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The Stokes-flow friction on a wedge surface by the sliding of a plate
Fluid Dynamics Research 30 (2002) 93–106
The Stokes- ow friction on a wedge surface by the sliding of a plate Jun Sang Parka , Jae Min Hyunb; ∗ a
Department of Mechanical Engineering, Halla University, San 66, HeungUp, Wonju, Kangwondo 220-712, South Korea b Department of Mechanical Engineering, Korea Advanced Institute of Science & Technology, 373-1 Kusong-Dong, Yusung-gu, Taejon 305-701, South Korea Received 4 June 2001; received in revised form 1 October 2001; accepted 6 October 2001
Abstract A theoretical analysis is conducted on the frictional stress in the Stokes ow in a channel. A triangular wedge is attached on the bottom plate, and the ow is induced by the top sliding plate. The governing harmonic equation is solved in the transformed -plane, which is obtained by applying conformal mappings. By use of the transformation technique, a closed-form expression is secured for the local frictional stress on the wedge surface. In general, increases as the wedge height h increases for a 5xed wedge apex angle 2. When h is held constant, decreases (increases) near the wedge apex (base), as increases. Iso-velocity lines are computed for two limiting cases of the wedge shape. These c 2002 Published by The Japan Society limiting-case solutions are consistent with the previous physical descriptions. of Fluid Mechanics and Elsevier Science B.V. All rights reserved.
1. Introduction A class of low-Reynolds number ows, which is referred to as the Stokes ow, is encountered when either the characteristic velocity or length becomes very small (Schlichting, 1979). In recent technological applications, the Stokes ow is of renewed interest in the design and operation of micro-machining techniques and micro-electric devices. For instance, the diameter and rotational speed of existing micromotors are typically 100 m and 20;000 rpm, respectively. If the working uid is air, the system Reynolds number is around 0.01 (Tai and Muller, 1989). In connection with the development of micro-machines, in-depth understanding of the detailed structure, in particular, the viscous resistance of the Stokes ow is warranted. In the present study, an analysis is made of the ow in a very narrow gap in which the top smooth plate slides with ∗
Corresponding author. Tel.: +82-42-869-3012; fax: +82-42-869-3210. E-mail address: [email protected] (J.M. Hyun).
c 2002 Published by The Japan Society of Fluid Mechanics and Elsevier Science B.V. 0169-5983/02/$22.00 All rights reserved. PII: S 0 1 6 9 - 5 9 8 3 ( 0 1 ) 0 0 0 4 0 - 5
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J.S. Park, J.M. Hyun / Fluid Dynamics Research 30 (2002) 93–106
Fig. 1. (a) Flow con5guration and (b) coordinate system.
a constant velocity and the bottom wedged plate is at rest (see Fig. 1). Obviously, if no wedge is present, the con5guration reduces to the standard Couette ow. The present ow model is relevant to a variety of practical situations, e.g., in labyrinth sealing, a rotating shaft with 5ns in electronic cooling, to name a few (e.g., Schey, 1983; Patankar and Murthy, 1984). In lubrication engineering, reduction of friction is sought by entrapping lubricating oil in the grooves, which are produced in the course of turning or milling processes. Also, the present problem setting provides a baseline description of the ow in a brush-type micro-actuation with oscillating 5ns (Tang et al., 1989). The Couette ows with protrusion have been extensively studied. The paint brush model was proposed by Taylor (1960, 1971) and Richardson (1971), in which the analysis was focused on the Stokes ow induced by the horizontally sliding plane over periodically placed two-dimensional vertical plates with in5nite height, i.e., h → ∞ in Fig. 1. Wang (1994) computed the Stokes drag due to the sliding of a smooth plate over a periodic 5nned plate with negligible thickness, i.e., → 0 in Fig. 1. The characteristics of drag were calculated by the methods of eigenfunction expansion and collocation. Several authors examined the reduction of turbulent drag by using a longitudinally grooved surface in the viscous sublayer (Bechert and Bartenwerfer, 1989; Luchini et al., 1991; Pozrikidis, 1993). In these models, the protrusion height was shown to be a major factor of drag reduction. In this paper, the conformal mapping technique is utilized to obtain comprehensive analytic solutions of friction in the Stokes ow on the wedge surface attached to the bottom wall induced by the top sliding plate. The eEects of wedge height h and wedge angle will be scrutinized. This has wide rami5cations in practical technological applications. Moreover, two limiting cases of the wedge shape, i.e., for the cases of h → 0 and=or → 0 in Fig. 1, will be discussed. Explicit theoretical solutions are secured which are consistent with the previous physical descriptions. These provide additional support to the validity of the present analysis.
