© 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advances in Fluid Mechanics IV, M Rahman, R Verhoeven and CA Brebbia (Editors). ISBN 1-85312-910-0
Stokes flow between sinusoidal walls
G. Bizzarri', V. Di Federico &L S.Cintoli2 'Dipavtimento di Architettwa, Univevsitti di Fevrava, Italy 2D.I.S.TA.R.T. - Idraulica, Universitcidi Bologna, Italy
Abstract
In this paper, we study two-dimensional Stokes flow between sinusoidal walls. A stream function is introduced, thustransforming the Stokes equation into a biharmonic one, whose solution is then derived for a single periodic cell of length equal to the wall fluctuation wavelength, and for a given pressure drop. Relevant boundary conditions arethe no-slip and no-flow conditions on the boundary, aswell as those deriving from the periodicity and an auxiliary condition based on an energy argument. For such a mathematical problem, an approximate solution is possiblevia a series expansion in terms of a small parameter equal to the ratio between the meanchannel width and the wavelength. We present closed-form second-order expressions for stream function, flow rate, and velocity components, and discuss the implications of the zero-order solution (lubrication approximation) for different values of two dimensionless parameters. Expressions derived for the velocity components show flow reversal for strong channel sinuosity; they will be useful for several purposes, such as study of solute transport in rough-walled fractures or of heat and mass transfer in conduitswith wavy walls.
Introduction
Laminar flow in complexgeometries is of interest in several fields, such as environmental engineering, bioengineering, and subsurface fluid mechanics. Among those, if special interest is modeling of fluid flow and solute transport in rock fractures, whichhas important practical implications for the design of nuclear waste repositories. The geometry of fractured media is best described as © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advances in Fluid Mechanics IV, M Rahman, R Verhoeven and CA Brebbia (Editors). ISBN 1-85312-910-0 324 Advmces irl Fluid Mechunks W a network of fractures; however, comprehension of the flow and transport mechanism at the single fracture scale is an essential step to modeling complex networks. Early attempts to model flow in a single fracture adopted the parallel plate model [l]; later, laboratory and field investigations [2] evidenced a strong degree of variability in the fracture aperture, i.e. the distance between the fracture walls. To represent this variability, the most general approach is to describe the aperture as a correlated random field [3 and references therein]; in the simplest case of a binary distribution fixed in space, this reduces to a modified parallel plate model including contact areas; this is especially appealing since contact areas are easily measurable, and flow in a fracture with cylindncal asperities is more easily amenable to numerical or analytical treatment [4-5 and references therein]. Other researchers adopted m aperture density distribution fixed in space rather than a statistical distribution, examining flow in channels with idealized wall profiles, such as sawtooth or sinusoidal [6-8]; the conceptual model most commonly adopted is the lubrication approximation. A more refined approach is adopted in Kitanidis and Dykaar [9], with the inclusion of higher- order terms in the Stokes equation to better describe the modification in the flow field due to channel sinuosity; their formulation is retaken here, deriving explicit expression for flow rate and velocity in a variable width channel with a sinusoidal wall profile.
Y
L .. ..~...... ~ . ~ X
Figure 1: Channel geometry.
