fluids ch 67

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Chapter 6

Stokes Flows

One of the earlier approximations to the Navier-Stokes equations goes back toStokes himself, who studied the limit of very small Reynolds number. Thishas applications to very viscous flows, suspensions and bubbles, and therecently important field of micro-fluid-dynamics. Highlights include Stokesparadox, far-field effects, kinematic reversibility, and the role of vorticity.(Fig. 6.1).

6.1 Stokes equation

We start from the Navier-Stokes equations, and use a velocity U and lengthL as scaling quantities. Notations:

ui = U u

i xj = L x

j

Then

tui + ujjui = 1

ip + 2jjui (6.1)

becomes

Utui + U2/Lujj ui =1

L i p + U/L22jj ui (6.2)

Since we expect the viscous term to be significant, we set its coefficient equalto 1 by multiplication by L2/U, yielding

L2

tu

i + ReLu

j

j u

i = L

Ui p +

2jj u

i (6.3)

145

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146 CHAPTER 6. STOKES FLOWS

G.I. Taylors movie: Creeping flows

Vanishing Reynolds number (large viscosity or small scale or lowvelocity) or convective terms vanish because of geometry (e.g.Poiseuille/Couette flow at moderate Re).

Kinematic reversibility (but stresses change sign). No inertia, no convective terms. Balls, drops and suspensions.

Long range effects: 2 everywhere. Vorticity essential (from B.C.).

Flow around a sphere: exact solution

Stokes paradox: flow around a cylinder

Stokes drag

Slender bodies

Lubrication Propulsion Hele-Shaw cell

Figure 6.1: A wonderful movie by G.I. Taylor: low Re flows

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6.1. STOKES EQUATION 147

Figure 6.2: Mind-map relative to scaling of pressure

In the limit of very small ReL, the nonlinear terms drop out. Mathemati-cally, this makes the equations linear in ui, hence more easily solvable. Also,the viscous time scale is L

2

. Flow disturbances associated with times larger

than this are in effect quasi-stationary, and the field is in equilibrium withthe current conditions imposed at the boundary (examples in the movie).

Furthermore, we see that U2 is no longer the correct scaling for pressure. Inthis instance, the pressure field scales as U/L, which may be representativeof the shear stress applied at a boundary. See Fig. 6.2

The resulting equation is very simple:

ip = 2jjui (6.4)

orp = 2u (6.5)

or again (using incompressibility, so that = 2u)p = (6.6)

This should be contrasted to Croccos result for inviscid flow.Since we assume incompressibility ( u = 0), it follows that

2p = 0. (6.7)

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Figure 6.3: Mind-map: Pressure and vorticity in Stokes flow: Laplace equa-

tion and far-field effects

Similarly, since p = 0 for any smooth scalar function, we also have2 = 0. (6.8)

Laplace equation everywhere! Note that in the case of (vector) vorticity, itapplies to each component. For vorticity, the stretching and advection termsare negligible in Stokes flow. As boundary conditions vary (slowly enough),diffusive equilibrium with the current boundary values is reached instantly(i.e. much faster).

Recall the important properties of the Laplace equation: the solution atany point is determined by boundary conditions over the entire boundary.The boundary conditions can be either for the field itself (e.g. pressure) orits normal gradient, at each point of the boundary. Actual dependence onremote boundary points is quantified by Greens function. Many of thenonlocal effects in Stokes flow (Taylor movies, see below) are related directlyto this analytical feature.

6.1.1 Kinematic reversibility

Go back to Stokes equation for velocity

ip = 2jjui (6.9)

If we reverse spatial directions, the pressure gradient changes sign, the Lapla-cian keeps its sign, but velocity changes sign as well (time, as we know, can-not be reversed!) All in all, the equation is unchanged when we change the

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6.2. STOKES TWO PROBLEMS 149

Figure 6.4: Reversal in Stokes flow.

sign of any (cartesian) coordinate. The illustrations in Taylors movie arespectacular!

See Panton Ch21 Fig. 21.1 p.641 for reversibility of a flow over a block(also note the secondary flows in the corners).

There is an important distinction to be made between kinematic re-versibility (above), dynamic reversibility (see the problem about potentialflow at the end of this chapter) and thermodynamic reversibility. Make sureyou dont mix them up!

