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Hydrodynamics and Contact Angle Relaxation during Unsteady Spreading Yue Suo,* Kroum Stoev, and Stephen Garoff Department of Physics and Center of Complex Fluids Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Enrique Rame´ National Center for Microgravity Research, c/o NASA Glenn Research Center, MS 110-3, Cleveland, Ohio 44135 Received March 26, 2001. In Final Form: June 24, 2001 We have studied the unsteady spreading of a liquid on a solid surface by observing the liquid-air meniscus shape and analyzing it with the hydrodynamic theory for Stokes’ flow. The unsteady spreading process exhibits both quasi-steady and unsteady regimes. The unsteadiness in our moderately high viscosity system arises from temporal relaxation of the contact angle rather than inertial effects or momentum diffusion. The quasi-steady regime can be accurately described by a model that ascribes a steady-state flow near the contact line. “Spontaneous” spreading, i.e., spreading to final state of zero contact line speed, is recognized as a special case of the general unsteady spreading process with the commonly used power laws for spreading occurring only under special conditions.
I. Introduction Experiments and analyses of unsteady spreading have been used extensively as a tool for studying wetting dynamics, in particular for the specific geometry of a spreading droplet, with the objective of determining the validity of hydrodynamic and nonhydrodynamic (i.e., molecular kinetics) spreading laws. Although the axisymmetric droplet is a relatively simple geometry, these studies are complicated by the fact that the spreading regime is inherently unsteady and so it does not allow one to study systematically the velocity dependence of material spreading parameters. Despite the limitations of studies of unsteady spreading, and to predict with confidence the majority of naturally occurring spreading processes, it is necessary to understand how to describe unsteady wetting. At one extreme of the spectrum are those processes where the time scale of the unsteadiness is much longer than that of the contact line flow itself. In these cases, it might be attractive to postulate that the flow near the contact line is quasisteady; i.e., at each instantaneous spreading velocity the flow is the same as that which would prevail in steady state with the same velocity. At the other extreme, the unsteadiness changes much faster than the characteristic spreading time. Then the contact line flow cannot be assumed to be quasi-steady and different analytical tools would be needed. Young and Davis1 and Murphy2 studied the unsteady dynamics of a meniscus formed on a flat plate moving in a liquid bath at controlled, time-varying velocity U. Using the quasi-steady approximation and assuming a simple dependence of the macroscopic dynamic contact angle, ω0, on the contact line velocity relative to the solid, Ucl,s, they derived analytical expressions for the time dependence of ω0 and the meniscus height, H, above the flat liquid level far from the solid. When the dynamic contact (1) Young, G. W.; Davis, S. H. J. Fluid Mech. 1987, 174, 327. (2) Murphy, E. D. M.S. Thesis, University of Pennsylvania, Philadelphia, PA, 1984.
angle exhibits hysteresis and the solid motion is oscillatory, they were able to generate cycles of the force vs solid displacement that represent the net work that must be done to move the contact line. In these studies they recognized that the slope, (dω0/dUcl,s)-1, represents a velocity scale for the time-varying part of the process. When the solid velocity U is nonzero, then U(dω0/dUcl,s) is a dimensionless group in the problem; when U ) 0, then (dω0/dUcl,s) -1 becomes the only relevant velocity scale. This additional velocity scale is a result of the competition of viscosity µ and surface tension σ. By written (dω0/dUcl,s)-1 ) (dω0/dCa)-1µ/σ and since the dependence of ω0 vs capillary number Ca ()Ucl,sµ/σ) is more or less universal, 3 it follows that the velocity scales σ/µ and (dω0/dUcl,s) -1 are equivalent. Using these concepts, de Gennes4 showed that the contact angle ω0(t) of a freshly formed meniscus of a perfectly wetting liquid of density F on a vertical flat plate evolves as ω0 ∼ (σt/(aµ))-1/2 when t . aµ/σ, where the capillary length is a ) (σ/Fg)1/2 and g is the acceleration due to gravity. Other power laws have been derived; e.g., when a droplet of volume V spreads to a zero-degree equilibrium contact angle, the radius of the wetted area R/V1/3 ∼ (tV-1/3σ/µ)1/10 for long times.5 We will show that only in the very specific and unusual cases of unsteady spreading to a zero-degree contact angle do power laws correctly describe the unsteady motion. DeConinck and co-workers 6-8 have studied droplet spreading in detail. They interpret the spreading processes by treating the energy dissipation near the contact line with both hydrodynamics and molecular kinetic theory. (3) Wilson, K. Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA, 1995. (4) Joanny, J. F.; de Gennes, P. G. J. Phys. 1986, 47, 121. (5) Tanner L. H. J. Phys. D: Appl. Phys. 1979, 12, 1473 (6) de Ruijter, M. J.; De Coninck, J.; Blake, T. D.; Clarke, A.; Rankin, A. Langmuir 1997, 13, 7293. (7) de Ruijter, M. J.; De Coninck, J.; Oshanin, G. Langmuir 1999, 15, 2209. (8) de Ruijter, M. J.; Charlot, M.; Voue´, M.; De Coninck, J. Langmuir 2000, 16, 2363.
