L. BOUNDARY CONDITIONS AT SOLID WALLS AND FLUID INTERFACES

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Basic Principles



molecular theories have also played a useful role. Our focus in this section is on boundary

conditions at either a solid boundary or at a fluid–fluid interface.

However, let us begin from a slightly more general perspective. There are really two

types of boundary conditions encountered in theoretical analyses of fluid flow or heat

transfer problems when (2–108)–(2–110) are used. In particular, when we are interested

in the temperature or velocity fields in the vicinity of an object of finite size in a much

larger fluid domain, it is often a convenient and reasonable approximation to assume that

the fluid domain is unbounded. This is particularly useful if the form of the temperature or

velocity field far from the object of interest is known in advance. In this case, the form of

the temperature or velocity field far from any boundaries is prescribed in lieu of boundary

conditions at an actual wall or surface. An example is the translation of a heated sphere

through a cooler, quiescent fluid that is held in a large container. Now, if the sphere is

much smaller than the container, and if it is not close to any of the container boundaries,

the velocity and temperature perturbations caused by the sphere will be relatively localized

in the vicinity of the sphere and be independent of where the sphere is located in relation

to the distant container walls. In this case, instead of solving for the temperature and

velocity fields in the complete fluid domain, with boundary conditions applied at the sphere

surface and container walls, an adequate approximation will be to treat the fluid domain

around the sphere as though it were unbounded (i.e., infinite in extent) and then require

that the temperature and velocity fields take the “ambient” form far from the sphere that

would exist in the container in the absence of any disturbance from the sphere. Although

the approximation of applying far-field boundary conditions “at infinity,” in lieu of actual

boundary conditions at some distant boundary, may at first seem questionable, the farfield conditions themselves are generally assumed to be known. Indeed, the unbounded

fluid approximation is useful only if we know the undisturbed form of the temperature or

velocity fields in advance.

The other class of boundary conditions is those applied at bounding surfaces, i.e., either

at solid surfaces or at an interface if there are two (or more) distinct “homogeneous” fluids in

the flow domain. We denote these surfaces with the generic symbol S. The transition between

two bulk materials occurs over a finite but thin region. In the continuum description, we

approximate this as a surface of discontinuity in material properties. An immediate question

that may arise is what surface we should choose within the finite, but thin, surface or interface

region for the purpose of applying the macroscale or continuum boundary conditions. For

that matter, we may equally ask whether it makes any difference. From a purely geometrical

point of view, there is no difference what choice we make for S as long as it remains within

the interfacial/surface zone. This whole region is vanishingly thin, in any case, compared

with the continuum scale of resolution L. Nevertheless, it has historically proven to be

extremely convenient to adopt a specific convention. To explain this convention, let us

initially limit ourselves to an interface separating two pure bulk fluids A and B. As we move

across the interfacial region there is a rapid, but smooth variation of the density from ρ A to

ρ B . However, from the continuum viewpoint, we model this as though there were simply

the two homogeneous fluids of density ρ A and ρ B right up to the surface S, across which

the density jumps discontinuously from ρ A to ρ B . The position of S is chosen so that the

total mass is the same in either description of the system, i.e.,

L

−L



ρ(z)dz = Lρ A + Lρ B ,



(2–111)



where z is the coordinate direction normal to S, the interface is (locally) at z = 0, and

the range −L to L covers a large but finite region such that ρ − ρ A → 0 as z → L and

ρ − ρ B → 0 as z → −L. If a different choice were made for S, there would be either more

or less mass in the idealized continuum description than in the real system, and it would be

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necessary to “account for” this excess (or deficit) mass by assigning any difference to the

surface itself, as a so-called surface-excess quantity. The convenience of the choice just

described is that the surface-excess mass is zero for a pure two-fluid system. In fact, the

exact same considerations can be applied at any phase or fluid–fluid boundary, including

that between a solid and a fluid (i.e., either gas or liquid). By convention, we assume that

the surface S where boundary conditions are applied is chosen such that there is no surfaceexcess mass in a pure fluid system.

