^{1}

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Flow through a channel bounded by a porous layer is considered when a transition layer exists between the channel and the medium. The variable permeability in the transition layer is chosen such that Brinkman’s equation governing the flow reduces to a generalized inhomogeneous Airy’s differential equation. Solution to the resulting generalized Airy’s equation is obtained in this work and solution to the flow through the transition layer, of the same configuration, reported in the literature, is recovered from the current solution.

In their recent study, Nield and Kuznetsov [

1) Introducing and reviving the implementation of classical integral functions in the porous media literature (as witnessed by their use of Airy’s differential equation and the Airy functions in providing an analytical solution to the flow in the transition zone);

2) Introducing a new integral function, Ni(x), to facilitate solution to the inhomogeneous Airy’s differential equation. This function is better known as the Nield-Kuznetsov function, and has been studied extensively by Hamdan and Kamel [

3) Initiating non-traditional models of permeability variations in porous media. While the classical use of elementary mathematical functions has served the subject matter well, the use of special functions in advancing the topic represents a new generation of models the computations of which is no longer a formidable task.

Many excellent reviews of flow through and over porous layers are available in the literature (cf. Nield and Bejan [

We consider fluid flow through a channel consisting of three layers, as shown in

layers is driven by a common pressure gradient. Permeability distribution in the configuration of

In the lower region, or Region 1 (the fluid zone):

K 1 → ∞ for 0 < y * < η H . (1)

In the middle region, or Region 2:

K 2 = K 0 [ ( ξ − η ) H ] n ( y * − η H ) n ; for η H < y * < ξ H . (2)

In the upper region, or Region 3:

K 3 = K 0 ; for ξ H < y * < H . (3)

The boundary value problem at hand can be formulated as follows:

μ 1 d 2 u 1 * d y * 2 + G = 0 for 0 < y * < η H . (4)

μ e 2 d 2 u 2 * d y * 2 − μ K 2 u 2 + G = 0 for η H < y * < ξ H . (5)

μ e 3 d 2 u 3 * d y * 2 − μ K 3 u 3 * + G = 0 for ξ H < y * < H . (6)

In the above equations, G = − d p d x is the constant pressure gradient. For

i = 1 , 2 , 3 , with i = 1 referring to layer 1, i = 2 referring to layer 2, and i = 3 referring to layer 3, the quantities in Equations (4), (5), and (6): u i * , K i , μ i , and μ e i denote the velocity, permeability, base fluid viscosity, and effective viscosity, respectively, in layer i .

Following Nield and Kozentsov, [

variables, where D a refers to Darcy number: y = y * H ; D a = K 0 H 2 ; u i = μ i G H 2 u i * for i = 1 , 2 , 3 ; and M i = μ e i μ i for i = 2 , 3 .

The permeability distributions, K 1 , K 2 and K 3 can be written in the following dimensionless form in terms of Darcy number, Da:

K 1 * = K 1 H 2 → ∞ (7)

K 2 * = K 2 H 2 = D a ( ξ − η y − η ) n (8)

K 3 * = K 3 H 2 = D a . (9)

Equations (4), (5), and (6) take the following forms, respectively:

d 2 u 1 d y 2 + 1 = 0 ; 0 < y < η . (10)

M 2 d 2 u 2 d y 2 − H 2 K 2 u 2 + 1 = 0 ; η < y < ξ . (11)

M 3 d 2 u 3 d y 2 − H 2 K 3 u 3 + 1 = 0 ; ξ < y < 1 (12)

and, upon substituting the permeability distributions, (7)-(9), in equations (10)-(12), we get

d 2 u 1 d y 2 + 1 = 0 ; 0 < y < η . (13)

d 2 u 2 d y 2 − 1 M 2 D a ( ξ − η ) n ( y − η ) n u 2 + 1 M 2 = 0 ; η < y < ξ . (14)

d 2 u 3 d y 2 − 1 D a M 3 u 3 + 1 M 3 = 0 ; for ξ < y < 1. (15)

Equation (13), (14) and (15) must be solved subject to the following boundary and matching conditions

u 1 = 0 at y = 0 (16a)

u 1 = u 2 at y = η (16b)

d u 1 d y = ϑ 1 d u 2 d y at y = η . (16c)

u 2 = u 3 at y = ξ . (16d)

ϑ 2 d u 2 d y = d u 3 d y at y = ξ . (16e)

u 3 = 0 at y = 1 , (16f)

where ϑ 1 = μ 2 μ 1 and ϑ 2 = μ 2 μ 3 are the viscosity ratios.

