Advertisement

Transport in Porous Media

, Volume 127, Issue 3, pp 631–642 | Cite as

Starting Darcy–Brinkman Flow in a Sector Duct Using the Method of Eigenfuction Superposition

  • C. Y. WangEmail author
Article
  • 37 Downloads

Abstract

The powerful method of eigenfunction superposition is applied to the starting flow in a sector duct filled with a porous medium. Using analytic eigenfunctions and eigenvalues of the Helmholtz equation, the solution can be expressed in a simple series. The properties of the velocity and the transient flow rate are found to depend on the sector geometry and a porous medium factor. The starting solution is then used to construct the solution to arbitrary unsteady flows.

Keywords

Unsteady Darcy–Brinkman Sector Eigenfunction 

List of Symbols

\( a_{j} \)

Coefficients Eq. (18)

A

Area

\( A_{j} \)

Amplitude Eq. (5)

Am

Amplitude of oscillation

b

Constant

\( c_{j} \)

Coefficents Eq. (8)

f

Pressure gradient function

\( F_{nj} \)

Integral Eq. (9)

\( G_{0} \)

Magnitude of pressure gradient (N/m3)

H

Unit step function

i

Integer

I

Input function Eq. (24)

j

Integer

J

Bessel function of first kind

k

Porous media factor \( \mu L^{2} /(\mu_{e} K) \)

K

Permeability (m2)

L

Length scale (radius of sector) (m)

n

Integer

N

Integer

p

Integer

\( p_{o} \)

Porosity

P

Pulse function Eq. (22)

r

Radial coordinate

R

Response function Eq. (23)

u

Axial velocity

U

Solution Eq. (25)

t

Time

\( \alpha_{n} \)

Constant Eq. (6)

\( \delta \)

Dirac delta

\( \phi_{j} \)

Eigenfunction

\( \theta \)

Angle coordinate

\( \theta_{0} \)

Half opening angle

\( \lambda_{j} \)

Eigenvalue

\( \mu \)

Viscosity of fluid (Ns/m2)

\( \mu_{e} \)

Effective viscosity of matrix (Ns/m2)

\( \rho \)

Density of fluid (kg/m3)

\( \omega \)

Frequency

\( \tau \)

Time variable

Dimensional quantity

Overbar

Steady state

Tilde

Transient state

Notes

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965)Google Scholar
  2. Alkam, M.K., Al-Nimr, M.A.: Solutions for classical fluid flow problems in porous domains. JSME Int. J. 42, 206–213 (1999)CrossRefGoogle Scholar
  3. Al-Nimr, M.A., Alkam, M.K.: Basic fluid flow problems in porous media. J. Por. Med. 3, 45–59 (2000)CrossRefGoogle Scholar
  4. Andersson, H.I., Holmedal, L.E.: Start-up flow in a porous medium channel. Acta Mech. 113, 155–168 (1995)CrossRefGoogle Scholar
  5. Ingham, D.B., Pop, I.: Transport in Porous Media. Pergamon, Oxford (2002)Google Scholar
  6. Kamish, F.: Analysis of laminar flow and forced convection heat transfer in a porous medium. Transp. Porous Med. 80, 345–371 (2009)CrossRefGoogle Scholar
  7. Khodadadi, J.M.: Oscillatory fluid flow through a porous medium channel bounded by two impermeable parallel plates. J. Fluids Eng. 113, 509–511 (1991)CrossRefGoogle Scholar
  8. Kim, S.Y., Kang, B.H., Hyun, J.M.: Heat transfer from pulsating flow in a channel filled with porous media. Int. J. Heat Mass Transf. 37, 2025–2033 (1994)CrossRefGoogle Scholar
  9. Kuznetsov, A.V., Nield, D.A.: Forced convection with laminar pulsating flow in a saturated porous channel or tube. Transp. Porous Med. 65, 505–523 (2006)CrossRefGoogle Scholar
  10. Leissa, A.W., Qatu, M.S.: Vibrations of Continuous Systems. McGraw-Hill, New York (2011)Google Scholar
  11. Morse, P.M., Feshbach, H.: Methods of Theoretical Physics. McGraw-Hill, New York (1953)Google Scholar
  12. Nield, D.A., Bejan, A.: Convection in Porous Media, 5th edn. Springer, New York (2017)CrossRefGoogle Scholar
  13. Riley, K.F., Hobson, M.P., Bence, S.J.: Mathematical Methods in Physics and Engineering, 3rd edn. Cambirdge University Press, Cambridge (2006)CrossRefGoogle Scholar
  14. Shah, R.K., London, A.L.: Laminar Flow Forced Convection in Ducts. Academic Press, New York (1978)Google Scholar
  15. Wang, C.Y.: The starting flow in ducts filled with a Darcy-Brinkman medium. Transp. Porous Med. 75, 55–62 (2008a)CrossRefGoogle Scholar
  16. Wang, C.Y.: Analytical solution for forced convection in a semi-circular channel filled with a porous medium. Transp. Porous Med. 73, 369–378 (2008b)CrossRefGoogle Scholar
  17. Wang, C.Y.: Analytical solution for forced convection in a sector duct filled with a porous medium. J. Heat Transf. 132, 084502 (2010)CrossRefGoogle Scholar
  18. Wang, C.Y.: Analytic solutions for pulsatile flow through annular, rectangular and sector ducts filled with a Darcy-Brinkmen medium. Transp. Porous Med. 112, 409–428 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Departments of Mathematics and Mechanical EngineeringMichigan State UniversityEast LansingUSA

Personalised recommendations