Abstract and Keywords
This chapter presents a pedagogical discussion of the surface tension and its manifestation in a number of fluid systems. Interfacial boundary conditions are derived and then applied in various settings. Particular attention is given to highlighting the role of curvature pressure in fluid statics, including fluid menisci and the floating of small bodies at interfaces. Dynamic settings influenced by capillary effects and capillary instability are also highlighted, including fluid jets, sheets, and hydraulic jumps. Marangoni flows (dominated by gradients of surface tension) are also considered, and the role of surface impurities in interfacial flows discussed. Simple mathematical developments are augmented with physical discussion with hopes of improving intuition for this class of problems.
(p.28) 2.1 Introduction
These lecture notes have been drawn from many sources, including textbooks, journal articles, and lecture notes from courses taken by the author as a student. They are not intended as a complete discussion of the subject, or as a scholarly work in which all relevant references are cited. Rather, they are intended as an introduction that will hopefully motivate the interested student to learn more about the subject. Topics have been chosen according to their perceived value in developing the physical insight of the students.
2.2 The definition and scaling of surface tension
2.2.1 Surface tension: A working definition
Discussions of the molecular origins of surface or interfacial tension may be found elsewhere; ours follows that of De Gennes et al. (2004).
Molecules in a fluid feel a mutual attraction. When this attractive force is overcome by thermal agitation, the molecules pass into a gaseous phase. Let us first consider a free surface, for example that between air and water. A water molecule in the fluid bulk is surrounded by attractive neighbors, while a molecule at the surface is attracted by a reduced number of neighbors and so is in an energetically unfavorable state. The creation of new surface is thus energetically costly, and a fluid system will act to minimize surface areas. It is thus that small fluid bodies tend to evolve into spheres; for example, a thin fluid jet emerging from the top in your kitchen sink will generally pinch off into spherical drops in order to minimize the total surface area (see Section 2.6).
If U is the total cohesive energy per molecule, then a molecule at a flat surface will lose U/2. Surface tension is a direct measure of this energy loss per unit area of surface. If the characteristic molecular dimension is R and its area is thus R 2, then the surface tension is σ ~ U/(2R 2). Note that surface tension increases as the intermolecular attraction increases and the molecular size decreases. For most oils, σ ~ 20 dyn/cm, while for water, σ ~ 70 dyn/cm. The highest surface tensions are for liquid metals; for example, liquid mercury has σ ~ 500 dyn/cm.
The origins of interfacial tension are analogous. Interfacial tension is a material property of a fluid–fluid interface whose origins lie in the different attractive intermolecular forces that act in the two fluid phases. The result is an interfacial energy per area that acts to resist the creation of new interface, and that is equivalent to a line tension acting in all directions parallel to the interface. Fluids between which no interfacial tension arises are said to be miscible. For example, salt molecules will diffuse freely across a boundary between fresh and salt water; consequently, these fluids are miscible, and there is no interfacial tension between them. Our discussion will be confined to immiscible fluid–fluid interfaces (or fluid surfaces), at which an effective interfacial (or surface) tension acts.
Surface tension σ has the units of force/length or, equivalently, energy/area, and so may be thought of as a negative surface pressure. Pressure is generally an isotropic force per area that acts throughout the bulk of a fluid: a small surface element dS will feel a total force p(x) dS owing to the local pressure field p(x). If the surface S (p.29) is closed, and the pressure uniform, the net pressure force acting on S is zero and the fluid remains static. Pressure gradients correspond to body forces (with units of force per unit volume) within a fluid, and so appear explicitly in the Navier–Stokes equations. Surface tension has the units of force per length, and its action is confined to the free surface. Consider, for the sake of simplicity, a perfectly flat interface. A surface line element dℓ will feel a total force σ dℓ owing to the local surface tension σ(x). If the surface line element is a closed loop C and the surface tension is uniform, the net surface tension force acting on C is zero, and the fluid remains static. If surface tension gradients arise, there may be a net force on the surface element that acts to distort it through driving flow.
2.2.2 Governing equations
The motion of a fluid of uniform density ρ and viscosity μ is governed by the Navier– Stokes equations, which represent a continuum statement of Newton's laws:
This represents a system of four equations in four unknowns (the fluid pressure p and the three components of the velocity field u). Here F represents any body force acting on the fluid; for example, in the presence of a gravitational field, F = ρ g, where g is the acceleration due to gravity.
