A WINDOWPANE SPECKLED WITH rain offers two subtle puzzles in the physics of fluids. How do the drops cling to the glass? When water runs down the glass in a stream, why does it often meander instead of going straight?

The clinging of drops is often attributed to the surface tension of water, yet under a common definition of surface tension the drops should not cling. Meandering is often attributed to contamination on the glass, yet contamination seems unlikely to account for the normally regular pattern of a meandering stream.

Imagine a small drop of water on a solid horizontal surface such as a piece of glass. If the drop does not spread over the surface as a film, it forms a roughly hemispherical, slightly flattened bead. The shape is determined in part by the mutual attraction of the water molecules; that force acts to minimize the surface area. The surface is effectively a stretched, elastic membrane.

Usually the tendency to minimize the surface area is described in terms of tension on the surface of the drop. Imagine a line running across that surface. The surface tension is represented by forces that pull perpendicularly to the line, causing it to be in a state of tension. The surface tension of the water is defined as the ratio of the force pulling on one side of the line to the length of the line.

The perimeter of the contact area between the drop and the solid surface is called the triple-phase line because Of the conjunction of water, air and solid. The angle at which the water touches the solid at the triple-phase line is called the contact angle. It is measured between the solid and a tangent to the water surface. If the contact angle is less than 90 degrees, the water tends to spread over the solid and is said to wet it. If the angle is more than 90 degrees, the water pulls itself into a bead and does not wet the solid.

In the early part of the l9th century Thomas Young, who is remembered for his pioneering work on optical interference, stated that the size of the contact angle is set by the tendency toward equilibrium of three tensions pulling on the triple-phase line. The tension of solid and water pulls in one direction along the interface between the solid and the water in what is properly called an interfacial tension. The tension of solid and air pulls in the opposite direction. The surface tension of the water pulls along a tangent to the surface of the water.

The horizontal component of the pull from the water's surface tension is, say, rightward and its size depends on the cosine of the contact angle. According to Young's argument, if the triple-phase line is in equilibrium, so that the drop is neither spreading nor contracting, the net horizontal force on the line must vanish. A specific value of the contact angle results in a balance of the horizontal forces on the line.

Young's argument is simple, but no one has verified it experimentally. Moreover, measurements of the contact angle of water on a solid surface differ among investigators. Hence Young's theory is open to question. Indeed, the common observation that water drops often cling to a windowpane is ample evidence that the theory is wrong.

One flaw in the theory is that no one has yet demonstrated how the surface of a solid can be in a state of tension of the kind required by Young. The concepts of the tension of solid and water and of solid and air are therefore in doubt, and so is the existence of a three-phase line. A better model might embody a three-dimensional zone in which the shape of the water surface within a few molecular widths of the solid surface may be quite different from what can be inferred with simple measurements.

You can readily demonstrate that Young's theory is in error by depositing a small drop of water on a horizontal plate of clean glass. According to the theory, the drop is in equilibrium and the contact angle must have a certain value. If you carefully add a small amount of water from a syringe, however, you can increase the size of the drop without making the triple-phase line move outward over the glass. The contact angle must then be larger than it was before, contrary to Young's theory. Similarly, you can remove a small amount of water from the drop to decrease the contact angle. Again the triple-phase line stays in place, contrary to the theory.

Obviously the drop is in equilibrium over a range of contact angles rather than at a single value. It spreads over the glass only when the contact angle exceeds some upper limit. It contracts only when the contact angle drops below some lower limit. After either event the contact angle is again within the range of values that make for stability. The upper limit to the contact angle is called the advancing limit, the lower one the receding limit. The ability of a drop to be in equilibrium at different values of the contact angle is referred to as the hysteresis of the contact angle. (Hysteresis describes a situation in which the forces acting on a body change but the resulting effect is delayed.)

It is hysteresis that enables a drop to cling to an inclined surface, including a vertical windowpane. Tilt the glass on which you have placed a drop. (Keep the tilt short of the amount that makes the entire drop slide.) The upper end of the drop remains in place while the lower end slides a short way down the slope. The contact angle is then small at the upper end of the drop and large at the lower end. If equilibrium resulted from only a single value of the contact angle, the drop would immediately slide down the glass.

Why a drop displays hysteresis is a subject of current research. One factor is the roughness of the surface holding the drop. Even glass is rough on a microscopic scale. The triple-phase line is thought by some workers to nestle in the hills and valleys of the surface. Perhaps the contact angle does have a specific value, but the microscopic tilting of the surface changes the apparent contact angle to some other value.

