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Diffusion in a fluid flow generated by a source at the apex of a wedge

We consider a particle diffusing inside a wedge with absorbing boundaries and driven by a radial flow of incompressible fluid generated by a source at the apex. The survival probability decays as (time)^{-b} with exponent depending on the opening angle of the wedge and the Reynolds number associated with the hydrodynamic flow. The computation of the decay exponent reduces to finding the ground state energy of the quantum particle in an infinitely deep potential well with shape determined by the radial flow velocity.


I. INTRODUCTION
The problem of survival of a Brownian particle inside a wedge with absorbing boundaries has been investigated by Sommerfeld [1] in the 19 th century. This and similar problems are encountered in various settings [2][3][4][5][6][7]. For instance, suppose we seek the probability that N onedimensional Brownian particles do not meet during the time interval (0, t). This is equivalent to the probability that a single Brownian particle remains confined to the conical region x 1 < x 2 < . . . < x N in R N . For three particles, the corresponding single Brownian particle remains inside a wedge formed by two intersecting planes (the angle of the wedge depends on the ratios of diffusion coefficients [2]). Other properties of Brownian particles often admit a similar interpretation in terms of a single Brownian particle confined to a conical region [8][9][10][11][12][13][14][15][16][17], and this conical region is a wedge in the case of three particles. The properties of diffusion inside a wedge with absorbing boundaries also explain the long-time kinetics of several one-dimensional reaction-diffusion processes [2,18,19].
In this paper we analyze the behavior of a Brownian particle that is advected by a flow of incompressible fluid generated by the source or sink at the apex of the wedge, and absorbed on the boundaries of the wedge. The velocity field is purely radial. The mathematical description of the processes in unidirectional flow fields is significantly simpler than in the case of generic flows. Therefore the phenomena in unidirectional flow fields have been investigated in numerous contexts such as microfluidics [20,21], transport in porous media [22], stochastic processes [2,23], biological reactions [24], electrokinetic phenomena [25], etc. Most analyses consider flows in the pipe [26][27][28] that are unidirectional and uniform, that is, independent on the longitudinal coordinate. The flow field in the wedge is unidirectional (radial) but non-uniform, albeit the r −1 variation with the distance from the apex is still sufficiently simple that helps in the analysis.
We consider the wedge with an opening angle 2α and absorbing boundaries. The particle eventually gets absorbed, and the probability that it has survived during time interval (0, t) has an algebraic t −β long-time tail. In the case of no-flow, Q = 0, the persistence exponent β = π/(4α) is well-known, see e.g. [2,18,19].
For the ideal incompressible fluid, the velocity field is v = Q/(2αr) where Q is the strength of the source. The exponent β in this situation is given by (Sec. II) where D is the diffusion constant. For the viscous incompressible fluid, the velocity remains radial, v = u(θ)/r. The Navier-Stokes equations are solvable in this situation [29][30][31][32]. The exact solution for the viscous flow in the wedge goes back to Jeffery [33] and Hamel [34]. The Jeffery-Hamel solution is tricky [30,[35][36][37] in the case of the source (Q > 0) when its validity is questionable for sufficiently large Reynolds numbers. Computing the decay exponent β is mathematically equivalent (Sec. III) to finding the ground state energy of the quantum particle in an infinitely deep potential well: U (±α) = ∞ and U (θ) proportional to u(θ) when |θ| < α. The analogy with Schrödinger equation describing stationary states of a quantum particle in a onedimensional potential is amusing, albeit we do not use it in computations. What is needed are the basic properties of the Jeffery-Hamel solution describing the viscous flow in the wedge. These properties are outlined in Sec. IV. In Secs. V-VI we employ perturbation techniques and deduce analytical predictions for β in the limiting cases of high and low Reynolds numbers. Three-dimensional analogs of the wedge problem, particularly the diffusionadvection in the jet flow, are briefly discussed in Sec. VII. Conclusions are presented in Sect. VIII.