Surface tension acts only at the free surface; consequently, it does not appear in the Navier–Stokes equations, but rather enters through the boundary conditions. The boundary conditions appropriate at a fluid–fluid interface are formally developed in Section 2.3. Here, we simply state them for the simple case of a free surface (such as between air and water, in which one of the fluids is not dynamically significant) in order to get a feeling for the scaling of surface tension.
The normal stress balance at a free surface must be balanced by the curvature force associated with the surface tension:
2.2.3 The scaling of surface tension
We consider a fluid of density ρ and viscosity μ = ρ ν with a free surface characterized by a surface tension σ. The flow is marked by characteristic length and velocity scales (p.30) of a and U, respectively, and evolves in the presence of a gravitational field g = −g ẑ. We thus have a physical system defined in terms of six physical variables (ρ, ν, σ, a, U, g) that may be expressed in terms of three fundamental units: mass, length, and time. Buckingham's Theorem thus indicates that the system may be uniquely prescribed in terms of three dimensionless groups. We choose
The Reynolds number prescribes the relative magnitudes of inertial and viscous forces in the system, while the Froude number prescribes those of inertial and gravity forces. The Bond number indicates the relative importance of forces induced by gravity and surface tension. Note that these two forces are comparable when Bo = 1, which arises on a length scale corresponding to the capillary length ℓc = (σ/(ρ g))1/2. For an air–water surface, or example, σ ≈ 70 dyn/cm, ρ = 1 g/cm3, and g = 980 cm/s2, so that ℓc ≈ 2 mm. Bodies of water in air are dominated by the influence of surface tension provided they are smaller than the capillary length. Roughly speaking, the capillary length prescribes the maximum size of pendant drops that may hang inverted from a ceiling, and the maximum size of water‐walking insects and raindrops. Note that as a fluid system becomes progressively smaller, the relative importance of surface tension and gravity increases; it is thus that surface tension effects are dominant in microscale engineering processes.
Finally, we note that other frequently arising dimensionless groups may be formed from products of B, Re, and Fr:
The Weber number indicates the relative magnitudes of inertial and curvature forces within a fluid, and the capillary number those of viscous and curvature forces. Finally we note that if the flow is marked by a Marangoni stress of characteristic magnitude Δσ/L, then an additional dimensionless group arises that characterizes the relative magnitude of the Marangoni and curvature stresses: aΔσ/(L σ).
We now demonstrate how these dimensionless groups arise naturally from the nondimensionalization of the Navier–Stokes equations and the surface boundary conditions. We first introduce a dynamic pressure p d = p − ρ g ∙ x, so that gravity appears only in the boundary conditions. We consider the special case of high‐Reynolds‐number flow, for which the characteristic dynamic pressure is ρ U 2. We define dimensionless primed variables according to
Nondimensionalizing the Navier–Stokes equations and appropriate boundary conditions yield the following system:
The normal‐stress condition assumes the dimensionless form
The importance of surface tension relative to gravity and viscous stresses is prescribed by the relative magnitudes of the Weber, Froude, and Reynolds numbers. In the high‐ Re limit of interest, the normal force balance requires that the dynamic pressure be balanced by either gravitational or curvature stresses, the relative magnitudes of which are prescribed by the Bond number.
The nondimensionalization scheme will depend on the physical system of interest. Our purpose here was simply to illustrate the manner in which the dimensionless groups arise in the theoretical formulation of the problem. Moreover, we see that those involving surface tension enter exclusively through the boundary conditions.
2.3 Stress conditions at a fluid–fluid interface
We proceed by deriving the normal‐ and tangential‐stress boundary conditions appropriate at a fluid–fluid interface characterized by an interfacial tension σ.
Consider an interfacial surface S bounded by a closed contour C (Figs. 2.1–2.3). One may think of there being a force per unit length of magnitude σ in the s‐direction at every point along C that acts to flatten the surface S. Perform a force balance on a volume element V enclosing the interfacial surface S defined by the contour C:
Here ℓ indicates an arc length, and so dℓ indicates a length increment along the curve C. t(n) = n ∙ T is the stress vector, the force/area exerted by the upper fluid (+) on the interface. The stress tensor is defined in terms of the local fluid pressure and velocity field as T = − p I + μ[∇u + (∇u)T]. Similarly, the stress exerted on the interface by the lower fluid − is t̂(n̂) = n̂ ∙ T̂ = −n ∙ T̂, where T̂, = −p I + μ̂[∇û + (∇û)T].