Problem formulation
Consider a two-dimensional, symmetric channel whose width is given bya sinusoidal,periodic function (Figure 1)
w(x)=< w > +2asin(2nx / L), (1) where
Steady-state isothermal slow flow of an incompressible Newtonian viscous fluid within a gravitational field of force is governedby the continuity and Navier-Stokes equations; forlow values of the Reynolds number, the latter reduces to the Stokesequation,yielding
v.u=o, (4
VP=pV%, (3) where U = (U, v) is the velocity field, P = pfpgz is the reduced pressure (z is elevation),p is the pressure, p and p are the fluid density and viscosity. For steady state flow betweentwo parallel plates separated by the mean width
where AP 0 is the pressure drop m the x direction in a periodic cell of dimensionL. For a variable distance between walls given by (l), we introduce the stream functionY, linked to the velocitycomponents by
zl=aY/ay, v=-a?/ax, (543) so that eqn (2) is automaticallyverified, and eqn (3) yields
v4YJ= 0. (6)
Relevant boundary conditions include the no-flux and no-slip boundary conditions,and those derivingfrom the periodicity
Finally, the auxiliary condition that the viscous dissipation within the unit cell of volume V equals the work done by surface forces on the same volume [1l]is expressedas
where © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advances in Fluid Mechanics IV, M Rahman, R Verhoeven and CA Brebbia (Editors). ISBN 1-85312-910-0 326 Advmces irl Fluid Mechunks W
Now, all distances are made dimensionless via the lengthscale
L])@dxdy = Q. 6L 0 -h
An approximate analytic solution is then derived as a series in terms of the parameter
X q = -= a, L where h = wl2 is equal to
h(q)= + asin(2nq), (13) 12 the solution is sought in terms of the rescaled variables in the form
Applying eqn (14) to eqn (6) and its boundary conditions, and separating terms of the same order in E, a sequence of boundary value problem is obtained. The auxiliary condition given by eqn (10) takes the form
1 ~h(q)@dxdy= Q 0-1
Resulting expressions for flow rate and stream function, to second-order in E, are respectively [9] © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advances in Fluid Mechanics IV, M Rahman, R Verhoeven and CA Brebbia (Editors). ISBN 1-85312-910-0
(€2)
(zz) ‘
‘ (12)
no!ssnas!p pm sqnsa~
(02)
(61)
(81)
(L11 ‘ [hPq3=9 1- (91) © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advances in Fluid Mechanics IV, M Rahman, R Verhoeven and CA Brebbia (Editors). ISBN 1-85312-910-0 328 Advmres irl Fluid Merhunks W yielding
1 EN?, 3Q07m cos(2nq) L 1+ 2a sen(2nq)
1 B?2 4-L5)=--= {~Q,,n’a h 35 1+2asin(2nq) 20
[2n+sin(2nq)+6acos2(2nq)~5~‘-6c2+1)+1Q2(1-<’~, 4
+ 2a [2~z+sin(2nq)+6acos~(2nq))&~-6k3+&)l+ 1+2asin(2nq)
Figure 2a shows the dimensionless flow rate Q as a function of dimensionless wall oscillation amplitude a (U 0.5for the channel opening to be larger than zero at throats), for various values of E (equal to the inverse of dimensionless channel wavelength); Figure 2b does the same for the relative second-order flow rate correction AQ21Qo = z2Q21Qo.The Figures show that the flow rate decreases for increasing a and E ; the sensitivity of flow rate to changes in the value of E is moderate; the relative importance of the second-order correction is maximum for a varyingbetween 0.30 and 0.40. Dimensionless longitudinal and transverse velocities are shown in Figures 3- 7 as a function of dimensionless transverse coordinate y, at different locations along the longitudinal axis, corresponding to different values of q (with q=O and q=O.50 corresponding to the average cross-section, q=O.25 corresponding to the maximum cross-section, or hollow, and q=O.75 corresponding to the minimum cross-section, or throat), and for various values of E and a. As a result of the variable channel width, the range of the transverse coordinate y varies according to the cross-section location and the amplitude of wall oscillations. Since the velocity field is symmetrical about the x axis, the velocity components are show along half of the channel width. Two different velocities are shown in each figure: the thin curve represents the zero-order approximation (uo or vo, corresponding to the lubrication approximation), while the thick curve shows the total velocity component (U or v), inclusive of the second-ordercorrection. © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advances in Fluid Mechanics IV, M Rahman, R Verhoeven and CA Brebbia (Editors). ISBN 1-85312-910-0 Ad~mcc~sill Fluid Mdxznics IV 329