II. IDEAL INCOMPRESSIBLE FLOWS IN WEDGES
At time t = 0, the particle is released inside the wedge. In polar coordinates (r, θ), the wedge is the region r ≥ 0 and |θ| ≤ α with 2α being the opening angle of the wedge. Let S(r, θ, t) be the probability that the particle released from (r, θ) has not touched the boundaries of the wedge during the time interval (0, t). This survival probability satisfies Here D is the diffusion constant and v the velocity field. The advection term (v · ∇)S is on the right-hand side because S(r, θ, t) depends on the initial position of the particle, and therefore Eq. (2) is the backward advectiondiffusion equation.
The velocity field of ideal incompressible fluid generated by a source of strength Q reads v(r) = Q 2αrr (3) In rotationally-symmetric situations with velocity field inversely proportional to the distance from the origin, advection can be absorbed into diffusion by an appropriate shift of the spatial dimension [38][39][40][41]. The advectiondiffusion equation (2) with velocity field (3) becomes where Pe = Q 2αD is the Péclet number. In a rotationallysymmetric situation in d dimensions, the diffusion equation reads Equations (4) and (5) coincide if we identify d = 2 + Pe. Thus the two-dimensional advection-diffusion process with a point sink or source of mass may be recast as a pure diffusion process in a fictitious space of dimension d = 2 + Pe. An exact analysis of the linear governing equation (5) subject to the initial condition S(r, θ, 0) = 1 and the absorbing boundary conditions S(r, ±α, t) = 0 is possible. Our chief goal, however, is the extracting of the large time behavior, and we shall take advantage of the useful feature of the survival probability in this limit, namely its algebraic decay: Substituting (6) into the advection-diffusion equation (4) and noting that the time derivative becomes negligible in the long-time limit we find that Φ(r, θ) satisfies The dependence of Φ(r, θ) on the radial coordinate can be determined using dimensional analysis. In principle, Φ(r, θ) = Φ(r, θ|D, Pe), so Φ depends on two dimensionfull quantities r and D. The dimension of Φ is T β where T denotes the dimension of time; this is obvious from (6) since the survival probability is dimensionless. The only variable with dimension T β which can be composed of r and D is (r 2 /D) β . Thus Plugging (8) into (7) we obtain ψ (θ) + 4β 2 + 2β Pe ψ(θ) = 0 (9) where prime denotes the derivative with respect to θ.
First passage characteristics of a Brownian particle advected by ideal incompressible flows have been investigated in several studies, particularly in two dimensions where one can use complex analysis, conformal mappings, the Wiener-Hopf technique, etc. (see, e.g., [42][43][44]). First passage characteristics of a Brownian particle advected by viscous incompressible flows can be investigated analytically in a few situations since exact solutions of the Navier-Stokes equations are rare [29][30][31][32]. Most of such exact solutions are unidirectional flows [27,31,32]. The viscous flow in the wedge is one such solvable case with purely radial flow.

III. VISCOUS INCOMPRESSIBLE FLOWS IN WEDGES: FIRST PASSAGE CHARACTERISTICS
In the case of incompressible viscous fluid, the velocity field generated by the source at the apex of the wedge is radial and inversely proportional to the distance from the apex. The chief distinction from the ideal flow is that the velocity now depends on θ. The incompressible viscous flow in the wedge was first studied by Jeffery [33] and Hamel [34]. This flow represents a rare exact solution of the Navier-Stokes equations of the incompressible viscous fluid. The solvability of the Navier-Stokes equations is the consequence of the uni-direction nature of the flow and simple dependence on the radial coordinate: The function u(θ) is dimensionless since we have used the kinematic viscosity ν as a pre-factor. With velocity field (19), the advection-diffusion equation (2) becomes where σ = ν/D is the Prandtl number.
Using the ansatz (6) we find that Φ(r, θ) satisfies Seeking again the solution in the form (8) we arrive at One can think about this linear ordinary differential equation as a Schrödinger equation describing stationary states of a particle in a one-dimensional potential [45]. Finding the exponent β constitutes a (non-linear) eigenvalue problem. The solution must be positive inside the wedge, so we are seeking the ground state of a quantum particle in an infinitely deep potential well −α < θ < α. The potential is where E = 4β 2 is the energy of the ground state. Inside the well, the potential is − √ E σu(θ). The function u(θ) is expressible in terms of elliptic integrals [33,34]. The potential is also proportional to √ E and it then determines the energy E of the ground state, so we effectively have a non-linear eigenvalue problem. Finding a ground state is analytically impossible due to the complicated nature of u(θ). In the limiting situations of high and low Reynolds numbers, one can employ asymptotic methods to obtain perturbative results as we show in Secs. V-VI.