10.1021/la0104572 CCC: $20.00 © 2001 American Chemical Society Published on Web 10/04/2001
Hydrodynamics and Contact Angle Relaxation
Figure 1. Schematic of ω0 vs Ucl,s during the spreading process. θs is the static advancing contact angle.
Both experiment and modeling are done for low viscosity liquids in regimes where the drop shape is a spherical cap and the equilibrium contact angle is at or near 0°. Two power law spreading regimes are derived from their theory, one corresponding to viscous flow and the other to energy dissipation. The contact angle shows an exponential relaxation to the final equilibrium value at long times for nonzero equilibrium angles. Sauer and Kampert’s 9 measurements of the wetting force during the spontaneous spreading on a Wilhelmy fiber also show that the spreading is influenced by the geometry of the wetting system. Ting and Perlin10 have studied experimentally the dynamic contact angle on oscillating solids. Their objective was to derive an empirical boundary condition for the dynamic contact angle vs spreading velocity. The objective of this study is to probe unsteady spreading dynamics beyond the scope of those spreading processes whose final state is one of static equilibrium with a zerodegree static contact angle. In this context, we will show that unsteady processes exhibit quasi-steady and unsteady regimes, and we will show that “spontaneous” spreading is a very special subset of a more general class of unsteady spreading flows. From an operational standpoint, unsteady spreading may be “forced” or “spontaneous”. Forced spreading has an externally imposed velocity scale (e.g., a plunging solid, a meniscus driven in a tube by a moving piston, and a droplet spreading radially by the injection of liquid at its center). On the other hand, spontaneous spreading does not have an external velocity scale. Instead, the velocity scale is σ/µ as discussed above (e.g., a meniscus rising in a capillary tube or a small droplet of constant volume spreading on a solid). In this study, we examine both cases by suddenly changing the forced spreading velocity. If our final forced velocity is zero, the spreading is “spontaneous”; if the final forced velocity is not zero, we are dealing with forced spreading. All transients, regardless of their evolution, are controlled by the variation of ω0 vs Ucl,s. Figure 1 shows a schematic of the dynamic variation of ω0. Consider a solid moving into a large liquid bath at steady speed Ui. At t ) 0 with the dynamic contact angle at ω0,i, the velocity is reduced suddenly to Uf. At t ) 0+, the solid velocity in the laboratory frame is Uf but the contact angle is still ω0,i. From Figure 1, the velocity Ucl,s must be Ui, which causes the contact line to rise in the laboratory frame. As a result, the meniscus contact angle is reduced to ω0′ < ω0i. At this point, the contact line velocity relative to the solid is U′ < Ui. Thus, the dynamics of the transient depend on the shape (specifically the slope) of ω0 vs Ucl,s. Since the final velocity Uf of the step change was assumed arbitrary, the evolution is controlled by dω0/dU for all final velocities including Uf ) 0. (9) Sauer, B. B.; Kampert, W. G. J. Colloid Interface Sci. 1998, 199, 28. (10) Ting, C.-L.; Perlin, M. J. Fluid Mech. 1995, 295, 263.
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Figure 2. Meniscus rise height (H) on a solid surface, being immersed at speed U relative to the laboratory.