Now, letting

τ n = 1 M 2 D a ( ξ − η ) n n + 2 (17)

λ 3 = 1 M 3 D a (18)

Equation (14) and Equation (15) become, respectively:

d 2 u 2 d y 2 − ( τ n ) n + 2 ( y − η ) n + 1 M 2 = 0 (19)

d 2 u 3 d y 2 − λ 3 2 u 3 + 1 M 3 = 0 . (20)

General solutions for Equation (13) and Equation (20) are given, respectively, by

u 1 ( y ) = c 1 y + d 1 − y 2 2 (21)

u 3 ( y ) = c 3 exp ( λ 3 y ) + d 3 exp ( − λ 3 y ) + 1 M 3 λ 3 2 . (22)

In order to solve Equation (19), we first use the following transformation y ˜ = τ n ( y − η ) and write u 2 ( y ) ≡ U 2 ( y ˜ ) . Equation (19) then becomes:

d 2 U 2 ( y ˜ ) d y ˜ 2 − ( y ˜ ) n U 2 ( y ˜ ) + 1 M 2 ( τ n ) 2 = 0 (23)

Equation (23) is the generalized inhomogeneous Airy’s differential equation. A fundamental pair of linearly independent solutions for the homogeneous part are the generalized Airy’s functions A n ( y ˜ ) and B n ( y ˜ ) , (cf. Swanson and Headley [

A n ( y ˜ ) = 2 p π sin ( p π ) ( y ˜ ) 1 / 2 K p ( ζ ) (24)

B n ( y ˜ ) = ( p y ˜ ) 1 2 ( I − p ( ζ ) + I p ( ζ ) ) (25)

The terms I p and K p are the modified Bessel functions defined as:

I p ( ζ ) = i − p J p ( i ζ ) = ∑ m = 1 ∞ 1 m ! Γ ( m + p + 1 ) ( ζ 2 ) 2 m + p (26)

K p ( ζ ) = π 2 ( I − p ( ζ ) − I p ( ζ ) ) sin ( p π ) (27)

with p = 1 n + 2 , ζ = 2 p Y 1 2 p , and Γ is the gamma function.

Solution to the homogeneous part of (23) is thus given by

U 2 h ( y ˜ ) = c 2 A n ( y ˜ ) + d 2 B n ( y ˜ ) . (28)

We find it convenient to introduce the following integral function:

Z n ( y ˜ ) = A n ( y ˜ ) ∫ 0 y ˜ B n ( t ) d t − B n ( y ˜ ) ∫ 0 y ˜ A n ( t ) d t . (29)

The function Z n ( y ˜ ) reduces to the Nield-Kuznetsov function N i ( y ˜ ) when n = 1.

The Wronskian of A n ( Y ) and B n ( Y ) is given by:

W ( A n ( y ˜ ) , B n ( y ˜ ) ) = 2 π p 1 2 sin ( p π ) (30)

and general solution of (23) is expressed, using variation of parameters, as:

U 2 ( y ˜ ) = c 2 A n ( y ˜ ) + d 2 B n ( y ˜ ) + σ n Z n ( y ˜ ) (31)

where p = 1 n + 2 and σ n = π 2 p s i n ( p π ) M 2 ( τ n ) 2 .