2.3.1 Physical interpretation of terms
inertial force associated with acceleration of fluid within V.
∫V f dV: body forces acting on fluid within V.
∫S t̂(n̂) dS: hydrodynamic force exerted at interface by fluid −.
∫C σ s dℓ: surface tension force exerted along perimeter C.
Now, if ϵ is the typical length scale of the element V, then the acceleration and body forces will scale as ϵ 3, but the surface forces will scale as ϵ 2. Hence, in the limit of ϵ → 0, we have that the surface forces must balance:
Now we have that
The surface force balance thus becomes
Now, since the surface element is arbitrary, the integrand must vanish identically. One thus obtains the interfacial stress balance equation.
2.3.3 Interpretation of terms
n ∙ T: stress (force/area) exerted by + on − (will generally have both normal and tangential components).
n ∙ T̂: stress (force/area) exerted by – on + (will generally have both normal and tangential components).
σ n(∇∙n): normal curvature force per unit area associated with local curvature of interface, ∇ ∙ n.
∇σ: tangential stress associated with gradients of surface tension.
Both the normal and the tangential stress must be balanced at the interface. We consider each component in turn.
2.3.4 Normal‐stress balance
Taking n ∙ eqn(2.13) yields the normal‐stress balance at the interface:
The jump in normal stress across the interface must balance the curvature force per unit area. We note that a surface with nonzero curvature (∇ ∙ n ≠ 0) reflects a jump in normal stress across the interface.
(p.34) 2.3.5 Tangential‐stress balance
Taking t ∙ eqn(2.13), where t is any unit vector tangent to the interface, yields the tangential‐stress balance at the interface:
The physical interpretation of this is that:
• the LHS represents the jump in the tangential components of the hydrodynamic stress at the interface;
• the RHS represents the tangential stress associated with gradients in σ, as may result from gradients in temperature or chemical composition at the interface;
• the LHS contains only velocity gradients, not pressure; therefore, a nonzero ∇σ at a fluid interface must always drive motion.
2.4 Fluid statics
We begin by considering static fluid configurations, for which the stress tensor reduces to the form T = −p I, so that n ∙ T ∙ n = −p, and the normal‐stress balance assumes the form
The pressure jump across the interface is balanced by the curvature force at the interface. Now, since n ∙ T ∙ s = 0 for a static system, the tangential‐stress balance equation indicates that 0 = ∇σ. This leads us to the following important conclusion.
There cannot be a static system in the presence of surface tension gradients. While pressure jumps can sustain normal‐stress jumps across a fluid interface, they do not contribute to the tangential‐stress jump. Consequently, tangential surface stresses can only be balanced by viscous stresses associated with fluid motion.
We proceed by applying eqn (2.17) to a number of static situations.
2.4.1 Stationary bubble
We consider a spherical bubble of radius R submerged in a static fluid. What is the pressure drop across the bubble surface?
The curvature of the spherical surface is computed simply:
So, the normal‐stress jump (eqn(2.17)) indicates that
The pressure within the bubble is higher than that outside by an amount proportional to the surface tension, and inversely proportional to the bubble size. It is thus that (p.35) small bubbles are louder than large ones when they burst at a free surface: champagne is louder than beer. We note that soap bubbles in air have two surfaces, which define the inner and outer surfaces of the soap film; consequently, the pressure differential is twice that across a single interface.
2.4.2 Static meniscus
Consider a situation where the pressure within a static fluid varies owing to the presence of a gravitational field: p = p 0 + ρ gz, where p 0 is the constant ambient pressure, and g⃗ = −g ẑ is the gravitational acceleration. The normal‐stress balance thus requires that the interface satisfy the Young–Laplace equation,
The vertical gradient in fluid pressure must be balanced by the curvature pressure; as the gradient is constant, the curvature must likewise increase linearly with z. Such a situation arises in a static meniscus (see Fig. 2.4).
The shape of the meniscus is prescribed by two factors: the contact angle between the air–water interface and the wall, and the balance between hydrostatic pressure and curvature pressure. We treat the contact angle θ as given; it depends on the physics of the log–water–air interaction. The normal‐force balance is expressed by the Young– Laplace equation, where now ρ = ρ w − ρ a ≈ ρ w is the density difference between water and air.