The longitudinal velocity U is depicted in Figures 3-5 for q=O (Figure 3), yo.25 (Figure 4), and q=O.75 (Figure 5), and forthe following sets of parameters: ~0.4,a=0.2 (column a)) and ~=0.4,a=0.4 (column b)). Inspection of the different velocity profiles shows that, as expected, the longitudinal velocity is larger at throats (q=O.75) than at hollows (q=O.25), and it markedly decreases for increasing valuesof the wall amplitude oscillations. Likewise, velocity values drop significantly for increasing valuesof E, corresponding to more rapid oscillations ofthe wall profile in space, as verified by plotting a second set of velocity profiles corresponding to ~=0.2(not shown). Moreover, the second-order term brings about a non-negligible reduction in the velocity with respect to the lubrication approximation. The latter effect is more significant for larger values of a and E; when both parameters are large (correspondmg to strong sinuosity of the channel), longitudinal velocities near the wall become negative if the second-order correction is taken into account (Figures 3b and 4b). The transverse velocity v is depicted in Figures 6-7 for q=O, adopting the following values of parameters: ~=0.2or 0.4 (Figure 6 or 7) and a=0.2 or 0.4 (column a) or b)). Inspection of the curves shows that the transverse velocity is about one order of magnitude less than the longitudinal one; like the latter, it significantly decreases for increasing values of a and E. Again, the second-order term implies a significant reduction in the velocity with respect to the lubrication approximation; for large sinuosity, transverse velocities near the wall are directed towards the channel axis (Figures 7a and 7b). The relative importance of tie second-order correction is enhanced for strong sinuosity of the channel; in any case, its influence is more significant for the transverse than for the longitudinal velocity.
Conclusions
The main goal of this study is to study higher-order effect in Stokes flow in a variable width channel. It is demonstratedthat the total flow rate and the velocity distribution within the channel are fimctions of the wall oscillation amplitude a, and of the inverse of channel wavelength E. Other major results are: 1. The sinuosity of the channel lowers the flow rate, more so for high values of a and E ; the sensitivity of flow rate to changes in the value of E is moderate; the relative importance ofthe second-order correction is maximum for a varyingbetween 0.30 and 0.40; 2. Velocity values markedly decrease for increasing values of a and E, more so in the transverse than in the longitudinal direction; when second-order effects are included, flow reversal is evident close to the channel wall, indicating the existence of nonturbulentvortices; 3. The second-order effect has a higher impact onthe distribution of point velocities within the channel than on the overall flow rate, since the flow tends to concentrate in the channel central section; in turn, this indicates that © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advances in Fluid Mechanics IV, M Rahman, R Verhoeven and CA Brebbia (Editors). ISBN 1-85312-910-0 330 Advmces irl Fluid Mechunks W
higher-order effects may be relevant in studying solute transport in variable width channels suchas subsurface fixtures, heat and mass transfer in conduitswith wavy walls, or physiology of humanbeings and animals [9].
Acknowledgements
The work was supported by the Italian Ministero dell’Universitae della Ricerca Scientifica e Tecnologica (MURST)40% “Metodologie innovative per il monitoraggio, la gestione ed il controllo quali-quantitativo delle acque sotterranee” and 60% “Ruolo dell’eterogeneitane1 controllo della circolazione sotterranea”.