Upon substituting y ˜ = τ n ( y − η ) , u 2 ( y ) ≡ U 2 ( y ˜ ) , and σ n = π 2 p sin ( p π ) M 2 ( τ n ) 2 , in (28) we obtain the following general solution to Equation (14):

u 2 ( y ) = c 2 A n ( τ n ( y − η ) ) + d 2 B n ( τ n ( y − η ) ) + π 2 p s i n ( p π ) M 2 ( τ n ) 2 Z n ( τ n ( y − η ) ) (32)

Derivatives of the functions A n ( y ˜ ) , B n ( y ˜ ) and Z n ( y ˜ ) are given by:

A ′ n ( y ˜ ) = − 2 p π [ sin p y ] y ˜ n + 1 2 K p − 1 ( ζ ) (33)

B ′ n ( y ˜ ) = p 1 2 y ˜ n + 1 2 [ I 1 − p ( ζ ) + I p − 1 ( ζ ) ] (34)

Z ′ n ( y ˜ ) = A ′ n ( y ˜ ) ∫ 0 y ˜ B n ( t ) d t − B ′ n ( y ˜ ) ∫ 0 y ˜ A n ( t ) d t . (35)

Shear stress expressions across the layers are obtained from Equations (21), (22) and (32), and take the following form:

d u 1 d y = c 1 − y (36)

d u 2 d y = c 2 τ n A ′ n ( τ n ( y − η ) ) + d 2 τ n B ′ n ( τ n ( y − η ) ) + π 2 p sin ( p π ) M 2 τ n Z ′ n ( τ n ( y − η ) ) (37)

d u 3 d y = c 3 λ 3 exp ( λ 3 y ) − d 3 λ 3 exp ( − λ 3 y ) . (38)

Upon using boundary and interfacial conditions, (16a)-(16f), in Equations (21), (22) and (32), we obtain the following system of linear equations that is to be solved for the arbitrary constants c 1 , d 1 , c 2 , d 2 , c 3 , d 3 :

d 1 = 0 (39)

c 1 η + d 1 − c 2 A n ( 0 ) − d 2 B n ( 0 ) = η 2 2 (40)

c 1 − c 2 ϑ 1 τ n A ′ n ( 0 ) − d 2 ϑ 1 τ n B ′ n ( 0 ) = η (41)

c 2 A n ( τ n ( ξ − η ) ) + d 2 B n ( τ n ( ξ − η ) ) − c 3 exp ( λ 3 ξ ) − d 3 exp ( − λ 3 ξ ) = 1 M 3 λ 3 2 − π 2 p sin ( p π ) M 2 ( τ n ) 2 Z n ( τ n ( ξ − η ) ) (42)

c 2 ϑ 2 τ n A ′ n ( τ n ( ξ − η ) ) + d 2 ϑ 2 τ n B ′ n ( τ n ( ξ − η ) ) − λ 3 c 3 exp ( λ 3 ξ ) + λ 3 d 3 exp ( − λ 3 ξ ) = − ϑ 2 π 2 p sin ( p π ) M 2 τ n Z ′ n ( τ n ( ξ − η ) ) (43)

c 3 exp ( λ 3 ) + d 3 exp ( − λ 3 ) = − 1 M 3 λ 3 2 (44)

Linear Equations (39)-(44) are cast in the following matrix-vector form

M x = c , (45)

where:

M = ( 0 1 0 0 0 0 η 1 − A n ( 0 ) − B n ( 0 ) 0 0 1 0 − ϑ 1 τ n A ′ n ( 0 ) − ϑ 1 τ n B n ( 0 ) 0 0 0 0 A ′ n ( τ n ( ξ − η ) ) B n ( τ n ( ξ − η ) ) − exp ( λ 3 ξ ) − exp ( − λ 3 ξ ) 0 0 ϑ 2 τ n A ′ n ( τ n ( ξ − η ) ) ϑ 2 τ n B ′ n ( τ n ( ξ − η ) ) − λ 3 exp ( λ 3 ξ ) λ 3 exp ( − λ 3 ξ ) 0 0 0 0 exp ( λ 3 ) exp ( − λ 3 ) ) (46)

x = [ c 1 d 1 c 2 d 2 c 3 d 3 ] (47)

c = ( 0 η 2 2 η 1 M 3 λ 3 2 − π 2 p sin ( p π ) M 2 ( τ n ) 2 Z n ( τ n ( ξ − η ) ) − ϑ 2 π 2 p sin ( p π ) M 2 τ n Z ′ n ( τ n ( ξ − η ) ) − 1 M 3 λ 3 2 ) . (48)

In solving the above linear system, we make use of the following values of the generalized functions A n , B n and Z n , and their first derivatives at zero (cf. [

