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New Trends in the Physics and Mechanics of Biological Systems$

Martine Ben Amar, Alain Goriely, Martin Michael Müller, and Leticia Cugliandolo

Print publication date: 2011

Print ISBN-13: 9780199605835

Published to Oxford Scholarship Online: September 2011

DOI: 10.1093/acprof:oso/9780199605835.001.0001

Surface tension

(p.27) 2 Surface tension
New Trends in the Physics and Mechanics of Biological Systems

John W. M. Bush

Oxford University Press

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.

Keywords:   surface tension, menisci, jets, sheets, Marangoni flows, surfactants, capillary instability

Surface tension

(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:

ρ ( u t   +   u · u ) = p   +   F   +   μ 2 u , (2.1)
· u = 0. (2.2)

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:

n · T · n =   σ   ( · n ) , (2.3)
where T = −p I + μ[∇u + (∇u)T] = −p I + 2μ E is the stress tensor, E = 1/2 [∇u + (∇u)T] is the deviatoric stress tensor, and n is the unit normal to the surface. The tangential stress at a free surface must balance the local surface tension gradient:
n · T · t =   σ · t , (2.4)
where t is the unit tangent to the interface.

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

R e = U a ν = INERTIA VISCOSITY = Reynolds number , (2.5)
F r = U 2 g a = INERTIA GRAVITY = Froude number , (2.6)
B o = ρ g a 2 σ = GRAVITY CURVATURE = Bond number . (2.7)

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:

W e = ρ U 2 a σ   =   INERTIA CURVATURE   =   Weber number , (2.8)
C a = ρ ν U σ   =   VISCOUS CURVATURE   =   capillary number . (2.9)

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 = pρ gx, 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

(p.31) u = U u ,     p d = ρ U 2 p ' ,     x = a x ,     t = a U t . (2.10)

Nondimensionalizing the Navier–Stokes equations and appropriate boundary conditions yield the following system:

( u t   +   u · u )   =   p d   +   1 R e 2 u ,     · u   =   0. (2.11)

The normal‐stress condition assumes the dimensionless form

p d   +   1 F r z   +   2 R e   n · E · n   =   1 W e   · n . (2.12)

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. 2.12.3). 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:

V ρ D u D t   d V   =   V f   d V   +   S   [ t ( n ) + t ^ ( n ^ ) ]   d S   +   C σ s   d .

Here ℓ indicates an arc length, and so dℓ indicates a length increment along the curve C. t(n) = nT 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 = −nT̂, where T̂, = −p I + μ̂[∇û + (∇û)T].

2.3.1 Physical interpretation of terms

V ρ D u D t d V : inertial force associated with acceleration of fluid within V.

V f dV: body forces acting on fluid within V.


Surface tension

Fig. 2.1 A surface S and bounding contour C on an interface between two fluids. The upper fluid (+) has density ρ and viscosity μ; the lower fluid – has corresponding values ρ̂ and μ̂. n represents the unit outward normal to the surface, and n̂ = −n the unit inward normal. m represents the unit tangent to the contour C, and s the unit vector normal to C but tangent to S.

Surface tension

Fig. 2.2 A Gaussian fluid pillbox of height and radius ϵ spanning the interface evolves under the combined influence of volume and surface forces.

Surface tension

Fig. 2.3 A definitional sketch of a fluid–fluid interface. Carets denote variables in the lower fluid.

S t(n) dS: hydrodynamic force exerted at interface by fluid +.

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:

S   [ t ( n ) + t ^ ( n ^ ) ]   d S   +   C σ   s   d   =   0.

Now we have that

t ( n )   =   n · T ,                 t ^ ( n )   =   n ^ · T ^   =   n · T ^ .

(p.33) Moreover, the application of Stokes' Theorem (see Appendix A) allows us to write

C   σ s   d   =   S   s σ     σ n   ( s · n )   d S ,
where the tangential gradient operator, defined by
s   =   [ I     n n ] ·   =       n n ,
appears because σ and n are defined only on the surface. We proceed by dropping the subscript s on ∇, with this understanding.

