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Analytic Solutions for Pulsatile Flow Through Annular, Rectangular and Sector Ducts Filled with a Darcy–Brinkman Medium

Abstract

Oscillatory (or pulsatile) flow in ducts filled with a Darcy–Brinkman medium is studied. Analytic solutions (some in closed form) are derived for the annular duct, the rectangular duct and the sector duct. These exact solutions also serve as accuracy standards for approximate and numerical solutions. The problem is governed by two parameters, a non-dimensional frequency and a porous medium factor, which is inversely proportional to the Darcy number. The effects of these parameters on the amplitude and phase lag of the flow are discussed.

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Abbreviations

\(a_n\) :

Coefficient defined by Eq. (23)

A :

Amplitude

\(A_{nl}\) :

Constant defined by Eq. (37)

b :

Normalized core radius or rectangle aspect ratio

\(b_n\) :

Coefficient defined by Eq. (30)

\(c_1, c_2\) :

Coefficients defined by Eq. (17)

\(C_{nl}\) :

Constant defined by Eq. (34)

D :

Darcy number \(K/L^{2}\)

f :

Function of y

F :

Function defined by Eq. (35)

g :

Function of y

\(G_0\) :

Oscillatory pressure amplitude \((\hbox {N/m}^2)\)

h :

Function of y

H :

Function of r

i :

\(\sqrt{-1}\)

\(I_n\) :

Modified Bessel function

\(j_n\) :

Function of r

\(J_\beta \) :

Bessel function

k :

Porous media factor defined in Eq. (4)

K :

Permeability of porous medium \((\hbox {m}^2)\)

\(K_n\) :

Modified Bessel function

L :

Length scale (m)

n :

integer

P :

Phase lag

Q :

Normalized flow rate

r :

Radial coordinate

t :

Normalized time

u :

Normalized velocity

xy :

Cartesian coordinates

X :

Function defined by Eq. (39)

Z :

Function defined in Eq. (27)

\(\alpha _n\) :

Constant defined by Eq. (22)

\(\beta _n\) :

Constant defined by Eq. (29)

\(\gamma \) :

Ratio of viscosities \(\mu /\mu _{e}\)

\(\delta _{BL}\) :

Boundary layer thickness

\(\theta \) :

Cylindrical coordinate

\(\theta _0\) :

Half opening angle of sector

\(\lambda _l\) :

Root of Bessel function

\(\mu \) :

Viscosity of fluid \((\hbox {Ns/m}^2)\)

\(\mu _{e}\) :

Effective viscosity of matrix \((\hbox {Ns/m}^2)\)

\(\rho \) :

Density of fluid \((\hbox {Kg/m}^3)\)

\(\sigma \) :

\(\sqrt{k+i\omega }\)

\(\tau _n\) :

Constant defined by Eq. (25)

\(\phi \) :

Porosity

\(\omega \) :

Normalized frequency

\(\Omega \) :

Frequency of oscillation (1/s)

\('\) :

Dimensional quantity

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Correspondence to C. Y. Wang.

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Wang, C.Y. Analytic Solutions for Pulsatile Flow Through Annular, Rectangular and Sector Ducts Filled with a Darcy–Brinkman Medium. Transp Porous Med 112, 409–428 (2016). https://doi.org/10.1007/s11242-016-0652-8

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Keywords

  • Darcy–Brinkman
  • Pulsatile
  • Duct
  • Annular
  • Rectangular
  • Sector