6.2 Stokes two problems

Two exact solutions of the equations can be derived in configurations thatreduce them to 1-D diffusion. They are part of a larger class of problemstreated systematically by Rayleigh (see e.g. Telionis Unsteady viscous flow,Springer)

The problems have in common an infinite flat plate, with a viscous fluidon one side. As the plate moves in its own plane, it induces motion in thefluid. The case of the rotating plate leads to the steady-state Karman pump .

The simpler case of rectilinear motion was solved by Stokes. It is noteworthythat these problems are unchanged if the Reynolds number (e.g. based onoscillation amplitude and frequency and on fluid viscosity) is not small. Seeproblem at the end of chapter.

Because of homogeneity (no privileged point), the motion of the fluid isalso rectilinear, with no pressure gradient in the homogenous direction of

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1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Figure 6.5: Stokes two problems: oscillating (dotted lines) and impulsivelystarted (solid line) plate; unit viscosity and frequency; times .1, .2 ... 1.

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6.3. STOKES FLOW AROUND A SPHERE 151

motion, and the equation is

tu = 2yyu (6.10)

The two Stokes problems have this diffusion equation in common; the differ-ence is in the boundary condition. For the first problem, we start from rest,and the plate is impulsively started at constant speed U for t > 0. For thesecond problem, the plate oscillates with frequency

U |y=0= U0e2it (6.11)The solutions are classics, and should be read carefully1. For the first prob-

lem, the motion progresses into the fluid, affecting a layer scaling as t.For the second problem, the cumulative effect of wall motion is partially can-celled by its periodic reversals, and the flow oscillates, out of phase with theforcing, with decreasing amplitude away from the wall.

See e.g. Panton Sections 11.2 and 1.3 p.266-279 for details.

6.3 Stokes flow around a sphere

With Stokes equation expressing momentum balance at vanishing Reynoldsnumber, we also need to satisfy mass balance. This can be done with the use

of a vector potential (review Ch2 !)

u = A or = 2Aas an alternative to Eq.6.7). Then, mass and momentum balance are com-bined in the single relation

22A = 0 (6.13)subject to boundary conditions that reflect the specifics of a given problem.

1One method of solution makes use of Greens functions, and (without going into thedetails of actual solution) points to interesting physics. Greens function for diffusion for

this case isG(y, t; y, t) =

1

2

(t t)e

(yy)2

4(tt) (6.12)

for t t, and zero otherwise. It shows that the effect of a source at (x, t) is felt instantlythroughout the field, but decreases very rapidly with distance. The equilibrium solutionfor infinite time corresponds to the Laplace equation: one obtains the Biot-Savart kernelby integrating the diffusion Greens function over time.

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Figure 6.6:

The flow around a sphere and around a cylinder are classical cases, forwhich the geometry indicates the need for spherical and cylindrical (polar)coordinates, respectively. (Make sure you know where to find the appropriateformulae of vector calculus.)

The flow around a sphere of radius R turns out to be simpler and ispresented first. In spherical coordinates (r, , ) (where is the longitudeand is the co-latitude), we have

u = A = 1r sin

[(sin A) A]er+

1

r[

1

sin Ar r(rA)]e +

1

r[r(rA) Ar]e (6.14)

Of course, we dont wish to handle this in the most general case! Withthe direction of the flow around the sphere as the polar axis, axisymmetryrequires that

u = 0, (6.15)

which implies that Ar and A should be constant (and we can take themequal to zero without loss of generality). Then, A = Ae, and the velocityfield simplifies into

u = A = 1r sin

(sin A)er 1

rr(rA)e (6.16)

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Then, introducing the streamfunction

= r sin A (6.17)

the velocity field is

u =1

r2 sin er

1

r sin re. (6.18)

(that wasnt so hard, was it?) What we have expressed so far is: massconservation, axisymmetry. Now, the BCs will make the solution specific tothe sphere. Since the sphere itself must coincide with a streamline, mustbe constant on the sphere, and we can take

|R= 0 (6.19)without loss of generality, with ur = u = 0 (no slip). At infinity, uniformflow requires

(r) |= 0 (6.20)and furthermore we know that

ur = Ucos and u = Usin so

r2

2 Usin2 as r (6.21)

Finally, we must get the right dynamics: Stokes equation takes the form:

[2rr +sin

r2(

1

sin )]

2 = 0 (6.22)

We know we are in luck (right combination of dynamics and boundary con-ditions) when separation of variables works: try it for practice. The resultis

= R 2U12

(r

R)2 sin2 [1 3

2

R

r+

1

2(

R

r)3] (6.23)

The velocity components are easily obtained, and vorticity is given by therelation

=U

R

3

2(

R

r)2 sin (6.24)

(which component is this, by the way?) maximum at the equator and zeroalong the polar axis.