The surface force balance thus becomes

S [ n · T     n · T ^ ]   d S   =   S σ n   ( · n )     σ   d S . (2.13)

Now, since the surface element is arbitrary, the integrand must vanish identically. One thus obtains the interfacial stress balance equation.

2.3.2 Stress balance equation

n · T     n · T ^   =   σ n   ( · n )     σ . (2.14)

2.3.3 Interpretation of terms

nT: stress (force/area) exerted by + on − (will generally have both normal and tangential components).

nT̂: 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:

n · T · n     n · T ^ · n   =   σ   ( · n ) . (2.15)

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 teqn(2.13), where t is any unit vector tangent to the interface, yields the tangential‐stress balance at the interface:

n · T · t     n · T ^ · t   . =   σ · t . (2.16)

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 nTn = −p, and the normal‐stress balance assumes the form

p ^ p   =   σ   · n . (2.17)

The pressure jump across the interface is balanced by the curvature force at the interface. Now, since nTs = 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:

· n   =   · r   =   1 r 2 r ( r 2 )   =   2 R . (2.18)

So, the normal‐stress jump (eqn(2.17)) indicates that

p ^ p   =   2 σ R . (2.19)

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,

ρ g z   =   σ   · n . (2.20)

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 wρ aρ 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) = zη (x) that vanishes on the surface. The normal to the surface is thus

n   =   f | f |   =   z ^ η ' ( x ) x ( 1 + η ' ( x ) 2 ) 1 / 2 . (2.21)

As deduced in Appendix C, the curvature of the free surface, ∇ ∙ n̂, may be expressed as

· n ^ = η x x ( 1 + η x 2 ) 3 / 2 η x x . (2.22)
Surface tension

Fig. 2.4 A definitional sketch of a planar meniscus at an air–water interface. The free surface is defined by z = η(x), varying from its maximum elevation at its point of contact with the wall (x = 0) to zero at large x. The shape is prescribed by the Young–Laplace equation.

(p.36) Assuming that the slope of the meniscus remains small, i.e. η x ≫ 1, allows one to linearize eqn (2.22) so that eqn (2.20) assumes the form

ρ g η   =   σ η x x . (2.23)

Applying the boundary condition η(∞) = 0 and the contact condition η x(0) = − cot θ and solving eqn (2.23) thus yields

η ( x ) = c cot ( θ ) e x / c , (2.24)
where c = σ / ρ g is the capillary length. The meniscus formed by a log in water is exponential, dying off on the scale of ℓc.

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.

Surface tension

Fig. 2.5 Schematic illustration of the geometry of a circular hydraulic jump. A jet of radius a impacts a reservoir of outer depth H at a speed U. The curvature force associated with the surface tension σ depends only on the geometry of the jump; specifically, on the radial distance Δ R = R 2R 1 and the arc length s between the two nearest points upstream and downstream of the jump at which the surface z = h(r) has vanishing slope.

(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:

F c   =   σ S   · n   ( n · r ^ )   d S . (2.25)

Now, defining the surface as z = h(r) allows us to express the area element as

d S   =   r   d θ ( d r 2 + d z 2 ) 1 / 2   =   r   d θ   ( 1 + h r ) 1 / 2   d r , (2.26)
so that our radial force assumes the form
F c   =   σ 0 2 π R 1 R 2   · n   ( n · r ^ )   r   ( 1 + h r ) 1 / 2   d r   d θ . (2.27)

Now we use the appropriate forms for n and ∇ ∙ n from Appendix C:

n   =   z ^ h r r ^ ( 1 + h r 2 ) 1 / 2 ,       · n   =   1 r d d r   r h r ( 1 + h r 2 ) 1 / 2 , (2.28)
so that
( · n ) ( n · r ^ )   =     h r ( 1 + h r 2 ) 1 / 2   1 r d d r   r h r ( 1 + h r 2 ) 1 / 2 , (2.29)
and the radial curvature force becomes
F c = 2 π σ R 1 R 2   h r ( 1 + h r 2 ) 1 / 2   1 r d d r ( r h r ( 1 + h r 2 ) 1 / 2 )   r   ( 1 + h r ) 1 / 2   d r (2.30)
= 2 π σ R 1 R 2   h r   d d r ( r h r ( 1 + h r 2 ) 1 / 2 )   d r . (2.31)

