STUDY ON FLOW OVER A ROTATING POROUS DISK WITH HEAT

TRANSFER

CHAPTER - VII

CHAPTER VII

STUDY ON FLUID FLOW OVER A ROTATING POROUS DISK

WITH HEAT TRANSFER

7.1 INTRODUCTION

Modern mathematical tools, supported by computers, allow interpreting,

analyzing and computing physical models with incredibly high precision. Boundary

layer theory is one of the most important theories of fluid mechanics. At first it was

developed for theoretically ideal incompressible fluids in the conditions of laminar

flow. Due to extended research through many years, scientists were to expand the

existing theory and find an application for many other non-incompressible fluids.

Prandtl was the first who introduced the theory in 1904. He was able to present the

solution neglecting viscosity but, as later research showed, his theory did not fully

explained practical experiments.

Hartnett and Eckert (1957) were among next who contributed into the

boundary layer theory. The problem of heat transfer from a rotating disk maintained

at a constant temperature was first considered by Milsaps and Pohlhausen (1952) for

a variety of prandtl numbers in the steady state. Sparrow and Gregg (1960) studied

the steady state heat transfer from a rotating disk maintained at a constant

temperature to fluids at any Prandtl number.

The rotating disk flow is one of the classical and important problems in fluid

mechanics. The rotating disk flows have practical applications in many areas, such as

helicopter rotor aerodynamics, chemical engineering, dynamic models, magnetic

energy propulsion systems, and lubrication, oceanography and computer storage

devices.

In this chapter, the work of Attia (2009) is studied and extended Here the

steady laminar flow of a viscous incompressible fluid due to the uniform rotation of a

disk of infinite extent in a porous medium is studied with heat transfer. Here the

problem is solved using Crank Nicholson method. The effect of the porosity of the

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medium on the steady flow and heat transfer is presented and the results are reported

for conclusion.

7.2 PHYSICAL DESCRIPTION OF THE PROBLEM

A rotating disk is immersed in a large amount of fluid. Motion within the

fluid is generated by rotating disk, which induces heat transfer phenomenon. Disk of

radius ‘R’ rotates around an axis perpendicular to the surface with uniform angular

velocity

‘Ω’. Due to the viscous forces n a layer of fluid is carried by the disk. Heat

transfer takes place at the surface of the disk where the layer near the disk is being

directed outward by centrifugal forces. Fluid motion is charactersied by velocity

components u – radial, v – circumferential, and w – axial.

7.3 MATHEMATICAL FORMULATION

Consider the system of cylindrical co-ordinates:

The mathematical statement of mass conservation is expressed by four

Navier – Stokes equations.

(7.1)

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(

)

(

)

u (7.2)

(

) (

)

v (7.3)

(

)

(

)

w (7.4)

Where u, v, w are velocity components in the directions of increasing r,

respectively. P is pressure, is the coefficient of viscosity, is the density of the

fluid, and K is the Darcy permeability.

Consider the disk in the plane is occupied by a viscous incompressible fluid.

The motion is due to the rotation of an insulated disk of infinite extent about an axis

perpendicular to its plane with constant angular velocity through a porous medium

where the Darcy model is assumed. Otherwise the fluid is at rest under pressure .

The equations of steady motions are given by

From physical and mathematical description, the boundary condition considering no

– slip condition at the wall of the disk can be determined

(7.5)

At first we need to define the thickness S of the fluid layer at the surface of

rotating disk, which is carried due to friction. That layer of fluid is spinning with

equal angular velocity . Thickness of the layer depends on angular velocity and

decreases when disk accelerate. Since in our experiment angular velocity is constant,

thickness of the fluid layer resting on the surface of the disk will also remain

unchanged.

The centrifugal force that acts on a fluid particle in the rotating layer at a

distance r from the axis can be presented as

= (7.6)

Therefore for a volume of area centrifugal force becomes:

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= (7.7)

Due to nature of physic the same element of fluid interacts with shearing

stress that points in direction in which the fluid is slipping. Angle Q is created

between shearing stress and circumferential velocity v. It is understood that the

radial component of the shearing stress must be equal with centrifugal force F.

(7.8)

Gradient of circumferential velocity at the wall has to be proportional to

circumferential component of shear stress.