IV. JEFFERY-HAMEL SOLUTION
Here we outline a few properties of the Jeffery-Hamel viscous flow necessary for the analysis of advectiondiffusion in the wedge. The velocity field (19) has a single radial component, so it manifestly satisfies the continuity equation for the incompressible fluid. Navier-Stokes equations [29,30] for purely radial two-dimensional flows. (The density of an incompressible fluid is constant, we have set it to unity without loss of generality.) Using (19), one re-writes (26b) as Substituting (19) and (27) into (26a) we obtain The left-hand side of (28) depends on r, while the righthand side depends on θ. Hence both sides must be equal to the same constant. Multiplying by 2u and integrating one obtains The boundary conditions are The solution of the boundary-value problem (39c)-(30b) can be written in terms of elliptic integrals. These results were already known to Jeffery [33] and Hamel [34]. The problem is more rich [30,[35][36][37] than it was initially believed. For instance, depending on α, the Reynolds number Re = |Q|/ν and the sign of Q there can be situations with infinitely many solutions; see [46,47] for the classification of solutions. Simpler behaviors occur if the volume flux is negative, that is, the flow is generated by a sink; flows in a wedge generated by sinks are also known as flows in converging channels. A convergent symmetric flow, that is a solution with everywhere negative u(θ) satisfying u(θ) = u(−θ), exists for any Q < 0 if α < π 2 . This has been established in [30,37] for convergent flows with arbitrary Reynolds numbers. For small Reynolds numbers, |Q|/ν 1, symmetric solutions with everywhere negative (when Q < 0) or positive (when Q > 0) velocity exist for all α < α * = 2.2467 . . ., see Sec. VI for details.
If Q > 0, the symmetric solutions with everywhere positive velocity do not exist when the Reynolds number Re = Q/ν is sufficiently large, although there may be symmetric solutions involving regions of inward and outward flow [37,46,47]. The stability of some of these solutions has been studied, see [48][49][50]. In the physically interesting case of large Reynolds numbers, Re = |Q|/ν 1, the main results can be summarized as follows: Let us now analyze diffusion in convergent channels at large Reynolds numbers.

V. CONVERGENT FLOWS AT HIGH REYNOLDS NUMBERS
For sink flows at high Reynolds numbers (Q < 0 and Re 1), the velocity field is close to the potential flow (3), so the exact expression (12) for the decay exponent in the ideal (non-viscous) fluid may provide a good approximation. Re-writing (12) as one deduces when Re 1. The leading term in (33) is asymptotically exact, but the sub-leading correction decaying as (Re) −1 is erroneous. The asymptotically exact sub-leading correction is an increasing function of the Reynolds number that scales as (Re) 1/2 as we show below. More precisely We derive (34) using perturbation techniques [51]. We need an expression for u(θ) when Re 1. An exact solution of the boundary-value problem (39c)-(30b) simplifies in the Re → ∞ limit: where The expression (35) for the flow field is uniformly valid in the entire |θ| < α region and it reveals the presence of the boundary layers near the walls. The thickness of these boundary layers is of the order of B −1 . Using (35) we compute Q = ν α −α dθ u(θ) to yield from which Due to the symmetry, u(θ) = u(−θ), it suffices to consider the half wedge: 0 ≤ θ ≤ α. In the outer region the function u(θ) given by (35) is independent on θ in the leading order: Indeed, the arguments of two hyperbolic tangents in (35) diverge as Re → ∞. The argument of the third hyperbolic tangent is also large, as B(α − θ) 1 in the outer layer. Thus we obtain (39b) in the leading order. A more precise approximation of (35) is showing that the deviation from u = −2B 2 is exponentially small as Θ 1 in the outer layer. Inside the boundary layer the velocity varies according to In the outer region (39a), the governing Eq. (22) simplifies to ψ + 4β 2 − 4βσB 2 ψ = 0, from which In the inner region (40a), the governing Eq. (22) becomes where Ψ(Θ) ≡ ψ(θ). The boundary condition ψ| θ=α = 0 yields Equation (33) implies that the term in the square brackets in Eq. (42a) scales as Re when the Reynolds number is large. This suggests to apply WKB techniques [51].
The WKB solution to (42a)-(42b) reads This solution can be re-written as In (44b) we used again the shorthand notation b defined in Eq. (41) and we also replaced the upper limit Θ by ∞ since the matching the inner solution (44a) with the outer solution (41) is made when Θ 1. Massaging the integral in (44b) we obtain where we made the transformation x → y = tanh x and used tanh φ = 2/3. We do not display an exact cumbersome expression for the last integral and only show the asymptotically exact result. To derive it we notice that b = O(1) in the Re → ∞ limit as we will confirm a posteriori. Hence we neglect b and arrive at with
Finally we show how to determine b using (47): Thus indeed b = O(1) in the Re → ∞ limit.

