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Stokes flow through a microchannel with periodical protuberances on the wall
Accepted Manuscript Stokes flow through a microchannel with periodical protuberances on the wall Jae-Tack Jeong, JeongSu Son PII: DOI: Reference:
S0997-7546(16)30225-4 http://dx.doi.org/10.1016/j.euromechflu.2016.09.022 EJMFLU 3071
To appear in:
European Journal of Mechanics B/Fluids
Received date: 31 May 2016 Accepted date: 27 September 2016 Please cite this article as: J.-T. Jeong, J. Son, Stokes flow through a microchannel with periodical protuberances on the wall, European Journal of Mechanics B/Fluids (2016), http://dx.doi.org/10.1016/j.euromechflu.2016.09.022 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Stokes Flow through a Microchannel with Periodical Protuberances on the Wall Jae-Tack Jeong*, JeongSu Son School of Mechanical Engineering, Chonnam National University, Gwangju 500-757, Korea ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Abstract In this study, we analyze the Stokes flow through a microchannel with semicircular periodic protuberances on the walls. The radius and period of semicircular protuberances and the stagger ratio of the staggered protuberance arrangement on both walls are arbitrary. Considering the periodicity and symmetry of the flow, the Fourier series expansion and least squared error method are applied. The stream function, pressure distribution in the flow region, and shear stress distribution on the semicircular protuberances are determined, and the results for some cases of arrangement of protuberances are presented in this paper. In particular, the average pressure gradient along the microchannel is calculated as a function of the size, period, and the stagger ratio of the protuberances on the walls. The results are compared with those obtained by lubrication analysis when the protuberances are close to the opposite protuberances or the opposite wall. As the limiting case of large periodicity, the induced pressure drops attributed to a single pair of semicircular protuberances on both walls and attributed to a single semicircular protuberance on a wall are also considered. Keywords: eigenfunction; lubrication analysis; microchannel; periodic flow; protuberance, pressure drop; stagger ratio; Stokes flow ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1. Introduction Microchannel flows have been studied for various applications, such as the micro heat exchanger for increasing the efficiency of heat transfer, micro nozzle for controlling the route of small satellites, and medical fields investigating synovial flow, capsular locomotive micro robots, and so on[1-3]. Wang analyzed the Stokes flow through a channel with longitudinal ribs[4], transverse fins[5], and transverse cylinders[6] of constant spacing and determined the flow resistance by using the method of eigenfunction expansion. Yang et al.[7] investigated the hydrodynamic forces and torques on a rotating cylinder in a narrow channel by lubrication analysis and scaling analysis and compared the results with numerical solutions for the case in which the gap between cylinder and wall is small. Pozrikidis[8] presented the change in shear stress on the surface of a circular tube with a permeable wall; viscous fluid flowed through to model the blood flow through a capillary vessel. Davis[9] studied two-dimensional creeping flow due to a periodic array of wall-attached barriers by the boundary singularity method. Kirsh[10] presented the flow along parallel cylinders with porous permeable shells by using Stokes and Brinkman equations. Jeong[11] theoretically analyzed the Stokes flow through a slit in a microchannel by using the slip boundary condition. Jeong and Yoon[12] and Jeong and Jang[13] analyzed the two-dimensional Stokes flow around a free-drifting circular cylinder inside a microchannel. As described above, many studies have been conducted on microchannel flow past a single obstacle or a series of obstacles. *
Corresponding author. Tel.: +82 62 530 1673, Fax.: +82 62 530 1689 E-mail address: [email protected]
Fig. 1. Geometry of a microchannel.
In this work, we considered the two-dimensional Stokes flow through a microchannel in which a series of semicircular protuberances is attached to each wall with constant spacing, as shown in Fig. 1. The presence of protuberances is considered to model the flow in the geometric complexity of microchannel. Previously, Son and Jeong[14] analyzed a specific microchannel flow where the protuberances attached to both walls of a microchannel were aligned with each other ( s = 0 in Fig. 1).
2. Problem description and analysis 2.1 Periodicity and symmetry of the flow Fig. 1 shows the two-dimensional viscous flow with a flow rate 2Q through a microchannel where semicircular protuberances are attached to each wall of the microchannel. The height of the channel is 2H ; the radius of the semicircular protuberances is a ; the constant spac-
ing between adjacent protuberances is 2b ; and the distance of stagger (positional dislocation) between the upper and lower protuberances is s . For convenience, we normalize the flow region with H and Q by setting H = 1 and Q = 1 . Now, if we define the stagger ratio as k≡
s , b
(1)
then it is enough to consider the range of 0 ≤ k ≤ 1 in our analysis. To avoid the cases in which the channel is blocked by protuberances, we should restrict the range of ( a, b)
where
0 ≤ a < min
{
}
( kb)2 + 4 / 2, 2
for
0 ≤ k ≤ 1 as shown in Fig. 2. The arrow in Fig. 2 indicates the possible range of a with fixed b for some value of k . For example, when k = 0 , the protuberances are aligned [14], and the range of a should be 0 ≤ a <1. For the flow in microchannel, the inertia terms in the Navier-Stokes equation are negligible, and the governing equation becomes the Stokes equation combined with the continuity equation:
∇ ⋅ v = 0,
(2)
∇p = μ∇ 2 v.