A n ( 0 ) = p 1 − p Γ ( 1 − p ) (49)

B n ( 0 ) = p 1 2 − p Γ ( 1 − p ) (50)

A ′ n ( 0 ) = − p p Γ ( p ) (51)

B ′ n ( 0 ) = p p − 1 2 Γ ( p ) (52)

Z n ( 0 ) = 0 (53)

Z ′ n ( 0 ) = 0 . (54)

Results have been obtained for a range of values of n and the range of Da = 1; 0.1; 0.001; 0.0001; and 0.00001 in order to illustrate the effects of changing n and Da on the generalized functions, on the permeability function, on the arbitrary constants, and on the velocity profiles. Thick and thin transition layers are considered. In particular, we choose η = 1 / 3 , ξ = 2 / 3 and η = 1 / 4 , ξ = 3 / 4 for thick transition layer, and η = 0.49 , ξ = 0.51 for thin layer.

Values of p = 1 n + 2 , A n , A ′ n , B n , B ′ n , Z n , Z ′ n , σ n and τ n at 0 and at

other specified points for different values of n and Da, have been computed using Maple. These values are important in the calculation of the arbitrary constants in the matrix-vector Equations (45), and in the determination and plotting of permeability functions and velocity profiles. Computational results are most accurate when Da = 1; 0.1; and 0.001.

For the sake of graphs we find it more convenient to plot the reciprocal of the dimensionless permeability functions (7)-(9), namely

1 K 1 * = 0 for 0 < y < η . (55)

1 K 2 * ( y ) = 1 D a ( y − η ξ − η ) n ; for η < y < ξ . (56)

1 K 1 * = 1 D a for ξ < y < 1. (57)

Dependence of the permeability distributions on the value of n is illustrated in

the three layers for n = 1, 2, 3 and 5, for η = 1 3 , ξ = 2 3 , and Da = 1.

demonstrates the increase of the reciprocal permeability of the transition layer (or decrease of permeability) with increasing n.

At the lower and upper interfaces, y = η and y = ξ , respectively, between layers, velocity expressions are obtained from Equations (21), (22), and (32), and take the form

u 1 ( η ) = c 1 η + d 1 − η 2 2 (58)

u 2 ( η ) = c 2 A n ( 0 ) + d 2 B n ( 0 ) (59)

u 2 ( ξ ) = c 2 A n ( τ n ( ξ − η ) ) + d 2 B n ( τ n ( ξ − η ) ) + π 2 p s i n ( p π ) M 2 ( τ n ) 2 Z n ( τ n ( ξ − η ) ) (60)

u 3 ( ξ ) = c 3 exp ( λ 3 ξ ) + d 3 exp ( − λ 3 ξ ) + 1 M 3 λ 3 2 . (61)

Velocity at the lower and upper interfaces for different middle layer thickness, different values of Da, and for n = 1 and n = 2 are given in

Da | Da = 1 | Da = 0.1 | Da = 0.01 | Da = 0.001 | |
---|---|---|---|---|---|

η = 1 / 3 ξ = 2 / 3 | u 1 ( η ) = u 2 ( η ) | 0.1068281 | 0.08214691 | 0.03908309 | 0.01714995 |

u 2 ( ξ ) = u 3 ( ξ ) | 0.1046975 | 0.06834714 | 0.01327264 | 0.00107072 | |

η = 4 ξ = 3 / 4 | u 1 ( η ) = u 2 ( η ) | 0.09037786 | 0.07064158 | 0.03455354 | 0.01513239 |

u 2 ( ξ ) = u 3 ( ξ ) | 0.0883398 | 0.0574775 | 0.0110710 | 0.00101 | |

η = 0.49 ξ = 0.51 | u 1 ( η ) = u 2 ( η ) | 0.1189297 | 0.08497432 | 0.0308578 | 0.01043350 |

u 2 ( ξ ) = u 3 ( ξ ) | 0.1186918 | 0.0833988 | 0.02721074 | 0.00630965 |

Da | Da = 1 | Da = 0.1 | Da = 0.01 | Da = 0.001 | |
---|---|---|---|---|---|

η = 1 / 3 ξ = 2 / 3 | u 1 ( η ) = u 2 ( η ) | 0.1078735 | 0.08819101 | 0.04952703 | 0.02688473 |

