Exact solutions and physical analogies for unidirectional flows

6.5 MB·

by  Martin Z. Bazant

Exact solutions and physical analogies for unidirectional flows
Martin Z. Bazant
Departments of Chemical Engineering and Mathematics,
Massachusetts Institute of Technology, Cambridge, MA 02139 USA and
Present address: Department of Materials Science and Engineering and SUNCAT
Interfacial Science and Catalysis, Stanford University, Stanford, CA 94305
(Dated: May 5, 2016)
Unidirectionalflowisthesimplestphenomenonoffluidmechanics. Itsmathematicaldescription,
theDirichletproblemforPoisson’sequationintwodimensionswithconstantforcing,arisesinmany
physical contexts, such as the torsion of elastic beams, first solved by de Saint-Venant for complex
shapes. Here, the literature is unified and extended by identifying seventeen physical analogies for
6
unidirectionalflowanddescribingtheircommonmathematicalstructure. Besidesclassicalanalogies
1
in fluid and solid mechanics, applications are discussed in stochastic processes (first passage in
0
two dimensions), pattern formation (river growth by erosion), and electrokinetics (ion transport in
2
nanochannels),whichalsoinvolvePoisson’sequationwithnon-constantforcing. Methodsaregiven

y to construct approximate geometries that admit exact solutions, by adding harmonic functions to
a quadratic forms or by truncating eigenfunction expansions. Exact solutions for given geometries
M
arealsoderivedbyconformalmapping. Weprovethattheremarkablegeometricalinterpretationof
Poiseuilleflowinanequilateraltriangularpipe(theproductofthedistancesfromaninteriorpoint

4 tothesides)isonlysharedbyparallelplatesandunboundedequilateralwedges(withthethirdside
hidden behind the apex). Wealso prove Onsager reciprocity for linearelectrokinetic phenomena in
] straight pores of arbitrary shape and surface charge, based on the mathematics of unidirectional
n
flow.
y
d
-
u I. INTRODUCTION In the twentieth century, pipe flows for different cross-
l sectional shapes were extensively characterized in the
f
s. PreciousfewexactsolutionsoftheNavier-Stokesequa- engineering literature, especially for heat transfer in
c tions are known, but they serve to guide our thinking ducts [17], and the number of physical analogies also
si about fluid mechanics [1–3]. Most students first en- grew. The common mathematical problem involves
y counter the parabolic profile of Poiseuille flow in a cir- Poisson’s equation from electrostatics [18],
ph cular pipe or between flat plates [4]. Some less familiar −∇2u=k (1)
examples are shown in Fig. 1. In these unidirectional
[
flows, inertia plays no role, leading to a simple balance typically with constant forcing k and Dirichlet (no-slip)

3 betweenviscousstressandtheappliedpressuregradient. boundary conditions on a two-dimensional domain. The
v Poiseuille’s law for the flow rate in a narrow capil- same problem arises in solid mechanics for beam tor-
3
lary [4, 5] was apparently first derived from the Navier- sion and bending [19–21], and myriad exact solutions
0
Stokes equations by Stokes himself [6] in 1845. He was have been derived. Beginning with the seminal paper
2
uncertainaboutboundaryconditionsandincludedanun- of de Saint-Venant [12], many complex shapes were an-
3
0 knownslipvelocity,latercalculatedbyButcher[7],build- alyzed by conformal mapping, notably by Morris [22],
. ing on ideas of Navier [8]. The derivation without slip is Muskhelishvili [19], and P´olya and Szeg¨o [23]. Mor-
1
normally attributed to Hagenbach [9] and Jacobson [10] ris also applied her general solution for beam torsion to
0
6 in 1860, although a decade earlier, Stokes had already Poiseuille flow [22], and Tao [24, 25] later solved related
1 used the no-slip boundary condition in his famous paper problems in forced convection. In contrast to the more
: on viscous drag [11]. familiar case of Laplace’s equation, however, conformal
v
In the original paper on viscous flow, Stokes remarked mapping cannot be as easily applied to Poisson’s equa-
i
X that it is “extremely easy” to derive the velocity profile tion, since it is not conformally invariant [26].