In section II we present a simple model based on the quasi-steady assumption for predicting the contact angle and contact line evolution after a change in spreading velocity has taken place. We also examine some characteristics needed to observe power law or exponential behaviors. In section III we explain our experimental methods. The results are presented and discussed in section IV. We conclude with a brief summary in section V. II. Model Quasi-Steady Model. Figure 2 illustrates the meniscus formed on the outside of a cylindrical solid surface. To construct a dynamical equation for ω0 vs t when the solid immersing velocity, U, changes suddenly from Ui to Uf, we note that U(>0) is related to the rate of change of the meniscus height, dH/dt, and to Ucl,s by
dH/dt ) Ucl,s - U
(1)
In the reference frame of Figure 2, H > 0 so that the contact angle 0 < ω0 < π/2. Putting dH/dt ) (dH/dω0)(dω0/dt) and Ucl,s ) F(ω0), where F is a known function derived from hydrodynamics to be discussed below, we write
( )
dω0 dH ) dt dω0
-1
[F(ω0) - Uf]
(2)
where ω0(0) ) F-1(Ui) and dH/dω0 ≡ aG(ω0, geometry) depends on the geometry of the spreading system. This model achieves its quasi-steady character by assigning G(ω0) and F(ω0) their steady-state values at each instantaneous velocity Ucl,s. Since G(ω0) and F(ω0) are determined by independent steady-state measurements, the model has no adjustable parameters. To specify F(ω0), we now discuss briefly the hydrodynamic theory that predicts the interface shape in the vicinity of the moving contact line in steady-state motion. In our experiments, we immerse a straight circular cylindrical tube of radius RT (.a) into the fluid. We choose a symmetric meniscus to simplify the wetting problem to 2-dimensions and expose more clearly to the fundamental nature of the relaxation. For this geometry, if we assume that the flow near the contact line is inertialess (i.e., it obeys the Stokes’ equation), satisfies the usual no-slip boundary condition, and that Ca , 1, then the steadystate liquid-air interface shape is described by 11,12
(
θ(r) ) g-1 g(ω0) + Ca ln
(ar)) + f(ar;ω ,Ra ) - ω 0
0
(3)
T
Here r is the distance from a point on the meniscus to the (11) Cox, R. G. J. Fluid Mech. 1986, 168, 169. (12) Dussan, V. E. B.; Rame´; E.; Garoff, S. J. Fluid Mech. 1991, 230, 97.
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identified later) for length, and replacing F(ω0) using eq 6, eq 4 becomes
dω0
)
dtˆ
Figure 3. Power law relation of ω0 and Ca at steady state over the range of Ca accessed in our unsteady experiments.
contact line, θ is the angle between the solid and the local tangent to the free surface, f(r/a; ω0,a/RT) is a static interface shape with contact angle ω0 (i.e., ω0 ) f(0;ω0,a/ RT)), and g(x) ≡ ∫0x{(y - cos y sin y)/2 sin y}dy. ω0 is the only fitting parameter and corresponds to the contact angle of the extrapolation to the solid of the static-like interface that forms far from the contact line. The success of eq 3 has been demonstrated experimentally for Ca e 0.113 on the same systems used in the experiments discussed in this paper. The model (eq 3) deals with the classic nonintegrable stress singularity at the contact line by assuming that a new hydrodynamic regime holds in an “inner region” near the contact line. Since eq 3 diverges at r ) 0, it cannot apply in the inner region. However, the physics of the inner region is irrelevant as long as it can be specified by two parameters (Li, θi), which may be functions of the spreading velocity Ucl,s ()U in steady state). For inner models which invoke slip to alleviate the stress singularity, θi is the microscopic contact angle and Li is the slip length. In terms of these inner parameters
g(ω0) ) g(θi(U)) + Ca ln
( ) a Li(U)
(4)
We may replace U with Ca(σ/µ) and write for Ca , 1
θi ) θs + Ca(A1) + ... Li ) L0 + Ca(L1) + ... where θs is the static contact angle and L0 is the slip length in the limit of zero velocity. If we assume that θs ) 0 and expand g(x) ∼ x3/9, eq 4 is approximated to a power law at low Ca over small ranges of Ca
ω0 ) RCaβ
(5)
whence
F(ω0) )
( )
γ ω0 µ R
1/β
(6)
β differs slightly from 1/3 owing to the influence of the higher order terms in g(x) and the velocity dependence of Li and θi.3,14,15 Figure 3 shows that the power law relation works well for our system in the range of Ca probed in our experiments. Using scales τ ) aµ/σ for time and a (to be (13) Chen, Q.; Rame, E.; Garoff, S. Phys. Fluids 1995, 7, 2631 (14) Hoffman R. L. J. Colloid. Interface Sci. 1975, 50, 228 (15) Marsh, J. A.; Garoff, S.; Dussan, V. E. B. Phys. Rev. 1993, 70, 2778
[( )
ω0 1 G(ω0) R
1/β
- Caf
]
(7)