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2. Problem formulation Consider the two-dimensional ow of a viscous uid (density ∗ and kinematic viscosity ∗ ) as sketched in Fig. 1. On the bottom wall (y∗ = 0), a triangular wedge (height H ∗ and half-wedge angle ) is attached. The ow is induced by the top wall (y∗ = D∗ ), which slides in the z-direction (perpendicular to the plane of Fig. 1(b)) with a constant velocity W0∗ . The well-documented governing equations, in nondimensional form, are (e.g., Wang, 1994) @2 w @ 2 w + 2 = 0: @x2 @y
(1)
In the above, nondimensionalization has been performed as ∗ ∗ H∗ x y w∗ ; ; w = ∗ ; h = ∗ ; (x; y) = W0 D D∗ D∗ in which w denotes the z-directional velocity, and ∗ indicates dimensional quantities. The associated boundary conditions are straightforward: w=1
at y = 1;
(2)
w=0
at y = 0; |x| ¿ h tan();
and at y = h − |x|=tan();
|x| 6 h tan():
(3)
3. Analysis The problem is symmetric with respect to the y-axis; therefore, only the semi-in5nite domain in the S(≡ √ x + iy)-plane [bounded by P1 P2 ; P2 P3 ; P3 P4 and P4 P5 in Fig. 2(a)] will be considered. Here, i ≡ −1. By undergoing a series of Schwarz–ChristoEel transformations, the domain is mapped onto an open-rectangular zone in the (≡ + i)-plane, as displayed in Fig. 2(c). The intermediate stages involve a Schwarz–ChristoEel transformation from the S(≡ x + iy)-plane to the S1 (≡ x1 + iy1 )-plane [see Fig. 2(b)]: dS = A(S1 − 2 )−1=2 (S1 − 3 )−= (S1 − 4 )−1=2+= : dS1
(4)
In the above, i indicates the S1 -plane locations of points Pi (i = 2; 3; 4) in the S-plane. Setting 2 = 0; 3 = 1 and 4 = , Eq. (4) gives S1 !−1=2 (! − 1)−= (! − )−1=2+= d! + B: (5) S =A 0
The transformation from the S1 -plane to the -plane [see Fig. 2(c)] is carried out by the relation 2 : (6) S1 = sin 2
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Fig. 2. Schematics of the conformal transformation.
In Eq. (5), A; B and can be determined by imposing the conditions that points (0,1), (0; h) and (h tan(); 0) in the S-plane are mapped, respectively, onto points (0,0), (1,0) and (; 0) in the S1 -plane. These exercises produce 1 !−1=2 (1 − !)−= ( − !)−1=2+= d!; (7) A = (1 − h) B=i and
1
(≡
√
0
−1);
(8)
!−1=2 (! − 1)−= ( − !)−1=2+= d!
=
1 h 1 − h cos()
0
1
!−1=2 (1 − !)−= ( − !)−1=2+= d!:
(9)
Based on Eqs. (7) – (9), is obtainable from the integral Eq. (9), and the constant A is determined by using Eq. (7). It is, therefore, useful to examine in detail the relationships underlying Eqs. (7) – (9). It is mentioned that the left-hand-side (LHS) of Eq. (9) is a monotonically increasing function
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97
Table 1 The numerically computed values of constant () (deg:)
h
10 20 30 45 60 70
0.01
0.1
0.3
0.5
0.7
1.00029 1.00035 1.00044 1.00069 1.00133 1.00266
1.02964 1.03600 1.04535 1.07117 1.14084 1.29677
1.31180 1.38712 1.50347 1.86032 3.10501 7.71095
2.24918 2.63688 3.30089 5.82843 20.3923 150.282
6.24137 8.71890 13.8146 42.1910 428.249 14868.1
of . It follows that, for the LHS of Eq. (9), the greatest lower bound is zero at = 1; and the least upper bound can be shown to be, by taking → ∞, !−1=2 (! − 1)−= ( − !)−1=2+= d! lim →∞
=
1
1 − 2
·
1 + 2
; (10) cos() in which (x) denotes the Gamma function. It is clear that the integral of the right-hand-side (RHS) of Eq. (9) decreases monotonically from the value h=(1 − h)·=cos() at = 1 to zero as → ∞. This implies that there will be a unique solution to the integral equation (10). The same conclusion may also be reached by using the mapping theorem of Riemann in the complex variable theory (Ahlfors, 1979). The solution pair (A; ) can be computed numerically by adopting the bisection method. The exemplary values of , up to six digits, are listed in Table 1. It is also worth mentioning that the values of the constant A is found to be A ∼ = 0:3183, which is independent of h and . It will be shown later that the theoretical value of A is 1=. When h is very small (see the column for h = 0:01 in Table 1), ∼ = 1:00, and this limiting value tends to become independent of the wedge angle . This is not unexpected in that, as the wedge height becomes very small, the overall eEect of the wedge on the global ow diminishes sharply, and the prevailing ow characteristics resemble the conventional Couette ow between the two parallel at plates. The above considerations yield the completed transformation formula from the S-plane to the -plane, i.e., ! ! −1=2+= S = −iA cos2 − sin2 dw + i: (11) 2 2 0 By use of Eq. (11), the solution to Eq. (1) is sought in the -plane. Because the above transformation is conformal, the governing equation in the transformed -plane is harmonic: @ 2 w @2 w + 2 =0 (12) @2 @ =
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with the boundary conditions w=1
at = 0;
(13)
w=0
at = 1;
(14)
and @w =0 @
at = 0:
(15)
The solution to Eq. (12), subject to boundary conditions (13) – (15), can be readily found: w(; ) = 1 − = Re(1 − )
(16)
in which Re denote the real part. Accordingly, by means of an inverse transformation from the -plane to the S-plane, the solution in the physical space can be obtained. A key physical variable of interest is the frictional force acting on the wedge surface and the bottom plate. The shear stress on the top and bottom surfaces are computed as
@w
=
@n at the top and bottom surfaces
@w d
=
@ dS at =0 or 1
1 − sin2 (=2) 1=2−=
=
; (17)
A cos2 (=2) at =0 or 1
