0

M f

m gm

m 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 7.44×10-8

7.31×10-9

2.71×10-8

127.0

Table 2: Individual averaged squared residual errors

using optimal values at m=10 from table 1.

RESULTS AND DISCUSSION

This portion of study involves the graphical

and numerical results of various significant

parameters on velocities, temperature, surface stress

coefficients and heat transfer coefficient. Such

variations have been observed in Figures (2) to (7).

Fig. 2 is sketched to display the behavior of primary

velocity '( )f for mixed convection parameter

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 and

boundary layer thickness increases with increasing

values of , further it is noticed that the primary

velocity '( )f has greater magnitude for

0.5 (i.e. in the presence of slip parameter). The

influence of slip parameter on primary velocity

'( )f is shown in Fig.3. It is devoted from the

figure that '( )f enhances its magnitude with an

increase in . The influence of mixed convection

parameter and slip parameter is to reduce the

secondary velocity ( )g respectively (See Fig. 4 and

5). Moreover it is seen that the secondary

velocity ( )g has least magnitude for 0.5 (i.e. in

the presence of slip parameter). Fig. 6 is devoted to

show the influence of Eckert number Ec on

temperature ( ) . The figure shows that the

temperature ( ) is an increasing function of .Ec

The influence of the Prandtl number Pr on the

temperature is drafted in figures7. It is clear from the

respective figure that ( ) 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. Fig. 8 is sketched to

observe the behavior of Nusselt number on mixed

convection parameter . It is depicted that '(0)

decreases with increasing . 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 0s Ec 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 12ReC xfx

increases for slip parameter , but the variation is

just opposite for azimuthal surface stress tensor

0.512ReC xfy . Mixed convection parameter and the

unsteady parameter 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 12ReNu x

decreases.

Similar behavior is observed for Eckert number.

Moreover, it is seen that the Prandtl number Pr

enhances the variation of heat transfer rate12ReNu x

.

Fig. 2 Variation of '( )f for .

M 0fc 0

gc 0c tm

CPU

time[s]

2 -1.16 -0.35 -1.28 5.49×10-4 3.37

4 -1.21 -0.29 -1.30 1.37×10-4

25.70

6 -0.99 -0.28 -1.33 2.27×10-5

150.28

8 -0.86 -0.28 -1.50 5.62×10-6

881.06

10 -1.03 -0.30 -0.85 1.83×10-6

2527.84

Fig. 3 Variation of '( )f for .

Fig. 4 Variation of ( )g for .

Fig. 5 Variation of ( )g for .

Fig. 6 Variation of ( ) for Ec.

Fig. 7 Variation of ( ) for Pr.

Fig. 8 Variation of

'(0) for .

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

Present Analytical results Numerical results[3]

Pr )0(''f )0('g )0(' )0(''f )0('g )0('

0.7 0 1.0255 0.6154 0.4299 1.0255*

1.0255

0.6158*

0.6158

0.4299*

0.4299

1 2.2010 0.8493 0.6121 2.2014*

2.2012

0.8497*

0.8496

0.6121*

0.6120

10 8.5042 1.3992 1.0098 8.5045

*

8.5041

1.3992*

1.3995

1.0099*

1.0097

10 0 1.0255 0.6158 1.4111 1.0255

*

1.0256

0.6158*

0.6158

1.4111*

1.4110

1 1.5630 6835.0 1.5661 1.5638

*

1.5636

0.6838*

0.6837

1.5663*

1.5662

10 5.0820 0.9845 2.3581 5.0825

*

5.0821

0.9841*

0.9840

2.3583*

2.3580

*values taken from Himasekhar et. al. [2]

Table 4: Values for surface shear stresses when Pr =

1.0 and Ec = 0.5

s 12ReC xfx

120.5 ReC xfy

0.0 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

0.0 5.0985 0.9602

0.5 5.0993 1.0949

Table 5: Values for reduced Nusselt number for

interesting physical parameters.

Ec Pr s 12ReNu x

0.0 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 0.2810

0.5 0.2798

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 and slip parameter respectively.

2. Surface stress tensor in x-direction 12ReC xfx

enhances its magnitude for mixed convection

parameter and unsteady parameter ,s but

possess opposite variation for slip

parameter .

3. Temperature field is an increasing function of

Eckert number Ec.

4. The heat transfer rate 12ReNu x

has opposite

variation Prandtl number Pr and Eckert

number Ec.

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