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2 0 2 4

10

8

6

4

2

0

2

2 0 2

4

6

8

10

12

14

Figure 6.7: Stokes flow around a sphere: (left) particle paths around a fixed

sphere, top to bottom; (right) seen from the fluid at rest, the motion of thesphere affects the motion of some particles (bottom to top).

It should be noticed that the streamlines exhibit symmetry upstream/downstream(dependence on sin(), not cosine!). Does this make sense to you? What isthe relevant concept earlier in this chapter?

In relation to kinematics (chapter 2), the difference between the steadyflow (as observed from the sphere) versus unsteady flow (as observed fromthe fluid away from the sphere) is of interest. On Fig. (6.7) Seen from thesphere, the streamlines, pathlines and streaklines are identical (steady flow).

From the fluid at rest, this is not the case. Edit and expand.The pressure distribution can be calculated exactly: separation of vari-

ables in the Laplace equation gives

p = p +3

2

U

R(

R

r)2 cos (6.25)

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Figure 6.8: Wall interference of rising bubble or falling sediment

As one would expect, pressure is maximum at the forward stagnation point( = 0) and minimum at the rear stagnation point (no dynamic reversibility!).Forcing an inertial scaling on this expression gives for the stagnation pressures

ps p = 12

U23

Re(6.26)

Finally, calculation of the normal and tangential stresses over the entire

surface can be carried out (e.g., do it in Maple, starting from the solutionfor velocity). The result (See Panton p647, Batchelor p233) is the classicalresult for Stokes drag:

F = 6RU (6.27)

This exact formula can be used to calculate viscosity from a measurementof terminal velocities of spherical objects (bubbles, etc.)

6.3.1 Nonlocal effects

Because of the ubiquitous Laplacians, and associated Greens functions, the

boundary condition at the sphere surface induces vorticity and relative ve-locity at large distances from the sphere. Two effects follow from this obser-vation.

First, consider a bubble rising near a vertical wall. The ball of influenceof the bubble (region where it disturbs appreciably the ambient liquid) inter-sects the wall, which prevents the induced motion from taking place. This

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constraint on the induced motion slows down the bubble, as the viscous fluid

needs to go around the bubble in some other way. So the bubble rises moreslowly near the wall than elsewhere in the bulk of the liquid.

A similar situation arises if two bubbles rise side-by-side. This corre-sponds to the method of images, a mainstay for the solution of Laplacesequation: the plane of symmetry is similar (though different in one impor-tant respect: do you see which?) to the impermeable wall. The lack of elbowroom slows down the bubbles, which rise more slowly than they would in-dividually. This is the reason why a cloud of rising bubbles of uniform size(only approximated by air bubbles in a vigorously shaken water container)will have a flat bottom: individual bubbles left behaind will soon catch up

the the cloud - but on the top side, any front-runner will run away from thecrowd.

6.3.2 Application: Slender bodies

See falling rod, Taylor movie.

6.4 Cylinder: Stokes paradox

With this background, Stokes flow around a cylinder appears as a simpler

variant of the above. Surprise! In cylindrical coordinates (r, , z), we have

u = A = ( 1r

Az zA)er+(zAr rAz)e +

1

r(r(rA) Ar)ez (6.28)

Then, symmetry can be imposed, and boundary conditions follow.The surprise is that it is impossible to match the boundary conditions

both at infinity (uniform flow) and at the cylinder surface (no-slip) withStokes flow dynamics. What the mathematics are telling us is the nonlocal

cumulative effect of vorticity near the cylinder does not vanish fast enoughin 2D, whereas they did in 3D (different Greens function!). The properReynolds number is the Re based on the region affected by vorticity and thisRe is always of O(1) for cylinder regardless ofU. Thus, Stokes flow arounda cylinder does not exist! Any flow around a cylinder shows effects of inertia(finite Re), e.g. a wake in which the velocity defect shown as wider spacing

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6.5. REYNOLDS LUBRICATION THEORY 157

Figure 6.9: Wake of a cylinder even at small Re shows that Stokes flow doesnot exist in this geometry.

of the streamlines on the downstream side. The corresponding solution wasfirst calculated by Oseen.