(3)
Here, v = (u , v ) is the velocity vector, and p is the pressure. We introduce the stream function ψ such that ∂ψ ∂ψ u= , v=− , ∂y ∂x
2
⎛ ∂2 ∂2 ⎞ ∇ 4ψ = ∇ 2 (∇ 2ψ ) = ⎜ 2 + 2 ⎟ ψ = 0 . ∂y ⎠ ⎝ ∂x
As the geometry of the flow field is periodic in the xdirection and the problem is linear, the flow is also periodic with a period of 2b in the x-direction. Therefore, we use the Fourier series expansion[16] for the stream function ψ ( x, y ) with period 2b about x , which can be expressed as follows:
ψ ( x, y ) = A0 + B0 y + C0 y 2 + D0 y 3 ∞
+
∑ ⎛⎜⎝ F ( y)cos n
n =1
nπ x nπ x ⎞ + Gn ( y )sin ⎟, b b ⎠
(6)
nπ , and the constants A0 , B0 , C0 , D0 and b the functions Fn ( y ) , Gn ( y ) (n = 1,2,3,") are unknowns.
where λn ≡
Note that it is sufficient to consider the flow in the range ( 0 ≤ x ≤ 2b , −1 ≤ y ≤ 1 ) only, because of the periodicity. Substituting Eq. (6) into Eq. (5), we obtain the ordinary differential equations for Fn ( y ) , Gn ( y ) :
Fn″″ ( y ) − 2λn 2 Fn″ + λn 4 Fn ( y ) = 0, G ″″ ( y ) − 2λ 2G ″ + λ 4G ( y ) = 0. n
(4)
(5)
n
n
n
n
(7a) (7b)
The solutions Fn ( y ) and Gn ( y ) of Eqs. (7a) and (7b) are expressed as follows:
Then, equations (3) and (4) indicate that the stream function satisfies the following biharmonic equation[15]:
Fn ( y ) = C1 cosh λn y + C2 sinh λn y + C3 y sinh λn y + C4 y cosh λn y ,
Gn ( y ) = C5 cosh λn y + C6 sinh λn y + C7 y sinh λn y + C8 y cosh λn y.
(8a) (8b)
Considering the symmetry of the flow in detail, we find that the stream function also has point symmetry about ( s / 2,0) and (b+ s / 2,0) :
ψ ( x, y ) = −ψ ( s − x, − y ) , ψ ( x, y ) = −ψ (2b + s − x, − y ) .
(9a) (9b)
Note that this symmetry comes from the linearity of the biharmonic equation (Eq. (5)). Substituting Eq. (9) in Eq. (8), the stream function in Eq. (6) can be expressed as follows: Fig. 2. The range of a, b with parameter k in our analysis to
ensure that the microchannel is not blocked.
ψ ( x, y ) = B0 y + D0 y 3 s ⎤ ⎡ ∞ ⎢ ( K n cosh λn y + Ln y sinh λn y )sin λn ( x − ) ⎥ (10) 2 + ⎢ ⎥, s n =1 ⎢ + ( P sinh λ y + Q y cosh λ y )cos λ ( x − ) ⎥ n n n n n 2 ⎦⎥ ⎣⎢
∑
where B0 , D0 and K n , Ln , Pn , Qn (n = 1,2,3,") are the unknown constants that must be determined from the boundary conditions.
s ⎤ ⎡ ⎢( Kn cosh λn y + Ln y sinh λn y)cos λn ( x − 2 ) ⎥ 0 = sin θ × λn ⎢ ⎥ ⎢−( P sinh λ y + Q y cosh λ y)sin λ ( x − s ) ⎥ n =1 n n n n ⎢⎣ n 2 ⎥⎦ ⎡ ⎤ ⎢ ⎥ 2 ⎢ B0 + 3D0 y ⎥ ⎢ N ⎥ K y λ sinh λ ⎧ n n ⎫ s ⎥ n x . − cosθ × ⎢+ ⎨ − sin λ ( ) ⎬ n ⎢ + Ln (sinh λn y + λn y cosh λn y) ⎭ 2 ⎥ ⎩ = 1 n ⎢ ⎥ ⎢ N ⎧λn Pn cosh λn y ⎫ s ⎥ ⎢+ ⎨ ⎬ cos λn ( x − ) ⎥ 2 ⎥⎦ ⎢⎣ n =1 ⎩+Qn (cosh λn y + λn y sinh λn y) ⎭ N
∑
∑
∑
2.2 Boundary conditions and numerical calculations Now, we apply the no-slip boundary condition to the surface of the semicircular protuberances and the walls of the microchannel. It is sufficient to consider the boundary conditions corresponding to 0 ≤ x ≤ 2b and 0 ≤ y ≤ 1 because of the point symmetry of the flow about ( s / 2,0) and (b + s / 2,0) expressed by Eq. (9). The boundary conditions are as follows.
(12b) ii) For a ≤ x ≤ 2b − a , y = 1 (on the wall of the microchannel),
1 = B0 + D0 s ⎤ ⎡ ⎢( K n cosh λn + Ln sinh λn )sin λn ( x − 2 ) ⎥ + ⎢ ⎥, s n =1 ⎢ + ( P sinh λ + Q cosh λ )cos λ ( x − ) ⎥ n n n n n ⎢⎣ 2 ⎥⎦ N
∂ψ ( x, y ) = 0 , ∂n x 2 + ( y − 1) 2 = a 2 and
i) ψ ( x, y ) = 1 , for
(11a)
( x − 2b) 2 + ( y − 1) 2 = a 2
∑
(12c)
(on the surface of the semicircular protuberances). ∂ψ ii) ψ ( x,1) = 1 , ( x,1) = 0 , (11b) ∂y for a ≤ x ≤ 2b − a (on the wall of the microchannel). By applying the stream function expression of Eq. (10) to the boundary conditions in Eq. (11a, b) and truncating the infinite series to N finite terms, the boundary conditions are expressed as follows: i) For x 2 + ( y − 1)2 = a 2 and ( x − 2b) 2 + ( y − 1) 2 = a 2 (on the surface of the semicircular protuberances), 1 = B0 y + D0 y 3 s ⎤ ⎡ ⎢( K n cosh λn y + Ln y sinh λn y )sin λn ( x − 2 ) ⎥ + ⎢ ⎥, s n =1 ⎢ + ( P sinh λ y + Q y cosh λ y )cos λ ( x − ) ⎥ n n n n n ⎢⎣ 2 ⎥⎦ N
∑
(12a)
0 = B0 + 3D0 s ⎤ ⎡ ⎢{λn Kn sinh λn + Ln (sinh λn + λn cosh λn )}sin λn ( x − 2 ) ⎥ + ⎢ ⎥. s n =1 ⎢ +{λ P cosh λ + Q (cosh λ + λ sinh λ )}cos λ ( x − ) ⎥ n n n n n n ⎢⎣ n n 2 ⎥⎦ N
∑
(12d) The boundary conditions in Eq. (12) involving (4 N + 2) unknown constants, namely B0 , D0 , and K n , Ln , Pn , Qn (n = 1,2,3,", N ) , must be satisfied for an infinite number of points on the boundary. If we apply boundary conditions (12) at finite M points almost evenly chosen on the boundary, then we have a system of 2M linear equations for (4 N + 2) unknowns. For the values M , N with 2M > 4 N + 2 , the number of equations is larger than the number of unknowns, and the method of least squares[17] can be applied to determine the unknowns B0 , D0 and Kn , Ln , Pn , Qn (n = 1,2,3,", N ) . The optimal choice of M , N should depend on the values of a, b and s , however, the numerical calculation shows that N ≈ 100 is sufficient to obtain a converged solution except the case of very narrow flow passages.