where Ca(i,f) )µU(i,f) /σ and tˆ ) t/τ. With eqs 3 and 7 we have two independent means of detecting the impact of unsteady motion on the hydrodynamics near the relaxing contact line. First, the interface may not obey eq 3 and the local curvature differs significantly from the one in the steady state. Second, the contact angle relaxation may deviate from the quasi-steady model (eq 7). Long-Time Behavior. When the final contact angle is not zero, the quasi-steady model predicts that the contact angle relaxes exponentially at long times. When a wetting system evolves to a nonzero final contact angle, ω0,f ) F-1(Uf), linearization of eq 7 gives a characteristic relaxation time (recall, G(ω0) < 0):
τf ≡ -
G(ω0f) F′(ω0f)
(aµσ)G(ω )Ca
) -Rβ
0f
β-1
f
(8)
Thus, at long times,
ω0(t) ) ω0f + (ω0i - ω0f) exp(-t/τf)
(9)
Equation 8 shows that the slope of the ω0 vs Ucl,s curve near the final contact angle controls the relaxation time. Alternatively,
τf ) -
|
µ dH σ dCa ω0f
(10)
We note that the relaxation time depends on dH/dCa, which includes geometrical terms rather than just dω0/ dCa as suggested in earlier analyses. Since β ≈ 1/3, eq 8 shows that the linearization only works for the forced spreading where the final velocity is not zero. Otherwise, the relaxation time τf becomes infinite.8 Power Law Behavior. It has often been suggested that contact angle relaxation behaves as a power law.5-8 However, the power law form is a special case of the general behavior of the contact angle relaxation. From eq 2, the contact angle relaxation can only be a power law if both G(ω0) and F(ω0) - Uf are constants or power laws over the ω0 range accessed by the system. Typical cases of power law behavior are a small, thin drop spreading with constant volume on a flat solid5-8 and a meniscus rising on a flat plate near the final state of zero-degree static contact angle.4 There are exceptional cases where G(ω0) is a constant. Except in the case of spontaneous spreading where Uf ) 0, the power law behavior for F - Uf would also be very unusual. Even when Uf ) 0, a power law relaxation for F only occurs in the cases where the static contact angle is zero, the inner parameters are only weakly dependent on contact line velocity, and the function g(x) defined below eq 3 may be approximated by a power law for small angles, g(x) ∼ x3/9. Since the latter behavior is equivalent to that derived in lubrication-type analyses, it is not surprising that all theories based on the assumption of small slopes near the contact line predict the same Ca ∼ 1/3 dependence. Unsteady Effects. The quasi-steady model assumes that deviations of the fluid flow near the contact line from the steady state are negligible. The relaxation is quasisteady in the sense that only the boundary conditions on the flow are time-dependent. Equation 3, where inertia
Hydrodynamics and Contact Angle Relaxation
is neglected, has been tested successfully in steady forced spreading under a range of Ca e 0.1 and Re e 10-3 for the material system we use.13,16 Thus, we may safely assume that any deviation from this theory that we may detect during transients in the experiments considered here is not due to the convective term in the Navier-Stokes equation. Further, the time-dependent term, F∂u/∂t, will not affect our results since the time of momentum diffusion, tmom ) R2F/µ e 0.002 s (where R may be considered the size of the region we observe), is much less than any time probed in our experiments. While the characteristic diffusion time in the macroscopic flow field, FRB2/µ ∼ 4 s (where RB is the radius of the bulk fluid bath), is on the time scale of the relaxation we observe, we show experimentally that this weak flow cannot be responsible for perturbing the interface shapes we observe near the contact line (see section IV). Thus, we should expect Stokes’ flow to govern the flow in the region of the meniscus and in the time frame we observe in our experiments. While Stokes flow may correctly describe the flow fields we observe during relaxation, the assumptions behind eq 3 must be considered. Cox11 argued that if an unsteadiness of characteristic time scale T takes place in the macroscopic flow, an additional unsteady normal stress, τu ∼ µ/T (where T ∼ dω0/dt)-1, is created on the free surface. In a nearcontact line region of characteristic size R, the “steadystate stress” is τs ∼ µU/R. If we require τu , τs, then the condition 1 , TU/R must be met for the flow to be quasisteady near the contact line. R may be considered the size of the region where we detect viscous bending in comparable steady-state experiments. This is on the order of 500 µm for our experiments. Most of the observed contact line evolution toward the new steady state can be described by assuming quasi-steady flow. However, we will see that a short initial transient in the evolution of contact angles is not consistent with the quasi-steady assumption. This deviation is not due to inertia or to the F∂u/∂t term in the conservation of linear momentum but rather due to a violation of the condition 1 , TU/R. III. Experiments We study the hydrodynamics near the unsteadily moving contact line by observing the behavior of the