r in a circular capillary [6], but he surely appreciated the Despite this extensive literature, it is worth revisiting
a
challengesposedbyothergeometries. Shapedependence the mathematics of uniform flow in various modern con-
was first analyzed by de Saint-Venant [12] in 1855 in the texts, such as microfluidics [27, 28], transport in porous
seeminglydifferentcontextoftorsionofelasticbeams. In media [29], stochastic processes [30, 31], chemical re-
1871, Boussinesq [13] recognized that de Saint-Venant’s actions [32], biological reactions [33, 34], architectural
theory of torsion is mathematically equivalent to Stokes’ structures [35], groundwater flow [36], river growth [37],
theory of Poiseuille flow. Ten years later Heaviside [14] and electrokinetic phenomena [38, 39], where approx-
noted the equivalence of beam torsion and the magnetic imate or numerical solutions have been used to treat
self-induction of an electrical wire, which was eventually complicated geometries. The initial motivation for this
recognized as another analogy for pipe flow [15], along workcamefromtheoreticalmicrofluidics[39]. Mortensen
with the “membrane analogy” identified by Prandtl [16] et al. [40] used boundary perturbation methods to ap-
in 1903. proximatethehydraulicresistanceofmicrochannelswith
2
(b)
(a)
(c) (d)
(e) (f)
FIG. 1. Exact solutions for unidirectional (Poiseuille) flow profiles in pipes of different cross sections, rendered as deformed
membranesorsoapbubbles. Thesamesolutionsarederivedandcontour-plottedbelow: (a)ellipse(Fig. 3(a)),(b)equilateral
triangle(Fig.6),(c)roundedpentagon(Fig. 7(a)),(d)seven-prongedstar(Fig. 7)(e),(e)off-centercoaxialpipes(Fig.9(b)),
and (f) grooved parallel plates (Fig. 11(c)).
near-circular cross sections, and these approximations II. PHYSICAL ANALOGIES
closely resemble exact solutions derived by de Saint-
Venant [12]. Similarly, eigenfunction expansions, such
Poisson’s equation in two dimensions describes a re-
as the well known Fourier series for a rectangular cross
markablevarietyofphysicalphenomena. Exactsolutions
section[39],canalsobeviewedassumofexactsolutions,
can be traced back (at least) to the seminal 1855 pa-
only for slightly different geometries.
per of de Saint-Venant [12] on the elastic deformation of
Inthisarticle,wedevelopthemathematicsofunidirec- straight,prismaticbeamsundertorsion. Asnotedabove,
tional flow through a variety of examples, both old and the mathematical equivalence of beam torsion and pipe
new. Although formal integral solutions can be derived flow was first recognized by Boussinesq [13] and later
foranygeometry, wefocusontheconstructionofspecial extended to convective heat transfer (Marco and Han,
geometriesthatapproximatedomainsofinterestandad- 1955 [41]). More easily visualized analogies are provided
mit simple exact solutions. We begin by reviewing the by the deflection of elastic membranes (Prandtl, 1903
history of this problem in fluid and solid mechanics and [16])orsoapbubbles(Griffith,1917[42]),andbythepo-
discussing many additional applications. tential profile of electrically conducting sheets (Waner,
3
u(x,y) ψ(x,y) h(x,y)2 φ(x,y) τ (x,y) h(x,y)
zz
P
θz
F
γ
ω
z I
κ
µ −∇p G Δp
(1) (2) (3) (4) (5) (6)
Fluid Mechanics Solid Mechanics
T(x,y) c(x,y) τ(x,y) A(x,y) φ(x,y) u(x,y) J (x,y)
s
!
B
+ ++ + ++ + + ++ + ++ + + ++ + ++ +
D ρe J
σ J kF D Rv µm J ε σ µ E ρe µ ∇pρe
(7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17)
Heat and Mass Transfer Stochas7c Processes Electromagne7sm Electrokine7c Phenomena
FIG.2. Seventeenanalogousphysicalphenomenafromsixbroadfields,alldescribedbyPoisson’sequationintwodimensions.
Fluid mechanics: (1) Poiseuille flow in a pipe, (2) circulating flow in tube of constant vorticity, and (3) groundwater flow fed
by precipitation. Solid Mechanics: (4) torsion or (5) bending of an elastic beam, and (6) deflection of a membrane, meniscus
or soap bubble; Heat and mass transfer: (7) resistive heating of an electrical wire, (8) viscous dissipation in pipe flow and (9)
reaction-diffusioninacatalystrod;Stochasticprocesses: (10)firstpassagetimeintwodimensions,(11)thechainlengthprofile
of a grafted polymer in a tube, and (12) the mean rate of a diffusion-controlled reaction; Electromagnetism: (13) vector
potential for magnetic induction in a shielded electrical wire, and the electrostatic potential in (14) a charged cylinder or (15)
a conducting sheet or porous electrode; Electrokinetic phenomena: (16) electro-osmotic flow and (17) streaming current in a
pore or nanochannel.