3. Results 3.1 Streamline pattern
After the values of unknown constants B0 , D0 , and K n , Ln , Pn , and Qn (n = 1,2,3,", N ) in Eq. (12) are determined numerically from the system of linear equations, the stream function ψ ( x, y ) can be obtained as a truncated form of Eq. (10):
ψ ( x, y ) = B0 y + D0 y 3 s ⎤ ⎡ ⎢( K n cosh λn y + Ln y sinh λn y )sin λn ( x − 2 ) ⎥ + ⎢ ⎥, s n =1 ⎢ + ( P sinh λ y + Q y cosh λ y )cos λ ( x − ) ⎥ n n n n n ⎢⎣ 2 ⎥⎦ N
∑
(13)
For a = 0 (no protuberances on the channel walls), B0 = 1.5, D0 = −0.5 , and K n , Ln , Pn , and Qn (n = 1,2,3,") vanish; the flow reduces to a plane Poiseuille flow through a smooth channel. Note that K n = Ln = 0 (n = 1,2,3,") for k = 0 (aligned arrangement), and K 2 n = L2 n = P2 n −1 = Q2 n −1 = 0 (n = 1,2,3,") for k = 1 (staggered arrangement). Streamlines can be drawn by calculating the stream function in Eq. (13) for any appropriate values of a , b , and s . It is convenient to show the streamlines in the range (0 ≤ x ≤ 2b , −1 ≤ y ≤ 1) by virtue of the periodicity. Some streamline patterns as the stagger ratio k increases for a = 0.7, b = 1 are shown in Fig. 3. In particular, Fig. 3(a) shows the case of aligned arrangement ( k = 0 ), and Fig. 3(f), the case of staggered arrangement ( k = 1 ). We can see the Moffatt viscous eddies[18] appearing in the corner between the protuberance and the wall. In this case, as k increases, the viscous eddies between the neighboring protuberances shrink and separate into two. Of course, this phenomenon depends on the magnitudes of a and b .
Fig. 3. Streamline patterns for a = 0.7, b = 1 with (a) k = 0 , (b) k = 0.2 , (c) k = 0.4 , (d) k = 0.6 , (e) k = 0.8 , and (f) k = 1 ( ∇ψ = 0.1 in the main stream, and ∇ψ = 0.002 in the viscous eddies).
3.2 Pressure distribution Pressure distribution is expressed using Eqs. (3), (4), and (13) as shown below.
p ( x, y ) s = 3D0 ( x − ) + 2μ 2
s ⎤ ⎡ ⎢Qn cosh λn y sin λn ( x − 2 ) ⎥ λn ⎢ ⎥. ⎢ − L sinh λ y cos λ ( x − s ) ⎥ n =1 n n ⎢⎣ n 2 ⎥⎦ (14) N
∑
In Eq. (14), the arbitrary constant is chosen to make the reference pressure p(s / 2,0) = 0 so that the pressure field in Eq. (14) has a point symmetry about ( s / 2,0) i.e., p ( x, y ) = − p ( s − x, − y ) . Fig. 4 shows the pressure distributions for the cases k = 0, 0.5,1 with a = 0.7, b = 2 . Note that for the cases k =0 and k =1, the pressure on the symmetric lines x = 0 and x = b is constant. The
pressure on the centerline y = 0 decreases monotonously as x increases. The magnitude of pressure drop in one period decreases as k increases probably because of the increase in the flow passage area.
We have checked that our results corresponding to b → 0 agree well with Eq. (16).
Fig. 5. The average pressure drop per unit length of the microchannel with k for a = 0.6,0.7,0.8 and b = 2 .
3.3 Lubrication analysis
Consider three flow passages of sizes d1 , d 2 , d 3 as shown in Fig. 6, where d1 ≡ s 2 + 4 − 2a ,
(17a)
d2 ≡ 2 − a ,
(17b)
d3 ≡ (2b − s ) + 4 − 2a . 2
(17c)
Fig. 4. The pressure distributions for a = 0.7, b = 2 with (a) k = 0 , (b) k = 0.5 , and (c) k = 1 .
The average pressure drop per unit length of microchannel is calculated as follows:
(ΔP) 2b p(0,0) − p(2b,0) = = −6 D0 , 2 μb 2μb
(15)
as (ΔP ) 2b ≡ p (0,0) − p (2b,0) = −12μbD0 from Eq. (14). Fig. 5 shows the graphs of the average pressure drop per unit length with k for a = 0.5,0.6,0.7 and b = 2 . This pressure drop per unit length of the microchannel decreases as k increases and increases as a increases. As b → 0 , the height of the channel becomes 2(1 − a ) , and the pressure gradient reduces to the well-known result as follows: ( ΔP ) 2b 3 → . (16) 2 μb (1 − a )3
Fig. 6. The sizes d1 , d 2 , d 3 of three flow passages between the protuberances and wall.
If these flow passages are reduced to small gaps, lubrication analysis[19] can be applied for the flow through these gaps; in such an analysis, the small gaps are regarded as a long narrow channel with slowly varying channel height. We derived the asymptotic expressions of the average pressure drop per unit length of the microchannel.