We define the free surface by z = η(x); equivalently we define a functional f (x, z) = z − η (x) that vanishes on the surface. The normal to the surface is thus
As deduced in Appendix C, the curvature of the free surface, ∇ ∙ n̂, may be expressed as
Applying the boundary condition η(∞) = 0 and the contact condition η x(0) = − cot θ and solving eqn (2.23) thus yields
2.4.3 Radial force on a circular hydraulic jump
Hydraulic jumps may be generated when a vertical jet strikes a flat plate. The jet spreads radially, giving rise to a fluid layer that generally thins with radius until reaching a critical radius, at which it increases dramatically (see Fig. 2.5). Here, we calculate the total radial force acting on the jump surface owing to the curvature of the jump between points A and B, located at radii R 1 and R 2, respectively. Assume that the points A and B are the points nearest the jump, upstream and downstream, respectively, at which the slope of the surface vanishes identically.
(p.37) We know that the curvature force per unit area is σ(∇ ∙ n)n. We must integrate the radial component of this force over the jump surface:
Now, defining the surface as z = h(r) allows us to express the area element as
Now we use the appropriate forms for n and ∇ ∙ n from Appendix C:
Integrating by parts, i.e. with , yields
Integrating by parts again, with , yields
We note that this relation yields reasonable results in two limits of interest. First, for a flat interface, S = Δ R, so that F c = 0. Second, for an abrupt jump in height of Δ H, Δ R = 0 and S = Δ H, so that F c = 2π σ Δ H. This result is commensurate with the total force exerted by a cylindrical annulus of radius R and height Δ H, which is deduced from the product of σ, the area 2π R Δ H, and the curvature 1/R:
The influence of this force has been found to be significant for small hydraulic jumps. Its importance relative to the hydrostatic pressure in containing the jump is given by
The simple result (2.37) was derived by Bush and Aristoff (2003), who also presented the results of an experimental study of circular hydraulic jumps. Their experiments indicated that the curvature force becomes appreciable for jumps with a characteristic radius of less than 3 cm.
While the influence of surface tension on the radius of a circular hydraulic jump is generally small, it may have a qualitative influence on the shape of the jump. In particular, it may prompt the axisymmetry‐breaking instability responsible for the polygonal hydraulic jump structures discovered by Ellegaard et al. (1998), and more recently examined by Bush et al. (2006). See also http://www-math.mit.edu/bush/jump.htm.
2.4.4 Floating bodies
Floating bodies must be supported by some combination of buoyancy and curvature forces. Specifically since the fluid pressure beneath the interface is related to the atmospheric pressure P 0 above the interface by
The buoyancy force
At the interface, the buoyancy and curvature forces must balance precisely so the Young–Laplace relation is satisfied:
Integrating the fluid pressure over the meniscus yields the vertical force balance:
The relative magnitudes of the buoyancy and curvature forces supporting a floating, nonwetting body are thus prescribed by the relative magnitudes of the volumes of the fluid displaced inside and outside the line of tangency:
For 2D bodies, we note that since the meniscus will have a length comparable to the capillary length ℓc = (σ/(ρ g))1/2, the relative magnitudes of the buoyancy and curvature forces,
Small objects such as paper clips, pins, or insects may reside at rest on a free surface provided the curvature force induced by their deflection of the free surface is sufficient to bear their weight. For example, for a body of contact length L and total mass M, static equilibrium on the free surface requires that
If body proportions were independent of size L, one would expect the body weight to scale as L 3 and the curvature force as L. Isometry would thus suggest a dependence of the form , represented as the dashed line in Fig. 2.9. The fact that the best‐fit line has a slope considerably larger than 1/3 indicates a variance from isometry: the legs of large water striders are proportionally longer.
2.5 Marangoni flows
Marangoni flows are flows driven by surface tension gradients. In general, the surface tension σ depends on both the temperature and the chemical composition at the interface; consequently, Marangoni flows may be generated by gradients in either temperature or chemical concentration at an interface.
Recall the tangential‐stress balance at a free surface,
2.5.1 Tears of wine
The first Marangoni flow considered was the tears‐of‐wine phenomenon (Thomson 1885); Thomson's treatment actually predates Marangoni's first published work on the subject by a decade. The tears‐of‐wine phenomenon is readily observed in a glass of any but the weakest wines following the establishment of a thin layer of wine on the walls of the wine glass.