1. The Kinematic Condition

Turning now to the question of boundary conditions, the solution of (2–108)–(2–110)

requires both thermal boundary conditions relating the temperature or its derivatives and

the velocity and its derivatives on the two sides of S. We begin with the so-called kinematic

boundary condition, which derives from the principle of mass conservation at any boundary

of the flow domain.

Let us denote the bulk-phase densities on the two sides of the interface as ρ and ρ and

ˆ

ˆ

the fluid velocities as u and u. The orientation of surface S is specified in terms of a unit

normal n. In general, the surface S is not a material surface. For example, if there is a phase

transition occurring between the two bulk phases (e.g., a solid phase is melting or a liquid

phase is evaporating), mass will be transferred across S. However, the surface S is not a

source or sink for mass, and thus mass conservation requires that the net flux of mass to (or

from) the surface must be zero.

It is probably useful to think of two specific situations. In one, there is no phase transformation occurring, and in this case S is in fact a material surface separating a viscous

fluid and a second medium that may either be solid or fluid. Hence, in the absence of phase

change at S, the normal component of velocity must be continuous across it and equal to

the normal velocity of the surface:

ˆ

u·n=u·n



at S.



(2–112)



ˆ

If the second phase is a solid wall, then u = Usolid , which is assumed to be known. In a

ˆ

frame of reference fixed to a solid wall, u · n = 0, and in this frame of reference

u·n=0



at S,



(2–113)



provided the wall is impermeable. This choice of reference configuration is often a convenient one when one of the materials is a solid. If the second medium is another fluid,

then the kinematic condition, (2–112), provides a single relationship between the two normal velocity components, both of which are unknowns to be determined in the solution

process.

A generalization of the condition (2–112) is required if there is an active phase transformation occurring at S, i.e., if the liquid is vaporizing or the solid is melting. In this case, we

must distinguish between the bulk fluid velocities in the limit as we approach the interface,

and the velocity of the interface itself, uI · n (where the interface is specified still by the

criteria of zero excess mass discussed earlier). The condition of conservation of mass then

requires that

ρ u − uI · n = ρ u − uI · n at S,

ˆ ˆ



(2.114a)



where the velocity components are all still measured with respect to fixed, “laboratory”

coordinates. The term on the left-handside is just the net mass flux of material from the first

fluid to (or from) the interface, and because mass cannot accumulate on the surface S, this

is balanced by an equal flux of mass away from (or to) S on the other side. It will be noted in

ˆ

this case that the normal velocity components u · n and u · n are no longer equal. Suppose,

for example, that one phase is a liquid with density ρ that is freezing to a solid phase with

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a different (generally larger) density ρ. There is no motion within the solid phase, and thus

ˆ

the velocity u · n, measured in the “fixed” laboratory reference frame, is zero. Hence, to

ˆ

satisfy mass conservation, the liquid must flow toward the interface with a velocity

u·n=



ρ−ρ

ˆ

ρ



uI · n



at S.



(2.114b)



In the absence of an active phase-transformation process, both sides of (2.114a) are

ˆ

zero, i.e. u · n = uI · n and u · n = uI · n, and the condition (2–114a) reduces to (2–112).

It is important to emphasize that the kinematic condition is a direct consequence of mass

conservation at S, and must always be satisfied, regardless of the specific fluid properties or

any details of the flow.

2. Thermal Boundary Conditions

Next, we consider the thermal boundary conditions, as these are also relatively straightforward to obtain.

The critical assumption about the bounding surface or phase boundary is that it is maintained at thermal equilibrium, independent of whatever mechanisms may exist for transport

to the surface or release of heat at the surface that is due to phase transformation. Hence it

follows that the two bulk (or “homogeneous”) materials will have the same temperature at

S, i.e.,

ˆ

θ =θ



at each point on S.