meniscus of silicone oil, PDMS (poly(dimethylsiloxane)), rising on the outside of a dry, clean Pyrex tube. Using the meniscus on the outside of a larger diameter tube serves three purposes: (1) It provides a definite focal plane of our optical technique, producing a sharp image of the liquid/vapor interface. (2) This geometry has the largest possible outer length scale, thus exposing the viscous bending near the contact line. (3) This outer length scale does not change during the contact angle relaxation, unlike in a spreading drop experiment where this length scale is continuously changing.17 The tube radius (1.25 cm) is much larger than the characteristic length scale of the macroscopic meniscus, the capillary length a (≡(σ/Fg)1/2 ∼ 0.15 cm for PDMS), so it introduces only a small perturbation away from the meniscus on a flat solid.12 Because of the precision demanded by our comparison between experiment and model, we use appropriate corrections for the meniscus shape owing to the cylindrical shape of the solid. Several high molecular weight PDMS are used as received from Dow Corning Corp. Their properties, listed in Table 1, change little with temperature during the experiments. 3 PDMS behaves as a Newtonian liquid at the shear rates encountered in the observed regions near the contact line ( 2 s) of the data. Though the fit is good even at shorter times, closer examination shows that it deviates more than the quasi-steady model of eq 10 at early times. Further evidence of the effectiveness of the exponential description of the contact angle relaxation is shown in Table 3. τ90% is the time for the contact angle to reach 90% of its total change, as directly measured from the experimental data. τe is obtained by fitting the relaxation curve with the exponential in eq 9. τf is the exponential relaxation time calculated from the linearization near the final state, eq 10, and using independently measured values for the quantities in that equation. τf@90% ≡ τf ln(10). The similarity of τe and τf indicates that exponential law is a good description of the relaxation curve at long times. The similarity of τ90% and τf@90% shows that the linearization is an effective way for estimating the time scale of the full contact angle relaxation. We ask whether the departure from quasi-steadiness found just after t ) 0 is due to the time scale of the unique physics of the region that relaxes over the first second of the evolution. To test this hypothesis, we repeat the same experiment with two rheologically identical PDMS (µ )10 P) but different -CH3 and -OH chain end terminations. Steady-state experiments show that their ω0 vs Ca relations differ significantly because the interaction of the -OH end group with the Pyrex surface changes the inner scale parameters.3 However, the relaxation experiments show a similar result in both the unsteady region and the quasi-steady region. The difference in scale factors
Figure 10. Contact angle relaxation during spontaneous spreading.
is significantly smaller than the other differences discussed so far. This suggests strongly that the departure from quasi-steadiness is not due to incompletely relaxed inner scale physics for this pair of fluids. We also show the contact angle relaxation for spontaneous spreading (Figure 10). In contrast to data in the literature for a spreading drop,6,8 no power law region is observed. This follows from the deviation of G(ω0) from a constant or power law in our geometry and demonstrates that, for spontaneous spreading, power law behavior only occurs under specific conditions. We also find no exponential region in the spontaneous spreading at later times. This is consistent with the predictions of the model and that of DeConinck.7 However, we note that, in spontaneous spreading to zero-degree equilibrium angle, the residual heterogeneity on any ambient surface will lead to an unsteady velocity of the contact line as the small static angle is approached with very little kinetic energy in the spreading system. This unsteady motion is due to pinning of the contact line on even minor heterogeneity and will affect the temporal evolution of the latter stages of the contact angle relaxation.22, 23 We detect unsteadiness at early times after the velocity change in our forced spreading experiments. The unsteadiness is perceived in two ways. First, as seen in Figures 4-6, it causes a departure of the data from the quasi-steady curve. Unlike the quasi-steady relaxation of the contact angle, the time scale over which the unsteadiness appears in our data does not scale with viscosity of the liquid. Rather, it is roughly independent of viscosity. Second, as shown in Figure 11a, it causes a departure of the interface shape away from eq 3. Compared to the steady interface shape at the same Ca, the unsteady shape is more curved near the contact line. These deviations are detectable over the same time period