(ΔP) 2b 9π a −5/ 2 d1 → + d3−5/ 2 + 2 2d 2 −5/ 2 2μb 2b
{
}
.
(18)
as d1 , d 2 , d 3 → 0 independently. Note that if 0 ≤ k < 1 , then d1 < d 3 , and (ΔP) 2b 9π a −5/ 2 d1 → + 2 2d 2 −5/ 2 2μb 2b
{
}
,
(19)
while if k = 1 , then d1 = d 3 , and (ΔP) 2b 9π a −5/ 2 d1 → + 2d 2 −5/ 2 2μb b
{
}
.
(20)
Fig. 7. The average pressure drop per unit length of the microchannel (I) with d1 ( a = 1.25, b = 4) and (II) with d 2 (b = 4, s = 4) . The solid lines correspond to the results of the present study, and the broken lines correspond to the lubrication analysis results.[19]
Fig. 7 shows the average pressure drop (ΔP ) 2 b / 2 μb with the distance d1 or d 2 . As the gap size d1 or d 2
de-
creases, the pressure drop increases rapidly and follows the results of the lubrication analysis (indicated by the broken lines). In the case of very small gaps, the present results diverge from the lubrication analysis probably because of the difficulty in carrying out the numerical calculation for the large N necessary to satisfy the boundary condition for small gaps.
Fig. 8. Induced pressure drop due to (a) a single pair of aligned semicircular protuberances on both walls, and (b) a single semicircular protuberance on a wall.
As b → ∞ , our problem becomes similar to a flow in a microchannel with a single pair of aligned protuberances on both walls or a single protuberance on a wall. Fig. 8(a) shows the pressure drop (ΔP )1 induced by a pair of semicircular protuberances, and Fig. 8(b) shows that induced by a single semicircular protuberance. Here, the dashed lines represent the results of the lubrication analysis[19] expressed as (ΔP)1
μ
→
9π 2a (1 − a) −5/ 2 as a → 1 8
(21)
for a single pair of aligned semicircular protuberances, and as ( ΔP )1
μ
→ 9π 2a (2 − a ) −5/ 2 as a → 2
(22)
for a single semicircular protuberance. We also confirmed the accuracy of our results by comparing these results.
3.4 Shear stress on the surface of a protuberance
The shear stress on the surfaces of semicircular protuberances can be expressed in polar coordinates, as shown in Fig. 1.
τ rθ = μ r
∂ ⎛ uθ ⎞ ∂ ⎛ 1 ∂ψ ⎞ ⎜ ⎟ = −μ r ⎜ ⎟ ∂r ⎝ r ⎠r = a ∂r ⎝ r ∂r ⎠r = a
(23)
Fig. 9 shows the shear stress distributions on the surface of a semicircular protuberance for k = 0, 0.5, 1 with a = 0.7 and b = 1 . The shear stress changes its sign as θ increases to π / 2 , which indicates flow separation and the existence of viscous eddies at the corner between the protuberance and the channel wall.[18]
pressure drop was maximum for the aligned arrangement ( k = 0 ) and minimum for the staggered arrangement ( k = 1 ). For the case in which there existed small gaps ( d1 or d 2 → 0 ) through which the viscous flow passed, the present results were compared with those of lubrication analysis. In addition, the pressure drops induced by a single pair of semicircular protuberances and by a single semicircular protuberance on one wall were also studied by considering the limiting case of b → ∞ . We confirmed that the present method is efficient for analyz ing flow through microchannel with protuberances, as the streamline patterns from the results show even the weak Moffatt eddies clearly and as the pressure drop past a small gap approaches the results of the lubric ation analysis. The present analysis method can be extended to flow in microchannel with protuberances of different shapes also and to the tribology research on the surface roughness of channel walls with a series of protuberances.
Acknowledgement This study was financially supported by Chonnam National University in 2015.
References
Fig. 9. The shear stress on a protuberance surface for k = 0, 0.5, 1 with a = 0.7 and b = 1 .
4. Conclusions In this study, we analyzed the Stokes flow in a microchannel with semicircular protuberances attached to each wall of the microchannel at constant spacing 2b and stagger distance s . Considering the symmetry and periodicity of the flow, the Fourier series expansion method was applied. The streamline patterns and pressure distributions for some values of a , b , and s were presented. We observed that viscous eddies exist at the corners between the semicircular protuberances and the walls of the microchannel, and that adjacent eddies tend to merge as s decreases. The average pressure drop per unit channel length, which is a physically important parameter of the microchannel, was obtained as a function of three parameters, namely, the protuberance size a , the spacing b , and the distance of stagger (positional dislocation) s (= kb) . For given values of a and b , the average
[1] N.-T. Nguyen, Micromixers : Fundamentals, Design and Fabrication, 2nd ed. Elsevier, Waltham, ENG., 2012. [2] J. Berthier, P. Silberzan, Microfluidics for Biotechnology, 2nd ed. Artech House Inc., Norwood, USA, 2010. [3] Z.-X. Sun, Z.-Y. Li, Y.-L. He, W.-Q. Tao, Coupled solid (FVM)-fluid (DSMC) simulation of micro-nozzle with unstructured-grid, Micro-Nano fluid 7 (2009) 621. [4] C. Y. Wang, Flow in a channel with longitudinal ribs, J. Fluids Eng.-Trans. ASME 116 (1994) 233. [5] C. Y. Wang, Stokes flow through a transversely finned channel, J. Fluids Eng.-Trans. ASME 119 (1997) 110. [6] C. Y. Wang, Stokes flow through a channel obstructed by horizontal cylinders, Acta Mech. 157 (2002) 213. [7] J. Yang, C.W. Wolgemuth, G. Huber, Force and torque on a cylinder rotating in a narrow gap at low Reynolds number: Scaling and lubrication analyses, Phys. Fluids 25 (2013) 051901. [8] C. Pozrikidis, Stokes flow through a permeable tube, Arch. Appl. Mech. 80 (2010) 323. [9] A. M. J. Davis, Periodic blocking in parallel shear or channel flow at low Reynolds number, Phys. Fluids A 5 (1993) 800. [10] V. A. Kirsh, Stokes flow in periodic systems of parallel cylinders with porous permeable shells, Colloid. J. 68 (2006) 173. [11] J-T. Jeong, Two-dimensional Stokes flow through a slit in a microchannel with slip, J. Phys. Soc. Jpn. 75 (2006) 094401. [12] J.-T. Jeong, S.-H. Yoon, Two-dimensional Stokes flow around a circular cylinder in a microchannel, J. Mech. Sci.
