220 The Navier-Stokes equations
6.13. Let x =x(X, t ) denote some fluid motion, as in Exercises 1.7, 5.18, and 5.21, and let J denote the determinant I " 3x1 3x1 1
ax1 ax2 ax,
Establish Euler's identity DJIDt = J V - u,
and use this to give a proof of Reynolds's transport theorem (6.6a). 6.14. If we apply the principle of moment of momentum (06.1) to a finite 'dyed' blob of some continuous medium occupying a region V(t) we obtain?
Use Reynolds's transport theorem, together with eqns (6.7) and (6.8), to write this in the form
where summation over 1, 2, 3 is implied for i , j, and k . Re-cast this equation into the form
and hence deduce that, subject to the provko in the footnote,
i.e. the stress tensor must be symmetric, whatever the nature of the deformable medium in question. (This famous requirement, to which eqn (6.9) conforms, is due to Cauchy .) t There is a proviso here, namely that the net torque on the blob is due simply to the moment of the stresses t on its surface and the moment of the body force g per unit mass. This is very generally the case, but there are exotic exceptions, as when the medium consists of a suspension of ferromagnetic particles, each being subject to the torque of an applied magnetic field (see Chap. 8 of Rosensweig 1985).
7.1. Introduction .
The character of a steady viscous flow depends strongly on the relative magnitude of the terms (u - V)u and vV2u in the equation of motion
We are here concerned with the 'very viscous' case in which the (u V)u term is negligible. There are two rather different ways in which this can happen.
First, the Reynolds number may be very small, i-e.
On the basis of the estimates (2.5) we then expect the slow flow equations
0 = -vp + p V2u, v - u = o (7.3)
to provide a good description of the flow, in the absence of body
We discuss the uniqueness and reversibility of solutions to these equations in 07.4, and some implications for the propulsion of biological micro-organisms follow in 97.5. In 07.3 we explore the so-called comer eddies that can occur at low Reynolds number, as in the superbly symmetric example of Fig. 7.l(a). First, however, we investigate in 07.2 the classical problem of slow flow past a sphere, and it is worth taking a moment to consider the kind of practical circumstances in which slow flow theory might apply in that case.
Suppose, for instance, that we tow a sphere of diameter D = l c m through stationary fluid at the quite modest speed
222 Very viscous flow
(a) ( b ) Fig. 7.1. Two very ,viscous flows: (a) flow at low Reynolds number past a square block on a plate; (b) a thin film of syrup on the outside of a
U = 2 cm s-'. Then according to Table 2.1 the Reynolds number UDlv will be about 200 for water, 2 for olive oil, 0.1 for glycerine, and 0.002 for golden syrup. Now, if we move the sphere at a speed of only 0.2cm s-', all these values will be reduced by a factor of 10,. but they are still not spectacularly small. Our point, then, is that while Reynolds numbers of order 10' or lo9 are not at all uncommon in nature, to get a genuinely small Reynolds number takes a bit more effort.
A second, quite different, way in which the (u . V)u term may be negligible in eqn (7.1) involves motion in a thin film of liquid, and in this case the 'conventional' Reynolds number need not be small. The key idea is, instead, that the velocity gradients across the film are so strong, on account of its small thickness, that viscous forces predominate. Thus if L denotes the length of the film, and h a typical thickness, the term (u V)u turns out to be negligible if
as we show in $7.6. The resulting thin-film equations are even simpler than eqn (7.3), and provide the opportunity of tackling some problems which would otherwise be unapproachable by elementary analysis. One example will be well known to patrons of Dutch pancake houses: if you dip a wooden spoon into syrup, withdraw it, and hold it horizontal, you can prevent the syrup
Very viscous flow 223 from draining off the handle by rotating the spoon (Fig. 7.1(6)). Moffatt (1977) used thin-film theory to show that there is a steady flow solution if
where A is the mean thickness of the film. If the peripheral speed U of the handle is below this critical value there is no steady solution, and the liquid slowly drains off. .
In the second half of this chapter we look at a number of thin-film flows of this kind, one of the most notable being that in a Hele-Shaw cell ($7.7). In this quite elementary apparatus it is possible to simulate many 2-D irrotational flow patterns that would, on account of boundary layer separation, be wholly unobservable at high Reynolds number.