in which n is the outward normal directional coordinate. As pointed out earlier, in the region far away from the wedge, i.e., x → ∞ (in the physical domain) or → ∞ (in the transformed domain), a simple Couette ow prevails. Consequently, by taking → ∞ in Eq. (17), the left-hand-side approaches → 1, which immediately leads to A → 1=. This is in line with the previously stated numerical solution A ∼ = 0:3183. It is now useful to make descriptions of the ow patterns. In particular, analytical approaches are capable of producing straightforward solutions in the special cases of: (a) h → 0, and (b) → 0. Obviously, case (a) models the limiting situation when the wedge simulates an in5nite plate parallel to the top wall, with a gap distance of 1. Case (b) models an extremely thin wedge with height h. First, for case (a), substituting h → 0; A = 1= and → 1 into Eq. (11), the complete transformation function from the S-plane to the -plane is obtained: S = i(− + 1);
(18)
which can be rewritten as x = ;
and
y = − + 1:
(19)
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The velocity w(x; y) in the physical domain is found, by using Eqs. (16) and (19): w(x; y) = y;
(0 6 y 6 1):
(20)
As anticipated, the above relation reproduces the standard Couette ow between parallel plates of gap distance 1. The derivations shown here also serve to check the correctness of the present conformal technique. Next, case (b), for which → 0, models a 5n of h, mounted perpendicularly to the bottom plate. Introducing → 0 and A = 1=, from Eqs. (7) and (9), √ 1 : (21) = cos(h=2) Combining these steps and substituting those into Eq. (11), the complete transformation function is secured: √ 2 = sin−1 (22) sin (1 − iS) : 2 Using Eq. (16), and, in the range 0 6 6 1 in the -plane, the velocity w can be expressed as w(; ) = 1 − : In view of Eq. (22), the iso-velocity lines in the physical plane satisfy the interrelationship shown below: cosh2 (x=2) cos2 (y=2) sinh2 (x=2) sin2 (y=2) − = sin2 (1 − h) ; (23) C 1−C 2 in which C [ ≡ sin2 (=2)] is a constant such that 0 ¡ C ¡ 1. Fig. 3 illustrates the iso-velocity lines for → 0. As h increases, the iso-velocity lines near the wedge apex are clustered. This leads to steep velocity gradients in the region, which give rise to enhancements of the local friction coeLcient. As can be recognized in Fig. 3(a), when h is small, the impact of the wedge on the upper portions of the ow 5eld is minimal. The in uence of the wedge is seen to decay as the distance from the wedge increases. Partly, as a test of the validity of the present analysis, the limiting behavior of the solution [Eq. (23)] at far distances from the wedge is examined. By introducing x → ∞ in Eq. (23), it is seen that cos
(y − ) (y + ) cos = 0; 2 2
(24)
in which 0 6 y 6 1 and 0 6 6 1. From Eq. (24), for arbitrary values of in the range 0 6 6 1; it follows that cos
(y + ) = 0; 2
i:e:; y = 1 − :
The above leads to, from Eq. (16), w(x → ∞; y) = y:
(25)
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Fig. 3. Plots of lines of iso-velocity ! for the case of → 0. The values of wedge height, h; are: (a) 0.3; (b) 0.6; (c) 0.9. The contour value between iso-velocity lines is 0.1.
Clearly, as expected, Eq. (25) depicts the standard Couette ow between two parallel plates with gap distance 1. This is consistent with the ow pattern at far distances from the wedge, as demonstrated in Fig. 3. 4. Frictional stress By executing algebraic manipulations, the local frictional stresses on the top and bottom surfaces can be found in a closed-form analytical expression: on the top surface:
+ sinh2 (=2) 1=2−=
t =
; (26a) cosh2 (=2)
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101
◦
Fig. 4. The friction stress distribution on the wedge surface. = 45 . The normalized distance, T , is de5ned as d=(h tan()), in which d denotes the distance from the apex (point P3 ) along the wedge surface.
on the bottom surface:
− cosh2 (=2) 1=2−=
b =
: sinh2 (=2)
(26b)
Finally, by using Eqs. (11) and (26), the frictional stresses on the top and bottom surfaces in the physical domain √ can be computed. It is remarked that, in the above Eq. (26b), the values = 0 and = (2=) cosh−1 ; respectively, correspond to the wedge apex (point P3 ) and the base (point P4 ) in the physical domain. The results of Eq. (26) for are plotted for graphical representation in Fig. 4. The wedge half-angle ◦ is 5xed at = 45 , and the wedge height h is varied in Fig. 4. The behavior of on the wedge surface (P3 P4 ) is of concern. As expected, the local frictional stress increases rapidly near the apex of the wedge (point P3 ); the velocity gradient goes to in5nity at a sharp convex point such as the wedge apex. Also, in the vicinity of the wedge base (point P4 ), diminishes rapidly. It is discernible in Fig. 4 that, in general, increases as the wedge height h increases. The variations of with , when h is 5xed (h = 0:3), is exhibited in Fig. 5. On the wedge surface close to the apex, increases as decreases, as easily anticipated. However, on the wedge surface near the base (point P4 ), a reverse trend is noticed, i.e., decreases as decreases. This may be explained by observing that, as the wedge angle is increased, the velocity gradient, which is determined by diEusion process as shown in Eq. (1), decreases (increases) on the wedge surface near the apex (near the base). To study the combined eEects of h- and -variations, Fig. 6 plots when the wedge base width (OP4 in Fig. 6) is held 5xed. In this case, as h is changed, is altered simultaneously. The qualitative character of -variations in Fig. 6 is akin to the combined features of Figs. 4 and 5. Finally, the total drag Dw over the entire wedge surface is calculated: Dw = dS : (27) wedge surface
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J.S. Park, J.M. Hyun / Fluid Dynamics Research 30 (2002) 93–106
Fig. 5. The friction stress distribution on the wedge surface. h = 0:3. The wedge half-angle, , are: ◦ ◦ ◦ ◦ (a) 10 ; (b) 30 ; (c) 45 ; (d) 70 . The normalized distance, T , is the same as in Fig. 4.
Fig. 6. The friction stress distribution on the wedge surface. h tan() = 0:01. The values of wedge height, h, are: (a) 0.01; (b) 0.02; (c) 0.1. The normalized distance, T , is the same as in Fig. 4.