Technol. 28 (2014) 573. [13] J.-T. Jeong, C.-S. Jang, Slow motion of a circular cylinder in a plane Poiseuille flow in a microchannel, Phys. Fluids 26 (2014) 123104. [14] J.S. Son, J.-T. Jeong, Stokes flow through a microchannel with protuberances of constant spacing, J. Mech. Eng. Trans. KSME. 39 (2015) 335. [15] P. N. Shankar, Slow Viscous Flows, Imperial College Press, London, 2007. [16] E. Kreyszig, Advanced Engineering Mathematics, 10th ed., Wiley & Sons, Inc., Hoboken, USA, 2011. [17] S. C. Chapra, Numerical Methods for Engineers, McGrawHill, New York, USA, 2012. [18] H. K. Moffatt, Viscous and resistive eddies near a sharp corner, J. Fluid Mech. 18 (1964) 1. [19] R. F. Day, H. A. Stone, Lubrication analysis and boundary integral simulations of a viscous micropump, J. Fluid Mech. 416 (2000) 197.
S0997-7546(16)30225-4 http://dx.doi.org/10.1016/j.euromechflu.2016.09.022 EJMFLU 3071
To appear in:
European Journal of Mechanics B/Fluids
Received date: 31 May 2016 Accepted date: 27 September 2016 Please cite this article as: J.-T. Jeong, J. Son, Stokes flow through a microchannel with periodical protuberances on the wall, European Journal of Mechanics B/Fluids (2016), http://dx.doi.org/10.1016/j.euromechflu.2016.09.022 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Stokes Flow through a Microchannel with Periodical Protuberances on the Wall Jae-Tack Jeong*, JeongSu Son School of Mechanical Engineering, Chonnam National University, Gwangju 500-757, Korea ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Abstract In this study, we analyze the Stokes flow through a microchannel with semicircular periodic protuberances on the walls. The radius and period of semicircular protuberances and the stagger ratio of the staggered protuberance arrangement on both walls are arbitrary. Considering the periodicity and symmetry of the flow, the Fourier series expansion and least squared error method are applied. The stream function, pressure distribution in the flow region, and shear stress distribution on the semicircular protuberances are determined, and the results for some cases of arrangement of protuberances are presented in this paper. In particular, the average pressure gradient along the microchannel is calculated as a function of the size, period, and the stagger ratio of the protuberances on the walls. The results are compared with those obtained by lubrication analysis when the protuberances are close to the opposite protuberances or the opposite wall. As the limiting case of large periodicity, the induced pressure drops attributed to a single pair of semicircular protuberances on both walls and attributed to a single semicircular protuberance on a wall are also considered. Keywords: eigenfunction; lubrication analysis; microchannel; periodic flow; protuberance, pressure drop; stagger ratio; Stokes flow ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1. Introduction Microchannel flows have been studied for various applications, such as the micro heat exchanger for increasing the efficiency of heat transfer, micro nozzle for controlling the route of small satellites, and medical fields investigating synovial flow, capsular locomotive micro robots, and so on[1-3]. Wang analyzed the Stokes flow through a channel with longitudinal ribs[4], transverse fins[5], and transverse cylinders[6] of constant spacing and determined the flow resistance by using the method of eigenfunction expansion. Yang et al.[7] investigated the hydrodynamic forces and torques on a rotating cylinder in a narrow channel by lubrication analysis and scaling analysis and compared the results with numerical solutions for the case in which the gap between cylinder and wall is small. Pozrikidis[8] presented the change in shear stress on the surface of a circular tube with a permeable wall; viscous fluid flowed through to model the blood flow through a capillary vessel. Davis[9] studied two-dimensional creeping flow due to a periodic array of wall-attached barriers by the boundary singularity method. Kirsh[10] presented the flow along parallel cylinders with porous permeable shells by using Stokes and Brinkman equations. Jeong[11] theoretically analyzed the Stokes flow through a slit in a microchannel by using the slip boundary condition. Jeong and Yoon[12] and Jeong and Jang[13] analyzed the two-dimensional Stokes flow around a free-drifting circular cylinder inside a microchannel. As described above, many studies have been conducted on microchannel flow past a single obstacle or a series of obstacles. *
Corresponding author. Tel.: +82 62 530 1673, Fax.: +82 62 530 1689 E-mail address: [email protected]
Fig. 1. Geometry of a microchannel.
In this work, we considered the two-dimensional Stokes flow through a microchannel in which a series of semicircular protuberances is attached to each wall with constant spacing, as shown in Fig. 1. The presence of protuberances is considered to model the flow in the geometric complexity of microchannel. Previously, Son and Jeong[14] analyzed a specific microchannel flow where the protuberances attached to both walls of a microchannel were aligned with each other ( s = 0 in Fig. 1).
2. Problem description and analysis 2.1 Periodicity and symmetry of the flow Fig. 1 shows the two-dimensional viscous flow with a flow rate 2Q through a microchannel where semicircular protuberances are attached to each wall of the microchannel. The height of the channel is 2H ; the radius of the semicircular protuberances is a ; the constant spac-
ing between adjacent protuberances is 2b ; and the distance of stagger (positional dislocation) between the upper and lower protuberances is s . For convenience, we normalize the flow region with H and Q by setting H = 1 and Q = 1 . Now, if we define the stagger ratio as k≡
s , b
(1)
then it is enough to consider the range of 0 ≤ k ≤ 1 in our analysis. To avoid the cases in which the channel is blocked by protuberances, we should restrict the range of ( a, b)
where
0 ≤ a < min
{
}
( kb)2 + 4 / 2, 2
for
0 ≤ k ≤ 1 as shown in Fig. 2. The arrow in Fig. 2 indicates the possible range of a with fixed b for some value of k . For example, when k = 0 , the protuberances are aligned [14], and the range of a should be 0 ≤ a <1. For the flow in microchannel, the inertia terms in the Navier-Stokes equation are negligible, and the governing equation becomes the Stokes equation combined with the continuity equation:
∇ ⋅ v = 0,
(2)
∇p = μ∇ 2 v.
