Fluid Flow on Rotating Porous Disk with Heat Transfer

7.1 Introduction

Modern mathematical tools, supported by computing machines, allow interpretation, analysing and calculating physical theoretical accounts with improbably high preciseness. Boundary bed theory is one of the most of import 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 old ages, scientists were to spread out the bing theory and happen an application for many other non-incompressible fluids. Prandtl was the first who introduced the theory in 1904. He was able to show the solution pretermiting viscousness but, as ulterior research showed, his theory did non to the full explained practical experiments.

J.P. Hartnett and E.R.G. Eckert were among following who contributed into the boundary bed theory. The job of heat transportation from a revolving disc maintained at a changeless temperature was foremost considered by Milsaps and Pohlhausen ( 1952 ) for a assortment of prandtl Numberss in the steady province. Sparrow and Gregg ( 1960 ) studied the steady province heat transportation from a revolving disc maintained at a changeless temperature to fluids at any Prandtl figure.

The rotating disc flow is one of the classical and of import jobs in unstable mechanics. The revolving disc flows have practical applications in many countries, such as chopper rotor aeromechanicss, chemical technology, dynamic theoretical accounts, magnetic energy propulsion systems, and lubrication, oceanology and computing machine storage devices.

In this chapter, the work of Attia ( 2009 ) is studied and extended. Here the steady laminar flow of a syrupy incompressible fluid due to the unvarying rotary motion of a disc of infinite extent in a porous medium is studied with heat transportation. Here the job is solved utilizing Crank Nicholson method. The consequence of the porousness of the medium on the steady flow and heat transportation is presented and the consequences are reported for decision.

7.2 Physical description of the job

A rotating disc is immersed in a big sum of fluid. Gesture within the fluid is generated by revolving disc, which induces heat transportation phenomenon. Disk of radius ‘R’ rotates around an axis perpendicular to the surface with unvarying angular speed

‘a„¦’ . Due to the syrupy forces n a bed of fluid is carried by the disc. Heat transportation takes topographic point at the surface of the disc where the bed near the disc is being directed outward by centrifugal forces. Fluid gesture is charactersied by speed constituents u – radial, v – circumferential, and w – axial.

7.3 Mathematical Formulation

See the system of cylindrical coordinates:

The mathematical statement of mass preservation is expressed by four Navier – Stokes equations.

( 7.1 )

U ( 7.2 )

V ( 7.3 )

tungsten ( 7.4 )

Where U, V, w are velocity constituents in the waies of increasing R,severally. P is force per unit area,is the coefficient of viscousness,is the denseness of the fluid, and K is the Darcy permeableness.

See the disc in the planeis occupied by a syrupy incompressible fluid. The gesture is due to the rotary motion of an insulated disc of infinite extent about an axis perpendicular to its plane with changeless angular speed i?- through a porous medium where the Darcy theoretical account is assumed. Otherwise the fluid is at remainder under force per unit area. The equations of steady gestures are given by

From physical and mathematical description, the boundary status sing no – faux pas status at the wall of the disc can be determined

( 7.5 )

At first we need to specify the thickness S of the fluid bed at the surface of revolving disc, which is carried due to clash. That bed of fluid is whirling with equal angular speed. Thickness of the bed depends on angular speed and lessenings when disc accelerate. Since in our experiment angular speed is changeless, thickness of the fluid bed resting on the surface of the disc will alsoremain unchanged.

The centrifugal force that acts on a unstable atom in the rotating bed at a distance R from the axis can be presented as

=( 7.6 )

Therefore for a volume of country centrifugal force becomes:

=( 7.7 )

Due to nature of physic the same component of fluid interacts with shearing emphasisthat points in way in which the fluid is stealing. Angle Q is created between shearing emphasisand circumferential speed v. It is understood that the radial constituent of the shearing emphasis must be equal with centrifugal force F.

( 7.8 )

Gradient of circumferential speed at the wall has to be relative to circumferential constituent of shear emphasis.