(2–115)



ˆ

If one of the two materials is a solid on which the temperature is known, θ = θsolid , the

condition (2–115) is sufficient to solve the thermal energy equation. If, on the other hand,

surface S is an interface, or phase boundary, the condition (2–115) provides only one

relationship between two unknown temperature fields, and an additional boundary condition

is required.

This second condition is the local statement of conservation of thermal energy at the

interface. In particular, assuming that there is no independent source of heat at the interface

other than that associated with the possibility of a phase transition (which is included as we

shall see), then conservation of thermal energy requires

j·n=ˆ·n

j



at S,



(2–116)



where the vectors j and ˆ are the total heat flux vectors. In a fluid for which Fourier’s law

j

applies, these take the form

j = −k∇θ + ρ(u − uI ) C p (θ − θref ),

ˆ ˆ ˆ ˆ

ˆ ˆ

ˆ = −k∇ θ + ρ(u − uI ) C p (θ − θref ).

j



(2–117)



With the convective term calculated with the heat capacity at constant pressure, the appropriate measure of heat content is the enthalpy H = C p θ per unit mass.

In the absence of a phase transition at S, the boundary S is a material surface so that

u · n = u · n = uI · n, and there is no convective flux of heat across it. Hence the condition

ˆ

(2–116) can be written in the form

ˆ ˆ

−k(∇θ · n) = −k(∇ θ · n)



at S.



(2–118)



ˆ

Because k = k in most cases, the temperature gradient is discontinuous across the boundary

S. In some cases, in which the second material is a solid, the heat flux at the solid surface

may be known, rather than the temperature. In this situation, instead of specifying θ in the

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ˆ

fluid in terms of a known temperature at the solid surface, θsolid , by means of (2–115) the

temperature gradient is specified in the fluid in terms of the known heat flux in the solid:

−k∇θ · n = Qsolid



at S,



(2–119)



where the right-hand side is assumed to be known. Either of the conditions (2–115) or

(2–118) with the right-hand side specified at a boundary is sufficient to determine the

temperature in the fluid (at least up to an arbitrary reference temperature if all boundary

conditions are of the gradient type), but both cannot be specified simultaneously unless the

ˆ

surface is an interface for which both the fields θ (x, t) and θ (x, t) are unknown.

If there is a phase change at S, the conservation of energy condition, (2–116), takes

a different form from (2–118). If we simply substitute (2–117) into (2–116), with C p θ

expressed in terms of the enthalpy H, we obtain

ˆ ˆ

ˆ

−k(∇θ · n) + [ρ(u − uI ) · n]H = −k(∇ θ · n) + (ρ(u − uI ) · n) H

ˆ ˆ



at S.



(2–120)



Using (2–114a), we can rearrange this to the form

ˆ ˆ

−k(∇θ · n) + k(∇ θ · n) = ρ( H − H )(u − uI ) · n

ˆ ˆ

ˆ



at S.



(2.121a)



Alternatively, this could be written as

ˆ ˆ

ˆ

−k(∇θ · n) + k(∇ θ · n) = ρ( H − H )(u − uI ) · n



at S.



(2.121b)



The change of enthalpy in transforming from one phase to the other is known as the “latent

heat” per unit mass, and may be denoted as either H or L. The thermal energy balance

at the interface thus requires that there be an imbalance in the heat flux to and from the

interface by conduction that is just equal to the rate of release (or adsorption) of enthalpy

as the material changes phase.