The “tears” or “legs” of wine are taken by sommeliers to be an indicator of the quality of wine. An illustration of the tears‐of‐wine phenomenon is presented in Fig. 2.10 (see also http://www-math.mit.edu/bush/tears.html). Evaporation of alcohol occurs everywhere on the free surface. The alcohol concentration in the thin layer is thus reduced relative to that in the bulk owing to the enhanced surface‐area‐to‐ volume ratio. As the surface tension decreases with alcohol concentration, the surface tension is higher in the thin film than in the bulk; the associated Marangoni stress drives upflow throughout the thin film. The wine climbs until it reaches the top of the film, where it accumulates in a band of fluid that thickens until eventually it becomes gravitationally unstable and releases the tears of wine. The tears or “legs” roll back to replenish the bulk reservoir, but with fluid that is depleted in alcohol.
Surfactants are molecules that have an affinity for interfaces; common examples include soap and oils. Owing to their molecular structure (often a hydrophilic head and hydrophobic tail), they find it energetically favorable to reside at the free surface. Their presence reduces the surface tension; consequently gradients in the surfactant concentration Γ result in surface tension gradients. Surfactants thus generate a special class of Marangoni flows. There are many different types of surfactants, some of which are insoluble (and so remain on the surface), and others which are soluble in the suspending fluid and so diffuse into the bulk. For a wide range of common surfactants, the surface tension is a monotonically decreasing function of Γ until a critical concentration is achieved, beyond which σ remains constant.
The concentration of surfactant Γ on a free surface evolves according to
The principal dynamical influence of surfactants is to impart an effective elasticity to the interface. Specifically the presence of surfactants will serve to alter not only the normal‐stress balance (through the reduction of σ) but also the tangential‐stress balance (through the generation of Marangoni stresses). The presence of surfactants will act to suppress any fluid motion characterized by nonzero surface divergence. For example, consider a fluid motion characterized by a radially divergent surface motion. The presence of a surfactant results in the redistribution of the surfactant: Γ is reduced near the point of divergence. The resulting Marangoni stresses act to suppress the surface motion, resisting it through an effective surface elasticity. Similarly, if the flow is characterized by a radial convergence, the resulting accumulation of surfactant in the region of convergence will result in Marangoni stresses that serve to resist it. It is this effective elasticity that gives soap films their longevity: the divergent motions that would cause a pure liquid film to rupture are suppressed by the surfactant layer on the soap film surface.
The ability of surfactants to suppress flows with nonzero surface divergence is evident throughout the natural world (Lucassen‐Reynders and Lucassen 1969). It was first remarked upon by Pliny the Elder, who rationalized that the absence of capillary waves in the wake of ships is due to the ships stirring up surfactant. This phenomenon was also widely known to spear‐fishermen, who poured oil on the water to increase their ability to see their prey, and to sailors, who would do similarly in an attempt to calm troubled seas. Finally, the suppression of capillary waves by surfactants is responsible for the “footprints of whales” (see Fig. 2.11). In the wake of whales, even in turbulent roiling seas, one sees patches on the sea surface (of characteristic width 5–10 m) that are perfectly flat. These are generally acknowledged to result from the whales sweeping biomaterial to the surface with their tails; this biomaterial serves as a surfactant that suppresses capillary waves.
2.5.3 The soap boat
Consider a floating body with perimeter C in contact with the free surface, which we assume for the sake of simplicity to be flat. Recalling that σ may be thought of as a force per length in a direction tangent to the surface, we see that the total surface tension force acting on the body is
A “soap boat” may be simply generated by coating one end of a toothpick with soap, which acts to reduce surface tension (see Fig. 2.12). The concomitant gradient in surface tension results in a net propulsive force that drives the boat away from the soap. We note that an analogous Marangoni propulsion technique arises in the natural world: certain water‐walking insects eject surfactant and use the resulting surface tension gradients as an emergency mechanism for avoiding predation. Moreover, when a pine needle falls into a lake or pond, it is propelled across the surface in an analogous fashion owing to the influence of the resin at its base decreasing the local surface tension.