6.5 Reynolds lubrication theory

Reynolds studied another important application at vanishing Re. The Couette-like shear flow between non-parallel surfaces is illustrated by Taylors littlegizmo in the movie; sliding a sheat of paper across a table, air-hockey andsimilar games, thrust bearings for marine propellers, and the circular ge-

ometry of excentric journal bearings, provide a wealth of illustrations. Thecombination of narrow gaps (small Re), shallow angles and moderate veloci-ties yield relatively large hydrodynamic forces on the solid surfaces.

The basic problem is formulated as follows (Batchelor p219, Panton p660), 2-D (plane) to simplify expressions. Fixed pad of length L above, flatplate sliding at velocity U, gap of thickness h(t, x) with h L everywhere.Then x L, y h and u U. It follows (x-momentum) that

p ULh2

(6.29)

which is independent of y (compare with BL approximation!) By continuity,

v UhL

U (6.30)

At small Re, we havexp =

2yyu (6.31)

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Figure 6.10: Definition sketch for the Reynolds lubrication problem.

with the no-slip conditions u |y=0= U and u |y=h= 0. The solution is readilyobtained (combination of Couette and Poiseuille) as

u =h2

2xp((

y

h)2 y

h) + U(1 y

h) (6.32)

It can be checked (see BL approximation) that in the y-momentum equa-tion, the pressure term is of order UL/h3, while the viscous term is only

of order U/hL. So the pressure term is unmatched in terms of order ofmagnitude, unless yp vanishes to eliminate its own scaling factor. So, the vcomponent is determined by mass balance, not by momentum balance.

Start from the continuity equation and integrate over the entire depth:h(x)0

xu dy +h(x)0

yv dy = 0 (6.33)

The second integral is straightforward, the first one calls for the Leibniztheorem (see integral method in related courses)2. The result is

1

x(h3

xp) = 6Uxh + 12th (6.34)

2In a nutshell,

dx

b(x)

a(x)

f(x, y) dy =

b

a

dxf dy + dxbf(x, b(x)) dxaf(x, a(x)).

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6.5. REYNOLDS LUBRICATION THEORY 159

0.2 0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

p.

4/3.

h02

/

L2

.

h1/2

/

U

x/L

Figure 6.11: Pressure distribution under a simple pad

known as Reynolds lubrication equation. It is a Poisson-type equation forpressure, with derivatives ofh as source terms and ambient pressure at eitherend of the pad as BCs. In the simple case of time-independent h and uniform

slope, we write = xh as the constant (small) pad angle, so that sin ,and the solution is

p(x) p0 = 6U

(h0 h)(h h1)h20(h0 + h1)

(6.35)

Writing h = h0 x, h1 = h0 L and h1/2 = (h0 + h1)/2, and keepingonly the leading term for small , we get

p(x) p0 = 6U x(L x)h20(h0 + h1)

=3

4

L2

h20

U

h1/2

2x

L(2 2x

L) (6.36)

This expression is informative. The spatial dependence is parabolic, withits maximum at the center of the pad. We factored out the average shearstress U

h1/2in accordance with Stokes flow scaling. The remaining factor

3L2/4h20 can be extremely large. The lower limit of h0 > L is determinedby surface tolerances.

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6.6 Lagrangian turbulence, chaotic mixing

Although reversibility is true for incremental time steps, and might be ex-pected to hold for finite times, there are instances of extreme sensitivity toinitial conditions such that chaotic mixing occurs even in Stokes flow. Seethe literature on Lagrangian turbulence for this.

6.7 Advanced topics and ideas for further read-

ing

Journal bearings: see introduction in Acheson, p 250. Hele-Shaw cell: flowviz of potential flow. See Acheson for a clear simple presentation.

Liquid adhesive: thin layer of fluid between matching surfaces. Separatingthe surfaces requires the creation of Poiseuille flow between them, which inturns requires very large pressure gradients.

The finite-Re effects for the flow around the sphere introduce weak non-linearities. The classic analysis of Oseen can be found in textbooks on viscousflows. The convective terms break the front/back symmetry of the flow.