3. The Dynamic Boundary Condition

The third type of boundary condition at the surface S involving the bulk-phase velocities

is known as the dynamic condition. It specifies a relationship between the tangential comˆ

ˆ

ponents of velocity, [u − (u · n)n] and [u − (u · n)n]. However, unlike the kinematic and

thermal boundary conditions, there is no fundamental macroscopic principle on which to

base this relationship. The most common assumption is that the tangential velocities are

continuous across S, i.e.,

ˆ

ˆ

u − (u · n)n = u − (u · n)n



at S,



(2–122)



and this is known as the no-slip condition. If the second medium is a solid, then again

ˆ

u = Usolid which is assumed to be known, and the condition (2–122) prescribes a specific

value for the tangential velocity of the fluid. A convenient frame of reference in this latter

case is one fixed to the solid wall, so that

u − (u · n)n = 0



at S.



(2–123)



This is the most common form of the no-slip condition.

It is generally accepted, based on empirical evidence, that the no-slip condition applies

under almost all circumstances for small-molecule (Newtonian) fluids at either solid surfaces or at a fluid–fluid interface and also applies under many circumstances for complex

liquids, such as polymer solutions or melts. This assertion is based primarily on comparisons of predictions from solutions of the equations of motion, which incorporate the no-slip

condition, and experimental data – we shall discuss one example of a problem for which

this kind of comparison has been done in the next chapter. Here, we simply note that these

comparisons with experiments are often between macroscopic quantities – such as overall

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flow rates – rather than detailed measurements of the velocities in the immediate vicinity

of boundaries or interfaces. There has, however, also been work in recent years, based on

molecular dynamics simulations of simple model fluids, in which detailed examination of

averaged velocities in the immediate vicinity of solid walls supports the relevance of the

no-slip at a solid wall.38 We will frequently use the no-slip condition for the solution of

Newtonian fluid flow problems in this book.

In spite of the frequent applicability of no-slip as the appropriate dynamical boundary

condition, however, there is no compelling theoretical argument for it, nor can kinetic theory

contribute much, in general, to understanding it because kinetic theory cannot cope with

the problem of dense liquids adjacent to a rigid solid. Scientists, for nearly 300 years, have

thus questioned whether “no-slip” is the fundamentally correct boundary condition at a

solid surface, or whether it is simply an approximation from some more general condition,

which just happens to work well for many fluids under many flow conditions.39 A more

general condition that is frequently suggested is the so-called Navier-slip condition. Before

this condition is stated, it should be emphasized that it has no more fundamental foundation

than the no-slip condition. It is only a somewhat more general condition, albeit still ad hoc

in origin, which happens to include the no-slip condition as a limiting case. In its most

general tensor form, the Navier-slip condition at a solid boundary is

ˆ

u − (u · n)n − β T · n − (T · n) · n n = 0.



(2–124)



Here n is the unit normal to the boundary, u and T are the (continuum) velocity and

ˆ

stress, and β is an empirical parameter known as the “slip coefficient.” The Navier-slip

condition says, simply, that there is a degree of slip at a solid boundary that depends on the

magnitude of the tangential stress. We note, however, that it is generally accepted that the

slip coefficient is usually very small, and then the no-slip condition (2–123) appears as an

excellent approximation to (2–124) for all except regions of very high tangential stress.

For a Newtonian fluid, an equivalent statement of the Navier-slip condition is

u − (u · n)n − β E · n − (E · n) · n n = 0.



(2–125)



In this case, β clearly has units of length and is known as the “slip length.” It is expected to

be of molecular dimensions.

Although the Navier-slip condition has been largely ignored for many years in favor

of the corresponding no-slip boundary condition, there has been a growing interest in

the Navier-slip condition in recent years, even for Newtonian fluids, driven by both new

experimental observations and by certain theoretical problems that arise from application

of the “no-slip” condition. The best known of the theoretical problems arises when a contact

line separating the two immiscible fluids moves with velocity −U along a solid surface at

which the no-slip condition is assumed to apply, as sketched in Fig. 2–11.