1953 [43]). These mathematical insights allowed Pois- free surfaces, nˆ·∇u =0, such as the upper free surface
z
son’sequationtobesolvedexperimentally, longbeforeit in simple models of rivers and glaciers, the application
could be solved numerically on a computer. for which the c.g.s. viscosity unit “Poise” was proposed
In this section, we survey the literature and expand to honor Poiseuille [4, 44].
the number of physical analogies to seventeen, sketched Thesamemathematicalproblemalsodescribesthecir-
in Fig. 2. culating flow in a tube of constant vorticity ω =∇×u
z
(Fig. 2(2)). As noted by Greenhill [45], the stream
function u = ψ(x,y), which defines the velocity field,
A. Fluid Mechanics u = −∇×ψzˆ, satisfies Eq. (1) with k = ω . In this
z
case, the boundary has perfect slip. Circulating flows
Theprimarymotivationofthisworkistostudyunidi- in acute-angle corners [46] have the same scaling as uni-
rectional(i.e. fullydeveloped,laminar)flowinastraight directional corner flows [47, 48] discussed below.
pipe of arbitrary cross section (Fig. 2(1)). The axial Here,wenoteanotheranalogyofpipeflowwithforced
velocity profile u (x,y) satisfies Eq. (1) with k = G/µ, gravitycurrentsinporousmedia[49],specificallyground-
z
where G is (minus) the axial pressure gradient and µ water flow [36]. In the Dupuit [50] approximation, the
the viscosity. On the boundary, we assume either no height h(x,y) of groundwater spreading over an flat im-
slip, u = 0 (Poiseuille flow), or, more generally, a pre- permeable rock through a porous soil of hydraulic per-
scribed velocity distribution, u = U(x,y), for moving meability κ, fed by a mean precipitation rate P, satisfies
z
walls (Couette flow). The model can also be extended Eq.(1)withu(x,y)=h(x,y)2 andk =2P/κ(Fig. 2(3)).
for hydrodynamic slip, b(nˆ ·∇u ) = u −U, or stress- Dirichlet boundary conditions, u=0, correspond to free
z z
4
drainage out of the soil, e.g. into a river network, and k =J2/σk . Boundary conditions could specify isother-
F
Neumann conditions represent an impermeable bound- mal (T=constant) or insulating (nˆ ·∇T = 0) surfaces.
aries or symmetry lines [37]. In mass transfer, the same problem could describe the
concentration profile c(x,y) of a chemical of diffusivity
D produced by a uniform volumetric reaction rate R in
v
B. Solid Mechanics a catalyst rod (Fig. 2(9)) [60].
Exact solutions of Poisson’s equation in two dimen-
D. Stochastic Processes
sions have an even longer history in solid mechanics
[19,21]. deSaint-Venant[12]firstformulatedandsolved
the general problem of torsion of a prismatic beam (Fig. Here, we identify some further analogies of unidirec-
2(4)). The beam is clamped on one end and twisted uni- tional flow in stochastic processes, or random walks,
formly with angle θz. The stress function, u = φ(x,y), which provide the microscopic basis for continuum mod-
of Prandtl [16] satisfies Eq. (1) with k = −2Gθ, where els of heat and mass transfer. The mean first passage
G is the shear modulus, and vanishes on the boundary. timeu=τ(x,y)ofaWienerprocess(i.e. arandomwalk
de Saint-Venant [51] also analyzed a prismatic beam un- with infinitesimal, independent, identically distributed
der pure bending, clamped at one end, with a transverse displacements with bounded variance) having diffusiv-
force F applied at the other end (Fig. 2(5)). The ax- ity D from a point in a two dimensional domain to its
ial normal stress, u = τ (x,y), satisfies Eq. (1) with boundary satisfies Eq. (1) with k = D−1 and an ab-
zz
k =F/I(1+ν), where I is the bending moment of iner- sorbing boundary condition u = 0 on the target bound-
tia and ν is Poisson’s ratio. ary [30, 31] (Fig. 2(10)). Reflecting boundaries can
Prandtl [16] also introduced the ‘membrane analogy’ also be included, with Neumann boundary conditions.
for these problems of beam elasticity (Fig. 2(6)). Equa- In finance, first passage processes arise in the pricing
tion (1) with k = ∆p/γ now describes the height of a of American options or other derivative securities [61],
membrane, u=h(x,y), under small elastic deflection by whereaboundedplanardomainwoulddescribetherange
a uniform pressure difference ∆p, and resisted by a con- of two underlying assets values where it is not yet prof-
stantsurfacetensionγ. Thesameanalogyalsoappliesto itable to exercise the option. There are also analogies in
the interface between two immiscible fluids [42], as dis- polymer physics [62]. The same random-walk problem
cussed in one of the earliest papers of G. I. Taylor. In describes the mean length of a polymer, fluctuating in-
equilibrium, the interface has constant mean curvature, side a tube or disk, from any interior point to a point
which describes many situations, such as a liquid menis- where it is attached to the wall (Fig. 2(11)).