That extra status( 7.9 )

From ( 7.8 ) and ( 7.9 )

( 7.10 )

Therefore,

( 7.11 )

After specifying thickness of the fluid bed, which rotates with the disc at no-slip status, we need to analyse system of Navier-Stokes equations. Solution for that system can be obtained much easier by transforming to cut down partial differential equations into system of ordinary differential equations.

Successful effort of work outing similar speed job for an impermeable disc revolving in a single-component fluid was achieved in 1921 by T. von Karman. In order to utilize similarity transform to cut down the partial differential equations to ordinary differential equations new variables need to be introduced:

Independent variable is given by

( 7.12 )

Dependent variables are given by

Where i?? is a non – dimensional distance measured along the axis of rotary motion, F, G, H and P are non-dimensional maps of i?? , andis the kinematic viscousness of the fluid,with these definitions, equations ( 7.1 ) – ( 7.4 ) takes the signifier

( 7.13 )( 7.14 )

( 7.15 )

( 7.16 )

Where M =is the porousness parametric quantity. The boundary conditions for the speed job are given by

,( 7.17 )

i??,( 7.18 )

The no-slip status of syrupy flow applied at the surface of the disc is indicated by the equation ( 7.17 ) . All the fluid speeds must be vanished aside the induced axial constituent 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 work out for the three constituents of the flow speed. Equations ( 7.16 ) can be used to work out for the force per unit area distribution.

Due to the difference in temperature between the wall and the ambient fluid, heat transportation takes topographic point. The energy equation without the dissipation footings takes the signifier

( 7.19 )

Where T is the temperature of the fluidis the specific heat at changeless force per unit area of the fluid, and K is the thermic conduction of the fluid.

The boundary conditions for the energy job are that, by continuity considerations, the temperature peersat the surface of the disc. At big distances from the disc, T tends towhereis the temperature of the ambient fluid.

In footings of the non-dimensional variableand utilizing von Karman transmutations, equation ( 10 ) takes the signifier ;

( 7.20 )

Whereis the Prandtl figure,=. The boundary conditions in footings ofare 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 constituents of the flow speed and temperature distribution, utilizing the Crank-Nicolson method.

1. Based on the boundary status and differential equation, the initial status will match to a adulthood or termination day of the month value status in our applications and T will denote clip left to adulthood.

2. The coefficients K, A, B, C may themselves be maps of clip. Any parametric quantities are presumed given, so that one can cipher a numerical value for these maps for any given clip.

3. See a solution is the numerical values that b takes on a grid of clip values placed over some sphere of involvement.

4. The sphere we will work on is Time Domain with Variable runing from xmin to xmax and t runing with aid of clip from 0 to T.

5. And Finally is what makes this process a finite difference method — is to come close the partial derived functions of map at each grid point by difference looks in the as yet unknown as Crank Nicolson Method.

6. The differences in the Current or following province, way have been centered around the point in to give ‘second order’ truth to the estimate.

7. The consequence of this is called an explicit finite difference solution for U. It is 2nd order accurate in the ten way, though merely first order accurate in the t way, and easy to implement.

1*e4

1.0000	1.0000	1.0000	1.0000
0.9000	0.8800	0.8800	0.8000
0.8000	0.7800	0.7800	0.7000
0.7500	0.7200	0.7200	0.6200
0.7000	0.6800	0.6800	0.5600
0.6600	0.6400	0.6400	0.5000
0.6200	0.6000	0.6000	0.4500
0.5700	0.5500	0.5500	0.4000
0.5300	0.5100	0.5100	0.3800
0.4800	0.4600	0.4600	0.3200