A more convenient, but entirely equivalent, problem for analysis is to consider the

position and shape of the interface to be fixed, with the boundary translating at a velocity

U. If we calculate the velocity and pressure fields for an incompressible, Newtonian fluid,

assuming no-slip at the solid wall and the kinematic plus no-slip conditions at the interface,

we find that the tangential stress component on the boundary exhibits a nonintegrable

singularity as the distance to the contact point goes to zero, i.e.,

1

T · n − (T · n) · n n ∼ m

r

as r → 0 with m ≥ 1. We also find that the viscous dissipation rate is divergent. Both results

lead to the conclusion that it would take an infinite force to maintain a finite relative velocity

between the contact line and the boundary. One proposed resolution of this physically

unacceptable result is to assume that the fluid slips at, and in the immediate vicinity of,

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U



Contact line



z

−U



Interface



y

Typical path of

fluid motion



(a)



(b)



Figure 2–11. A sketch of the moving contact line problem. In (a) the interface is assumed to be moving

with velocity −U along the solid wall. In (b), the equivalent problem is shown in which the interface is

viewed as fixed and the wall moving in its own plan with velocity U.



the contact line. If we impose the Navier-slip condition at the solid boundary, say in the

form of (2–125) with β a very small but nonzero parameter, then the fluid will come

very close to satisfying the no-slip condition everywhere except within a region of length

proportional to β around the contact line, where the tangential stress (and hence also the

velocity gradient) becomes extremely large so that the slip term in (2–125) is nonnegligible

and the tangential velocity is significantly different from zero. It has been shown that this

eliminates the singular behavior of the solution associated with a moving contact line.40

The rationale is that even a Newtonian fluid may slip at a boundary if it is acted on by

a sufficiently large tangential stress. Again, molecular dynamic simulations of a moving

contact line also appear to be consistent with the presence of a local “slip” region.38

The example of slip in the vicinity of a moving contact line may appear a rather special

circumstance, involving slip in a region of very high tangential stress. For the majority

of Newtonian fluids and boundary materials, the idea that slip manifests itself under only

extreme stress conditions is most likely correct. However, recent experimental results, some

coming from the emerging literature on small-scale flows [microelectromechanical system

(MEMS) devices], suggest that slip may be relevant under more general flow conditions for

water at highly hydrophobic walls.41 At the same time these observations may shed some

light on the underlying physical factors that are responsible for the no-slip condition at a

solid wall. For example, Watanabe et al.42 showed that a Newtonian fluid (water) showed

evidence of slip at the walls in flow through a pipe when the pipe was made from a class

of materials that they termed “water repellent.” By this they meant a material for which

the contact angle θc with water was greater than 150◦ (see Fig. 2–12). To understand the

significance of this, we must digress briefly to discuss the contact angle.

The contact angle exhibited at a stationery contact line, where a gas–liquid interface

intersects a solid surface, is a unique characteristic of the three materials involved; namely,

the gas, the liquid, and the solid. It reflects the nature of their interaction across the various

surfaces that intersect at the contact line. It is known from thermodynamics that a fluid–fluid

Interface



Figure 2–12. A sketch of a three-phase contact

line region at a solid wall illustrating the definition of the contact angle.



Liquid

Gas

θc



Solid



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interface (i.e., gas–liquid or liquid–liquid) can be characterized by its surface free energy

per unit area.

This quantity is called the interfacial tension (sometimes also “surface tension” when

the interface involves a liquid and a gas). It provides a measure of the work required for

increasing the area of an interface (i.e., to form new interface by bringing molecules from

the bulk fluids to the interface), and in this context it can also be viewed as the force per

unit length acting at the boundaries of an element of surface that is required to keep it from

shrinking in area (thus decreasing the surface free energy). We shall discuss the significance

of this latter interpretation of interfacial tension in a more general fluid dynamical context

in the next subsection in which we expand on our discussion of boundary conditions at

a fluid–fluid interface. For present purposes, it is sufficient to note that we can apply the

concept of interfacial tension, viewed as surface free energy per unit area, to the solid–liquid

and/or solid–gas (or vapor) surfaces that meet the fluid–fluid interface at the contact line.