7.2. Low Reynolds number flow past a sphere
We now seek a solution to the slow flow equations (7.3) for uniform flow past a sphere, and using appropriate spherical polar coordinates we therefore want an axisymmetric flow
We may automatically satisfy V u = 0 by introducing a Stokes stream function V(r, 8 ) such that
1 a v 1 a v ur =--
r2sin 8 a8 ' u e = - - - r sin 8 a r (7.6) (cf. $5.5). Then
3: $4 where E2 denotes the differential operator
,52=-+-- -- ar2 r2 38 sin 8 38
.2< ,:$ Writing eqn (7.3) in the form $2
224 Very vkcous flow
(see eqn (6.12)) we obtain
and eliminating the pressure by cross-differentiation we find that E ~ ( E ~ Y ) = 0, i.e.
The boundary conditions are
together with the condition that as r+m the flow becomes uniform with speed U:
u,-Ucos8 and u,--Usin8 a s r + a . This infinity condition may be written
Y - 4Ur2 sin28 as r+ m, which suggests trying a solution to eqn (7.7) of the form
Y = f (r)sin28. This turns out to be possible provided that
Fig. 73. Low Reynolds number flow past a sphere.
Very vircous flow 225
This equation is homogeneous in r and has solutions of the form r " provided that
[(a - 2)(a - 3) - 2][a(a - 1) - 21 = 0, so that
A f (r) = - + Br + c r 2 f ~ r ~ , r
where A, B, C, and D are arbitrary constants. The condition of uniform flow at infinity implies that C = +U and D = 0. The constants A and B are then determined by applying the boundary conditions on r = a , which reduce to f (a) = f '(a) = 0. We thus find that
The streamlines are symmetric fore and aft of the sphere (Fig. 7.2).
A quantity of major interest is the drag D on the sphere. By computing E2Y = $ ~ a r - ' sin28 and then integrating the equa- tions above for the pressure p we obtain
where pm denotes the pressure as r+m. The stress components on the sphere are
(see eqn (A.44) and $6.4). Having found Y?, we may calculate u,, and u,, and hence
226 Very viscous flow
By symmetry we expect the net force on the sphere to be in the direction of the uniform stream, and the appropriate component of the stress vector is
The drag on the sphere is therefore
D = Lb [ to2 sin 8 dB d$ = 6npUa. Laboratory experiments confirm the approximate validity of
this formula at low Reynolds number R = Ualv. One such experiment involves dropping a steel ball into a pot of glycerine; the ball accelerates downwards until it reaches a terminal velocity U, such that the viscous drag exactly balances the (buoyancy- reduced) effect of gravity:
The above theory, due to Stokes (1851), is not without its problems. Stokes himself knew that a similar analysis for 2-D flow past a circular cylinder does not work (Exercise 7.4). Later, in 1889, Whitehead attempted to improve on Stokes's theory for flow past a sphere by taking account of the (u V)u term as a small correction, but his 'correction' to the flow inevitably became unbounded as r + m.
In 1910 Oseen identified the source of the difficulty. The basis for the neglect of the (u . V)u term at low Reynolds number lies in eqn (2.6), where the ratio of J(u V)ul to lv V2ul is estimated to be of order ULIv. But L here denotes the characteristic length scale of the flow, i.e. a typical distance over which u changes by an amount of order U. Now, in the immediate vicinity of the sphere, L will be of order a , so if the Reynolds number based on the radius of the sphere R = Ua/v is small, then the term (u . V)u will certainly be negligible in that vicinity. The trouble is that the further we go from the sphere, the larger L becomes, for the flow becomes more and more uniform. Inevitably, then, sufficiently far from the sphere the neglect of
Very viscous flow 227
the (u V)u term becomes unjustified, and the basis for using eqn (7.3) as an approximation breaks down. Compare this with what often happens in flow problems at high Reynolds number; the viscous terms are small throughout most of the flow but inevitably become important in boundary layers, where velocity gradients are untypically high. Here the viscous terms are assumed to be large, but inevitably cease to dominate in regions of the flow where velocity gradients are untypically low.
Oseen provided an ingenious (partial) resolution of the difficulty, but it was not until 1957 that Proudman and Pearson thoroughly clarified the whole issue by using the method of matched asymptotic expansions, which subsequently proved to be one of the most effective techniques in theoretical fluid mechanics. We provide here only the briefest sketch of this work, hoping simply to convey some idea of what the 'matching7 entails. For this purpose it is helpful to work with dimensionless variables
based on the sphere radius a and the speed at infinity U. Then, dropping primes in what follows, substitution of eqn (7.6) into the full Navier-Stokes equations