References
[l] Bear, J., Dynamics ofJluids in porous media, Elsevier:Amsterdam, 1972. [2] Johns R.A., Steude, J.S., Castanier, L.M., & Roberts, P.V., Nondestructive measurements of fracture aperture in crystalline rock cores using X ray computedtomography,J Geophys.Res., 92, pp. 1889-1900,1993. [3] Di Federico, V., Non-Newtonianflow in a variable aperture fracture, Transport in Porous Media, 30(1), pp. 75-86, 1998. [4] Muralhidar, K., Flow and transport in single rock fixtures, J. Fluid Mech., 215, pp. 481-502, 1990. [5] Tsay, R.-Y., & Weinbaum,S., Viscous flow in a channel with periodic cross-bridging fibers: exact solution and Brinkman approximation, J. Fluid Mech., 226, pp. 125-148, 1991. [6] Zimmerman, R.W., Kumar,S., & Bodvarsson, G.S., Lubrication theory analysis of the permeability of rough-walled fractures, kt. J. Rock Mech. Min. Sci. B Geomech.Abstr., 28, pp. 325331, 1991. [7] Brown, S.R., Stockman, H.W., & Reeves, S.J., Applicability of the Reynolds equation formodeling fluid flow between rough surfaces, Geophys. Res. Letters, 22, pp. 2537-2540, 1995. [8] Zimmerman, R.W., & Bodvarsson, G.S., Hydraulic conductivity of rock fixtures, Transport in Porous Media, 23, pp. 1-30, 1996. [S] Kitanidis, P.K., Dykaar, B.B., Stokes flow in a slowly varying two- dimensionalperiodicpore, Transport in Porous Media, 26, pp. 89-98, 1997. [lo]Happel, J. &L Brenner, E., Low-Reynolds Number Hydrodynamics, Kluwer: Boston, 1983. [ll]Bird, R.B., Stewart, W.E. & Lightfoot, E.N., Transport Phenomena, J. Wiley:New York, 1961. 1121Hinch, E.J., Perturbation methods, CambridgeUniv. Press: Cambridge, 1991. [13]Gradshteyn, IS., & Ryzhik, I.M., Table of Integrals, Series, and Products, ed. A.Jeffrey,Academic Press:New York, 1994. © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advances in Fluid Mechanics IV, M Rahman, R Verhoeven and CA Brebbia (Editors). ISBN 1-85312-910-0 Admcc~sill Fluid MdxznicsIV 331
1.0 / Q 0.8
0.6
0.4
0.2
0.0 0 0.1 0.20.30.4 a 0.5 0.0 0.10.2 0.3 0.4 a 0.5
Figure 2: a) Dimensionless flow rate as a function of a for various values of E (0.05, 0.1, 0.2, 0.4 from top to bottom); b) Same for relative flow rate second- order correction.
...... ,. __ ,, . - .,..,,, ,. Y b) a = 0.4, E = 0.4; 0.4 -
0.3 - 0.3 -
0.2 - 0.2 -
0.1 - 0.1 -
0 0.80.60.40.2 1 -0.04 0 0.040.080.12 Figure 3: Dimensionless longitudinal velocity as a fimction ofyfor q=O.
0.7 b) a = 0.4, E = 0.4 0.6 ::: j Y 0.7 - 0.5 0.6 - 0.4 0.5 - 0.3 0.4 - 0.3 - 0.2 0.2 - 0.1 0.1 - 0 O.l0 L 7 0 0.60.40.2 0.8 -0.02 0 0.060.040.02 Figure 4: Dimensionless longitudinal velocity as a function ofy for q=O.25. © 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Web: www.witpress.com Email [email protected] Paper from: Advances in Fluid Mechanics IV, M Rahman, R Verhoeven and CA Brebbia (Editors). ISBN 1-85312-910-0 332 Advmces irl Fluid Mechunks W
0.3 0.1 Y Y 0.08 0.2 0.06
0.04 0.1 0.02
0 0 0 1.20.80.4 " 1.6 0 0.20.1 0.3 0.50.4 Figure 5: Dimensionless longitudinal velocity as a function ofy for q=O.75.
0.5 0.5 Y Y 0.4 0.4 ,
0.3 0.3
0.2 0.2
0.1 0.1
0 1 0 0 0.030.06 ' 0.09 0 0.006 0.012 '0.018 Figure 6: Dimensionless transverse velocity as a fimction ofyfor q=O and ~=0.2.
0.5 0.5 Y Y 0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 , I 0 I -0.020.02 0.06 0.1 0.14" 0.18-0.01 0 0.01 0.02 0.03' 0.04 Figure 7: Dimensionless transverse velocity'las a fimction ofyfor q=O and ~=0.4.