Integrating by parts, i.e. a b u d v = u v | a b a b v d u with u = h r , v = r h r / ( 1 + h r 2 ) 1 / 2 , yields

F c = 2 π σ   [   r h r 2 1 ( 1 + h r 2 ) 1 / 2 | R 1 R 2   R 1 R 2 r h r h r r ( 1 + h r 2 ) 1 / 2   d r ] (2.32)
= 2 π σ   R 1 R 2   r h r h r r ( 1 + h r 2 ) 1 / 2   d r . (2.33)

Integrating by parts again, with u = r , v = ( 1 + h r 2 ) 1 / 2 , yields

F c = 2 π σ [ r ( 1 + h r 2 ) 1 / 2 | R 1 R 2   R 1 R 2 ( 1 + h r 2 ) 1 / 2 d r ] (2.34)
=   2 π σ [ ( R 2 R 1 )     R 1 R 2 ( 1 + h r 2 ) 1 / 2   d r   ] , (2.35)
(p.38) since h r = 0 at r = R 1, R 2 by assumption. We note also that
R 1 R 2 ( 1 + h r 2 ) 1 / 2   d r   =   A B ( d r 2 + d z 2 ) 1 / 2   =   A B d   =   S , (2.36)
where S is defined as the total arc length of the surface between points A and B. We define ΔR = R 1R 2 in order to obtain the simple result
F c   =   2 π σ ( S Δ R ) . (2.37)

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:

F c   =   σ   ( 2 π R   Δ H   1 R )   =   2 π σ Δ H . (2.38)

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

Curvature Gravity   =   σ / R ρ g Δ H   =   H R B o 1 , (2.39)
where the Bond number is defined here as
B o   =   ρ g H 2 σ . (2.40)

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

p   =   P 0   +   ρ g z   +   σ · n ,
(p.39) one may express the vertical force balance as
M g   =   z · C p   n   d   =   F b   +   F c . (2.41)

The buoyancy force

F b   =   z · C ρ g z   n   d   =   ρ g V b (2.42)
is thus simply the weight of the fluid displaced above the object and inside the line of tangency (Fig. 2.6). A simple expression for the curvature force may be deduced using the first of the Frenet–Serret equations (see Appendix B),
F c   =   z · C σ ( · n ) n   d   =   σ z · C   d t d   d   =   σ z · ( t 1 t 2 )   =   2 σ   sin θ . (2.43)

At the interface, the buoyancy and curvature forces must balance precisely so the Young–Laplace relation is satisfied:

0   =   ρ g z   +   σ · n . (2.44)

Integrating the fluid pressure over the meniscus yields the vertical force balance:

F b m   +   F c m   =   0 , (2.45)
F b m = z · C m ρ g z n   d   =   ρ g V m , (2.46)
F c m = z · C m σ ( · n ) n   d = σ z · C m   d t d   d   =   σ z · ( t x t 2 )   =   2 σ   sin θ , (2.47)
where we have again used the Frenet–Serret equations to evaluate the curvature force.
Surface tension

Fig. 2.6 A nonwetting two‐dimensional body of radius r and mass M floats on a free surface with surface tension σ. In general, its weight M g must be supported by some combination of curvature and buoyancy forces. V b and V m denote the fluid volumes displaced, respectively, inside and outside the line of tangency.

