# Approximate solution using momentum integral in estimating the shear stress over a porous rotating disk.

Introduction

The estimation of the wall shear stress on a porous rotating disk is of great importance particularly in rotating machines, lubrication, droplet generators, filtering systems, computer storage devices as well as operation units for heat and mass transfer. The von Karman  was the first to derive a theoretical expression for the shear stress on a smooth disk rotating in an infinite fluid when the flow is laminar. Von Karman describing similarity transformation that enables the Navier-Stokes equation to be reduced to a system of coupled ordinary equation differential equation. A disk in housing was investigated theoretically by [2, 3] for laminar and turbulent case. The fluid in the boundary layer on the rotating disk is centrifuged outward, and this is compensated by a flow inward in the boundary layer on the housing at rest. There is no appreciable radial component in the intermediate layer of fluid which rotates with about half the angular velocity of the disk. Early characterization of the hydro dynamics of a rotating disk in an enclosed vessel was reported by [4, 5, and 6].

This paper indicates a procedure to estimate the wall shear stress on an enclosed porous rotating disk in laminar regime.

A Fundamental Equations

Consider an enclosed porous rotating disk which has a constant angular velocity [omega], with uniform suction velocity normal to the disk [v.sub.0] (see Figure 1). The Prandtles boundary layer equations in cylindrical polar coordinates near an infinite rotating disk are given by.

[FIGURE 1 OMITTED]

* r- component of momentum equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

* [theta] - component of momentum equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

* z- component of momentum equation:

0= -1 / p [partial derivative]p / [partial derivative]z (3)

* Equation of continuity:

[partial derivative][v.sub.r]/[partial derivative]r + [v.sub.r]/r + [partial derivative][v.sub.z]/[partial derivative]z = 0 (4)

The no-slip condition for non porous rotating disk at the wall can be expressed as:

z = 0 [v.sub.r] = [v.sub.[theta]] = [v.sub.z] = 0 (5)

z = [delta] [v.sub.r] = [v.sub.z] = 0, [v.sub.[theta]] = k x [omega]x r (6)

Rotating non Porous Disk

Assumptions should be considered for the following estimation. These are (i) the boundary layer thickness is the same for the radial flow and tangential, (ii) secondary flow arises just inside the boundary layer, in the core, between the boundary layers, (iii) the medium rotate like a solid body with constant tangential velocity, (iv) the radial pressure gradient in the core is dominate between the gap, where the axial pressure gradient is 0 and (v) the radial and tangential velocity profile in the boundary layer which was assumed by  are used.

[v.sub.r] = [v.sub.0r] (1 - [(2 z / [delta]-1).sup.2]) (7)

[v.sub.[theta]] = [v.sub.0[theta]] [(1 - (1-z / [delta]).sup.2]) (8)

Where [v.sub.0r] is the maximum radial velocity and [v.sub.0[theta]] is the velocity in core flow, [v.sub.0[theta]] = k x [omega] x r

By integrate the radial momentum over the boundary layer thickness we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Integrating the third term on the left - hand side by parts,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

This becomes after using the continuity equation (equation 4) as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Substitute this equation in 9 we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

If there is no suction ([v.sub.z0] = 0) and the gradient pressure inside the boundary layer is the same outside we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

By integrate the tangential momentum equation over the boundary layer thickness we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

Integrating the third term on the left--hand side by parts yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Substitution of the above equation into equation 14 and rearrange yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

Using the radial and tangential velocity profile in the boundary layer which was assumed by Shultz-Grunow to calculate:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

The radial and tangential shear stress calculated by:

[[tau].sub.z,r] = [eta] x [partial derivative][v.sub.r]/[partial derivative]z[|.sub.0] = 4[eta] x [v.sub.0r]/[delta]; [[tau].sub.z,[theta]] = [eta] x [partial derivative][v.sub.[theta]]/[partial derivative]z[|.sub.0] = 2[eta] [v.sub.0[theta]]/[delta]; (18)

By assuming [v.sub.0r] = [c.sub.4] x w x r and substituting equations 17 and 18 into equations 13 and 14 we obtain:

[c.sub.4] = k x [([1-[c.sub.3]]/[8.[c.sub.2]-3.[c.sub.1]]).sup.1/2] (19)

[c.sub.4] = 0.468 x k (20)

[delta] = 1.513 x [(V/[k x [omega]]).sup.1/2] (21)

The radial and tangential shear stresses for non porous rotating disk are obtained like:

[[tau].sub.z,r] = 1.237 x [rho] x [(v).sup.1/2] x [(k x [omega]).sup.3/2] x r (22)

[[tau].sub.z,[theta]] = 1.32 x [rho] x [(v).sup.1/2] x [(k x [omega]).sup.3/2] x r (23)

The resultant local shear stress for non porous rotating disk is:

[[tau].sub.z,R] = 1.81 x [rho] x [(v).sup.1/2] x (k x [[omega]).sup.3/2] x r (24)