u 2 ( ξ ) = u 3 ( ξ ) | 0.1057217 | 0.07335809 | 0.01652092 | 0.0012043 | |

η = 4 ξ = 3 / 4 | u 1 ( η ) = u 2 ( η ) | 0.0915251 | 0.07740496 | 0.04608594 | 0.0252177 |

u 2 ( ξ ) = u 3 ( ξ ) | 0.0894578 | 0.0628103 | 0.0134685 | 0.00108 | |

η = 0.49 ξ = 0.51 | u 1 ( η ) = u 2 ( η ) | 0.119023 | 0.0854829 | 0.03166939 | 0.01131475 |

u 2 ( ξ ) = u 3 ( ξ ) | 0.1187851 | 0.08390017 | 0.02796018 | 0.00694503 |

1) Computations render reasonable results for Da as low as 0.001 when n = 1. Inaccuracies start creeping when Da < 0.001. For n = 2, results are reasonable for Da as low as 0.0001 and inaccuracies creep when Da < 0.0001. This behavior may be attributed to both round-off errors for small Da and inaccuracies in evaluation of Airy’s functions for small Da. This behavior persists for thick layers, and is less noticeable for thin middle layer calculations.

2) Computations of velocity at the lower interface using expressions (58) and (59) agree up to within seven significant digits. The same is true for computations of velocity at the upper interface using expressions (60) and (61). This is indicative of appropriate matching of the velocity at the interfaces, used in this work.

3) For a given Da, velocity at the lower interface decreases with increasing middle layer thickness. Similarly, velocity at the upper interface decreases with increasing middle layer thickness. This behavior persists for both n = 1 and n = 2.

4) For a given middle layer thickness, velocity at each of the lower and upper interfaces increases with increasing Da. This is expected, as Da is defined as a dimensionless reference permeability and accompanied with increasing permeability is a velocity increase.

5) The effect of increasing n on the velocity at the interfaces for a given thickness and Da is that the velocity at each interface increases with increasing n. This is true for both thin and thick transition layers, and for the range of Da used.

At the lower and upper interfaces, y = η and y = ξ , respectively, between layers, shear stress expressions are obtained from Equations (36), (37), and (38), and take the form

d u 1 ( η ) d y = c 1 − η (62)

d u 2 ( η ) d y = c 2 τ n A ′ n ( 0 ) + d 2 τ n B ′ n ( 0 ) (63)

d u 2 ( ξ ) d y = c 2 τ n A ′ n ( τ n ( ξ − η ) ) + d 2 τ n B ′ n ( τ n ( ξ − η ) ) + π 2 p sin ( p π ) M 2 τ n Z ′ n ( τ n ( ξ − η ) ) (64)

d u 3 ( ξ ) d y = c 3 λ 3 exp ( λ 3 ξ ) − d 3 λ 3 exp ( − λ 3 ξ ) . (65)

Shear stress at the lower and upper interfaces for different middle layer thickness, different values of Da, and for n = 1 and n = 2 are given in

1) Computations render reasonable results for Da as low as 0.001 when n = 1 and n = 2. Inaccuracies start creeping when Da < 0.001. This behavior may be attributed to the inaccuracies reported earlier when computing velocity at the interfaces using Maple’s built-in functions. When dealing with a thin transition layer, results are accurate for as low as Da = 0.00001.

2) Computations of shear stress at the lower interface using expressions (62) and (63) agree up to within a minimum of five significant digits. The same is true for computations of velocity at the upper interface using expressions (64) and (65).

3) For a given Da, the absolute value of the shear stress at the lower interface increases with increasing transition layer thickness. Similarly, at the upper interface. This behavior persists for both n = 1 and n = 2.