The relation in Eq. (26b) is used, which yields
−1
d
d Dw = 2
dS
0 √ √ √ 4 4 −1 = cosh ( ) ≡ ln ( + − 1) :
√
(2=) cosh−1 ( )
@w d
@ dS
(28)
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103
Fig. 7. Total drag force Dw on the wedge surface.
The total drag Dw is plotted in Fig. 7. In general, Dw increases as h and increase. It is noted that these represent two separate eEects: (1) as h increases, the local skin frictional stress increases, which is referred to as the stress eEect; (2) as increases, the surface area of the wedge increases, which is termed as the area eEect. The above-stated stress eEect (1) is obvious in the -plots of Fig. 4. Over the entire wedge surface, increases as h increases with 5xed. In this situation, the total wedge surface area also increases; however, this increase is linear with h. The nonlinear increases of Dw with h, as displayed in Fig. 7, are caused more by the stress eEect. On the other hand, when increases with h held 5xed, increases (decreases) near the wedge base (apex), as illustrated in Fig. 5. Therefore, the contribution by spatial variations of to Dw tend to cancel each other. Consequently, the increase in Dw is largely due to the area eEect. As remarked earlier, the case → 0 is of special interest. The local skin frictional stress acting on a perpendicularly attached 5n can be obtained by letting → 0 in Eqs. (26a) and (26b): at the top sliding surface, 1=2 1 + sinh2 (x=2) ; (29a) t = cos(h=2) + sinh2 (x=2) at the 5n and bottom surface, 1=2 1 − cos2 (y=2) ; (x = 0; 0 6 y 6 h); cos2 (y=2) − cos2 (h=2) b = 1=2 2 cosh (x=2) − 1 ; (y = 0): cosh2 (x=2) − cos2 (h=2)
(29b)
The plots of t are shown in Fig. 8(a). Near the 5n (x small), t increases as h increases, and at far distances from the 5n (x1), t ∼ = 1 regardless of the value of h. This recon5rms the earlier
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J.S. Park, J.M. Hyun / Fluid Dynamics Research 30 (2002) 93–106
Fig. 8. Skin frictional stress in the case of → 0. (a) t versus x. Height, h, are: a, 0.1; b, 0.3; c, 0.5; d, 0.7; e, 0.9. (b) b versus surface coordinate s. Height, h; are: a, 0.2; b, 0.5; c, 0.8.
5nding that the Couette ow is recovered far away from the 5n. It is also interesting to note that in Eq. (29a), as x → 0 and h → 1; t goes to in5nity. This simulates the ow in the vanishing clearance (h → 1) between the 5n and the plate. Furthermore, as elucidated previously, by letting h → 0 in Eq. (29a), t → 1; which indicates the Couette ow limit when the 5n height is negligibly small. The results of Fig. 8(a) for the special case of = 0 indicate the lower bounds of on the top wall for a triangular-shaped wedge of arbitrary value of . Fig. 8(b) depicts the behavior of b . Clearly, b in Fig. 8(b) for 0 6 s=h 6 1:0 denotes the frictional stress on the 5n surface, and for s=h ¿ 1:0 b represents the stress on the base plate. At the top point of the 5n (s=h → 0:0), b → ∞; and at the base of the 5n (s=h = 1:0), b → 0. The frictional stress decreases rapidly from the apex of the 5n to the base. Along the bottom plate, b increases slowly toward the value b → 1; as the afore mentioned Couette ow is approaching. These observations are in line with the characteristic pictures presented earlier for 5nite .
J.S. Park, J.M. Hyun / Fluid Dynamics Research 30 (2002) 93–106
105
Fig. 9. Total drag force on the 5n in the case of → 0.
The total frictional drag on the 5n can be computed in Eq. (29b): h Df = 2 (b )x=0 dy 0
=
4 1 + sin(h=2) ln : cos(h=2)
(30)
Fig. 9 exhibits Df versus h. Clearly, it is seen that Df → ∞ as h → 1; and Df → 0 as h → 0. When the 5n height approaches the gap of channel (h → 1), the drag increases in5nitely because in the vanishing clearance between the 5n and the plate goes to in5nity. In the other extreme case, when the 5n itself vanishes (h → 0), it is natural that no frictional force acts on the 5n. 5. Concluding remarks The conformal mapping techniques have been applied to describe the Stokes ow between the top sliding plate and the stationary bottom wedged wall. Closed-form analytical solutions were obtained for the ow and local frictional stress . As h increases, increases on the wedge surface, which gives rise to an increased frictional drag. On the other hand, as increases, increases (decreases) near the wedge apex (base). The changes in total drag force on the wedge stem from both the stress and area eEects. The limiting-behavior analysis for → 0 recovers the characteristics of a perpendicularly placed 5n, and the theoretical results are consistent with the previously established pictures. Acknowledgements This work was supported by Korea Research Foundation Grant (KRF-2000-042-E00004).
