The Foundations of Fluid Mechanics
1.8 Boundary Conditions
Such a dependence arises when temperature or entropy variations can be neglected.
In the general case, such relations do not exist. The flows are calledbaroclinic because isobars and isotherms are inclined with respect to each other. Barotropicity and baroclinicity may have important consequences on the nature of the flows as it will be shown in Chaps.2,3or7.
1.8 Boundary Conditions 33 boundary conditions) and experiment showed that the no-slip hypothesis was certainly quite relevant.9
We may note that if the fluid is perfect (no viscosity), no adherence is possible on the wall. Of the three conditions (1.62), only a single one remains, namelyvnD0 wherenis the normal to the solid wall. The component of the velocity perpendicular to the wall vanishes, while the other components are unspecified.
1.8.1.2 On a Free Surface
The other type of boundary conditions on the velocity is the one called the free surfaceor free interface. This is the condition to be used when the fluid defines itself the surface, just like the sea surface is defined by that of the water. Let
S.r; t /DCst
be the equation of this surface. At any point of this surface dSD0D @S
@t dtCdx@S
@x Cdy@S
@y Cdz@S
@z
Similarly as (1.3),.dx=dt;dy=dt;dz=dt/represents the velocity of the surface, which, by definition, is also the fluid velocity. Hence, a first boundary condition is
@S
@t Cvx@S
@x Cvy@S
@y Cvz
@S
@z D0 on S.r; t /DCst or
DS
Dt D0 on S.r; t /DCst (1.63) This last relation shows that the material derivative of the surface is zero at the surface. In other words, the surface is fixed for a fluid particle at the surface, or, a fluid particle initially at the surface remains attached to it.
We note that this boundary condition is purely geometrical. We did not use any physical law to write it down. In many situations, it is simplified because the surface is time-independent. In such a case it reads
vrSD0
9A rather complete account of the history of the quest of the correct boundary conditions at a solid wall may be found in Goldstein (1938, 1965). The irony of the story is that scientists are presently looking for materials that let the fluid slipping on the walls. This is especially important when dealing with small pipes in microfluidic (see Tabeling,2004).
ButrS is a vector perpendicular to the surface. Settingnas the unit normal, the preceding relation is just
vnD0 (1.64)
expressing that the flow is tangential to the surface at the fluid boundary.
At this stage it is worth pointing out that this condition, much simpler than (1.63), is often used even if the surface is not strictly steady. This approximation is physically acceptable when the time scales or the length scales of the problem at hands are far larger than the ones arising from surface waves (capillarity or gravity waves).
As may be guessed, condition (1.63) is not sufficient to fully specify the solution of a problem. We need now expressing the continuity of the stress when crossing the surface. In other words, on each side of the surface the stress must be the same (up to the sign). For instance, if the surface separates the fluid from the vacuum, we write
ŒnD0 on S.r; t /DCst
Together with (1.63), this relation constitutes the free-surface boundary conditions.
If we compare to (1.62), we may note that these boundary conditions are four. The additional equation is in fact the one that determines the surfaceS.r; t / which is also an unknown of the problem. We shall dwell on this problem more thoroughly when discussing the propagation of surface waves in Chap.5.
1.8.1.3 The Stress-Free Boundary Conditions
In many situations the bounding surface is known and it is useful to assume that the fluid slips freely along the boundary, either because this boundary separates fluids of very different densities, or because in a first approach of a complex problem, one wishes to avoid boundary layers generated by a solid–fluid interface or waves allowed by a moving surface.
A fluid freely slipping on a surface does not exert any tangential stress.
Mathematically, this is expressed by
n.Œn/D0 on S (1.65)
This vectorial condition in fact amounts to two scalar conditions and needs to be completed by the kinematic one (1.64). Conditions (1.65) together with (1.64) now give three scalar conditions, just like (1.62). These conditions are known asstress- free or free-slip conditions.
1.8 Boundary Conditions 35
1.8.2 Boundary Conditions on Temperature
The foregoing boundary conditions described the dynamics of the interaction of the fluid with its environment. They are related to the momentum equation and mass conservation. We should now ask for the conditions which are associated with the equation of energy. Such conditions express the way energy is exchanged through a bounding surface. Since we restrict our discussion to the case where the boundary does not allow for mass exchanges, fluxes of energy are only of microscopic origin, namely from thermal conduction. Generally speaking, these conditions require the continuity of temperature and energy flux, namely
T DText and nFDnFext: (1.66)
For a fluid with constant conductivities, the second condition is also a condition on the temperature gradient.