(p.40) Equations (2.44)–(2.47) thus indicate that the curvature force acting on the floating body is expressible in terms of the fluid volume displaced outside the line of tangency:

F c   =   ρ g V m . (2.48)

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:

F b F c   =   V b V m . (2.49)

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,

F b F c     r c , (2.50)
are prescribed by the relative magnitudes of the body size and capillary length. Very small floating objects (r ≪ ℓc) are supported principally by curvature rather than buoyancy forces. This result has been extended to three‐dimensional floating objects by Keller (1998).

2.4.5 Water‐walking insects (Figs. 2.7 and 2.8)

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

M g 2 σ L sin θ     1 , (2.51)
where θ is the angle of tangency of the floating body.
Surface tension

Fig. 2.7 Water‐walking insects deflect the free surface, thus generating curvature forces that bear their weight. The water strider has characteristic length of 1 cm and weight of 1–10 mg. See http://www-math.mit.edu/dhu/Striderweb/striderweb.html.


Surface tension

Fig. 2.8 The legs of a water strider are covered with hair, rendering them effectively nonwet‐ ting. The tarsal segments of its legs rest on the free surface. The free surface makes an angle θ with the horizontal, resulting in an upward curvature force per unit length 2σ sin θ that bears the insect's weight.

This simple criterion is an important geometric constraint on water‐walking insects. Figure 2.9 indicates the dependence of contact length on body weight for over 300 species of water striders, the most common water‐walking insect. Note that the solid line corresponds to the requirement (2.51) for static equilibrium. Smaller insects (p.42)
Surface tension

Fig. 2.9 The relation between the maximum curvature force F s = σ P and body weight F g = Mg for 342 species of water striders. P = 2(L 1 + L 2 + L 3) is the combined length of the tarsal segments (see strider B). From Hu et al. (2003).

maintain a considerable margin of safety, while the larger striders live close to the edge. The size of water‐walking insects is limited by the constraint (2.51).

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 F c F g 1 / 3 , 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,

n · T · t   =   t · σ , (2.52)
where n is the unit outward normal to the surface, and t is any unit tangent vector. The tangential component of the hydrodynamic stress at the surface must balance the tangential stress associated with gradients in σ. Such Marangoni stresses may result from gradients in temperature or chemical composition at the interface. For a static system, since nTt = 0, the tangential‐stress balance equation indicates that 0 = ∇σ. As seen in Section 2.4, a static situation cannot arise in the presence of Marangoni stresses.

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.


Surface tension

Fig. 2.10 The tears of wine. Fluid is drawn from the bulk up the thin film on the walls of the glass by Marangoni stresses induced by evaporation of alcohol from the free surface.

The flow relies on the transfer of chemical potential energy to kinetic and ultimately gravitational potential energy. The process continues until the alcohol is completely evaporated.

2.5.2 Surfactants

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

Γ t   +   s · ( Γ u s )   +   Γ ( s · n ) ( u · n )   =   J ( Γ , C s )   +   D s s 2 Γ , (2.53)
(p.44) where u s is the surface velocity, ∇s is the surface gradient operator, and D s is the surface diffusivity of the surfactant (Stone 1990). J is a surfactant source term associated with adsorption onto or desorption from the surface, and depends on both the surface surfactant concentration Γ and the concentration in the bulk C s. Tracing the evolution of a contaminated free surface requires the use of the Navier–Stokes equations, relevant boundary conditions, and the surfactant evolution equation (2.53). The dependence of the surface tension on the surfactant concentration, σ(Γ), requires the coupling of the flow field and surfactant field. In certain special cases, the system may be made more tractable. For example, for insoluble surfactants, J = 0, and many surfactants have a sufficiently small D s that the surface diffusivity may be safely neglected.

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

F c   =   C   σ s   d , (2.54)
Surface tension

Fig. 2.11 The “footprint” of a whale, caused by the whale sweeping biomaterial to the sea surface. The biomaterial acts as a surfactant in locally suppressing the capillary waves evident elsewhere on the sea surface. Observed in the wake of a whale on a whale watch from Boston Harbor.

where s is the unit vector tangent to the free surface and normal to C, and dℓ is an incremental arc length along C. If σ is everywhere constant, then this line integral vanishes identically by the Divergence Theorem. However, if σ = σ(x), then it may result in a net propulsive force.