Rotating Porous Disk

Schlichting found an approximate solution for a flat plate with uniform suction which can be given as:

[delta] = -v/[v.sub.z0] (25)

Now we implement the Schlichting result for a flat plate with uniform suction into the tangential momentum equation (equation14) subject to the following boundary conditions:

z = 0 [v.sub.r] = [v.sub.[theta]] = 0, [v.sub.z] = [v.sub.z0] (26)

z = [delta] [v.sub.r] = [v.sub.z] = 0, [v.sub.[theta]] = k x [omega] x r (27)

The boundary layer thickness over the porous rotating disk [delta] and [c.sub.4] will be

[delta] = 1.378 x [(v/k x [omega]).sup.1/2]}

[c.sub.4] = k x [([1-[c.sub.3]]/[16.[c.sub.2]-3.[c.sub.1]]).sup.1/2]}

[c.sub.4] = 0.282 x k} (28)

The radial and tangential shear stress over the porous rotating disk becoming:

[[tau].sub.z,r] = 0.819 x [rho] x [(v).sup.1/2] x [(k x [omega]).sup.3/2] x r (29)

[[tau].sub.z,[theta]] = 1.451 x [rho] x [(v).sup.1/2] x [(k x [omega]).sup.3/2] x r (30)

The resultant local shear stress for porous rotating disk becomes:

[[tau].sub.z,R] = 1.666 x [rho] x [(v).sup.1/2] x [(k x [omega]).sup.3/2] x r (31)

Results and Discussion

It is interesting to note that Equation (31) has the same form as Equation (24), except for the numerical coefficient which is lower by factor 1.08. Figure (2) show the tangential shear stress for rotating porous and non porous disk under the same consideration. The shear stress increases linearly and enhanced by applying uniform suction. This is due to form thinner boundary layer. Figure (3) depict the radial shear stress for rotating porous and non porous disk. It can be seen that applying uniform suction decrease the shear stress over the disk since the amount of the fluid in the boundary layer centrifuged outward decreased. Figure (4) show the resultant shear stress for rotating porous and non porous disk under the same condition. It is clear that the resultant shear stress for non porous rotating disk is larger than for porous disk since the radial shear stress component is larger.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Conclusion

The wall shear stresses for porous rotating disk are approximated using the momentum integral approach following the same technique of Schiele for non porous rotating disk. The results showed clear evidence of tangential wall shear stress increase when suction is applied due to thinner boundary layer. The radial wall shear stress decrease when suction is applied because the amount of the fluid in the boundary layer centrifuged outward decreased.

Nomenclature

k [-] velocity coefficient

r,[theta],z [-] cylindrical coordinates

[v.sub.0] [m/s] uniform suction velocity

[v.sub.0,[theta]] [m/s] velocity in core flow

[v.sub.0,r] [m/s] maximum radial velocity

[v.sub.r] [m/s] fluid velocity in r-direction

[v.sub.[theta]] [m/s] fluid velocity in [theta]-direction

[v.sub.z] [m/s] fluid velocity in z-direction

Greek Symbols

[[tau].sub.z,[theta]] [Pa] tangential wall shear stress

[[tau].sub.z,r] [Pa] radial wall shear stress

[T.sub.z,R] [Pa] resultant wall shear stress

[mu] [Pa.s] fluid dynamic viscosity

[rho] [kg/[m.sup.3]] fluid density

V [[m.sup.2]/s] fluid kinematics viscosity

[omega] [rpm] disk angular velocity

[delta] [m] boundary layer thickness

References

 Von Karman, T. 1921. Uber laminare und turbulente reibung. Zeitschrift fur Angewandte Mathematik und Mechanik., 1 (4): 233-252.

 Bodewadt, U.T., 1940. Die drehstromung uber festen grunde. ZAMM. J. Applied Math. Mech., 20: 241-253

 Schiele, B. Untersuchung zur Filtration feindisperser Suspensionen und zur Stromung im dynamischen Druckfilter, PhD thesis, Universitat Stuttgart, 1977.

 Murkes, J. and C.G. Carlsson, 1988. Cross Flow Filtration: Theory and Practice. John Willy and Sons Ltd., New York.

 Jonsson, A.S., 1993. Influence of shear rate on the flux during ultra filtration of colloidal substances. J. Membr. Sci., 79: 93-99.

 Schlichting, H. Boundary Layer Theory, McGraw-Hill, New York, 1968.

 Brady, J.F. and L. Durlofsky, 1987. On rotating disk flow. J. Fluid Mech., 175: 363-394.

Yazan Taamneh (1) * and Siegfried Ripperger (2)

(1) Faculty of Engineering, Department of Mechanical Engineering, Tafila Technical University P.O. Box 92, 66110 Tafila, Jordan

(2) Technische Universitat Kaiserslautern, Fachbereich Maschinenbau und Verfahrenstechnik Lehrstuhl fur Mechanische Verfahrenstechnik, D- 67663 Kaiserslautern, Germany

(1) * Corresponding Author E-mail: ytaamneh@yahoo.com
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