cus [52], a soap bubble with a pinned contact line [53],
or the Cassie-Baxter state of a textured superhydropho-
bic surface [54]. For small deflections, |∇h| (cid:28) 1, the E. Diffusion-Controlled Reactions
Young-Laplace equation,
The first passage time (or “escape time” or “survival
(cid:32) (cid:33)
∇h time”) describes many phenomena in science and en-
−∆p=γ ∇·nˆ =γ ∇· ≈γ ∇2h (2)
(cid:112) gineering, such as the mean reaction time in Smolu-
1+|∇h|2
chowski’s theory of diffusion-controlled homogeneous re-
actions[30,32,63]. Thetwo-dimensionalcaseconsidered
again reduces to Eq. (1) with k =∆p/γ.
herecoulddescribeadsorbedreactantsonasurfaceasin
heterogeneous catalysis [60, 64] (Fig. 2(12)). Many ap-
plicationsariseinbiology,suchasligandbindingoncells
C. Heat and Mass Transfer or ligand accumulation in cell culture assays [33, 34].
The connection between Poisson’s equation and the rate
The same problem also naturally arises in transport of diffusion-controlled reactions was perhaps first noted
phenomena, such as heat transfer in a pipe heated uni- by Reck and Prager [65] in 1965. A general statistical
formly by viscous dissipation or another constant heat homogenization theory based on the first passage time
source [41, 55, 56]. A peculiarity of pipe flow is that the wasformulatedbyRubinsteinandTorquatofordiffusion-
mathematical problem for the fluid velocity is equivalent controlled reactions[66] and viscous flow [67], leading
to that of the temperature profile, u = T(x,y), gener- to many mathematical results on reaction-diffusion pro-
ated by viscous dissipation (Fig. 2(7)), where k = g/k cesses in porous media [29, 68].
F
in Eq. (1) and k is the heat conductivity in Fourier’s In1990,Torquato[69]unifiedstochasticprocesseswith
F
law [17, 41, 57, 58]. viscous flow in arbitrary three-dimensional geometries, a
Another important application is the resistive heating sweeping generalization that has not yet been appreci-
ofastraightwireofconductivityσandFouriercoefficient ated for many of the other analogies discussed in this
k passing a uniform current density J [59] (Fig. 2(8)). paper. He established a rigorous link between the mean
F
Thesteadytemperatureu=T(x,y)satisfiesEq.(1)with survivaltimefordiffusion-controlledreactionsinaporous
5
medium and the Darcy fluid permeability tensor, k for visualize the profile of elastic deformation of beams in
D
Stokes flow in the same porous medium, which relates torsion [43].
the mean velocity to the mean pressure gradient,
k
u=− D∇p. (3) H. Electrochemistry
µ
Here, we note that same mathematical problem also
In particular, Torquato proved the general inequality,
describes some problems in electrochemistry. The
k ≤τφ DI (4) steady-stateelectrostaticpotentialprofileφ(x,y)satisfies
D 1
Eq. (1) in a planar porous electrode or electrochromic
(i.e. τφ DI−k ispositivesemi-definite),whereφ isthe glass [73, 74] (Fig. 2(15)), where Dirichlet boundary
1 D 1
porosityandIistheidentitymatrix,andheshowedthat conditions apply at the current collectors and Neumann
equality holds, k =τφ D, in the limit of unidirectional boundary conditions at the separator. This is another
D 1
flowinstraightparallelpores(asalsonotedabove). This variation on the conducting sheet analogy, where again
analogy was exploited by Hunt et al. [15] for the prob- k =J/σbutnowσisthemacroscopicconductivityofthe
abilistic computation of Poiseuille flow fields by Monte phase with rate limiting transport (electronic or ionic)
Carlo simulations of random walks, similar to the algo- and J is the Faradaic reaction rate, assumed to be uni-
rithmofTorquatoandKim[70]tocalculatetheeffective form for a perfect analogy.
conductivity of porous media [71, 72]. The stochastic al-
gorithm enables the efficient approximation of the flow
field in pipes with complicated cross sections, including I. Electrokinetics
rough, fractal shapes [15].