If we then apply a force balance at the contact line in the plane of the solid boundary, we

obtain the so-called Young equation, first proposed in 1805:

γLG cos θc = γSG − γS L ,



(2–126)



where γLG is the interfacial tension at the liquid–gas interface, γSG and γS L are the corresponding interfacial energy densities at the solid–gas and solid–liquid surfaces, respectively,

and θc is the contact angle. If we interpret interfacial tension loosely as specifying the work

required to create a unit of surface area (as is the case at a fluid–fluid interface), then a small

value of γ X Y indicates a strong attractive interaction between X and Y. Now, it can be seen

from (2–126) that a small value of θc implies that γSG > γS L , whereas a large value of θc

means that γS L > γSG . Hence, when θc is small, it implies that the liquid is strongly attracted

to the solid, whereas the opposite is true if γS L > γSG . In fact, the issue of “adhesion” between

a solid and liquid is more complex than suggested by this simplified discussion, and the

Young equation is only qualitatively useful as it is nearly impossible to directly measure

either γSG or γS L .

The conclusion from the experimental observations of Watanabe et al.,42 when coupled

with the preceding discussion about the contact angle, is that the strength of “adhesion”

between a fluid and a wall is an important factor in determining the applicability of the

slip versus no-slip boundary conditions. Indeed, more recent investigations41 have used

direct (“micro-particle image velocimetry”) measurements of the velocity profile in small

channels to confirm that highly hydrophobic (high-contact-angle) walls allow for slip and

to provide an estimate of the slip length. For water with walls coated by hydrophobic

octadecyltrichlorosilane, the slip length was found to be 1 μm (i.e., 10−4 cm). This implies

that slip will tend to be important only for small-scale flows, for example, measurable

only for channels smaller than about 1 mm even when the walls are highly hydrophobic

(“repelling”) for the liquid in question. Presumably β

10−4 for a wall that is not highly

hydrophobic. In fact measurements in the same channel without the hydrophobic coating

showed no evidence of slip even though the channel cross section was only 30 × 300 μm.

For a solid boundary, we conclude that the no-slip approximation is generally quite good

except in special regions of very high tangential stress, such as the vicinity of the moving

contact line or with wall/fluid combinations in which the adherence between the wall and

the fluid is very low (for water such walls are called hydrophobic).

There is also the possibility of slip at a fluid–fluid interface, especially for a pair of

thermodynamically incompatible fluids. However, we are not aware of any evidence of slip

at an interface between two “small-molecule” Newtonian fluids. One reason is that there are

no examples of flow conditions analogous to the moving contact line at solid boundaries that

can lead to very large tangential stress. Hence, because β is usually very small for Newtonian

fluids, conditions that would produce any significant slip are absent, even assuming that

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we should apply the Navier-slip condition in the first place. Although the potential exists

for liquid pairs that exhibit relatively larger β in analogy with the liquid–solid surface, no

examples have yet been found.

The body of evidence on Newtonian fluids is that significant departures from the no-slip

conditions are uncommon. Thus, though we should always keep in mind the possibility of

significant slip for some materials or in some local regions of the flow, we will follow the

usual custom of applying no-slip conditions in the analysis of Newtonian fluid flows.

In spite of this, it is perhaps useful to briefly consider the conditions at solid boundaries

and fluid interfaces for complex/non-Newtonian fluids. One reason for doing this is that

it provides additional emphasis to the idea from the proceeding paragraphs that there will

be conditions when the commonly applied no-slip condition breaks down. It should be

stated, at the outset, that the question of slip or no-slip is still a matter of current research

interest for complex fluids. Nevertheless, the occurrence of “slip” is generally accepted to be

much more common for complex/non-Newtonian fluids than for Newtonian/small molecule

liquids. In the latter case, we have seen that “slip” generally involves either extreme shear

stresses or solid walls that exhibit extremely weak attractive interactions with the liquids,

and the issue is primarily one of basic scientific interest. Polymer melts, on the other hand,

commonly exhibit apparent manifestations of “slip” that play a critical role in the success

or failure of certain types of commercial processing applications.43

It appears that slip can occur for polymer melts by means of two distinct mechanisms.