(3)
Here, v = (u , v ) is the velocity vector, and p is the pressure. We introduce the stream function ψ such that ∂ψ ∂ψ u= , v=− , ∂y ∂x
2
⎛ ∂2 ∂2 ⎞ ∇ 4ψ = ∇ 2 (∇ 2ψ ) = ⎜ 2 + 2 ⎟ ψ = 0 . ∂y ⎠ ⎝ ∂x
As the geometry of the flow field is periodic in the xdirection and the problem is linear, the flow is also periodic with a period of 2b in the x-direction. Therefore, we use the Fourier series expansion[16] for the stream function ψ ( x, y ) with period 2b about x , which can be expressed as follows:
ψ ( x, y ) = A0 + B0 y + C0 y 2 + D0 y 3 ∞
+
∑ ⎛⎜⎝ F ( y)cos n
n =1
nπ x nπ x ⎞ + Gn ( y )sin ⎟, b b ⎠
(6)
nπ , and the constants A0 , B0 , C0 , D0 and b the functions Fn ( y ) , Gn ( y ) (n = 1,2,3,") are unknowns.
where λn ≡
Note that it is sufficient to consider the flow in the range ( 0 ≤ x ≤ 2b , −1 ≤ y ≤ 1 ) only, because of the periodicity. Substituting Eq. (6) into Eq. (5), we obtain the ordinary differential equations for Fn ( y ) , Gn ( y ) :
Fn″″ ( y ) − 2λn 2 Fn″ + λn 4 Fn ( y ) = 0, G ″″ ( y ) − 2λ 2G ″ + λ 4G ( y ) = 0. n
(4)
(5)
n
n
n
n
(7a) (7b)
The solutions Fn ( y ) and Gn ( y ) of Eqs. (7a) and (7b) are expressed as follows:
Then, equations (3) and (4) indicate that the stream function satisfies the following biharmonic equation[15]:
Fn ( y ) = C1 cosh λn y + C2 sinh λn y + C3 y sinh λn y + C4 y cosh λn y ,
Gn ( y ) = C5 cosh λn y + C6 sinh λn y + C7 y sinh λn y + C8 y cosh λn y.
(8a) (8b)
Considering the symmetry of the flow in detail, we find that the stream function also has point symmetry about ( s / 2,0) and (b+ s / 2,0) :
ψ ( x, y ) = −ψ ( s − x, − y ) , ψ ( x, y ) = −ψ (2b + s − x, − y ) .
(9a) (9b)
Note that this symmetry comes from the linearity of the biharmonic equation (Eq. (5)). Substituting Eq. (9) in Eq. (8), the stream function in Eq. (6) can be expressed as follows: Fig. 2. The range of a, b with parameter k in our analysis to
ensure that the microchannel is not blocked.
ψ ( x, y ) = B0 y + D0 y 3 s ⎤ ⎡ ∞ ⎢ ( K n cosh λn y + Ln y sinh λn y )sin λn ( x − ) ⎥ (10) 2 + ⎢ ⎥, s n =1 ⎢ + ( P sinh λ y + Q y cosh λ y )cos λ ( x − ) ⎥ n n n n n 2 ⎦⎥ ⎣⎢
∑
where B0 , D0 and K n , Ln , Pn , Qn (n = 1,2,3,") are the unknown constants that must be determined from the boundary conditions.
s ⎤ ⎡ ⎢( Kn cosh λn y + Ln y sinh λn y)cos λn ( x − 2 ) ⎥ 0 = sin θ × λn ⎢ ⎥ ⎢−( P sinh λ y + Q y cosh λ y)sin λ ( x − s ) ⎥ n =1 n n n n ⎢⎣ n 2 ⎥⎦ ⎡ ⎤ ⎢ ⎥ 2 ⎢ B0 + 3D0 y ⎥ ⎢ N ⎥ K y λ sinh λ ⎧ n n ⎫ s ⎥ n x . − cosθ × ⎢+ ⎨ − sin λ ( ) ⎬ n ⎢ + Ln (sinh λn y + λn y cosh λn y) ⎭ 2 ⎥ ⎩ = 1 n ⎢ ⎥ ⎢ N ⎧λn Pn cosh λn y ⎫ s ⎥ ⎢+ ⎨ ⎬ cos λn ( x − ) ⎥ 2 ⎥⎦ ⎢⎣ n =1 ⎩+Qn (cosh λn y + λn y sinh λn y) ⎭ N
∑
∑
∑
2.2 Boundary conditions and numerical calculations Now, we apply the no-slip boundary condition to the surface of the semicircular protuberances and the walls of the microchannel. It is sufficient to consider the boundary conditions corresponding to 0 ≤ x ≤ 2b and 0 ≤ y ≤ 1 because of the point symmetry of the flow about ( s / 2,0) and (b + s / 2,0) expressed by Eq. (9). The boundary conditions are as follows.