The variation in the contact angle may also be due to microscopic variations in the composition of the solid surface. Adsorbed films and other contamination would also contribute to variations in the contact angle. In some cases the triple-phase line might in fact move, but so slowly as not to be noticed.

If you tilt the glass plane enough, the drop may begin to slide downward. Much research is directed at predicting the conditions that give rise to this kind of movement. In applications such as the spraying of plant leaves one wants the drops to remain on the inclined surface even if it is appreciably tilted. In other applications one wants the drops to run off, leaving the solid surface dry.

For a given size of drop what is the largest angle of tilt at which the drop will stay in place? For a given angle of tilt what is the largest drop that will cling to the substrate?

Suppose the glass is tilted enough to put the drop on the verge of sliding. The force tending to dislodge the drop is the component of the drop's weight that is directed along the glass. The force working against the slide is due to the surface tension on the drop. In the simplest model the force resisting the slide is equal to the product of the water's surface tension and the horizontal width of the drop. When you tilt the glass more, the weight component increases until it overwhelms the force due to surface tension; the drop then slides down the glass.

In another simple model the sliding begins when the weight component causes the contact angle on the lower edge to grow beyond its advancing limit. Even more sophisticated models for resistance to sliding have been constructed by incorporating the contact angles on both the lower and the upper edges of the drop.

I investigated the behavior of a water drop on a plate of Plexiglas. Since the water was from a tap, both the drops and the Plexiglas were undoubtedly contaminated. When the plate was horizontal, the drop was a bead. As I tilted the plate the bottom of the drop slid slightly, narrowing the width of the drop. As the tilt increased, the drop began to slide down the Plexiglas, leaving a thin trail.

By tilting the plate so that some drops traveled slowly enough for me to follow them with a magnifying glass I found that the sliding consisted of a series of swift, short advances. Sometimes I spotted narrow fingers that darted ahead. I could not see what caused them. When I made a drop advance on a visible speck of contamination or even on a hair, no fingers appeared. Perhaps both the fingers and the sudden advance of an entire drop result from microscopic contamination or tiny variations in the topography of the Plexiglas.

My second puzzle concerns meandering. Since water on a tilted surface is pulled downward by gravity, the path of the stream should be straight down. Yet in many experiments the stream meanders, thus developing bends that connect relatively straight stretches slanting to one side or the other. A meander may remain stable for hours or may be so unstable that the stream thrashes over the surface like a snake. What forces give rise to meandering?

Much of the puzzle seems to have been solved recently by Takeo Nakagawa of Kanazawa University in Japan and John C. Scott, who was then at the University of Essex. They made a stream of water flow along a smooth plate of Plexiglas one meter long and .6 meter wide. A grid was attached to the bottom face of the plate. The slope of the plate could be varied between five and 85 degrees in five-degree increments. The water flowed from a reservoir down onto the plate through a vinyl tube whose inside diameter was one centimeter.

After the total stream length was measured from a tracing of the stream, Nakagawa and Scott computed the sinuosity, which is the ratio of the total stream length to the nonmeandering length. The depth of the stream was measured with a needle at intervals of five millimeters down the slope of the plate and one millimeter across the width of the stream. From these data the shape of the stream's cross section was constructed.

The stream, seemingly free of turbulence, emerged from the tube 20 centimeters from the top edge of the plate. To reveal the flow within a stream a hypodermic needle was used to inject a small amount of methylene-blue solution into the stream. The discharge rate of the water was kept steady by maintaining a constant head of water in the reservoir. The rate was measured by collecting the water for a certain period of time at the lower end of the plate and then weighing the water. After each experiment the Plexiglas was cleaned with soft tissues and dried for 30 minutes.

To repeat some of these experiments I built a similar apparatus. I bought a sheet of Plexiglas from a hardware store and taped it to a sturdy wood shelf so that it could not flex. The upper end of the shelf was supported by two laboratory jacks with which I could change the tilt of the Plexiglas. I measured the tilt with a protractor, attaching to it a small weight on a string so that I could keep track of the true vertical.

In a few trials I added a short length of roof gutter at the bottom edge of the Plexiglas. The water ran into the gutter and out through an opening that normally would be connected to a downspout. By timing the collection of water in a graduated beaker under the opening I computed the volume flow rate (the volume of water flowing past a checkpoint each second).