We also note that Poisson’s equation arises in three
different ways in the theory of electrokinetic phenomena
F. Magnetostatics
[38, 75]. Besides determining the electrostatic potential
profileφ(x,y)fromthechargedensityρ (i.e. electrostat-
e
Heaviside[14]discoveredoneoftheearliestanalogiesof ics, above), it also determines the electro-osmotic flow
beamtorsion,tothemagneticself-inductionofashielded u = u (x,y) in response to an axial electric field with
E
electrical wire (Fig. 2(13)). In modern terminology [18], k = ρ E/µ (Fig. 2(16)), as well as the streaming cur-
e
Heaviside expressed the magnetic induction in terms of rent u = j (x,y) = ρ u (x,y) in response to a pressure-
s e p
the vector potential, B = ∇ × A, chose the Coulomb driven flow (Fig. 2(17)). A perfect analogy with uni-
gauge, ∇·A = 0, and considered uniform current den- form flow requires a constant charge density ρ , which
e
sity J in a wire of constant magnetic permeability µm, approximates the diffuse charge profile in a nanochan-
shielded by a perfectly conducting metal sheath. The nel with uniform surface charge and thick, overlapping
magnetic induction circulates around the current in the double layers. In the absence of flow, the linearized
cross section, and the vector potential plays the role of potential profile, satisfying the Debye-Huckel equation
the stream function in a vortex tube (Fig. 2(2)). The foranydouble-layerthickness,hasbeenanalyzedbyDu-
axial vector potential amplitude u = A(x,y) satisfies plantier [76] for different two-dimensional shapes, also
Eq. (1) with k = µmJ and Dirichlet boundary con- taking advantage of conformal invariance and connec-
ditions. Heaviside exploited this analogy to apply de tions with random walk theory noted above. At the
Saint-Venant’s results for torsion to the self-induction of end of this article, we shall return to the general situa-
shieldedwiresofdifferentshapes,butapparentlydidnot tion of non-uniform charge density, ρ (x,y), in the full
e
recognize the analogy with Poiseuille flow [14]. nonlinear problem with electro-osmotic flow.
G. Electrostatics III. PARTICULAR SOLUTIONS FOR CONIC
SECTION DOMAINS
Of course, the eponymous application of Poisson’s
equation is in electrostatics, e.g. for a two dimensional In all of these applications, there is a subtle physical
cylinder (wire, cavity, etc.) of constant charge density compatibility constraint on the solutions of Eq. (1),
ρ and permittivity ε (Fig. 2(14)). The electrostatic po-
e
tential u = φ(x,y) satisfies Eq. (1) with k = ρe/ε with uk >0 (5)
a prescribed potential profile on the boundary. Another
variation on this problem with k = J/σ is the potential which requires that u and k have the same sign at each
profile of a conducting sheet of conductivity σ, cut to pointinthedomain. Here,wewillassumek >0,sothat
a certain shape with bus bars at the edges, sustaining physical solutions are positive, u>0. In the case of uni-
a uniform normal current density J (Fig. 2(15)). This directionalflow,thefluidvelocitymustbedirecteddown
problem has been used as an analogy to experimentally the pressure gradient; for resistive heating, a heat source
6
1.2 1
0.8
0.5
0.4
-2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2 -0.5 0 0.5 1 1.5 2 2.5 3
-0.4
-0.5
-0.8
(a) -1.2 (b) -1
FIG.3. Dimensionlessunidirectionalflowin(a)anellipticalpipe(u˜=1−(x˜/2)2−y˜2)and(b)aparabolicgroove(u˜=x˜−y˜2).
mustleadtoariseintemperature;inelectrokinetics,the Unbounded conic section domains are also included in
flow of positive charge is in the direction of the electric Eq. (6). The simple solution
field; in soap bubbles, the Laplace pressure is larger on
k
the concave side of the interface; etc. Since the litera- u= (x−κy2) (11)
2κ
ture has mostly focused on bounded domains, such as
pipe cross sections, this constraint has not been empha- satisfies no-slip on a parabola (x = κy2). The flow do-
sized, but it becomes important when selecting physical main (u>0) is the region inside the parabola where the
solutions in unbounded domains, such as conic sections. walls provide enough viscous drag to balance the pres-
A particular solution of Equation (1) is the quadratic sure gradient and reach a steady state, as shown in Fig.
form, 3(b).
Another simple solution
(cid:88)2 (cid:88)2 (cid:20)(cid:16)x(cid:17)2 (cid:16)y(cid:17)2(cid:21)
u(x,y)= Amnxmyn (6) u=A − +B, (12)
a b
m=0n=0
withA=k/(2(a−2−b−2))anda(cid:54)=b,satisfiesno-slipona
whichsatisfiesnosliponaconicsectiondefinedbyu=0,
hyperbola(B (cid:54)=0)orwedge(B =0). Fora<bandB =
subject to the constraints
0 the flow domain lies inside a wedge of acute opening
angle 2tan−1(a/b) < π along the x axis with |x/a| >
2(A +A )=−k and A =A =A =0. (7) 2
20 02 22 12 21 |y/b|, as shown in Fig. 4(a). For a > b and B = 0, the
flow domain is again in an acute-angle wedge, but now
The allowable domains satisfying Eq. (5) shed light on
alongtheyaxiswith|x/a|<|y/b|. Physically,thereisno
the physics of unidirectional flow, as well as the other
allowable solution for an obtuse angle wedge because it
applications sketched in Figure 2.
cannotexertenoughviscousdragtobalancethepressure
The most important and well known case is that of an
gradient. Mathematically, the local similarity solution
elliptical cross section ((x/a)2+(y/b)2 < 1), introduced
only exists for an acute corner [47].