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)

Surface tension

Fig. 2.12 The soap boat. A floating body (length 2.5 cm) contains a small volume of soap, which serves as its fuel for propelling it across the free surface. The soap exits the rear of the boat, decreasing the local surface tension. The resulting fore‐to‐aft surface tension gradient propels the boat forward. The water surface was covered with thymol blue, which parts owing to the presence of the soap, which is visible as a white streak.

rates appropriate to for rigid spheres with equivalent buoyancy in all but the most carefully cleaned fluids. This discrepancy may be rationalized through consideration of the influence of surfactants on the surface dynamics. The flow generated by a clean spherical bubble of radius a rising at low Re = Ua/ν is intuitively obvious. The interior flow is toroidal, while the surface motion is characterized by divergence and convergence at the leading and trailing surfaces, respectively. The presence of surface contamination changes the flow qualitatively.

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

We consider here the form and stability of fluid jets falling under the influence of gravity.

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,

1 2 ρ U 0 2   +   ρ g z   +   P A   =   1 2 ρ U 2 ( z )   +   P B . (2.55)
Surface tension

Fig. 2.13 A fluid jet extruded from an orifice of radius a accelerates under the influence of gravity. Its shape is influenced both by the gravitational acceleration g and by the surface tension σ.

(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:

· n   =   1 R 1 + 1 R 2     1 r .

Thus, the fluid pressures within the jet at the points A and B may be simply related to that of the ambient, P 0:

P A P 0   +   σ a ,                     P B P 0   +   σ r . (2.56)

Substituting into eqn (2.55) thus yields

1 2 ρ U 0 2   +   ρ g z   +   P 0   +   σ a   =   1 2 ρ U 2 ( z )   +   P 0   +   σ r , (2.57)
from which one finds
U ( z ) U 0 = [ 1 + 2 F r   z a + 2 W e ( 1 a r ) ] 1 / 2 , (2.58)
where we define the dimensionless groups
F r = U 0 2 g a = INERTIA GRAVITY = Froude number , (2.59)
W e = ρ U 0 2 a σ = INERTIA CURVATURE = Weber number . (2.60)

Now flux conservation requires that

Q = 2 π 0 r U ( z ) r ( z )   d r = π a 2 U 0 = π   r 2 U ( z ) , (2.61)
from which one obtains
r ( z ) a = ( U 0 U ( z ) ) 1 / 2 = [ 1 + 2 F r   z a + 2 W e ( 1 a r ) ] 1 / 4 . (2.62)

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

r a = ( 1 + 2 g z U 0 2 ) 1 / 4 ,                 U ( z ) U 0 = ( 1 + 2 g z U 0 2 ) 1 / 2 .

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)

Surface tension

Fig. 2.14 The capillary‐driven instability of a water thread falling under the influence of gravity. The initial jet diameter is approximately 3 mm.

that is responsible for the pinch‐off of thin water jets emerging from kitchen taps (see Fig. 2.14).

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

p 0 = σ · n         p 0 = σ R 0 . (2.63)

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

R ˜ = R 0 + ϵ e ω t + i k z , (2.64)
where the perturbation amplitude ϵR 0, ω is the growth rate of the instability and k is the wavenumber of the disturbance in the z‐direction. The corresponding (p.50)
Surface tension

Fig. 2.15 A cylindrical column of initial radius R 0, comprising an inviscid fluid of density ρ and bound by surface tension σ.

wavelength of the varicose perturbations is necessarily 2π/k. We denote the radial component of the perturbation velocity by ũr the axial component by ũz, and the perturbation pressure by p̃. Substituting these perturbation fields into the Navier– Stokes equations and retaining terms only to order ϵ yields
u ˜ r t = 1 ρ p ˜ r , (2.65)
u ˜ z t = 1 ρ p ˜ z . (2.66)