Above the Plexiglas I put a large beaker, which was continuously filled with water from a garden hose. A plastic tube ran from a side arm on the beaker to the Plexiglas, where it was taped in position with its opening facing down the slope. I kept the beaker overflowing to maintain a constant flow of water onto the Plexiglas. (I did my experiments outdoors, but a room with a floor drain would have served.) The beaker was taped to a laboratory jack so that I could vary the flow rate by altering the height of the beaker and thus the head of water.

Nakagawa and Scott discovered that when they tilted their Plexiglas more than 30 degrees, the stream formed a stable meander if the volume flow rate was between an upper and a lower limit. If the flow exceeded the upper limit, the meander was unstable and changed constantly. If the flow dropped below the lower limit, the stream broke up into drops that slid separately down the Plexiglas. Between the limits the stream always meandered.

As the tilt was increased beyond 30 degrees the value of the upper limit decreased, narrowing the range of flow rates within which the meander was stable. At a tilt of less than about 30 degrees the lower limit disappeared; consequently the meander was stable at any small flow rate. The sinuosity of the meander increased whenever either the slope of the plate or the flow rate increased.

Nakagawa and Scott measured the cross-sectional area of a stream in order to monitor the speed of the water One can determine speed in this way because the volume flow rate through a cross section equals the product of the cross-sectional area and the average speed of the water through the cross section. Since a constant flow of water requires that the volume flow rate be the same everywhere along the stream, the cross-sectional area varies inversely with any change in speed. For example, if a stream were to flow directly down a Plexiglas slope while being accelerated by gravity, it should become narrower as the length of travel increases. (In the course of about 10 centimeters, however, the viscous drag from the Plexiglas overcomes the acceleration; the cross-sectional area is then constant.)

Nakagawa and Scott presented data on the area and shape of the cross sections along a stream displaying a stable meander. From the discharge tube the stream went almost straight down the slope for about seven centimeters, becoming narrower and maintaining a symmetrical cross section with the highest point at the center. Beyond seven centimeters the stream meandered several times. The bends connected relatively straight sections slanted to one side or the other of the plate. Within the meander the cross-sectional area varied, being large just before a bend and smaller within the bend. Hence the flow was slow just before a bend and faster in the bend.

The cross sections within the meander were asymmetrical because the highest point was off-center. I call the deeper side of the cross section a ridge. In straight, diagonal stretches the ridge was on the lower side of the stream. Near the upper end of a bend the ridge crossed the stream, so that it ran down along the outside of the bend and then along the lower side of the next diagonal stretch.

Nakagawa and Scott maintain that the ridge causes the meanders. In a stream with a symmetrical cross section the forces from the surface tension along the opposite sides cancel. If the cross section is asymmetrical, the force on the ridge side is larger than the one on the shallow side. This imbalance yields a net force toward the shallow side.

The ridge first appears at the point where the stream flows approximately straight down from the discharge tube. Presumably it is created by contamination or microscopic roughness of the plate or by undetected turbulence in the stream. The net force arising from the ridge deflects the stream, sending it into the first diagonal stretch of a meander.

The diagonal stretch can be stable against the pull of gravity because the ridge is then on the lower side and the force from it is upward. This arrangement is similar to that of a drop clinging to an inclined surface, held in place by the net force due to surface tension. As in the case of a drop, the contact angle is large on the lower side and small on the upper side.

As the water flows along the diagonal stretch, however, it is no longer accelerated as much by gravity as it was when it ran straight down. Viscous drag from the plate slows the water, increasing the cross-sectional area and also the contact angle on the lower side. Just before a bend the contact angle on the lower side exceeds the advancing limit and the water begins to run straight down the plate.

The stream would continue in the same direction except for the continued presence of the ridge. At the beginning of the fall the ridge shifts across the stream. The force due to the ridge then deflects the stream into another straight, diagonal section. The turn is the bend in the meander.

In doing my experiments I observed several more characteristics of the flow of water down a slope. Usually a drop slid down, meandering slightly or not at all. Similarly, streams with low volume flow rates often did little meandering. Strong meanders had prominent loops at the top of the plate but indistinct ones lower down. Occasionally I had a stream that meandered strongly at the top of the plate and weakly near the bottom, with a virtually straight run between the meanders. How does one explain these phenomena?