2.5.4 Bubble motion
Theoretical predictions of the rise speed of small drops or bubbles do not adequately describe the observations. Specifically air bubbles rise at low Reynolds number at (p.46)
The effective surface elasticity imparted by a surfactant acts to suppress the surface motion. Surfactant is generally swept to the trailing edge of the bubble, where it accumulates, giving rise to a local decrease in surface tension. The resulting fore‐to‐ aft surface tension gradient results in a Marangoni stress that resists surface motion, and so rigidifies the bubble surface. The air bubble thus moves as if its surface were stagnant, and it is thus that its rise speed is commensurate with that predicted for a rigid sphere: the no‐slip boundary condition is more appropriate than the free‐slip condition. Finally, we note that the characteristic Marangoni stress Δ σ/a is most pronounced for small bubbles. It is thus that the influence of surfactants is greatest on small bubbles.
(p.47) 2.6 Fluid jets
2.6.1 The shape of a falling fluid jet
Consider a circular orifice of radius a ejecting a flux Q of fluid of density ρ and kinematic viscosity ν (Fig. 2.13). The resulting jet is shot downwards, and accelerates under the influence of gravity −g ẑ. We assume that the Reynolds number Re = Q/(a ν) of the jet is sufficiently high that the influence of viscosity is negligible; furthermore, we assume that the speed of the jet is independent of radius, and so is adequately described by U(z). We proceed by deducing the shape r(z) and speed U(z) of the evolving jet.
Applying Bernoulli's Theorem at the points A and B,
(p.48) The local curvature of a slender thread may be expressed in terms of the two principal radii of curvature, R 1 and R 2:
Thus, the fluid pressures within the jet at the points A and B may be simply related to that of the ambient, P 0:
Substituting into eqn (2.55) thus yields
Now flux conservation requires that
This may be solved algebraically to yield the thread shape r(z)/a, and then this result may be substituted into eqn (2.61) to deduce the velocity profile U(z). In the limit of W e → ∞, one obtains
2.6.2 The Plateau–Rayleigh instability
Here, we summarize the work of Plateau and Rayleigh on the instability of cylindrical fluid jets bound by surface tension. It is precisely this Rayleigh–Plateau instability (p.49)
The equilibrium base state consists of an infinitely long, quiescent, cylindrical, inviscid fluid column of radius R 0, density ρ, and surface tension σ (Fig. 2.15). The influence of gravity is neglected. The pressure p 0 is constant inside the column and may be calculated by balancing the normal stresses with surface tension at the boundary. Assuming zero external pressure yields
We consider the evolution of infinitesimal varicose perturbations on the interface, which enables us to linearize the governing equations. The perturbed columnar surface takes the form
The linearized continuity equation becomes
We anticipate that the disturbances in velocity and pressure will have the same form as the surface disturbance (2.64), and so write the perturbation velocities and pressure as
Eliminating Z(r) and P(r) yields a differential equation for R(r):
This corresponds to modified Bessel equation of order 1, whose solutions may be written in terms of the modified Bessel functions of the first and second kind, I 1(kr) and K 1(kr), respectively. We note that K 1(kr) → ∞ as r → 0; therefore, the well‐ behavedness of our solution requires that R(r) take the form
We proceed by applying appropriate boundary conditions. The first is the kinematic condition on the free surface:
Substitution of eqn (2.73) into this condition yields
Second, we require a normal‐stress balance on the free surface:
We write the curvature as σ∇ ∙ ṉ = (1/R 1 + 1/R 2), where R 1 and R 2 are the principal radii of curvature of the jet surface:
We note first that unstable modes are possible only when
The column is thus unstable to disturbances whose wavelengths exceed the circumference of the cylinder. A plot of the dispersion relation is shown in Fig. 2.16.
The fastest‐growing mode occurs for kR 0 = 0.697, i.e. when the wavelength of the disturbance is
By inverting the maximum growth rate ω max, one may estimate the characteristic breakup time:
When a vertical water jet impinges on a horizontal reservoir of water, a field of standing waves may be excited on the base of the jet (see Fig. 2.17). The wavelength is determined by the requirement that the wave speed correspond to the local jet speed: U = −ω/k. Using our dispersion relation (2.82) thus yields
Provided the jet speed U is known, this equation may be solved in order to deduce the wavelength of the waves that will travel at U and so appear to be stationary in the lab frame. For jets falling from a nozzle, the result (2.86) may be used to deduce the local jet speed.