When we study the equilibrium or the motion of fluids in presence of temperature gradient, we shall use the notion of perfect conductor. Such a medium is an idealization of a material that can accept any heat flux. Thus, when a fluid is in contact with a perfect conductor its temperature is fixed to that of the conductor.
The other extreme case is also useful: it is theperfect insulator. For this medium the heat flux is set to zero (or fixed to a given value), while the temperature can take any value. An example is given in Chap.7.
1.8.3 Surface Tension
Free surface boundary conditions are often taken at the interface of two immiscible fluids. A complete description of free-surface boundary conditions thus calls for the introduction of surface tension. This phenomenon is the consequence of the fact that some energy must be spent to increase the surface of contact between two immiscible fluids. Only liquids own a surface tension at their boundaries because the liquid phase is characterized by an attractive interaction between the molecules (a van der Waals type force). The energy of the liquid is therefore minimized when each of its molecule is surrounded by other similar molecules. Those molecules on the boundary have a higher energy. Hence, a larger bounding surface demands more energy.
If we introduce, the ratio of the energy variation to the surface variation, namely
dEDdS; (1.67)
we note that is both an energy per unit surface and a force per unit length. Let us therefore consider a surface, delimited by a contour C, taken on the surface separating two immiscible fluids. If we decomposedSintodldn,dlbeing locally
parallel toC anddnperpendicular to it,dldncan be interpreted as the work done by a forcedlen to extend the surface byLdn (Lis the length ofC). Thus, the surface supports a resulting force
RD I
.C /
dlen
whereenis the outer normal unit vector ofC. The use of the divergence theorem in two dimensions (see Sect.12.2.3) allows us to transform this integral into
RD Z
.S /
rdS (1.68)
which shows now that variations of surface tension are sources of a surface force, or, in other words, of a stress. This stress has the peculiarity of being purely tangential, which implies that if the surface separating two Newtonian fluids experiences variations of the surface tension, some flow will appear for no static constraint can compensate this stress. Such a phenomenon is at the origin of Marangoni–Bénard convection which is an instability coming from the dependance ofwith respect to temperature (see Sect.6.3.5for a detailed presentation).
The foregoing discussion focused on a first effect of surface tension. Indeed, we restricted the surface variationdSto the local tangent plane ofC. This is just like the case where one pulls on a piece of rubber to increase its size. However, another simple way of extending the surface exists: this is by pushing it in a direction perpendicular to its actual surface. An easy way to make this idea quantitative is to consider a drop of liquid. If its radius varies ofdRits surface varies ofdSD8RdR, and the energydE D 8RdRmust be spent. As above, this energy may also be interpreted as the work of the surface tension F D 8R, which has a surface density
fD 8R
4R2er D 2 Rer
It works like a normal stress. Hence, inside a liquid drop at equilibrium, the pressure is slightly higher than outside the drop since
PextD PintC 2 R
” PintDPextC 2 R as demanded by the continuity of normal stress.
1.9 More About Rheological Laws: Non-Newtonian Fluids 37 The foregoing formula is however specific to the sphere. With more general surfaces, two radii of curvature (R1andR2) are necessary to describe the surface variations associated with a normal motion. This leads to the famous Laplace formula
PintDPextC 1
R1 C 1 R2
(1.69) which is demonstrated in Landau and Lifchitz (1971) for instance.
Finally, the two effects of surface tension that we just described can be gathered in a single formula which states the dynamic boundary condition at a liquid–gas interface
ŒliqnC 1
R1 C 1 R2
nCrDŒgasn (1.70)
Here, n is the normal of the surface that is oriented from the liquid to the gas.
Curvature radii are positive if the centre of curvature is inside the liquid.
We shall come back to surface tension in a few occasions: first, for some aspects of fluids equilibria, and then when considering the propagation of surface waves.
1.8.4 Initial Conditions
Finally, we should say a few words about the boundary conditions on time, in other words the initial conditions. The equations of motions are all of first order in time. This means that the initial state of the fluid completely determines its future evolution. This is true only in principle. The example of meteorology just shows that the behaviour of the fluid is unpredictable beyond a few days, essentially because the initial state is always imperfectly known and imperfections are amplified by the nonlinearities of the equations of motion.