The linearized continuity equation becomes

u ˜ r r + u ˜ r r + u ˜ z z = 0. (2.67)

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

u ˜ r = R ( r ) e ω t + i k z ,     u ˜ z = Z ( r ) e ω t + i k z , and   p ˜ = P ( r ) e ω t + i k z . (2.68)

Substituting eqn (2.68) into eqns (2.65) –(2.67) yields the linearized equations governing the perturbation fields:

Momentum equations:

ω R = 1 ρ d P d r , (2.69)
ω Z = i k ρ P ; (2.70)

(p.51) Continuity:

d R d r + R r + i k Z = 0. (2.71)

Eliminating Z(r) and P(r) yields a differential equation for R(r):

r 2 d 2 R d r 2 + r d R d r ( 1 + ( k r ) 2 ) R = 0. (2.72)

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

R ( r ) = C I 1 ( k r ) , (2.73)
where C is an as yet unspecified constant to be determined later by application of appropriate boundary conditions.

The pressure may be obtained from eqns (2.73) and (2.69) and by using the Bessel function identity I 0 ( ξ ) = I 1 ( ξ ) :

P ( r ) = ω ρ C k I 0 ( k r ) . (2.74)

We proceed by applying appropriate boundary conditions. The first is the kinematic condition on the free surface:

R ˜ t = u ˜ ¯ · n ¯ u ˜ r . (2.75)

Substitution of eqn (2.73) into this condition yields

C = ϵ ω I 1 ( k R 0 ) . (2.76)

Second, we require a normal‐stress balance on the free surface:

p 0 + p ˜ = σ · n ¯ . (2.77)

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:

1 R 1 = 1 R 0 + ϵ e ω t + i k z 1 R 0 ϵ R 0 2 e ω t + i k z , (2.78)
1 R 2 = ϵ k 2 e ω t + i k z . (2.79)

Substitution of eqns (2.78) and (2.79) into eqn (2.77) yields

p 0 + p ˜ = σ R 0 ϵ σ R 0 2 ( 1 k 2 R 0 2 ) e ω t + i k z . (2.80)

(p.52) Cancellation via eqn (2.63) yields an equation for p̃ accurate to order ϵ:

p ˜ = ϵ σ R 0 2 ( 1 k 2 R 0 2 ) e ω t + i k z . (2.81)

Combining eqns (2.74), (2.76), and (2.81) yields the dispersion relation, which indicates the dependence of the growth rate ω on the wavenumber k:

ω 2 = σ ρ R 0 3 k R 0 I 1 ( k R 0 ) I 0 ( k R 0 ) ( 1 k 2 R 0 2 ) . (2.82)

We note first that unstable modes are possible only when

k R 0 〈1 . (2.83)

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

λ max 9.02 R 0 . (2.84)

By inverting the maximum growth rate ω max, one may estimate the characteristic breakup time:

t breakup 2.91 ρ R 0 3 σ . (2.85)
Surface tension

Fig. 2.16 The dependence of the growth rate ω on the wavenumber k for the Rayleigh–Plateau instability.


Surface tension

Fig. 2.17 The field of stationary capillary waves excited on the base of a water jet impinging on a horizontal water reservoir. The grid at the right is millimetric.

A water jet of diameter 1 cm has a characteristic breakup time of about 1/8 s, which is consistent with casual observation of jet break‐up in a kitchen sink.

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

U 2   =   ω 2 k 2   =   σ ρ k R 0 2 I 1 ( k R 0 ) I 0 ( k R 0 ) ( 1 k 2 R 0 2 ) . (2.86)

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

Surface tension

Fig. 2.18 A fluid pipe generated by a falling water jet impinging on a contaminated water reservoir. The field of stationary capillary waves is excited above the fluid pipe. The grid at the right is millimetric.

(p.55) jet surface becomes stagnant at its base: the Marangoni stresses associated with the surfactant gradient are balanced by the viscous stresses generated within the jet. The quiescence of the jet surface may be simply demonstrated by sprinkling a small amount of talc or lycopodium powder onto the jet. The fluid jet thus enters a contaminated reservoir as if through a rigid pipe.

