Mathematical Modelling of the Laminar Entrance Flow into a Circular Tube Using a Fourier Series Approximation

Abstract

In this work, the permanent laminar flow in the entrance region of a cylindrical tube is analyzed using a new Fourier series-based approximation technique of similarity for velocity profiles. The similarity method used in this work is well known in the literature on fluid mechanics and can be applied in other engineering education fields. The potential of the original modified Fourier cosine series is compared with the mathematical modelling of the entrance flow. Different physical entrance flow parameters of the interest are obtained through the integral formulation of differential equations for conservation laws. The results of this mathematical approach were found to be compatible with those reported in the literature.

Keywords

entrance region laminar flow circular tube similarity method Fourier series

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References

Liang

, ‘Determination of the entry region length of viscoelastic fluid flow in a channel’, Chemical Engineering Science, 53(17) (1998), 3185–3187.

Shipkowitz

Rodgers

V. G. J.

Frazin

L. J.

and Chandran

K. B.

, ‘Numerical study on the effect of steady axial flow development in the human aorta on local shear stresses in abdominal aortic branches’, Journal of Biomech., 31(11) (1998), 995–1007.

Beretta

Ferrari

Galbiati

and Andreini

P. A.

, ‘Horizontal oil—water flow in small diameter tubes. Pressure drop’, International Communications in Heat and Mass Transfer, 24(2) (1997), 231–239.

Deen

W. M.

, Analysis of Transport Phenomena (Oxford University Press, New York, 1998).

Von Kármán

, ‘Über laminare and turbulente Reibung’, ZAMM, 1 (1921), 244–247.

Pohlhausen

, ‘Der Wärmeaustausch zwischen festen Körpern und Flüssigkeiten mit kleiner Reibung und kleiner Wärmeleitung’, ZAMM, 1 (1921), 121.

Schiller

, ‘Die Entwicklung der laminaren geschwindigkeitsverteilung und ihre bedeutung für zahigkeitsmessungen’, ZAMM, 2 (1922), 96–106.

Shah

R. K.

and London

A. L.

, Laminar Flow Forced Convection in Ducts (Academic Press, New York, 1978).

Sparrow

E. M.

Lin

S. J.

and Lundgren

T. S.

, ‘Flow development in the hydrodynamics entrance region of tubes and ducts’, Journal of Physics Fluids, 7 (1964), 338–347.

10.

Campbell

W. D.

and Slaterry

J. C.

, ‘Flow in the entrance of a tube’, Journal of Basic Engineering, 85 (1963), 41–46.

11.

Fargie

and Martin

B. W.

, ‘Developing laminar flow in a pipe of circular cross-section’, Proceedings of the Royal Society of London, A321 (1971), 461–476.

12.

Al-Nassry

S. A.

and Unny

, ‘Developing laminar flow in the inlet length of a smooth pipe’, Applied Science Research, A36 (1980), 313–332.

13.

Tao

L. N.

and Gupta

S. D.

, ‘Variational analysis of the flow development in the entrance region of circular tubes and parallel-plate channels’, Proceedings of the 13^th Midwestern Mechanics Conference. Developments in Mechanics, 7 (1964), 247–263.

14.

Comini

Giudici

and Strada

, ‘Finite element analysis of laminar flow in the entrance region of ducts’, International Journal for Numerical Methods in Engineering, 15 (1980), 507–517.

15.

McComas

S. T.

and Eckert

E. R. G.

, ‘Coeffient K for the pressure drop caused by laminar flow through the inlet section of a tube’, Transactions ASME, Journal Applied Mechanics, 32 (1965), 765–770.