One is believed to result from adhesive “failure” at the walls, in which “adsorbed” polymers

are literally pulled off of the walls when the tangential stress exceeds some critical value.

Empirical evidence from the polymer-processing literature indicates that this mechanism

for slip can be “tuned” by the choice of wall materials. A second mechanism for “slip” is

believed to be active for systems that exhibit strong adherence between the polymer and the

walls. Because of the forces on the polymer caused by flow, the polymer, which exists in

an “entangled” state under normal circumstances, can become (temporarily) disentangled,

leaving an adsorbed layer of polymer on the wall, but creating a local region of effectively

low viscosity between the polymer layer on the wall and the bulk (entangled) polymer fluid.

In this case, there is empirical evidence that the fluid may alternate between apparent “slip”

and “no-slip,” suggesting that the entanglement/disentanglement process may be a dynamic

one in which the fluid spontaneously alternates between being entangled and unentangled.

In either case, when the polymer either loses adherence at the wall or becomes disentangled

very near to the wall, there is a fundamental change in the “efficiency” with which the wall

is able to inhibit the motion of the polymer, and the bulk polymer motion is consistent with

a boundary condition of slip that is at least qualitatively similar to the Navier-slip condition

discussed earlier.

Polymeric liquids at an interface between two immiscible polymers may also exhibit

an “apparent” slip that is due to a similar disentanglement mechanism, driven again by the

applied shear stress, but also facilitated by the incompatibility between the two bulk-phase

polymers, which minimizes the amount of chain intermingling at the interface. Indeed,

statistical mechanical theories of the interfacial region between two bulk polymers show

that the condition of “slip” or at least “partial-slip” is likely to be relevant whenever the

flow system is one involving relatively small length scales.44 Two examples are (1) motions

involving thin superposed fluid layers and (2) the motions of very small drops. The thermodynamic incompatibility of the two bulk polymers leads to an interfacial region with

a modified polymer density. Slip comes from the fact that this interfacial fluid region has

an apparent viscosity that is less than that of the two bulk fluids. Hence, when there is a

velocity gradient across (i.e., normal to) this region, it tends to be higher than in the two

bulk fluids and there is an apparent jump in the bulk-phase velocities across “the interface.”

The magnitude of this jump depends on the ratio of the viscosity of the interfacial region

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to that of the bulk fluids and on the width of the interfacial region. Hence if the width of

the interfacial region becomes comparable with the length scales characteristic of the bulk

fluids, the apparent slip associated with this jump is increased.

In summary, we have so far seen that there are two types of boundary conditions that

apply at any solid surface or fluid interface: the kinematic condition, (2–117), deriving

from mass conservation; and the dynamic boundary condition, normally in the form of

(2–122), but sometimes also in the form of a Navier-slip condition, (2–124) or (2–125).

ˆ

When the boundary surface is a solid wall, then u is known and the conditions (2–117) and

(2–122) provide a sufficient number of boundary conditions, along with conditions at other

boundaries, to completely determine a solution to the equations of motion and continuity

when the fluid can be treated as Newtonian.

M. FURTHER CONSIDERATIONS OF THE BOUNDARY CONDITIONS AT THE

INTERFACE BETWEEN TWO PURE FLUIDS – THE STRESS CONDITIONS



When a bounding surface is a fluid–fluid interface instead of the surface of a solid, the kinematic and dynamic boundary conditions can be seen, from (2–112) and (2–122), to provide

either two (or three) independent relationships between the unknown velocity vectors, u

ˆ

ˆ

and u. However, there are a total of either four or six unknown components of u and u (the

number depending on whether the flow is 2D or fully 3D), and thus additional conditions

must be imposed at an interface to completely specify the solutions of the Navier–Stokes

and continuity equations. In this section, we assume that there is no phase change at the

interface.