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References Ahlfors, L.V., 1979. Complex Analysis. McGraw-Hill Inc., New York. Bechert, D.W., Bartenwerfer, M., 1989. The viscous ow on surfaces with longitudinal ribs. J. Fluid Mech. 206, 105–129. Luchini, P., Manzo, F., Pozzi, A., 1991. Resistance of a grooved surface to parallel ow and cross- ow. J. Fluid Mech. 228, 87–109. Patankar, J.Y., Murthy, S.V., 1984. Analysis of heat transfer from a rotating cylinder with circumferential 5ns. In: Heat and Mass Transfer in Rotating Machinery. Hemisphere Publishing Corp., Washington, DC, pp. 155 –166. Pozrikidis, C., 1993. Unsteady viscous ow over irregular boundaries. J. Fluid Mech. 255, 11–34. Richardson, S., 1971. A model for the boundary condition of a porous material. Part 2. J. Fluid Mech. 49, 327–336. Schey, J.A., 1983. Tribology in Metalworking-Friction, Lubrication and Wear. American Society for Metals, Metal Park, OH. Schlichting, H., 1979. Boundary Layer Theory. McGraw-Hill Inc., New York. Tai, Y.C., Muller, R.S., 1989. IC-processed electrostatic synchronous micromotors. Sensors Actuators 20, 49–56. Tang, W.C., Nguyen, T.C.H., Howe, R.T., 1989. Laterally driven polysilicon resonant microstructures. Sensors Actuators 20, 25–32. Taylor, G.I., 1960. Deposition of a viscous uid on a plane surface. J. Fluid Mech. 9, 218–225. Taylor, G.I., 1971. A model for the boundary condition of a porous material. Part 1. J. Fluid Mech. 49, 319–326. Wang, C.Y., 1994. The Stokes drag due to the sliding of a smooth plate over a 5nned plate. Phys. Fluids 6 (7), 2248–2252.
The Stokes- ow friction on a wedge surface by the sliding of a plate Jun Sang Parka , Jae Min Hyunb; ∗ a
Department of Mechanical Engineering, Halla University, San 66, HeungUp, Wonju, Kangwondo 220-712, South Korea b Department of Mechanical Engineering, Korea Advanced Institute of Science & Technology, 373-1 Kusong-Dong, Yusung-gu, Taejon 305-701, South Korea Received 4 June 2001; received in revised form 1 October 2001; accepted 6 October 2001
Abstract A theoretical analysis is conducted on the frictional stress in the Stokes ow in a channel. A triangular wedge is attached on the bottom plate, and the ow is induced by the top sliding plate. The governing harmonic equation is solved in the transformed -plane, which is obtained by applying conformal mappings. By use of the transformation technique, a closed-form expression is secured for the local frictional stress on the wedge surface. In general, increases as the wedge height h increases for a 5xed wedge apex angle 2. When h is held constant, decreases (increases) near the wedge apex (base), as increases. Iso-velocity lines are computed for two limiting cases of the wedge shape. These c 2002 Published by The Japan Society limiting-case solutions are consistent with the previous physical descriptions. of Fluid Mechanics and Elsevier Science B.V. All rights reserved.
1. Introduction A class of low-Reynolds number ows, which is referred to as the Stokes ow, is encountered when either the characteristic velocity or length becomes very small (Schlichting, 1979). In recent technological applications, the Stokes ow is of renewed interest in the design and operation of micro-machining techniques and micro-electric devices. For instance, the diameter and rotational speed of existing micromotors are typically 100 m and 20;000 rpm, respectively. If the working uid is air, the system Reynolds number is around 0.01 (Tai and Muller, 1989). In connection with the development of micro-machines, in-depth understanding of the detailed structure, in particular, the viscous resistance of the Stokes ow is warranted. In the present study, an analysis is made of the ow in a very narrow gap in which the top smooth plate slides with ∗
Corresponding author. Tel.: +82-42-869-3012; fax: +82-42-869-3210. E-mail address: [email protected] (J.M. Hyun).
c 2002 Published by The Japan Society of Fluid Mechanics and Elsevier Science B.V. 0169-5983/02/$22.00 All rights reserved. PII: S 0 1 6 9 - 5 9 8 3 ( 0 1 ) 0 0 0 4 0 - 5
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J.S. Park, J.M. Hyun / Fluid Dynamics Research 30 (2002) 93–106
Fig. 1. (a) Flow con5guration and (b) coordinate system.
a constant velocity and the bottom wedged plate is at rest (see Fig. 1). Obviously, if no wedge is present, the con5guration reduces to the standard Couette ow. The present ow model is relevant to a variety of practical situations, e.g., in labyrinth sealing, a rotating shaft with 5ns in electronic cooling, to name a few (e.g., Schey, 1983; Patankar and Murthy, 1984). In lubrication engineering, reduction of friction is sought by entrapping lubricating oil in the grooves, which are produced in the course of turning or milling processes. Also, the present problem setting provides a baseline description of the ow in a brush-type micro-actuation with oscillating 5ns (Tang et al., 1989). The Couette ows with protrusion have been extensively studied. The paint brush model was proposed by Taylor (1960, 1971) and Richardson (1971), in which the analysis was focused on the Stokes ow induced by the horizontally sliding plane over periodically placed two-dimensional vertical plates with in5nite height, i.e., h → ∞ in Fig. 1. Wang (1994) computed the Stokes drag due to the sliding of a smooth plate over a periodic 5nned plate with negligible thickness, i.e., → 0 in Fig. 1. The characteristics of drag were calculated by the methods of eigenfunction expansion and collocation. Several authors examined the reduction of turbulent drag by using a longitudinally grooved surface in the viscous sublayer (Bechert and Bartenwerfer, 1989; Luchini et al., 1991; Pozrikidis, 1993). In these models, the protrusion height was shown to be a major factor of drag reduction. In this paper, the conformal mapping technique is utilized to obtain comprehensive analytic solutions of friction in the Stokes ow on the wedge surface attached to the bottom wall induced by the top sliding plate. The eEects of wedge height h and wedge angle will be scrutinized. This has wide rami5cations in practical technological applications. Moreover, two limiting cases of the wedge shape, i.e., for the cases of h → 0 and=or → 0 in Fig. 1, will be discussed. Explicit theoretical solutions are secured which are consistent with the previous physical descriptions. These provide additional support to the validity of the present analysis.