That additional condition

(7.9)

From (7.8) and (7.9)

(7.10)

Therefore,

√

(7.11)

After defining thickness of the fluid layer, which rotates with the disk at no-slip

condition, we need to analyze system of Navier-Stokes equations. Solution for that

system can be obtained much easier by transforming to reduce partial differential

equations into system of ordinary differential equations.

Successful attempt of solving similar velocity problem for an impermeable

disk rotating in a single-component fluid was achieved in 1921 by T. von Karman. In

order to use similarity transform to reduce the partial differential equations to

ordinary differential equations new variables need to be introduced:

Independent variable is given by

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√

(7.12)

Dependent variables are given by

( )

( )

( )

( )

Where is a non – dimensional distance measured along the axis of rotation, F, G, H

and P are non-dimensional functions of , and is the kinematic viscosity of the

fluid,

with these definitions, equations (7.1) – (7.4) takes the form

(7.13)

(7.14)

(7.15)

(7.16)

Where M = is the porosity parameter. The boundary conditions for the

velocity problem are given by

, (7.17)

, (7.18)

The no-slip condition of viscous flow applied at the surface of the disk is indicated

by the equation (7.17). All the fluid velocities must be vanished aside the induced

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axial component as indicated in equation (7.17). The above system of equations

(7.13) - (7.15) with the prescribed boundary conditions given by equations (7.17) and

(7.18) are sufficient to solve for the three components of the flow velocity.

Equations (7.16) can be used to solve for the pressure distribution.

Due to the difference in temperature between the wall and the ambient fluid, heat

transfer takes place. The energy equation without the dissipation terms takes the form

(

) (

) (7.19)

Where T is the temperature of the fluid is the specific heat at constant pressure of

the fluid, and k is the thermal conductivity of the fluid.

The boundary conditions for the energy problem are that, by continuity

considerations, the temperature equals at the surface of the disk. At large

distances from the disk, T tends to where is the temperature of the ambient

fluid.

In terms of the non-dimensional variable ( )

( ) and using von Karman

transformations, equation (10) takes the form;

(7.20)

Where is the Prandtl number, = . The boundary conditions in terms

of are expressed as

( ) ( ) (7.21)

The system of non-linear ordinary differential equations (7.13) – (7.15) and (7.20) is

solved under the conditions given by equations (7.17), (7.18) and (7.21) for the three

components of the flow velocity and temperature distribution, using the Crank-

Nicolson method.

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The resulting system of difference equations has to be solved in the infinite domain

A finite domain in the -direction can be used instead with chosen

large enough to ensure that the solutions are not affected by imposing the asymptotic

conditions at a finite difference. The independence of the results from the length of

the finite domain and the grid density was ensured and successfully checked by

various trial and error numerical experimentations. Computations are carried out for

7.4 RESULTS AND DISCUSSION

In this section the numerical and graphical results obtained through MATLAB using

Crank – Nicolson method is presented.

The influence of some governing parameters on the velocity and temperature

fields is described in this section. In particular, attention has been focused on

the variations of the porosity parameter M and for Pr = 0.07.

The current results are well in agreement with the results given in the

reference. Figure 7.1 - 7.4 represent the variation of the profiles of the

velocity components G, F, and H and the temperature , respectively, for

various values of the porosity parameter M and for Pr = 0.07.

Figures 7-3 indicate the restraining effect of the porosity of the medium on

the flow velocity in the three directions. Increasing the porosity parameter M

decreases G, F, and H and the thickness of the boundary layer.

Figure 7.4 represents the influence of the porosity parameter M in increasing

the temperature as a result of the effect of the porosity in preventing the

fluid at near-ambient temperature from reaching the surface of the disk.

Consequently, increasing M increases the temperature as well as the thermal

boundary layer thickness. The absence of fluid at near-ambient temperature

close to the surface increases the heat transfer.

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Effect of the porosity parameter M on the velocity profile of

Figure 7.1

G

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Effect of the porosity parameter M on the velocity profile of F

Figure 7.2

F

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Effect of the porosity parameter M on the velocity profile of H

Figure 7.3

H

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Effect of the porosity parameter M on the velocity profile of

Figure 7.4

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7.5 CONCLUSION

In this study the steady flow induced by a rotating disk with heat transfer in a

porous medium has been studied.

The results indicate the restraining effect of the porosity on the flow

velocities and the thickness of the boundary layer.

Increase of the porosity parameter increases the temperature and thickness of

the thermal boundary layer

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