Abstract
Uniform Darcy–Brinkman flow over a surface with periodic rectangular grooves is studied by domain decomposition and matching. It is found that the effect of corrugations is equivalent to replacing the rough surface with a smooth surface with an apparent slip for the bulk flow. Such equivalence would greatly simplify the boundary conditions for porous flow bounded by a rough surface. The slip velocity is larger along the grooves than transverse to the grooves, and is increased by the porous media parameter k.
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Abbreviations
- a :
-
Normalized half width of groove
- a n :
-
Constants defined by Eq. 7
- A n :
-
Coefficients
- b :
-
Normalized depth of groove
- B n :
-
Coefficients
- C n :
-
Coefficients
- D n :
-
Coefficients
- E n :
-
Coefficients
- F n :
-
Functions of y
- G n :
-
Coefficients
- H :
-
Number of terms
- i :
-
Integer
- j :
-
Integer
- k :
-
\({L\sqrt{\mu /\mu_{\rm e} K}}\)
- K :
-
Permeability (m2)
- L :
-
Half period of grooves (m)
- M :
-
Number of terms
- N :
-
Number of terms
- R i :
-
Functions of y, Eq. 17
- S :
-
Normalized apparent slip velocity
- T :
-
Function of x, Eq. 17
- \({\vec{u}}\) :
-
Normalized velocity vector
- U :
-
Uniform velocity at infinity (m/s)
- w :
-
Normalized velocity in the z direction
- x, y, z:
-
Normalized Cartesian coordinates
- α n :
-
n π
- \({\tilde {\alpha}_n}\) :
-
\({\sqrt{\alpha_n^2 +k^{2}}}\)
- β n :
-
(n − 0.5)π/a
- \({\tilde {\beta }_n}\) :
-
\({\sqrt{\beta_n^2 +k^{2}}}\)
- μ :
-
Fluid viscosity (Ns/m2)
- μ e :
-
Effective viscosity of matrix (Ns/m2)
- ψ :
-
Stream function
- ′:
-
Dimensional quantity
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Wang, C.Y. Darcy–Brinkman Flow over a Grooved Surface. Transp Porous Med 84, 219–227 (2010). https://doi.org/10.1007/s11242-009-9496-9
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DOI: https://doi.org/10.1007/s11242-009-9496-9