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

ρ ν V δ H   ~   Δ σ H , (2.87)
where Δσ is the surface tension differential between the jet and the reservoir, V is the jet speed at the top of the fluid pipe, and δ H is the boundary layer thickness at the base of the fluid pipe. We assume that the boundary layer thickness increases with the distance z from the inlet according to classical boundary layer scaling:
δ a   ~   ( ν z a 2 V ) 1 / 2 . (2.88)

Substituting for δ(H) from eqn(2.88) into eqn(2.87) yields

H   ~   ( Δ σ ) 2 ρ μ V 3 . (2.89)

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

F c   =   C   σ ( · n )   n   d . (2.90)

(p.56) Using the first Frenet–Serret equation (Appendix B),

( · n )   n   =   d t d , (2.91)
this yields
F c   =   C   σ   d t d   d   =   σ ( t 1 t 2 )   =   2 σ x . (2.92)

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:

ρ u 2 h   =   2 σ . (2.93)
Surface tension

Fig. 2.19 A circular fluid sheet generated by the impact of a water jet on a circular impactor (bottom). The impacting circle has a diameter of 1 cm.

(p.57) Continuity requires that the sheet thickness h depend on the speed u, the jet flux Q, and the radius r as

h   =   Q 2 π r u . (2.94)

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,

R T   =   ρ Q u 4 π σ . (2.95)

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

Surface tension

Fig. 2.20 A water sheet generated by collision of water jets at the left. The fluid streams radially outward in a thinning sheet; once the fluid reaches the sheet rim, it is ejected radially in the form of droplets. From Taylor (1960).

(p.58) fluid drops are continuously ejected from it. The sheet shape is computed in a similar manner, but now depends explicitly on the flux distribution within the sheet, Q(θ). The normal‐force balance on the sheet edge now depends on the normal component of the sheet speed, u n:
ρ u n 2 h   =   2 σ . (2.96)

The sheet thickness is again prescribed by eqn (2.94), but now Q = Q(θ), so the sheet radius R(θ) is given by

R ( θ )   =   ρ u n 2 Q ( θ ) 4 π σ u . (2.97)

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:

0 = u n h s ( v π R 2 ) . (2.98)

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:

ρ u n 2 h + ρ π R 2 v 2 r c = 2 σ . (2.99)


Surface tension

Fig. 2.21 Schematic illustration of a fluid sheet bounded by a stable rim.

Note that the force balance (2.96) is augmented here by the centripetal force. The tangential‐force balance at the rim requires a balance between tangential gradients of tangential momentum flux, tangential gradients of curvature pressure, viscous resistance to stretching of the rim, and the tangential momentum flux arriving from the sheet. For most applications involving high‐speed sheets, the Reynolds number characterizing the rim dynamics is sufficiently large that viscous resistance may be safely neglected. Moreover, the curvature term ∇ ∙ n̂ generally depends on θ; however, with an accuracy to O(R/r c), we may use ∇ ∙ n̂ = 1/R. One thus obtains
s ( π R 2 v 2 )   =   h u t u n π R 2 σ ρ s ( 1 R ) . (2.100)

Equations (2.98)–(2.100) must be supplemented by the continuity relation,

h ( r , θ ) = Q ( θ ) u 0 r , (2.101)
in addition to a number of relations that follow directly from the system geometry:
u n = u 0 sin Ψ , u t = u 0 cos Ψ , (2.102)
1 r c = sin Ψ r   ( Ψ θ + 1 ) . (2.103)

(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)

Surface tension

Fig. 2.22 A water bell produced by the impact of a descending water jet and a solid impactor. The impactor radius is 1 cm. Fluid is splayed radially by the impact, and then sags under the influence of gravity. The sheet may close on itself owing to the azimuthal curvature of the bell.

line, and s is the arc length along a meridional line measured from the origin. Finally, ΔP is the pressure difference between the outside and inside of the bell.