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2. Problem formulation Consider the two-dimensional ow of a viscous uid (density ∗ and kinematic viscosity ∗ ) as sketched in Fig. 1. On the bottom wall (y∗ = 0), a triangular wedge (height H ∗ and half-wedge angle ) is attached. The ow is induced by the top wall (y∗ = D∗ ), which slides in the z-direction (perpendicular to the plane of Fig. 1(b)) with a constant velocity W0∗ . The well-documented governing equations, in nondimensional form, are (e.g., Wang, 1994) @2 w @ 2 w + 2 = 0: @x2 @y
(1)
In the above, nondimensionalization has been performed as ∗ ∗ H∗ x y w∗ ; ; w = ∗ ; h = ∗ ; (x; y) = W0 D D∗ D∗ in which w denotes the z-directional velocity, and ∗ indicates dimensional quantities. The associated boundary conditions are straightforward: w=1
at y = 1;
(2)
w=0
at y = 0; |x| ¿ h tan();
and at y = h − |x|=tan();
|x| 6 h tan():
(3)
3. Analysis The problem is symmetric with respect to the y-axis; therefore, only the semi-in5nite domain in the S(≡ √ x + iy)-plane [bounded by P1 P2 ; P2 P3 ; P3 P4 and P4 P5 in Fig. 2(a)] will be considered. Here, i ≡ −1. By undergoing a series of Schwarz–ChristoEel transformations, the domain is mapped onto an open-rectangular zone in the (≡ + i)-plane, as displayed in Fig. 2(c). The intermediate stages involve a Schwarz–ChristoEel transformation from the S(≡ x + iy)-plane to the S1 (≡ x1 + iy1 )-plane [see Fig. 2(b)]: dS = A(S1 − 2 )−1=2 (S1 − 3 )−= (S1 − 4 )−1=2+= : dS1
(4)
In the above, i indicates the S1 -plane locations of points Pi (i = 2; 3; 4) in the S-plane. Setting 2 = 0; 3 = 1 and 4 = , Eq. (4) gives S1 !−1=2 (! − 1)−= (! − )−1=2+= d! + B: (5) S =A 0
The transformation from the S1 -plane to the -plane [see Fig. 2(c)] is carried out by the relation 2 : (6) S1 = sin 2
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Fig. 2. Schematics of the conformal transformation.
In Eq. (5), A; B and can be determined by imposing the conditions that points (0,1), (0; h) and (h tan(); 0) in the S-plane are mapped, respectively, onto points (0,0), (1,0) and (; 0) in the S1 -plane. These exercises produce 1 !−1=2 (1 − !)−= ( − !)−1=2+= d!; (7) A = (1 − h) B=i and
1
(≡
√
0
−1);
(8)
!−1=2 (! − 1)−= ( − !)−1=2+= d!
=
1 h 1 − h cos()
0
1
!−1=2 (1 − !)−= ( − !)−1=2+= d!:
(9)
Based on Eqs. (7) – (9), is obtainable from the integral Eq. (9), and the constant A is determined by using Eq. (7). It is, therefore, useful to examine in detail the relationships underlying Eqs. (7) – (9). It is mentioned that the left-hand-side (LHS) of Eq. (9) is a monotonically increasing function
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Table 1 The numerically computed values of constant () (deg:)
h
10 20 30 45 60 70
0.01
0.1
0.3
0.5
0.7
1.00029 1.00035 1.00044 1.00069 1.00133 1.00266
1.02964 1.03600 1.04535 1.07117 1.14084 1.29677
1.31180 1.38712 1.50347 1.86032 3.10501 7.71095
2.24918 2.63688 3.30089 5.82843 20.3923 150.282
6.24137 8.71890 13.8146 42.1910 428.249 14868.1
of . It follows that, for the LHS of Eq. (9), the greatest lower bound is zero at = 1; and the least upper bound can be shown to be, by taking → ∞, !−1=2 (! − 1)−= ( − !)−1=2+= d! lim →∞
=
1
1 − 2
·
1 + 2
; (10) cos() in which (x) denotes the Gamma function. It is clear that the integral of the right-hand-side (RHS) of Eq. (9) decreases monotonically from the value h=(1 − h)·=cos() at = 1 to zero as → ∞. This implies that there will be a unique solution to the integral equation (10). The same conclusion may also be reached by using the mapping theorem of Riemann in the complex variable theory (Ahlfors, 1979). The solution pair (A; ) can be computed numerically by adopting the bisection method. The exemplary values of , up to six digits, are listed in Table 1. It is also worth mentioning that the values of the constant A is found to be A ∼ = 0:3183, which is independent of h and . It will be shown later that the theoretical value of A is 1=. When h is very small (see the column for h = 0:01 in Table 1), ∼ = 1:00, and this limiting value tends to become independent of the wedge angle . This is not unexpected in that, as the wedge height becomes very small, the overall eEect of the wedge on the global ow diminishes sharply, and the prevailing ow characteristics resemble the conventional Couette ow between the two parallel at plates. The above considerations yield the completed transformation formula from the S-plane to the -plane, i.e., ! ! −1=2+= S = −iA cos2 − sin2 dw + i: (11) 2 2 0 By use of Eq. (11), the solution to Eq. (1) is sought in the -plane. Because the above transformation is conformal, the governing equation in the transformed -plane is harmonic: @ 2 w @2 w + 2 =0 (12) @2 @ =
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with the boundary conditions w=1
at = 0;
(13)
w=0
at = 1;
(14)
and @w =0 @
at = 0:
(15)
The solution to Eq. (12), subject to boundary conditions (13) – (15), can be readily found: w(; ) = 1 − = Re(1 − )
(16)
in which Re denote the real part. Accordingly, by means of an inverse transformation from the -plane to the S-plane, the solution in the physical space can be obtained. A key physical variable of interest is the frictional force acting on the wedge surface and the bottom plate. The shear stress on the top and bottom surfaces are computed as
@w
=
@n at the top and bottom surfaces
@w d
=
@ dS at =0 or 1
1 − sin2 (=2) 1=2−=
=
; (17)
A cos2 (=2) at =0 or 1