Da | Da = 1 | Da = 0.1 | Da = 0.01 | Da = 0.001 | |
---|---|---|---|---|---|

η = 1 / 3 ξ = 2 / 3 | d u 1 d y ( η ) = d u 2 d y ( η ) | 0.1538176 | 0.07977407 | −0.04941739 | −0.1152168 |

d u 2 d y ( ξ ) = d u 3 d y ( ξ ) | −0.1604999 | −0.1231173 | −0.0399533 | −0.00225 | |

η = 1 / 4 ξ = 3 / 4 | d u 1 d y ( η ) = d u 2 d y ( η ) | 0.2365115 | 0.1575664 | 0.01321413 | −0.06447045 |

d u 2 d y ( ξ ) = d u 3 d y ( ξ ) | −0.2363372 | −0.1570521 | −0.027384 | −0.003 | |

η = 0.49 ξ = 0.51 | d u 1 d y ( η ) = d u 2 d y ( η ) | −0.002286399 | −0.071583 | −0.1820249 | −0.2237071 |

d u 2 d y ( ξ ) = d u 3 d y ( ξ ) | −0.02109832 | −0.08318887 | −0.1736158 | −0.167906 |

Da | Da = 1 | Da = 0.1 | Da = 0.01 | Da = 0.001 | |
---|---|---|---|---|---|

η = 1 / 3 ξ = 2 / 3 | d u 1 d y ( η ) = d u 2 d y ( η ) | 0.1569537 | 0.09790637 | −0.01808557 | −0.08601249 |

d u 2 d y ( ξ ) = d u 3 d y ( ξ ) | −0.1636852 | −0.1433454 | −0.0725202 | −0.00648 | |

η = 1 / 4 ξ = 3 / 4 | d u 1 d y ( η ) = d u 2 d y ( η ) | 0.2411004 | 0.1846199 | 0.05934378 | −0.02412919 |

d u 2 d y ( ξ ) = d u 3 d y ( ξ ) | −0.2409034 | −0.1826534 | −0.051689 | −0.0032 | |

η = 0.49 ξ = 0.51 | d u 1 d y ( η ) = d u 2 d y ( η ) | −.002095967 | −0.07054511 | −0.1803686 | −0.2219087 |

d u 2 d y ( ξ ) = d u 3 d y ( ξ ) | −0.02130357 | −0.08492409 | −0.1811111 | −0.1886142 |

4) For a given transition layer thickness, absolute value of shear stress at each of the lower and upper interfaces increases with increasing Da.

5) The effect of increasing n on the absolute value of shear stress at the interfaces for a given thickness and Da is that at each interface, this absolute value increases with increasing n. This is true for both thin and thick transition layers, and for the range of Da used.

A quantity of interest is the negative of the shear stress term at the interface

between the channel and the transition layer, namely c f = − d u 1 d y at y = η .

This has been analyzed and defined by Nield and Kuznetsov, [

c f = − d u 1 ( η ) d y = η − c 1 . (66)

Values of − c f for different Da, layer thickness and n = 1 are listed in

The dimensionless mean velocities across the layers are defined as

u ¯ 1 = ∫ 0 η u 1 d y = 1 2 c 1 η 2 + d 1 η − 1 6 η 3 (67)

u ¯ 2 = ∫ η ξ u 2 d y = ∫ η ξ c 2 A n ( τ n ( y − η ) ) + d 2 B n ( τ n ( y − η ) ) + π 2 p s i n ( p π ) M 2 ( τ n ) 2 Z n ( τ n ( y − η ) ) d y (68)

Da | 1 | 0.1 | 0.01 | 0.001 | 0.0001 | 0.00001 | |
---|---|---|---|---|---|---|---|

η = 1 / 3 ξ = 2 / 3 | c f | −0.1538176 −0.153* | −0.07977407 | 0.04941739 0.049* | 0.1152168 | −0.1436413 | −0.1562024 |

η = 1 / 4 ξ = 3 / 4 | c f | −0.2365115 | −0.1575664 | −0.01321413 | 0.06447045 | 0.0981652 | 0.1129008 |

η = 0.49 ξ = 0.51 | c f | 0.0022864 0.002* | 0.0715832 | 0.1820249 0.182* | 0.2237071 | 0.2361822 | 0.2409651 |

*Nield and koznetsov results.