(12b) ii) For a ≤ x ≤ 2b − a , y = 1 (on the wall of the microchannel),
1 = B0 + D0 s ⎤ ⎡ ⎢( K n cosh λn + Ln sinh λn )sin λn ( x − 2 ) ⎥ + ⎢ ⎥, s n =1 ⎢ + ( P sinh λ + Q cosh λ )cos λ ( x − ) ⎥ n n n n n ⎢⎣ 2 ⎥⎦ N
∂ψ ( x, y ) = 0 , ∂n x 2 + ( y − 1) 2 = a 2 and
i) ψ ( x, y ) = 1 , for
(11a)
( x − 2b) 2 + ( y − 1) 2 = a 2
∑
(12c)
(on the surface of the semicircular protuberances). ∂ψ ii) ψ ( x,1) = 1 , ( x,1) = 0 , (11b) ∂y for a ≤ x ≤ 2b − a (on the wall of the microchannel). By applying the stream function expression of Eq. (10) to the boundary conditions in Eq. (11a, b) and truncating the infinite series to N finite terms, the boundary conditions are expressed as follows: i) For x 2 + ( y − 1)2 = a 2 and ( x − 2b) 2 + ( y − 1) 2 = a 2 (on the surface of the semicircular protuberances), 1 = B0 y + D0 y 3 s ⎤ ⎡ ⎢( K n cosh λn y + Ln y sinh λn y )sin λn ( x − 2 ) ⎥ + ⎢ ⎥, s n =1 ⎢ + ( P sinh λ y + Q y cosh λ y )cos λ ( x − ) ⎥ n n n n n ⎢⎣ 2 ⎥⎦ N
∑
(12a)
0 = B0 + 3D0 s ⎤ ⎡ ⎢{λn Kn sinh λn + Ln (sinh λn + λn cosh λn )}sin λn ( x − 2 ) ⎥ + ⎢ ⎥. s n =1 ⎢ +{λ P cosh λ + Q (cosh λ + λ sinh λ )}cos λ ( x − ) ⎥ n n n n n n ⎢⎣ n n 2 ⎥⎦ N
∑
(12d) The boundary conditions in Eq. (12) involving (4 N + 2) unknown constants, namely B0 , D0 , and K n , Ln , Pn , Qn (n = 1,2,3,", N ) , must be satisfied for an infinite number of points on the boundary. If we apply boundary conditions (12) at finite M points almost evenly chosen on the boundary, then we have a system of 2M linear equations for (4 N + 2) unknowns. For the values M , N with 2M > 4 N + 2 , the number of equations is larger than the number of unknowns, and the method of least squares[17] can be applied to determine the unknowns B0 , D0 and Kn , Ln , Pn , Qn (n = 1,2,3,", N ) . The optimal choice of M , N should depend on the values of a, b and s , however, the numerical calculation shows that N ≈ 100 is sufficient to obtain a converged solution except the case of very narrow flow passages.
3. Results 3.1 Streamline pattern
After the values of unknown constants B0 , D0 , and K n , Ln , Pn , and Qn (n = 1,2,3,", N ) in Eq. (12) are determined numerically from the system of linear equations, the stream function ψ ( x, y ) can be obtained as a truncated form of Eq. (10):
ψ ( x, y ) = B0 y + D0 y 3 s ⎤ ⎡ ⎢( K n cosh λn y + Ln y sinh λn y )sin λn ( x − 2 ) ⎥ + ⎢ ⎥, s n =1 ⎢ + ( P sinh λ y + Q y cosh λ y )cos λ ( x − ) ⎥ n n n n n ⎢⎣ 2 ⎥⎦ N
∑
(13)
For a = 0 (no protuberances on the channel walls), B0 = 1.5, D0 = −0.5 , and K n , Ln , Pn , and Qn (n = 1,2,3,") vanish; the flow reduces to a plane Poiseuille flow through a smooth channel. Note that K n = Ln = 0 (n = 1,2,3,") for k = 0 (aligned arrangement), and K 2 n = L2 n = P2 n −1 = Q2 n −1 = 0 (n = 1,2,3,") for k = 1 (staggered arrangement). Streamlines can be drawn by calculating the stream function in Eq. (13) for any appropriate values of a , b , and s . It is convenient to show the streamlines in the range (0 ≤ x ≤ 2b , −1 ≤ y ≤ 1) by virtue of the periodicity. Some streamline patterns as the stagger ratio k increases for a = 0.7, b = 1 are shown in Fig. 3. In particular, Fig. 3(a) shows the case of aligned arrangement ( k = 0 ), and Fig. 3(f), the case of staggered arrangement ( k = 1 ). We can see the Moffatt viscous eddies[18] appearing in the corner between the protuberance and the wall. In this case, as k increases, the viscous eddies between the neighboring protuberances shrink and separate into two. Of course, this phenomenon depends on the magnitudes of a and b .
Fig. 3. Streamline patterns for a = 0.7, b = 1 with (a) k = 0 , (b) k = 0.2 , (c) k = 0.4 , (d) k = 0.6 , (e) k = 0.8 , and (f) k = 1 ( ∇ψ = 0.1 in the main stream, and ∇ψ = 0.002 in the viscous eddies).
3.2 Pressure distribution Pressure distribution is expressed using Eqs. (3), (4), and (13) as shown below.
p ( x, y ) s = 3D0 ( x − ) + 2μ 2
s ⎤ ⎡ ⎢Qn cosh λn y sin λn ( x − 2 ) ⎥ λn ⎢ ⎥. ⎢ − L sinh λ y cos λ ( x − s ) ⎥ n =1 n n ⎢⎣ n 2 ⎥⎦ (14) N
∑
In Eq. (14), the arbitrary constant is chosen to make the reference pressure p(s / 2,0) = 0 so that the pressure field in Eq. (14) has a point symmetry about ( s / 2,0) i.e., p ( x, y ) = − p ( s − x, − y ) . Fig. 4 shows the pressure distributions for the cases k = 0, 0.5,1 with a = 0.7, b = 2 . Note that for the cases k =0 and k =1, the pressure on the symmetric lines x = 0 and x = b is constant. The
pressure on the centerline y = 0 decreases monotonously as x increases. The magnitude of pressure drop in one period decreases as k increases probably because of the increase in the flow passage area.
We have checked that our results corresponding to b → 0 agree well with Eq. (16).
Fig. 5. The average pressure drop per unit length of the microchannel with k for a = 0.6,0.7,0.8 and b = 2 .
