Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Some exact solutions for the flow of fluid through tubes with uniformly porous walls

  • 32 Accesses

Summary

This paper considers an incompressible fluid flowing through a straight, circular tube whose walls are uniformly porous. The flow is steady and one dimensional. The loss of fluid through the wall is proportional to the mean static pressure in the tube. Several formulations of the wall shear stress are considered; these formulations were motivated by the results from Hamel's radial flow problem, boundary layer flows/and boundary layer suction profiles. For each of these formulations exact solutions for the mean axial velocity and the mean static pressure of the fluid are obtained. Sample results are plotted on graphs. For the constant wall shear stress problem, the theoretical solutions compare favorably with some experimental results.

Abbreviations

A, B, D, E :

constant parameters

a, b :

constant parameters

Ai(z), Bi(z) :

Airy functions

Ai′, Bi′ :

derivatives of Airy functions

k :

constant of proportionality betweenV andp

L :

length of pores

p,p :

mean static pressure

p 0 :

static pressure outside the tube

p 0 :

value ofp atx=0

Q :

constant exponent

R :

inside radius of the tube

T :

wall shear stress

T 0 :

shear parameter

t :

wall thickness

U :

free stream velocity

ū,u :

mean axial velocity

u 0 :

value ofu atx=0

V,V :

mean seepage velocity through the wall

v 0 :

mean seepage velocity

x,x :

axial distance along the tube

z :

transformed axial distance

z 0 :

value ofz atx=0

α:

mean outflow angle through the wall

β:

cos α

ϱ:

density of the fluid

\(\bar \tau _0 \) :

wall shear stress

μ:

dynamic viscosity of the fluid

‘over-bar’:

dimensional terms

‘no bar’:

nondimensional terms

References

  1. [1]

    M. Abramowitz andI. A. Stegun,Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover Publ. Inc., New York, 1965), pp. 446–452, 475–477.

  2. [2]

    H. T. Davis,Introduction to Nonlinear Differential and Integral Equations (Dover Publ. Inc., New York, 1960), pp. 57–76.

  3. [3]

    G. G. Havens andD. B. Guy,The Use of Porous Fiber Glass Tubes for Reverse Osmosis, Chem. Eng. Prog. Symp. Ser.64 (90), (1968), 299–305.

  4. [4]

    C. Orr, Jr.,Particle Technology (Macmillan Company, New York, 1966), pp. 179–228.

  5. [5]

    H. Schlichting,Boundary Layer Theory (McGraw-Hill Book Co., Inc., New York, 1955), Trans byJ. Kestin, pp. 80–82, 108, 229–240, 433.

  6. [6]

    T. H. Shaffer, III,Porous Tube Flow Phenomena, Drexel Tech. J.29 (1967), 9–15 and 76.

  7. [7]

    S. W. Yuan,Cooling by Protective Fluid Films, Section G,Turbulent Fows and Heat Transfer, High Speed Aerodynamics and Jet Propulsion, Vol. V (Princeton Univ. Press, Princeton, N.J., 1959), pp. 428–488.

Download references

Author information

Additional information

The National Center for Atmospheric Research is sponsored by the National Science Foundation

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Drake, R.L. Some exact solutions for the flow of fluid through tubes with uniformly porous walls. PAGEOPH 94, 248–259 (1972). https://doi.org/10.1007/BF00875685

Download citation

Keywords

  • Boundary Layer
  • Shear Stress
  • Exact Solution
  • Fluid Flow
  • Static Pressure