The difference between large Re (inertial) and small Re (viscous) propul-sion is well illustrated in Taylors movie.

The field of micro fluid mechanics has evolved recently under combinedpressures of MEM-actuators for flow control strategies and the growth ofinterest in biomedical applications of fluid dynamics. This is beyond thescope of this course.

Problems

1. Discuss the contrast between the reversal of dye motion in Taylorsmovie and the lack of reversal for the piece of thread. Map out the

relevant ideas.

2. We saw that Stokes flow is kinematically reversible. Carry out a similaranalysis for potential flow, and comment on the differences. How aboutdynamic similarity? In which is pressure independent of flow direction?why?

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6.7. ADVANCED TOPICS AND IDEAS FOR FURTHER READING 161

3. In irrotational flows, we had 2 = 0; in axisymmetric Stokes flow, wehave

2

= 0. Discuss analytical and phenomenological similaritiesand differences.

4. Identify the sources of pressure in Stokes flow.

5. Outline differences and similarities between Stokes first problem (im-pulsively started plate) and boundary layer development.

6. Taylors two problems, as well as classical Poiseuille and Couette flows,use the Stokes flow approximation in spite of Reynolds numbers pos-sibly of the order of 1000. Resolve this apparent discrepancy.

7. Analyze the vertical motion of Taylors teetotum, to determine howthe elevation of the pads above the table surface is affected by angularspeed (other obvious parameters: pad angle and area, weight per pad,typical radius, weight,...)

8. Consider two extreme cases of the flow past a rectangular block (VanDyke Figs. 5 and 11, sketched on Fig. 6.12) Discuss kinematic and dy-

Figure 6.12: Potential and creeping flow past a blocknamic reversiblity; discuss similarities and differences relative to pres-sure, vorticity, wall stress, velocity magnitudes, etc.

9. Consider Stokes second problem: a Reynolds number based on ampli-tude of plate oscillation, frequency and viscosity, is easily constructed.

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We not assume this Re to be small, yet Stokes equation applies. Ex-

plain why, and list familiar examples where the same situation occurs.

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Chapter 7

Interlude

This is a good time to organize some lines of thought. The student is urgedto go back over the material covered, and collect facts and equations (andmind maps) associated with every topic mentioned more than once: no topicshould be without context. Here, the emphasis is put on three main themes:the approximations, non-local effects, and vorticity.

7.1 The approximationsThe approximations to the intractable Navier-Stokes equations are based onscaling analysis and dimensionless numbers: depending on the correct ordersof magnitude, the ability to neglect some terms is of great importance foranalysis, computation and experimentation alike.

Just as important is the awareness of the discarded physics. When wedrop the convective or the viscous term, a number of phenomena becomeinconsistent with the new equations. Even experienced fluid dynamicists canoverlook these inconsistencies on occasion: context should be ever present as

the best safeguard.The key to the correct use of approximations is to remember that they

correspond to extreme cases. The results and insights obtained in theseextreme cases cannot be taken blindly into the more complicated cases ofpractical interest, but they can help our thinking, if only to treat resultswith caution.

163

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164 CHAPTER 7. INTERLUDE

7.2 Non-local effects

Non-local effects are not emphasized explicitly in many fluid mechanics texts.Although Biot-Savart and induced velocities are generally mentioned, and thepressure equation (divergence of Navier-Stokes) is sometimes listed, the gen-eral properties of the Poisson and Laplace equations is generally overlookedexcept for numerical work. These elliptic equations are mathematically andcomputationally challenging in practical geometries; of interest at this levelis the concept of non-locality. Velocity in potential flow, pressure in largeand small Reynolds and small Rossby number flows, vorticity in Stokes flows,and others, provide the opportunity for remote diagnostics and therefore forflow control strategies. Students should make non-local effects part of theirthinking.

7.3 The role of vorticity

Vorticity can be taken as the leitmotiv for this course. Aside from the ele-gance of potential flow, with its unique solutions and easy phenomenology,vorticity is responsible for the complexity and/or beauty of most of fluidmechanics.

Vorticity comes in many contexts: kinematics with two of the Helmholtz

theorems, inviscid dynamics with Kelvins theorem and Helmholtzs othertheorem, and viscous flows of all types, instabilities and secondary flows androtating flows in the following chapters, and so many more topics.

fluids ch 67

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