by de Saint-Venant [12]. The solution is
For B > 0 in Eq. (12), the flow domain lies between
(cid:26) (cid:20)(cid:16)x(cid:17)2 (cid:16)y(cid:17)2(cid:21)(cid:27) two branches of a hyperbola, as shown in Fig. 4(b). The
u(x,y)=K 1− + (8) velocity profile is a saddle surface, which is a growing
a b
parabolaalongthecenterlinebetweenthehyperbolicsur-
faces, and a decaying parabola along the line of closest
where K = k/(2(a−2 +b−2)). An example is shown in
approach. Thissolutiondemonstratesthegeneralprinci-
Fig. 3(a). The limit a → ∞ describes pressure-driven
ple that viscous drag in unidirectional flow is dominated
flow between parallel plates (|y|<b),
by the narrowest constriction, where the transverse flow
profile is approximately parabolic.
k
u(x,y)= (b2−y2) (9) Thehyperbolic(orwedge)particularsolutionscanalso
2 be written in polar coordinates [46–48],
and the case a = b = R corresponds to the Hagen- kr2 (cid:18)cos(2θ) (cid:19)
u(r,θ)= −1 +c. (13)
Poiseuille flow in a circular pipe (r <R),
4 cos(2α)
u(r)= k (cid:0)R2−r2(cid:1) (10) Tschaelems aaxsimuu∼mxv2e.locTithyealtaregaecrhsrcaadliinugs iesxaploonnegnθt=th0ananind
4
7
1.5
1.6
1
0.8
0.5
-3.2 -2.4 -1.6 -0.8 0 0.8 1.6 2.4 3.2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3
-0.5
-0.8
-1
-1.6
-1.5
(a) (b)
FIG. 4. Unidirectional flow in (a) a narrow wedge (u˜=(x˜2−4y˜2)/6) and (b) a hyperbolic constriction (u˜=1−x2+y2/2).
the case of a parabolic corner, u∼x, reflects the weaker tion vectors {nˆ } and distances {c >0} from the origin.
i i
geometrical confinement of the flow in the wedge. The InspiredbythecasesN =2,3, letusseeksolutionofthe
physical constraint, Eq. (5), implies α < π. In the case form,
4
of a wedge (c=0), we see again that a physical solution
is only possible for acute opening angles, 2α< π. (cid:89)N
2 u=a nˆ ·((cid:126)x−c nˆ ) (14)
i i i
i=1
IV. EQUILATERAL DOMAINS which is proportional to the product of all the distances
from the point (cid:126)x to the N sides and thus automatically
satisfies the no-slip Dirichlet boundary conditions.
A. Geometrical ansatz
As beautifully demonstrated by Needham [77], solu-
B. Equilateral pipe
tionstoLaplace’sequationintwodimensionshavesimple
geometricalinterpretations,inheritedfromtheiranalytic
First, let us show that for a bounded domain with
complex potentials. For example, the harmonic function
N = 3, Eq. (14) only solves Eq. (1) for the case of
intheupperhalfplanewithpiecewiseconstantboundary
an equilateral triangle:
conditions is a weighted sum of angles subtended at the
discontinuities, and its harmonic conjugate is the log- k (cid:34)1(cid:18) 1(cid:19) √3(cid:32) √3(cid:33)(cid:35)
arithm of a product of distances to the discontinuities u= (x−c) x+ − y−
3c 2 2 2 2
raisedtocertainpowers. Theseconstructionsarerelated
to the Schwartz-Christoffel mapping of the upper half (cid:34)1(cid:18) 1(cid:19) √3(cid:32) √3(cid:33)(cid:35)
plane to a polygon. Even more remarkable is Schwarz’ x+ + y+
2 2 2 2
geometrical interpretation of Poisson’s integral formula,
whichsolvestheDirichletproblemforLaplace’sequation = k (x−c)(cid:2)(2x+4c)2−12y2(cid:3)
in a disk, by averaging the boundary data after circular 12c
inversion through an interior point [78]. This well known solution [24, 39, 79] is shown in Fig. 6.
Incontrast,solutionsofPoisson’sequationdonotseem
to have simple geometrical interpretations – with a no- Proof: Substituting (14) into (1), we obtain
table exception, the case of an equilateral triangular do-
(cid:88)(cid:88) (cid:89)
main. Textbooks on fluid mechanics leave it as an ex- ∇2u=a (nˆ ·nˆ ) ((nˆ ·(cid:126)x)−c )=−k (15)
i j l l
ercise to show that the velocity at an interior point is
i j(cid:54)=i l(cid:54)=i,j
proportional to the product of the distances to the three
sides [39, 79]. Below, we show that the same construc- Since k is constant, all terms involving (cid:126)x must vanish.