I believe that when a sliding drop forms an asymmetrical cross section, the force due to the asymmetry quickly restores the symmetry. The drop is momentarily deflected to one side but does not travel far along this slanted path. Since it has less mass than a stream, it may also be affected more by contamination and irregularities along the surface. The result is a wiggly path.

The pattern developed by a stream with a low volume flow rate seems to depend greatly on the initial straight run. I created such a stream and blocked it briefly by holding a cloth near the discharge tube. Each time I released the water the stream formed a different pattern: sometimes a strong meander, at other times little or no meandering. The pattern seemed to depend on how the pool of water just above the cloth first broke free. Apparently if no significant ridge developed in the initial straight run, the stream meandered weakly, whereas a significant ridge or strong turbulence caused a definite meander. The path of the initial descent persisted, indicating that the first wetted section of the plate determined the direction of flow.

In many trials I began with a low volume flow rate and a slight meander. As I increased the flow rate the first meander shifted down the slope, presumably because the force due to a ridge required more time to deflect the stream. The stream also began to reshape itself into stronger meanders with longer diagonals and greater distances along the slope between bends. This pattern seems to be due to the flow of water within the ridge on the diagonal stretches. Water approaching a bend overshot the bend before it began to fall along the plate. As overshoot persisted the bend migrated down the plate and to one side, and the diagonals lengthened.

After a while the diagonal stretch leading to a bend began to slide down the plate, apparently because the viscous drag in the stretch slowed the flow, thereby increasing the cross-sectional area and also the lower contact angle. Once that angle exceeded the advancing limit the diagonal stretch began to slide. If it slid faster than the bend, it collapsed, breaking off from the bend and forming a new bend nearer the middle of the plate. When an old bend broke up, the water in it quickly ran down the plate.

When I further increased the volume flow rate, the stream began to collapse, often in the diagonal stretches; sometimes it released not only an old bend but also several parts of a diagonal. The intact portion of the stream rapidly developed a new bend, which might curve in the same direction as the old one or in the opposite direction. The rapid collapse and release of isolated stretches of the stream gave the illusion that the main part was thrashing about over the plate, propelling water into a fan of angles toward the bottom.

In one set of trials involving a stable meander I used a syringe to inject food coloring into the stream. Without color the stability of the meander gives the impression that the flow is sluggish. The coloring shows how rapidly the water is flowing.

Not long after I put in the color the stream cleared ex sides of the diagonals of meanders. These stretches kept the color several seconds longer, suggesting that the flow is slower in them than it is through the ridges along the lower sides.

Next I put in a smaller amount of coloring at a steady rate. Where the flow was fast the color was too weak to see, but in the regions where the flow seemed to be sluggish small amounts of color were visible.

During these trials I noticed another peculiar feature of a stable meander. At times the stream developed wide sections I call slugs. A slug would remain stationary for a while and then creep along the stream, often causing a meander to collapse.

With a magnifying glass I examined the flow of color through a slug. I discovered that in the upper side the water flow formed a vortex: colored threads looped around, flowing both upstream and down. I then understood how a slug could make a meander collapse. The formation of the vortex at the upper side of the slug forces the water to flow faster and with more turbulence along the lower side. The contact angle on the lower side could then increase beyond its advancing limit, leading to a collapse. Alternatively the vortex could suddenly increase in size, disrupting the flow through the slanted section and deflecting the flow downward.

Much can be learned from further analysis of clinging drops and meandering streams. You might investigate how putting alcohol in the water alters the results I have described. (Alcohol lowers the surface tension of the water.) Do viscous fluids such as cold syrup meander? Does the meandering of streams increase or decrease if the roughness of the slanted surface is increased?

Bibliography

CONTACT ANGLE HYSTERESIS. T. D. Blake and J. M. Haynes in Progress in Surface and Membrane Science, Vol. 6, pages 125-138; 1973.

STREAM MEANDERS ON A SMOOTH HYDROPHOBIC SURFACE. Takeo Nakagawa and John C. Scott in Journal of Fluid Mechanics, Vol. 149, pages 89-99; December, 1984. :

ON THE ABILITY OF DROPS OR BUBBLES TO STICK TO NON-HORIZONTAL SURFACES OF SOLIDS, PART 2: SMALL DROPS OR BUBBLES HAVING CONTACT ANGLES OF ARBITRARY SIZE. E. B. Dussan V. in Journal of Fluid Mechanics, Vol. 151, pages 1-20; February, 1985.

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