in which n is the outward normal directional coordinate. As pointed out earlier, in the region far away from the wedge, i.e., x → ∞ (in the physical domain) or → ∞ (in the transformed domain), a simple Couette ow prevails. Consequently, by taking → ∞ in Eq. (17), the left-hand-side approaches → 1, which immediately leads to A → 1=. This is in line with the previously stated numerical solution A ∼ = 0:3183. It is now useful to make descriptions of the ow patterns. In particular, analytical approaches are capable of producing straightforward solutions in the special cases of: (a) h → 0, and (b) → 0. Obviously, case (a) models the limiting situation when the wedge simulates an in5nite plate parallel to the top wall, with a gap distance of 1. Case (b) models an extremely thin wedge with height h. First, for case (a), substituting h → 0; A = 1= and → 1 into Eq. (11), the complete transformation function from the S-plane to the -plane is obtained: S = i(− + 1);
(18)
which can be rewritten as x = ;
and
y = − + 1:
(19)
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The velocity w(x; y) in the physical domain is found, by using Eqs. (16) and (19): w(x; y) = y;
(0 6 y 6 1):
(20)
As anticipated, the above relation reproduces the standard Couette ow between parallel plates of gap distance 1. The derivations shown here also serve to check the correctness of the present conformal technique. Next, case (b), for which → 0, models a 5n of h, mounted perpendicularly to the bottom plate. Introducing → 0 and A = 1=, from Eqs. (7) and (9), √ 1 : (21) = cos(h=2) Combining these steps and substituting those into Eq. (11), the complete transformation function is secured: √ 2 = sin−1 (22) sin (1 − iS) : 2 Using Eq. (16), and, in the range 0 6 6 1 in the -plane, the velocity w can be expressed as w(; ) = 1 − : In view of Eq. (22), the iso-velocity lines in the physical plane satisfy the interrelationship shown below: cosh2 (x=2) cos2 (y=2) sinh2 (x=2) sin2 (y=2) − = sin2 (1 − h) ; (23) C 1−C 2 in which C [ ≡ sin2 (=2)] is a constant such that 0 ¡ C ¡ 1. Fig. 3 illustrates the iso-velocity lines for → 0. As h increases, the iso-velocity lines near the wedge apex are clustered. This leads to steep velocity gradients in the region, which give rise to enhancements of the local friction coeLcient. As can be recognized in Fig. 3(a), when h is small, the impact of the wedge on the upper portions of the ow 5eld is minimal. The in uence of the wedge is seen to decay as the distance from the wedge increases. Partly, as a test of the validity of the present analysis, the limiting behavior of the solution [Eq. (23)] at far distances from the wedge is examined. By introducing x → ∞ in Eq. (23), it is seen that cos
(y − ) (y + ) cos = 0; 2 2
(24)
in which 0 6 y 6 1 and 0 6 6 1. From Eq. (24), for arbitrary values of in the range 0 6 6 1; it follows that cos
(y + ) = 0; 2
i:e:; y = 1 − :
The above leads to, from Eq. (16), w(x → ∞; y) = y:
(25)
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Fig. 3. Plots of lines of iso-velocity ! for the case of → 0. The values of wedge height, h; are: (a) 0.3; (b) 0.6; (c) 0.9. The contour value between iso-velocity lines is 0.1.
Clearly, as expected, Eq. (25) depicts the standard Couette ow between two parallel plates with gap distance 1. This is consistent with the ow pattern at far distances from the wedge, as demonstrated in Fig. 3. 4. Frictional stress By executing algebraic manipulations, the local frictional stresses on the top and bottom surfaces can be found in a closed-form analytical expression: on the top surface:
+ sinh2 (=2) 1=2−=
t =
; (26a) cosh2 (=2)
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101
◦
Fig. 4. The friction stress distribution on the wedge surface. = 45 . The normalized distance, T , is de5ned as d=(h tan()), in which d denotes the distance from the apex (point P3 ) along the wedge surface.
on the bottom surface:
− cosh2 (=2) 1=2−=
b =
: sinh2 (=2)
(26b)
Finally, by using Eqs. (11) and (26), the frictional stresses on the top and bottom surfaces in the physical domain √ can be computed. It is remarked that, in the above Eq. (26b), the values = 0 and = (2=) cosh−1 ; respectively, correspond to the wedge apex (point P3 ) and the base (point P4 ) in the physical domain. The results of Eq. (26) for are plotted for graphical representation in Fig. 4. The wedge half-angle ◦ is 5xed at = 45 , and the wedge height h is varied in Fig. 4. The behavior of on the wedge surface (P3 P4 ) is of concern. As expected, the local frictional stress increases rapidly near the apex of the wedge (point P3 ); the velocity gradient goes to in5nity at a sharp convex point such as the wedge apex. Also, in the vicinity of the wedge base (point P4 ), diminishes rapidly. It is discernible in Fig. 4 that, in general, increases as the wedge height h increases. The variations of with , when h is 5xed (h = 0:3), is exhibited in Fig. 5. On the wedge surface close to the apex, increases as decreases, as easily anticipated. However, on the wedge surface near the base (point P4 ), a reverse trend is noticed, i.e., decreases as decreases. This may be explained by observing that, as the wedge angle is increased, the velocity gradient, which is determined by diEusion process as shown in Eq. (1), decreases (increases) on the wedge surface near the apex (near the base). To study the combined eEects of h- and -variations, Fig. 6 plots when the wedge base width (OP4 in Fig. 6) is held 5xed. In this case, as h is changed, is altered simultaneously. The qualitative character of -variations in Fig. 6 is akin to the combined features of Figs. 4 and 5. Finally, the total drag Dw over the entire wedge surface is calculated: Dw = dS : (27) wedge surface
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Fig. 5. The friction stress distribution on the wedge surface. h = 0:3. The wedge half-angle, , are: ◦ ◦ ◦ ◦ (a) 10 ; (b) 30 ; (c) 45 ; (d) 70 . The normalized distance, T , is the same as in Fig. 4.
