Abstract
Analytical solutions are found for the transient starting flow due to a sudden pressure gradient in cylindrical, rectangular, and parallel plate ducts fill with a Darcy– Brinkman porous medium. It is found that, for all geometries, the initial velocity front is flat. It eventually becomes more parabolic for small porous media parameter s but remains flat for large s. The boundary layer thickness is of order (1/s). The transient is also shorter (proportional to exp(−s 2 t)) for large s.
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Abbreviations
- A n , A mn :
-
Coefficients
- b :
-
Aspect ratio
- G :
-
Pressure gradient (N/m3)
- G 0 :
-
Constant pressure gradient (N/m3)
- H :
-
Unit step function
- I :
-
Modified Bessel function
- J :
-
Bessel function
- k n ,k mn :
-
Parameters defined by Eq.12 or Eq. 23
- K :
-
Permeability (m2)
- L :
-
Length scale (m)
- n :
-
Integer
- Q :
-
Normalized flow rate
- r :
-
Normalized radial coordinate
- s :
-
Non-dimensional parameter \(L\sqrt{\mu /K\mu_e}\)
- t :
-
Normalized time
- u :
-
Normalized axial velocity
- ū :
-
Normalized steady-state velocity
- ũ :
-
Normalized transient velocity
- x, y :
-
Normalized Cartesian coordinates
- λ n :
-
Zero of J 0
- α n :
-
(n−1/2)π
- β m :
-
(n−1/2)π /b
- γ :
-
(n−1/4)π
- μ :
-
Fluid viscosity (N s/m2)
- μ e :
-
Effective viscosity (N s/m2)
- ρ :
-
Fluid density (kg/m3)
- θ :
-
Polar angle coordinate
- \(\phi\) :
-
Porosity
- ′:
-
Dimensional quantity
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Wang, C.Y. The Starting Flow in Ducts Filled with a Darcy–Brinkman Medium. Transp Porous Med 75, 55–62 (2008). https://doi.org/10.1007/s11242-008-9210-3
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DOI: https://doi.org/10.1007/s11242-008-9210-3