tionalsoappliestoanunboundedequilateralwedge. Itis For N =3:
interestingtonotethatPoiseuilleflowbetweenflatplates ∇2u=2a((cid:126)b·(cid:126)x+b )=−k
isalsoproportionaltotheproductofthedistancestothe 0
two sides. where
These cases, shown in Fig. 5, suggest that the same
b =c (nˆ ·nˆ )+c (nˆ ·nˆ )+c (nˆ ·nˆ )
geometrical ansatz might also apply to other situations. 0 1 2 3 2 1 3 3 1 2
Consider a polygonal domain of N sides having orienta- (cid:126)b=nˆ (nˆ ·nˆ )+nˆ (nˆ ·nˆ )+nˆ (nˆ ·nˆ )
1 2 3 2 1 3 3 1 2
8
(a) (b) (c)
R1 R2
R
1 R
2
u=RR R
R2 1 2 R1 3 u=R1R2R3
R
3 u=RR R
1 2 3
FIG. 5. Geometrical construction of special unidirectional flows, obeying Eq. (14). Famous solutions for parallel plates (a)
and equilateral pipes (b) are proportional to the product of the distances to the boundaries. The latter solution also holds for
equilateral wedges (c), where the third boundary lies beyond the apex of the wedge.
Without loss of generality, let nˆ = (1,0), nˆ = Proof: InordertosatisfyEq. (15)forall(cid:126)xinthedomain
1 2
(cosθ ,sinθ ), nˆ = (cosθ ,−sinθ ). Setting (cid:126)b = (cid:126)0 (wherexandy varyindependently)thecoefficientsofall
2 2 3 3 3
yields terms am,nxmyn for m,n = 0,1,..,N − 2 must vanish
except for the constant term, which must satisfy a =
0,0
sinθ2cosθ2 =sinθ3cosθ3 −k. We must therefore satisfy (N − 2)2 equations by
sinθ sinθ =cosθ cosθ +cos2θ +cos2θ choosing parameters Eq. (14). There are N coefficients
2 3 2 3 2 3
c andN orientationanglesfortheunitvectorsnˆ ,which
Requiringθ ,θ ∈(0,π)impliesθ =θ =θandtan2θ = l i
2 3 2 3 are independent, except for overall rotation and dilation
3. and thus θ = π/3 or 2π/3. The only possible finite
(and rescaling of u). Solvability thus requires 2N −2 ≥
domain is an equilateral triangle, which can be centered √
(N −1)2, or N <1+ 7<4. (cid:3)
ontheoriginbychoosingθ =2π/3andc =c =c =c.
1 2 3
Finally, b =−k/a implies a=k/3c. (cid:3)
0
Therefore, while the ansatz (14) holds in general for
N = 2, it only works for the equilateral geometries for
C. Equilateral wedge N > 2, where the unique solution (up to translation,
rotation,ordilation)isgivenbyEq. (15). Evenisosceles
triangular domains have different solutions [80].
As shown in Fig. 6, the same solution (15) also holds
with u > 0 in an unbounded 60◦ wedge-shaped domain.
Thegeometricinterpretationisthesame(productofdis-
V. DOMAINS FOR GIVEN UNIDIRECTIONAL
tances to three surfaces of the equilateral triangle) only
FLOWS
now the third surface is hidden behind the apex of the
wedge, as shown in Fig. 5(c). Since it is the product of
A. General solution of Poisson’s equation
three distances, the velocity along the centerline of the
equilateral wedge has cubic scaling, u∝|x|3:
For any geometry and Dirichlet boundary condition,
k kx3 u = U(x,y), the solution to Poisson’s equation (1) for
u(x,0)= (x−c)(2x+4c)2 ∼ , as x→−∞. (16)
12c 3c unidirectional flow can be expressed as the sum of a
pressure-driven Poiseuille flow (particular solution) u
Below in Eq. (27), we shall explain why this scaling p
and a Couette shear flow (homogeneous solution):
differs from that of the hyperbolic wedge solution, u ∼
x2, derived above in Eq. (13). u(x,y)=u (x,y)+u (x,y) (17)
p s
where we can choose u to be any of the exact solutions
p
D. Uniqueness of the equilateral solution above, such as Hagen-Poiseille flow (10), or the general
conicsectionflow(6). Theshearflowisaharmonicfunc-
tion, which solves Laplace’s equation,
It is clear that the ansatz (14) cannot hold in gen-
eral. For example, for N = 4, the bi-parabolic form ∇2u =0 (18)
s
u ∝ (a2−x2)(b2−y2) violates Eq.(1). The ansatz also
fails for parabolic Hagen-Poiesuille flow for a circular with a Dirichlet boundary condition,
cross section or any other smoothly curved shape, which
u (x,y)=U(x,y)−u (x,y). (19)
can be viewed as a limit of infinitely many infinitesimal s p
sides. In fact, no solutions exist of the form (14) for whichisgenerallynon-constanteveninthecaseofnoslip,
N >3. U =0. Asusual,theharmonicfunctioncanbeexpressed
9
1.5
1
0.5
-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-0.5
-1
-1.5
FIG.6. ExactsolutionforPoiseuilleflowindomainswith60◦ anglesbetweenflatno-slipsurfaces. Center: Closedequilateral
triangle. Left: Unbounded60◦ wedge. ThesolutioninbothcasesisgivenbyEq.(14)forN =3inthedomainswhereu>0.