VI. ADVECTION-DIFFUSION AT LOW REYNOLDS NUMBERS
In the low Reynolds number limit the inertial terms in the Navier-Stokes equations are omitted. In the present situation, we drop the term v∂ r v on the left-hand side of (26a). Instead of (29) we obtain u + 4u = const (48) A solution of Eq. (48) satisfying the boundary conditions (30b) reads where the amplitude was fixed by (31). The velocity has the same sign as Q inside the wedge when α ≤ π 2 , see Fig. 2. This is no longer true when π 2 < α < α * , with α * defined by by Eq. (50) This root is α * ≈ 2.2467047, or α * ≈ 128.727 • . The solution (49) explodes at α = α * , and it is well-defined only when α < α * . Thus the Jeffery-Hamel solutions have rich structure even in the low Reynolds number limit.
A. Wedge with α = π 4 As a concrete example, consider the wedge with right opening angle, 2α = π 2 . The dependence of the velocity on θ is particularly simple in this case: The governing equation (22) turns into an ordinary differential equation known as the Mathieu equation. Here we shortly write We seek a symmetric solution, ψ(θ) = ψ(−θ), of Eq. (52) which is positive for all |θ| < π 4 and vanishes on the boundaries: ψ(±π/4) = 0. In principle, one can solve the problem analytically by using Mathieu functions which arise in numerous problems, see e.g. [52,53]. However, the velocity field (51) is applicable only at small Reynolds numbers, i.e., | | 1, and it suffices to determine a perturbative solution of (52). The form of the unperturbed solution, ψ 0 = cos(2θ) and β 0 = 1, suggests to seek the perturbative solution in the form Substituting (54a)-(54b) into (52) we obtain The symmetric solution to this equation reads where we omitted C cos(2θ) term which can be absorbed into the unperturbed solution. The boundary condition ψ 1 (±π/4) = 0 fixes the amplitude A = −1/(3π). Thus the decay exponent is Recalling (53), we thus have In the general case of arbitrary opening angle in the range α ∈ (0, α * ), the governing equation (22) with velocity field (49) is again the Mathieu equation with a slightly modified small parameter We seek the perturbative solution in the form The perturbation ψ 1 (θ) now obeys The boundary condition ψ 1 (±α) = 0 fixes the amplitude A. The decay exponent becomes β = π 4α + 16α 2α cos(2α) + sin(2α) (2α/π) 2 − 1 + . . . (64) or equivalently

C. Mean exit time
The governing equation (25) for the mean exit time becomes with given by (60). Plugging the perturbative solution into (66) we find The boundary conditions are τ 0 (±α) = 0, τ 1 (±α) = 0 (69) Solving for τ 0 and then for τ 1 we obtain τ 0 = cos(2θ) − cos(2α) 4 cos(2α) Thus in the Stokes regime a physically sensible solution for the mean exit time is only possible for acute opening angles 2α < π 2 . The mean exit time in this situation reads with τ 0 , τ 1 given by (70) and given by (60). Interestingly, there are a few exact viscous similarity solutions in the wedge that also exist only for acute opening angles [54,55].