Fig. 6. The friction stress distribution on the wedge surface. h tan() = 0:01. The values of wedge height, h, are: (a) 0.01; (b) 0.02; (c) 0.1. The normalized distance, T , is the same as in Fig. 4.
The relation in Eq. (26b) is used, which yields
−1
d
d Dw = 2
dS
0 √ √ √ 4 4 −1 = cosh ( ) ≡ ln ( + − 1) :
√
(2=) cosh−1 ( )
@w d
@ dS
(28)
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Fig. 7. Total drag force Dw on the wedge surface.
The total drag Dw is plotted in Fig. 7. In general, Dw increases as h and increase. It is noted that these represent two separate eEects: (1) as h increases, the local skin frictional stress increases, which is referred to as the stress eEect; (2) as increases, the surface area of the wedge increases, which is termed as the area eEect. The above-stated stress eEect (1) is obvious in the -plots of Fig. 4. Over the entire wedge surface, increases as h increases with 5xed. In this situation, the total wedge surface area also increases; however, this increase is linear with h. The nonlinear increases of Dw with h, as displayed in Fig. 7, are caused more by the stress eEect. On the other hand, when increases with h held 5xed, increases (decreases) near the wedge base (apex), as illustrated in Fig. 5. Therefore, the contribution by spatial variations of to Dw tend to cancel each other. Consequently, the increase in Dw is largely due to the area eEect. As remarked earlier, the case → 0 is of special interest. The local skin frictional stress acting on a perpendicularly attached 5n can be obtained by letting → 0 in Eqs. (26a) and (26b): at the top sliding surface, 1=2 1 + sinh2 (x=2) ; (29a) t = cos(h=2) + sinh2 (x=2) at the 5n and bottom surface, 1=2 1 − cos2 (y=2) ; (x = 0; 0 6 y 6 h); cos2 (y=2) − cos2 (h=2) b = 1=2 2 cosh (x=2) − 1 ; (y = 0): cosh2 (x=2) − cos2 (h=2)
(29b)
The plots of t are shown in Fig. 8(a). Near the 5n (x small), t increases as h increases, and at far distances from the 5n (x1), t ∼ = 1 regardless of the value of h. This recon5rms the earlier
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Fig. 8. Skin frictional stress in the case of → 0. (a) t versus x. Height, h, are: a, 0.1; b, 0.3; c, 0.5; d, 0.7; e, 0.9. (b) b versus surface coordinate s. Height, h; are: a, 0.2; b, 0.5; c, 0.8.
5nding that the Couette ow is recovered far away from the 5n. It is also interesting to note that in Eq. (29a), as x → 0 and h → 1; t goes to in5nity. This simulates the ow in the vanishing clearance (h → 1) between the 5n and the plate. Furthermore, as elucidated previously, by letting h → 0 in Eq. (29a), t → 1; which indicates the Couette ow limit when the 5n height is negligibly small. The results of Fig. 8(a) for the special case of = 0 indicate the lower bounds of on the top wall for a triangular-shaped wedge of arbitrary value of . Fig. 8(b) depicts the behavior of b . Clearly, b in Fig. 8(b) for 0 6 s=h 6 1:0 denotes the frictional stress on the 5n surface, and for s=h ¿ 1:0 b represents the stress on the base plate. At the top point of the 5n (s=h → 0:0), b → ∞; and at the base of the 5n (s=h = 1:0), b → 0. The frictional stress decreases rapidly from the apex of the 5n to the base. Along the bottom plate, b increases slowly toward the value b → 1; as the afore mentioned Couette ow is approaching. These observations are in line with the characteristic pictures presented earlier for 5nite .
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105
Fig. 9. Total drag force on the 5n in the case of → 0.
The total frictional drag on the 5n can be computed in Eq. (29b): h Df = 2 (b )x=0 dy 0
=
4 1 + sin(h=2) ln : cos(h=2)
(30)
Fig. 9 exhibits Df versus h. Clearly, it is seen that Df → ∞ as h → 1; and Df → 0 as h → 0. When the 5n height approaches the gap of channel (h → 1), the drag increases in5nitely because in the vanishing clearance between the 5n and the plate goes to in5nity. In the other extreme case, when the 5n itself vanishes (h → 0), it is natural that no frictional force acts on the 5n. 5. Concluding remarks The conformal mapping techniques have been applied to describe the Stokes ow between the top sliding plate and the stationary bottom wedged wall. Closed-form analytical solutions were obtained for the ow and local frictional stress . As h increases, increases on the wedge surface, which gives rise to an increased frictional drag. On the other hand, as increases, increases (decreases) near the wedge apex (base). The changes in total drag force on the wedge stem from both the stress and area eEects. The limiting-behavior analysis for → 0 recovers the characteristics of a perpendicularly placed 5n, and the theoretical results are consistent with the previously established pictures. Acknowledgements This work was supported by Korea Research Foundation Grant (KRF-2000-042-E00004).
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