2.6.3 Fluid pipes
The following system may be readily observed in a kitchen sink. When the volume flux exiting the tap is such that the falling stream has a diameter of 2–3 mm, obstructing the stream with a finger at a distance of several centimeters from the tap gives rise (p.54) to a stationary field of varicose capillary waves upstream of the finger. If the finger is dipped in a liquid detergent (or soap) before insertion into the stream, the capillary waves begin at some critical distance above the finger, below which the stream is cylindrical. Closer inspection reveals that the surface of the jet's cylindrical base is quiescent.
An analogous phenomenon arises when a vertical fluid jet impinges on a deep water reservoir (Figs. 2.17 and 2.18). When the reservoir is contaminated by a surfactant, the surface tension of the reservoir is diminished relative to that of the jet. The associated surface tension gradient draws the surfactant a finite distance up the jet, prompting two salient alterations in the jet surface. First, the surfactant suppresses surface waves, so that the base of the jet surface assumes a cylindrical form (Fig. 2.18). Second, the
A detailed theoretical description of the fluid pipe has been given by Hancock and Bush (2002). We present here a simple scaling that yields the dependence of the vertical extent H of the fluid pipe on the governing system parameters. We assume that, once the jet enters the fluid pipe, a boundary layer develops on its outer wall owing to the no‐slip boundary condition appropriate there. Balancing viscous and Marangoni stresses on the pipe surface yields
The pipe height increases with the surface tension differential and pipe radius, and decreases with fluid viscosity and jet speed.
2.7 Fluid sheets
The dynamics of high‐speed fluid sheets was first considered by Savart after his early work on electromagnetism with Biot, and was subsequently examined in a series of papers by Taylor (1959). Fluid sheets have recently received a great deal of attention owing to their relevance in a number of spray atomization processes. Such sheets may be generated from a variety of source conditions, for example the collision of jets with rigid impactors, and jet–jet collisions.
There is generally a curvature force acting on the sheet edge which acts to contain the fluid sheet. For a 2D (planar) sheet, the magnitude of this curvature force is given by
There is thus an effective force per unit length 2σ along the length of the sheet rim acting to contain the rim.
We now consider how this result may be applied to compute sheet shapes for three distinct cases: (i) a circular sheet, (ii) a lenticular sheet with an unstable rim, and (iii) a lenticular sheet with a stable rim.
2.7.1 Circular sheet
We consider the geometry considered in Savart's original experiment. A vertical fluid jet strikes a small, horizontal circular impactor. If the flow rate is sufficiently high that gravity does not influence the shape of the sheet, the fluid is ejected radially, giving rise to a circular, free fluid sheet (Fig. 2.19). The sheet radius is prescribed by a balance of radial forces; specifically the inertial force must balance the curvature force:
(p.57) Continuity requires that the sheet thickness h depend on the speed u, the jet flux Q, and the radius r as
Experiments (specifically, tracking of particles suspended within the sheet) indicate that the sheet speed u is independent of radius; consequently, the sheet thickness decreases as 1/r. Substituting the form (2.94) for h into the force balance (2.93) yields the sheet radius, or Taylor radius,
The sheet radius increases with source flux and sheet speed, but decreases with surface tension. We note that the fluid proceeds radially to the sheet edge, where it accumulates until it goes unstable via a modified Rayleigh–Plateau instability, often referred to as the Rayleigh–Plateau–Savart instability, as it was first observed on a sheet edge by Savart.
2.7.2 Lenticular sheet with an unstable rim
We now consider a nonaxisymmetric fluid sheet, such as may be formed by the oblique collision of water jets (see Fig. 2.20), a geometry originally considered by Taylor in 1960. Fluid is ejected radially from the origin into a sheet with a flux distribution given by Q(θ), so that the volume flux flowing into the sector between θ and θ + d θ is Q(θ) d θ. As in the previous case of the circular sheet, the sheet rim is unstable, and
The sheet thickness is again prescribed by eqn (2.94), but now Q = Q(θ), so the sheet radius R(θ) is given by
Computing sheet shapes thus relies on measurement of the flux distribution Q(θ) within the sheet.
2.7.3 Lenticular sheet with a stable rim
In a certain region of parameter space, specifically for fluids more viscous than water, one may encounter fluid sheets with a stable rims (see http://www-math.mit.edu/bush/bones.html). The force balance describing the sheet shape must change accordingly. When the rim is stable, fluid entering the rim proceeds along the rim. As a result, there is a centripetal force normal to the fluid rim associated with flow along the curved rim that must be considered in order to correctly predict the sheet shape.