as the deviations from our quasi-steady model. Neither momentum diffusivity in the region of the meniscus observed nor relaxation of the macroscopic flow throughout the beaker of fluid causes the observed unsteadiness. The time for diffusion of momentum tmom is ∼0.002 s for 10 P PDMS and ∼0.0004 s for 50 P, much less than the duration of the unsteadiness (∼1 s). Thus, the unsteadiness detected in our experiments cannot be caused by the term F∂u/∂t in the Navier-Stokes equation. The time of the relaxation of recirculation flow is about 4 s for the container we used. However, we get identical relaxation curves when performing experiments in beaker sizes of RB ) 2 cm and RB ) 6 cm. Thus, the weak recirculation flow has little influence to the relaxation curve and its relaxation is not the origin of the unsteadiness we see. The observed unsteadiness arises from added stress on the interface due to the evolving contact angle. Figure 12 (22) Decker, E. L.; Garoff, S. Langmuir 1996, 12, 2100 (23) Decker E. L.; Garoff, S. Langmuir 1997, 13, 6321
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added fluid motion and stress on the interface caused by the time variation of the contact angle. V. Conclusion
Figure 11. Liquid-air interface shape at (a) t ) 0.5 s and (b) t ) 6 s in the spreading process (Ca ) 4.8 × 10-4 f 2.4 × 10-2). Key for (a): dots, experimental data; solid line, theoretical curve with Ca ) 1 × 10-3; dashed line, theoretical curve with Ca ) 2 × 10-3, the extreme instantaneous values of Ca allowed by the data. Key for (b): dots, experimental data; solid line, theoretical curve with Ca ) 1 × 10-2.
Figure 12. Time variation of the steady-state stress (τs), unsteady stress (τu), and contact angle deviation from the quasisteady model for Ca ) 4.8 × 10-4 f 2.4 × 10-2 (O, deviations from the quasi-steady model; solid line, steady stress; dashed line, unsteady stress).
shows the evolution of the steady-state stress (τs) and the unsteady stress (τu) during a typical relaxation (see section IV). We see that τu > τs (where τu ∼ dω0/dt is calculated from the relaxation curve) during the same time that the interface shapes deviate from the steady-state model and the contact angle relaxation deviates from the quasi-steady model. Thus, the unsteadiness we observe arises from
When the dynamic contact angle is forced away from its steady-state condition (either by contacting a meniscus with a surface or causing a change in the surface velocity), it relaxes back to steady state first through an unsteady fluid motion and then through a quasi-steady regime. For our intermediate viscosity system, the quasi-steady regime begins fairly rapidly after the system is perturbed from steady state. This quasi-steady regime is accurately described by a model which is the same as would exist at the same surface velocity in a completely steady-state condition. This model shows that a number of factors control the characteristic time of the relaxation, including the viscosity, capillary length, the form of the dynamic contact angle’s dependence on velocity, geometric factors, and the final contact angle. At long times for nonzero final contact angles, the relaxation is very well approximated by an exponential. The relaxation of the contact angle is only a power law under very special conditions, even beyond the requirement that the final state is a static, completely wetting condition. The unsteady regime of the relaxation is on the order of 1 s in our system. During this regime, the curvature of the interface near the contact line is smaller than that of a similarly moving interface in steady state. The dynamic contact angle relaxes more slowly than the quasi-steady behavior would predict, and the maximum deviation from the quasi-steady behavior does not occur immediately after the system is driven out of steady-state motion. This unsteady motion is due neither to inertial or timedependent effects nor to slow relaxation of inner scale physics in our system. Rather, the observed unsteady motion arises within Stokes flow from the added stress on the interface due to the contact angle relaxation. Our study leads to a somewhat new picture of unsteady spreading, a condition ubiquitous in nature and technology. We find that quasi-steady relaxation of typical systems will not follow the often cited power laws but will be well approximated by an exponential relaxation. The time constant of this relaxation depends not only on the variation of the dynamic contact angle with capillary number but also on the geometry of the system. The unsteady motion occurring just after the contact angle is driven away from steady state is not due to deviations from Stokes flow. Stokes flow prevails, but the flow field is modified by stresses on the interface caused by the relaxation process. Acknowledgment. The authors wish to acknowledge the support of NASA, Grant No. NCC3-465. LA0104572