as real (or imaginary) part of an analytic function, the unidirectional flow by relaxing constraints on the precise
complex potential, u =ReΦ(z), where z =x+iy. shape of the boundary. By choosing any harmonic func-
s
In summary, the general solution of Eq. (1) has the tion, u , we can generate a new exact solution of the
s
form form, u = u +u , from any of the particular solutions
p s
above, where the no-slip boundary is defined implicitly
2 2
u(x,y)= (cid:88) (cid:88)A xmyn+Re Φ(x+iy) (20) by u=0.
mn
m=0n=0
The basic idea of adding a harmonic function to a
B. Deformed pipes
quadratic particular solution can be traced to de Saint-
Venant [12], and general formulae were perhaps first de-
Power-law complex potentials, Φ(z)∝zm, in Eq. (21)
veloped by Muskhelishvili in the 1920s [19, 21]. Using
yield Saint-Venant’s original solutions for m−sided
Hagen-Poiseuille flow (10) as the particular solution, we
polygonal domains with rounded edges [21]:
recover the representation of Tao [24],
k u˜=1−r˜2+amr˜msin(mθ) (23)
u(z,z)=− zz+Ψ(z)+Ψ(z) (21)
4
where we use polar coordinates, z =reiθ and define r˜=
where z = x−iy is the complex conjugate and 2Ψ = r/Randu˜=4u/kR2. Forexample,theno-slipboundary
(k/4)R2+Φ. Thisformofthesolutioncanalsobederived in the case m = 5 and a = 0.11 resembles a rounded
5
by integrating Poisson’s equation (1) after a change of pentagon,andthecasem=4anda =−0.21iarounded
4
variables [24, 56, 58] square, as shown in Fig. 7(a)-(b). By varying a , the
m
pipe cross section can have more pointed corners, like a
∂2u k
− = (22) star, as shown in Fig. 7(e) for m=7 and a =0.117.
7
∂z∂z 4
Interestingly, these exact solutions of de Saint-Venant
related to the complex gradient operator [26, 77]. This [12] resemble the approximate solutions for slightly dif-
solutionmayseemlessgeneral,butitisequivalenttoEq. ferentgeometriesobtainedbyMortensenet al.[40]using
(20). The bilinear terms in the sum are harmonic, and boundary perturbation methods. Such approximations
thequadratictermshavethesameformasEq. (21),since have a long history in fluid mechanics, also based on
2x2 =zz+Rez2 and2y2 =zz+Re(iz2). Nevertheless,as conformal mapping in two dimensions [81, 82]. These
illustratedbelow,itiseasiertostartwithotherparticular comparisons suggest that, rather than deriving approx-
solutions u in Eq. (20) when constructing solutions for imate solutions for a given geometry [39], it could be
p
domains that do not resemble pipes. advantageous to derive exact solutions for approximate
Finding the complex potential Φ(z) for non-constant geometries,guidedbythemanyexampleshereandinthe
Dirichlet boundary conditions in a given domain is pos- literature.
sibleintheory(SectionVI),butoftenchallenginginprac- A wide variety of other shapes can be created from
tice. However, we can easily produce new solutions for the general Laurent series, Φ(z) = (cid:80)∞ a zn, to ar-
n=−∞ n
10
1.6 1.6
1.2
0.8 0.8
0.4
-2.4 -2 -1.6 -1.2 -0.8 -0.4 0 0-.24.4 0.8 1-.12.6 1.6 -20.8 2.4 0 0.8 1.6 2.4
-0.4
(a) -0.8 (b) -0.8
1.6
-1.2
0.5
-1.6 -1.6
0.8
0.25
-2.4 -1.6 -0.8 0 0.8 1.6 -1 2.4 -0.75 -0.5 -0.25 0 0.25 0.5
-0.25
-0.8
(c) (d)
-0.5
-1.6
1.2
0.8
0.4
-2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2
-0.4
-0.8
(e) -1.2
FIG. 7. Exact solutions for unidirectional flow in deformed pipes: (a) a pentagonal pipe and (b) a square pipe given by
Eq.(23); (c)aroundedslitgivenbyEq.(24); (d)anasymmetric“flattened”pipe,givenbyEq. (25); and(e)aseven-pointed
star pipe, as well as seven smoothed wedge domains, given by Eq. (23), with parameters in the main text.