Exact solutions and physical analogies for unidirectional flows

Unidirectional flow is the simplest phenomenon of fluid mechanics. Its mathematical description, the Dirichlet problem for Poisson's equation in two dimensions with constant forcing, arises in many 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 17 physical analogies for unidirectional flow and describing their common mathematical structure. Besides classical analogies in fluid and solid mechanics, applications are discussed in stochastic processes (first passage in two dimensions), pattern formation (river growth by erosion), and electrokinetics (ion transport in nanochannels), which also involve Poisson's equation with nonconstant forcing. Methods are given to construct approximate geometries that admit exact solutions, by adding harmonic functions to quadratic forms or by truncating eigenfunction expansions. Exact solutions for given geometries are also derived by conformal mapping. We prove that the remarkable geometrical interpretation of Poiseuille flow in an equilateral triangular pipe (the product of the distances from an interior point to the sides) is only shared by parallel plates and unbounded equilateral wedges (with the third side hidden behind the apex). We also prove Onsager reciprocity for linear electrokinetic phenomena in straight pores of arbitrary shape and surface charge, based on the mathematics of unidirectional flow.

Figure 1

Exact solutions for unidirectional (Poiseuille) flow profiles in pipes of different cross sections, rendered as deformed membranes or soap bubbles. The same solutions are derived and contour plotted below: (a) ellipse [Fig. 3], (b) equilateral triangle (Fig. 6), (c) rounded pentagon [Fig. 7], (d) seven-pronged star [Fig. 7], (e) off-center coaxial pipes [Fig. 9], and (f) grooved parallel plates [Fig. 11].

Figure 2

Seventeen analogous physical phenomena from six broad fields, all described by Poisson's equation in two dimensions. Fluid mechanics: (a) Poiseuille flow in a pipe, (b) circulating flow in a tube of constant vorticity, and (c) groundwater flow fed by precipitation. Solid mechanics: (d) torsion or (e) bending of an elastic beam, and (f) deflection of a membrane, meniscus, or soap bubble. Heat and mass transfer: (g) resistive heating of an electrical wire, (h) viscous dissipation in pipe flow, and (i) reaction-diffusion process in a catalyst rod. Stochastic processes: (j) first-passage time in two dimensions, (k) the chain length profile of a grafted polymer in a tube, and (l) the mean rate of a diffusion-controlled reaction. Electromagnetism: (m) vector potential for magnetic induction in a shielded electrical wire, and the electrostatic potential in (n) a charged cylinder or (o) a conducting sheet or porous electrode. Electrokinetic phenomena: (p) electro-osmotic flow and (q) streaming current in a pore or nanochannel.

Figure 3

Dimensionless unidirectional flow in (a) an elliptical pipe [$\tilde{u}=1-{(\tilde{x}/2)}^2-{\tilde{y}}^2$] and (b) a parabolic groove ($\tilde{u}=\tilde{x}-{\tilde{y}}^2$).

Figure 4

Unidirectional flow in (a) a narrow wedge [$\tilde{u}=({\tilde{x}}^2-4{\tilde{y}}^2)/6$] and (b) a hyperbolic constriction ($\tilde{u}=1-x^2+y^2/2$).

Figure 5

Geometrical construction of special unidirectional flows, obeying Eq. (14). Famous solutions for (a) parallel plates and (b) equilateral pipes are proportional to the product of the distances to the boundaries. The latter solution also holds for (c) equilateral wedges, where the third boundary lies beyond the apex of the wedge.

Figure 6

Exact solution for Poiseuille flow in domains with ${60}^{\circ }$ angles between flat no-slip surfaces: center, closed equilateral triangle; left, unbounded ${60}^{\circ }$ wedge. The solution in both cases is given by Eq. (14) for $N=3$ in the domains where $u>0$.

Figure 7

Exact solutions for unidirectional flow in deformed pipes: (a) a pentagonal pipe; (b) a square pipe given by Eq. (23); (c) a rounded slit given by Eq. (24); (d) an asymmetric flattened pipe, given by Eq. (25); and (e) a seven-pointed star pipe, as well as seven smoothed wedge domains, given by Eq. (23), with parameters in the main text.

Figure 8

Geometry dependence of the centerline velocity scaling $u\sim x^p$ near a corner in unidirectional flow. Particular solutions grow faster (larger $p$) with decreasing confinement, as illustrated by (a) parallel plates with $p=0$, (b) parabolic corners with $p=1$, and (c) acute-angle wedges with $p=2$. No solutions exist for (d) obtuse angle wedges. For acute-angle wedges, homogeneous solutions instead grow faster with increasing confinement, as shown for opening angles (e) $\pi /2$ with $p=2$, (f) $\pi /3$ with $p=3$, and (g) $\pi /4$ with $p=4$.

Figure 9

Exact solutions for (a) coaxial circular pipes [Eq. (28)] as well as (b) off-center near-circular pipes, (c) a rounded square inside a circle, and (d) a single $\mathsf{C}$-shaped single pipe, given by various cases of Eq. (29) described in the text.

Figure 10

Exact solutions for flow between deformed parallel plates with a central bulge, given by Eq. (32) with $a=4$ and (a) $\gamma =2$ and (b) $\gamma =6$.

Figure 11

Exact solutions for flow between deformed parallel plates with a sequence of twists, given by Eq. (33) with $a=4, b=4$, and $\gamma =10$ for (a) one, (b) two, (c) four, and (d) six terms in Eq. (34).

Figure 12

Exact solutions for flows between periodically grooved plates, given by (35) with (a) $\sigma =0.3$ and (b) $\sigma =0.84$. These solutions in the lower half plane could also represent the free-surface flow of a river over a grooved bottom, or of a glacial ice cap over a mountain range.

Figure 13

Exact solutions for flows in microchannels with (a) slipping or (b) no-slip lower walls, obtained by adding constants to the solutions in (a) Fig. 11 and (b) Fig. 12.

Figure 14

Exact solutions for nearly rectangular corners with $M$ terms in Eq. (41).

Figure 15

Exact solutions for near rectangles with $M$ terms in Eq. (42).

Figure 16

(a) Exact solution to Poisson's equation for unidirectional flow between flat plates with a parallel half plate inserted at the center [Eqs. (47) and (48)]. (b) Taking a square root, the same solution also describes the height of the water table in a strip of land between two rivers or lakes, where another river grows by erosion down the center of the strip.

Figure 17

Unidirectional electrokinetic phenomena for a liquid electrolyte in a straight nanochannel or pore having irregular cross section and nonuniform surface charge density. In the absence of applied concentration gradients, the Onsager linear-response matrix in Eq. (61) relates two fluxes, the flow rate $Q$ and electrical current $I$, to two forces, the axial pressure gradient $G$ and electric field $E$.