The relevant geometry is presented in Fig. 2.21. r(θ) is defined to be the distance from the origin to the rim centerline, and u n(θ) and u t(θ) are the normal and tangential components of the fluid velocity in the sheet where it contacts the rim. v(θ) is defined to be the velocity of flow in the rim, R(θ) is the rim radius, and ψ(θ) is the angle between the position vector r and the local tangent to the rim centerline. Finally, r c(θ) is defined to be the radius of curvature of the rim centerline, and s is the arc length along the rim centerline. The differential equations governing the shape of a stable fluid rim bounding a fluid sheet may be deduced by consideration of conservation of mass in the rim and the local normal and tangential force balances at the rim.
For a steady sheet, continuity requires that the volume flux from the sheet balances the tangential gradient of the volume flux along the rim:
The normal‐force balance requires that the curvature force associated with the rim's surface tension balances the force resulting from the normal flow into the rim from the fluid sheet and the centripetal force resulting from the flow along the curved rim:
(p.60) This resulting system of equations may be nondimensionalized, and reduces to a set of coupled ordinary equations in the four variables r(θ), v(θ), R(θ), and Ψ(θ). Given a flux distribution Q(θ), the system may be integrated to deduce the sheet shape.
2.7.4 Water bells
All of the fluid sheets considered thus far have been confined to a plane. In Section 2.7.1, we considered a circular sheet generated from a vertical jet striking a horizontal impactor. The sheet remains planar only if the flow is sufficiently fast that the fluid reaches its Taylor radius before sagging substantially under the influence of gravity. Decreasing the flow rate will make this sagging more pronounced, and the sheet will no longer be planar. While one might expect the sheet to fall along a parabolic trajectory, the toroidal curvature of the bell induces curvature pressures that act to close the sheet. Consequently, the sheet may actually close upon itself, giving rise to a water bell, as illustrated in Fig. 2.22. We proceed by outlining the theory required to compute the shapes of water bells. See Clanet (2007).
We consider a fluid sheet extruded radially at a speed u 0 and subsequently sagging under the influence of a gravitational field g = −g ẑ. The inner and outer sheet surfaces are characterized by a constant surface tension σ. The sheet has a constant density ρ and a thickness t(r, z), and is assumed to be inviscid. Q is the total volume flux in the sheet.
We define the origin to be the center of the impact plate; r and z are the radial and vertical distances, respectively, from the origin. u is the sheet speed, and ϕ is the angle between the sheet and the vertical. r c is the local radius of curvature of a meridional (p.61)
Flux conservation requires that
The total curvature force acting normal to the bell surface is given by
The factor of two results from there being two free surfaces. Therefore, the force balance normal to the sheet takes the form
2.8 Appendix A
Recall Stokes' Theorem,
Along the contour C, d⃗ℓ = m dℓ, so that we have
Now let F = f ∧ b, where b is an arbitrary constant vector. We thus have
We now use standard vector identities to see that
Since b is arbitrary, we thus have
We now choose f = σ n, and recall that n ∧ m = −s. We thus obtain
We note that ∇σ ∙ n = 0 since ∇σ must be tangent to the surface S; moreover, . We thus obtain the desired result:
2.9 Appendix B: The Frenet–Serret equations
Differential geometry yields relations that are often useful in computing curvature forces on 2D interfaces:
Note that the LHS of eqn (2.108) is proportional to the curvature pressure acting on an interface.
2.10 Appendix C: Computing curvatures
We see the appearance of the divergence of the surface normal, ∇ ∙ n, in the normal‐ stress balance. We proceed by briefly reviewing how to formulate this curvature term in two common geometries.
In Cartesian coordinates (x, y, z), we consider a surface defined by z = h(x, y). We define a functional f (x, y, z) = z − h(x, y) that necessarily vanishes on the surface. The normal to the surface is defined by
In the simple case of a 2D interface z = h(x), these results assume the simple forms
Note that n is dimensionless, while ∇ ∙ n has the units of 1/L.
In 3D polar coordinates (r, θ, z), we consider a surface defined by z = h(r, θ). We define a functional g(r, θ, z) = z − h(r, θ) that vanishes on the surface, and compute the normal:
In the case of an axisymmetric interface z = h(r), these results reduce to
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