Flux conservation requires that

Q   =   2 π r t u , (2.104)
while Bernoulli's Theorem indicates that
u 2   =   u 0 2   +   2 g z . (2.105)

The total curvature force acting normal to the bell surface is given by

2 σ   · n   =   2 σ   ( 1 r c + cos ϕ r ) . (2.106)

The factor of two results from there being two free surfaces. Therefore, the force balance normal to the sheet takes the form

2 σ r c   +   2 σ cos ϕ r     Δ P   +   ρ g t   sin ϕ     ρ t u 2 r c   =   0. (2.107)

Equations (2.104), (2.105), and (2.107) may be appropriately nondimensionalized and integrated to determine the shape of the bell.

2.8 Appendix A

Recall Stokes' Theorem,

C   F · d   =   S   n · ( F )   d S .

Along the contour C, d⃗ℓ = m dℓ, so that we have

C   F · m   d   =   S   n · ( F )   d S .

Now let F = fb, where b is an arbitrary constant vector. We thus have

C   ( f b ) · m   d   =   S   n · ( ( f b ) )   d S .

We now use standard vector identities to see that

( f b ) · m = b · ( f m ) , ( f b ) = f ( · b ) b ( · f ) + b · f f · b = b ( · f ) + b · f ,
(p.62) since b is a constant vector. We thus have
b · C ( f m )   d   =   b · S   [ n ( · f ) ( f ) · n ]   d S .

Since b is arbitrary, we thus have

C ( f m )   d   =   S   [ n ( · f ) ( f ) · n ]   d S .

We now choose f = σ n, and recall that nm = −s. We thus obtain

C   σ s   d = S   [ n · ( σ n )     ( σ n ) · n ]   d S . = S   [ n σ · n + σ n ( · n ) σ σ ( n ) · n   ]   d S .

We note that ∇σn = 0 since ∇σ must be tangent to the surface S; moreover, ( n ) n = 1 2 ( n n ) = 1 2 ( 1 ) = 0 . We thus obtain the desired result:

C   σ s   d   =   S [ σ     σ n   ( · n ) ] d S .

2.9 Appendix B: The Frenet–Serret equations

Differential geometry yields relations that are often useful in computing curvature forces on 2D interfaces:

( · n ) n  = d t d , (2.108)
( · n )   t = d n d . (2.109)

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) = zh(x, y) that necessarily vanishes on the surface. The normal to the surface is defined by

n   =   f | f |   =   z ^ h x x ^ h y y ^ ( 1 + h x 2 + h y 2 ) 1 / 2 , (2.110)
(p.63) and the local curvature may thus be computed:
· n   =   ( h x x + h y y ) ( h x x h y 2 + h y y h x 2 ) + 2 h x h y h x y ( 1 + h x 2 + h y 2 ) 3 / 2 . (2.111)

In the simple case of a 2D interface z = h(x), these results assume the simple forms

n   =   z ^ h x x ^ ( 1 + h x 2 ) 1 / 2 , · n   =   h x x ( 1 + h x 2 ) 3 / 2 . (2.112)

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) = zh(r, θ) that vanishes on the surface, and compute the normal:

n   =   g | g |   =   z ^ h r r ^ ( 1 / r ) h θ θ ^ ( 1 + h r 2 + 1 / r 2 h θ 2 ) 1 / 2 , (2.113)
from which the local curvature is computed:
· n   =   h θ θ h r 2 h θ θ + h r h θ r h r ( 2 / r ) h r h θ 2 r 2 h r r h r r h θ 2 + h r h θ h r θ r 2 ( 1 + h r 2 + ( 1 / r 2 ) h θ 2 ) 3 / 2 . (2.114)

In the case of an axisymmetric interface z = h(r), these results reduce to

n   =   z ^ h r r ^ ( 1 + h r 2 ) , · n   =   r h r r 2 h r r r 2 ( 1 + h r 2 ) 3 / 2 . (2.115)


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