3.3 Lubrication analysis
Consider three flow passages of sizes d1 , d 2 , d 3 as shown in Fig. 6, where d1 ≡ s 2 + 4 − 2a ,
(17a)
d2 ≡ 2 − a ,
(17b)
d3 ≡ (2b − s ) + 4 − 2a . 2
(17c)
Fig. 4. The pressure distributions for a = 0.7, b = 2 with (a) k = 0 , (b) k = 0.5 , and (c) k = 1 .
The average pressure drop per unit length of microchannel is calculated as follows:
(ΔP) 2b p(0,0) − p(2b,0) = = −6 D0 , 2 μb 2μb
(15)
as (ΔP ) 2b ≡ p (0,0) − p (2b,0) = −12μbD0 from Eq. (14). Fig. 5 shows the graphs of the average pressure drop per unit length with k for a = 0.5,0.6,0.7 and b = 2 . This pressure drop per unit length of the microchannel decreases as k increases and increases as a increases. As b → 0 , the height of the channel becomes 2(1 − a ) , and the pressure gradient reduces to the well-known result as follows: ( ΔP ) 2b 3 → . (16) 2 μb (1 − a )3
Fig. 6. The sizes d1 , d 2 , d 3 of three flow passages between the protuberances and wall.
If these flow passages are reduced to small gaps, lubrication analysis[19] can be applied for the flow through these gaps; in such an analysis, the small gaps are regarded as a long narrow channel with slowly varying channel height. We derived the asymptotic expressions of the average pressure drop per unit length of the microchannel.
(ΔP) 2b 9π a −5/ 2 d1 → + d3−5/ 2 + 2 2d 2 −5/ 2 2μb 2b
{
}
.
(18)
as d1 , d 2 , d 3 → 0 independently. Note that if 0 ≤ k < 1 , then d1 < d 3 , and (ΔP) 2b 9π a −5/ 2 d1 → + 2 2d 2 −5/ 2 2μb 2b
{
}
,
(19)
while if k = 1 , then d1 = d 3 , and (ΔP) 2b 9π a −5/ 2 d1 → + 2d 2 −5/ 2 2μb b
{
}
.
(20)
Fig. 7. The average pressure drop per unit length of the microchannel (I) with d1 ( a = 1.25, b = 4) and (II) with d 2 (b = 4, s = 4) . The solid lines correspond to the results of the present study, and the broken lines correspond to the lubrication analysis results.[19]
Fig. 7 shows the average pressure drop (ΔP ) 2 b / 2 μb with the distance d1 or d 2 . As the gap size d1 or d 2
de-
creases, the pressure drop increases rapidly and follows the results of the lubrication analysis (indicated by the broken lines). In the case of very small gaps, the present results diverge from the lubrication analysis probably because of the difficulty in carrying out the numerical calculation for the large N necessary to satisfy the boundary condition for small gaps.
Fig. 8. Induced pressure drop due to (a) a single pair of aligned semicircular protuberances on both walls, and (b) a single semicircular protuberance on a wall.
As b → ∞ , our problem becomes similar to a flow in a microchannel with a single pair of aligned protuberances on both walls or a single protuberance on a wall. Fig. 8(a) shows the pressure drop (ΔP )1 induced by a pair of semicircular protuberances, and Fig. 8(b) shows that induced by a single semicircular protuberance. Here, the dashed lines represent the results of the lubrication analysis[19] expressed as (ΔP)1
μ
→
9π 2a (1 − a) −5/ 2 as a → 1 8
(21)
for a single pair of aligned semicircular protuberances, and as ( ΔP )1
μ
→ 9π 2a (2 − a ) −5/ 2 as a → 2
(22)
for a single semicircular protuberance. We also confirmed the accuracy of our results by comparing these results.
3.4 Shear stress on the surface of a protuberance
The shear stress on the surfaces of semicircular protuberances can be expressed in polar coordinates, as shown in Fig. 1.
τ rθ = μ r
∂ ⎛ uθ ⎞ ∂ ⎛ 1 ∂ψ ⎞ ⎜ ⎟ = −μ r ⎜ ⎟ ∂r ⎝ r ⎠r = a ∂r ⎝ r ∂r ⎠r = a
(23)
Fig. 9 shows the shear stress distributions on the surface of a semicircular protuberance for k = 0, 0.5, 1 with a = 0.7 and b = 1 . The shear stress changes its sign as θ increases to π / 2 , which indicates flow separation and the existence of viscous eddies at the corner between the protuberance and the channel wall.[18]
pressure drop was maximum for the aligned arrangement ( k = 0 ) and minimum for the staggered arrangement ( k = 1 ). For the case in which there existed small gaps ( d1 or d 2 → 0 ) through which the viscous flow passed, the present results were compared with those of lubrication analysis. In addition, the pressure drops induced by a single pair of semicircular protuberances and by a single semicircular protuberance on one wall were also studied by considering the limiting case of b → ∞ . We confirmed that the present method is efficient for analyz ing flow through microchannel with protuberances, as the streamline patterns from the results show even the weak Moffatt eddies clearly and as the pressure drop past a small gap approaches the results of the lubric ation analysis. The present analysis method can be extended to flow in microchannel with protuberances of different shapes also and to the tribology research on the surface roughness of channel walls with a series of protuberances.
Acknowledgement This study was financially supported by Chonnam National University in 2015.
References
Fig. 9. The shear stress on a protuberance surface for k = 0, 0.5, 1 with a = 0.7 and b = 1 .
4. Conclusions In this study, we analyzed the Stokes flow in a microchannel with semicircular protuberances attached to each wall of the microchannel at constant spacing 2b and stagger distance s . Considering the symmetry and periodicity of the flow, the Fourier series expansion method was applied. The streamline patterns and pressure distributions for some values of a , b , and s were presented. We observed that viscous eddies exist at the corners between the semicircular protuberances and the walls of the microchannel, and that adjacent eddies tend to merge as s decreases. The average pressure drop per unit channel length, which is a physically important parameter of the microchannel, was obtained as a function of three parameters, namely, the protuberance size a , the spacing b , and the distance of stagger (positional dislocation) s (= kb) . For given values of a and b , the average
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