1*e4

Ki = 1.0000 0.9000 0.8000 0.7500 0.7000 0.6600 0.6200 0.5700 0.5300 0.4800

K2 = 1.0000 0.8800 0.7800 0.7200 0.6800 0.6400 0.6000 0.5500 0.5100 0.4600

K3 = 1.0000 0.8800 0.7800 0.7200 0.6800 0.6400 0.6000 0.5500 0.5100 0.4600

K4 = 1.0000 0.8000 0.7000 0.6200 0.5600 0.5000 0.4500 0.4000 0.3800 0.3200

1*e4

C1 = 1.0000 0.9000 0.8000 0.7500 0.7000 0.6600 0.6200 0.5700 0.5300 0.4800

C2 = 1.0000 0.8800 0.7800 0.7200 0.6800 0.6400 0.6000 0.5500 0.5100 0.4600

C3 = 1.0000 0.8600 0.7600 0.7000 0.6600 0.6200 0.5800 0.5300 0.4900 0.4400

C4 = 1.0000 0.8400 0.7400 0.6800 0.6400 0.6000 0.5600 0.5100 0.4700 0.4300

1*e4

B1 = 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9800 0.9700 0.9500 0.9500

B2 = 1.0000 0.9800 0.9600 0.9000 0.8500 0.8500 0.8500 0.8500 0.8500 0.8500

B3 = 1.0000 0.9500 0.8300 0.7500 0.6900 0.6000 0.6000 0.6000 0.6000 0.6000

B4 = 1.0000 0.9000 0.8000 0.6500 0.6200 0.5100 0.4500 0.4200 0.3500 0.3000

1*e4

A1 = 0 0.2800 0.3800 0.4600 0.5300 0.5500 0.5100 0.4600 0.4100 0.3500

A2 = 0 0.2000 0.3000 0.3400 0.3500 0.3200 0.2800 0.2600 0.2400 0.2100

A3 = 0 0.2000 0.2400 0.2500 0.2300 0.2100 0.2000 0.1900 0.1800 0.1700

A4 = 0 0.0700 0.1200 0.1500 0.1300 0.1200 0.1100 0.1000 0.0950 0.0900

The ensuing system of difference equations has to be solved in the space sphereA finite sphere in the-direction can be used alternatively withchosen big plenty to guarantee that the solutions are non affected by enforcing the asymptotic conditions at a finite difference. The independency of the consequences from the length of the finite sphere and the grid denseness was ensured and successfully checked by assorted test and mistake numerical experimentations. Calculations are carried out for

7.3 Consequences and Discussion

In this subdivision the numerical and graphical consequences obtained through MATLAB utilizing Crank – Nicolson method is presented. The influence of some regulating parametric quantities on the speed and temperature Fieldss is described in this subdivision. In peculiar, attending has been focused on the fluctuations of the porousness parametric quantity M and for Pr = 0.07. The current consequences are good in understanding with the consequences given in the mention. Figure 7.1 – 7.4 represent the fluctuation of the profiles of the speed constituents G, F, and H and the temperature, severally, for assorted values of the porousness parametric quantity M and for Pr = 0.07. Figures 7-3 indicate the keeping consequence of the porousness of the medium on the flow speed in the three waies. Increasing the porousness parametric quantity M decreases G, F, and H and the thickness of the boundary bed. Figure 7.4 represents the influence of the porousness parametric quantity M in increasing the temperature i?± as a consequence of the consequence of the porousness in forestalling the fluid at near-ambient temperature from making the surface of the disc. Consequently, increasing M increases the temperature every bit good as the thermic boundary bed thickness. The absence of fluid at near-ambient temperature stopping point to the surface increases the heat transportation.

Consequence of the porousness parametric quantity M on the speed profile of

Figure 7.1

Consequence of the porousness parametric quantity M on the speed profile of F

Figure 7.2

Consequence of the porousness parametric quantity M on the speed profile of H

Figure 7.3

Consequence of the porousness parametric quantity M on the speed profile of

Figure 7.4

7.4 Decision

In this survey the steady flow induced by a revolving disc with heat transportation in a porous medium has been studied.

The consequences indicate the keeping consequence of the porousness on the flow speeds and the thickness of the boundary bed.

Increasing the porousness parametric quantity increase the temperature and thickness of the thermic boundary bed.