Da | 1 | 0.1 | 0.01 | 0.001 | 0.0001 | 0.00001 | |
---|---|---|---|---|---|---|---|

η = 1 / 3 ξ = 2 / 3 | c f | −0.156954 | −0.0979064 | 0.0180856 | 0.0860125 | 0.122236 | 0.142053 |

η = 1 / 4 ξ = 3 / 4 | c f | −0.241100 | −0.184620 | −0.0593438 | 0.0241292 | 0.0696405 | 0.0944781 |

η = 0.49 ξ = 0.51 | c f | 0.0020959 | 0.0705451 | 0.180369 | 0.221909 | 0.234224 | 0.239089 |

By letting t = τ n ( y − η ) in (7.61) we get

u ¯ 2 = c 2 τ n ∫ 0 τ n ( y − η ) A n ( t ) d t + d 2 τ n ∫ 0 τ n ( y − η ) B n ( t ) d t + π 2 p s i n ( p π ) M 2 ( τ n ) 3 ∫ 0 τ n ( y − η ) Z n ( t ) d t (69)

u ¯ 3 = ∫ ξ 1 u 3 d y = ∫ ξ 1 [ c 3 exp ( λ 3 y ) + d 3 exp ( − λ 3 y ) + 1 M 2 λ 3 2 ] d y = 1 λ 3 [ c 3 { exp ( λ 3 ) − exp ( λ 3 ξ ) } − d 3 { exp ( − λ 3 ) − exp ( − λ 3 ξ ) } + 1 − ξ M 2 λ 3 ] (70)

The dimensionless mean velocity across the configuration (that is, across the three layers) is defined as

u ¯ = u ¯ 1 + u ¯ 2 + u ¯ 3 = ∫ 0 η u 1 d y + ∫ η ξ u 2 d y + ∫ ξ 1 u 3 d y (71)

and represents a measure of the overall volume flux through the channel configuration.

The above expressions are evaluated using Maple and the values are listed in for n = 1 and n = 2 in

Da | Da = 1 | Da = 0.1 | Da = 0.01 | Da = 0.001 | |
---|---|---|---|---|---|

η = 1 / 3 ξ = 2 / 3 | u ¯ | 0.079395 0.0794* | 0.05692682 | 0.02071123 0.0207* | 0.007936325 |

η = 1 / 4 ξ = 3 / 4 | u ¯ | 0.07938781 | 0.05659447 | 0.01881449 | 0.00553387 |

η = 0.49 ξ = 0.51 | u ¯ | 0.07940428 0.0794* | 0.05735128 | 0.02355505 0.0236* | 0.01315206 |

Da | Da = 1 | Da = 0.1 | Da = 0.01 | |
---|---|---|---|---|

η = 1 / 3 ξ = 2 / 3 | u ¯ | 0.08744141 | 0.06403606 | 0.02670879 |

η = 1 / 4 ξ = 3 / 4 | u ¯ | 0.09902008 | 0.06954515 | 0.02749235 |

η = 0.49 ξ = 0.51 | u ¯ | 0.07945951 | 0.05759342 | 0.0238459 |

For a given transition layer thickness and a given value of n, the total volume flux decreases with decreasing Da. This is due to flow retardation for smaller permeability.

It should be noted that since the velocity computations for thick layers are accurate for Da as low as 0.001, so are the computations of the mean velocity.

Velocity profiles across the three-layered channel have been obtained for the various flow parameters and are graphed in Figures 3-9.

η = 1 3 , ξ = 2 3 , when Da = 1 and n = 1. This velocity profile is parabolic and is

close to what one might expect in the case of Poiseuille flow and the solution of the Navier-Stokes equations due to the high value of permeability. This parabolic shape is lost when Da is reduced, as shown in

For the case of n = 1, and thick transition layer,

The effect of varying n is illustrated in

attributed to higher momentum transfer from the adjacent layers that tend to increase the flow velocity there.

In this work, we considered the problem of flow through a porous medium over a free-space channel in the presence of a transition layer. This problem was treated by Nield and Kuznetsov [

Abu Zaytoon, M.S., Alderson, T.L. and Hamdan, M.H. (2018) A Study of Flow through a Channel Bounded by a Brinkman Transition Porous Layer. Journal of Applied Mathematics and Physics, 6, 264-282. https://doi.org/10.4236/jamp.2018.61025