One additional new feature, if the bounding surface is an interface, is that it will generally

deform (i.e., change shape) when the fluids undergo motion. Hence we cannot specify the

interface shape, but must determine the shape as part of the solution of a flow problem. In

other words, the shape of any interface that appears in a flow problem must be considered

as an additional unknown. If we go back and examine the kinematic and dynamic boundary

conditions, we see that these also involve the interface shape implicitly, through the normal

unit vector n. In fact, we shall see shortly that the kinematic condition in the form (2–112)

is not complete, because it should involve the interface shape explicitly. This condition

ˆ

is to be applied at the interface, but if u · n = u · n are not zero, then the interface itself

must be moving. Only two possibilities are consistent with this. The first is that the whole

ˆ

interface is translating with the same velocity u · n(= u · n) at every point relative to the

chosen reference frame. In this case, the shape of the interface does not change, and we

ˆ

could adopt an alternative reference frame such that u · n(= u · n) = 0. The more likely

ˆ

case is that u · n(= u · n) = 0 and varies from point to point on the interface. In this case,

the shape of the interface must change in time, and we may anticipate that there should be

ˆ

a direct relationship between u · n(= u · n) and the rate of change of the unknown function

that describes the interface shape.

To incorporate the unknown interface shape into the boundary conditions, the most

convenient approach is to introduce a scalar function F(x, t), which describes the interface

shape in the sense that the interface is the set of points xs , such that

F(xs , t) ≡ 0.



(2–127)



It may be useful to introduce a couple of examples to illustrate the use of F to describe

interface geometry. Let us start with the simplest (trivial) example of an interface that is a

flat surface at equilibrium. This would correspond to the interface between two stationary

fluids of different density, with the larger density fluid on the bottom. In this case, we can

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M. Further Considerations of the Boundary Conditions

z



z



ho

x



h(x, y)

y



y



x



Figure 2–13. A sketch of a flat and deformed interface, showing the definition of the coordinates and the

interface shape function as discussed in the text.



introduce a Cartesian coordinate system, with x and y in the plane of the interface and z

normal to it, as illustrated in Fig. 2–13. The choice for F in this case is

F ≡ z − h0



all x, y.



Now, let us suppose that there is flow present in the two fluids. In this case, the interface shape

is expected to change, again as illustrated in Fig. 2–13, and we would need to generalize

to

F ≡ z − h(x, y, t)



all x, y,



if we have chosen to specify the problem (i.e., equations of motion plus boundary conditions)

in the same Cartesian coordinates.

Now, in general, with the interface defined in terms of a functionF(x, t), it is known

from analytic geometry that the unit normal to S can be defined in terms of F as

n=±



∇F

,

|∇ F|



(2–128)



where the sign is chosen so that n is (by convention) the outer unit normal vector. For an

interface that is a closed surface (such as the interface of a liquid drop) this means that the

sign is chosen so that n points from the interface outward into the external fluid. For an

open interface, such as the flat surface pictured in Fig. 2–13, we can choose n to be in either

direction.

1. Generalization of the Kinematic Boundary Condition for an Interface

Now, because F is a scalar function that is always equal to zero at any point on the fluid

interface, its time derivative following any material point on the interface [which means

ˆ

that there is no phase transformation occuring so that the velocity u = u on S according to

(2–112) and (2–122)] is obviously equal to zero, that is

∂F

+ u · ∇F = 0

∂t



for any material point on S.



Thus, rearranging slightly, we obtain

1 ∂F

+u·n=0

|∇ F| ∂t



on S.



(2–129)



This is the most general form of the kinematic condition. Obviously, in view of (2–112),

ˆ

it can be written in terms of either u or u. The reader should note that if the shape of the

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