D. Wide wedges: α ≥ α *
The singularity of the velocity at α = α * has been noted by Fraenkel [46]. The same angular dependence as in Eq. (49) has appeared in several problems concerning wedges, e.g., in a problem [56] related to elastic wedges and in fluid dynamics problems [55,57]. A possible resolution of the singularity at α = α * relies on a more physical realization of the Jeffery-Hamel flow. Indeed, we have assumed that the size of the input region is equal to zero. We have used the boundary condition v| θ=±α = 0 (72) for the radial velocity and the boundary condition for the tangential velocity. To ensure that the flux has a non-zero strength Q and emerges from an apex one must rely on generalized functions (distributions). Physically, the size of the input region is finite. Intriguingly, relying on this realistic property, one can overcome the singularity of the velocity at α = α * . To model the finiteness of the input region, it proves convenient [55] to replace the boundary condition (73) by The flux is thus introduced through the boundaries in the input region r < a near the apex. The total flux is The Jeffery-Hamel problem is recovered when the flux is introduced in the tiny region, more precisely in the limit a → 0 and ω → ∞ with Q = ωa 2 constant. If α < α * , the flow field far away from the input region (r a) is the Jeffery-Hamel solution (49) in the leading order. The stream function defining the radial and tangential velocity components via v = r −1 ∂ θ ψ, w = −∂ r ψ is given by in the leading order. The details of the input region, viz. the parameters a and ω, do not affect the behavior. For α > α * , however, the leading term is different, it depends on the details of the input region even far away from it. The stream function is [55] in the leading order. Here p = p(α) is the proper root of There are many roots of W (p) = 0, real and complex [55]; the proper root, p = p(α), is a decreasing function of angle in the α * < α < π range, with p(α * ) = 0 and p(π) = − 1 2 . The dependence of the stream function on the angular coordinate is [55] The details of the hydrodynamic solution, such as the precise form of the angular dependence, Eq. (79), do not affect the ultimate fate of the diffusing particle. The only relevant features are the sign of the input Q and the negativity of p(α) when in the α * < α < π range. For source flows, Q > 0, the radial displacement grows as t 1/(p+2) , while the tangential scales diffusively as t 1/2 . Since p < 0, the radial displacement dominates, t 1/(p+2) t 1/2 . Thus we effectively have a diffusing particle in a growing interval with absorbing walls receding faster than diffusively. In this situation, the diffusing particle survives with a finite probability [11,12]. The computation of S ∞ (r, θ) is difficult, but the chief property, S ∞ (r, θ) > 0 if Q > 0, is clear. In the case of sink flows, Q < 0, the particle is hovering on distances r ∼ R with (We tacitly assume that |Q| D 1 which is consistent with the low Reynolds number limit, |Q| ν 1, in the natural situations when transport coefficients are comparable, ν ∼ D.) The survival probability is then exponential in time, S ∼ e −Dt/R 2 , that is Finally, a word of caution regarding the low Reynolds number limit. The effective local Reynolds number eventually becomes large since − 1 2 < p < 0. Therefore when α > α * the low Reynolds number treatment of the Jeffery-Hamel flow is inconsistent far away from the apex even if the source Reynolds number Q/ν is small.

VII. THREE-DIMENSIONAL ANALOGS OF THE WEDGE FLOWS
An obvious three-dimensional analog is a flow in a circular cone. Unfortunately, the Navier-Stokes equations do not admit an analytical solution for the flow inside a cone. Fortunately, the decay exponent β is independent of the flow. Indeed, if the particle has survived for a long time, it is far away from the apex. On large distances, the velocity field is v ∼ Q/r 2 , so advection is negligible compared to diffusion and the decay exponent β = β(α) is the same as if there were no flow [2,16]. This decay exponent is the smallest root of the Legendre function: We rely on the Stokes approximation valid far away from the apex: r Q/ν. The analysis in Appendix A reveals that the hydrodynamic solution changes depending on whether α smaller, equal, or larger than α * = 2π 3 . A more interesting three-dimensional analog of the flow in the wedge is the jet flow caused by the point source of force (rather than mass as in the Jeffery-Hamel flow). For the jet flow the velocity components are also inversely proportional to the distance from the origin Although the velocity field (84) is not unidirectional, the r −1 dependence of the velocity components makes the hydrodynamic problem solvable [58][59][60].
The angular dependence of the velocity components is and the pressure is given by [30,32] The parameter A is related to the momentum of the jet The quantity ν −1 √ M plays the role of the Reynolds number of the jet flow.
Overall, the jet flow is a much better analog of the Jeffery-Hamel flow than the flow in the cone. The dimensionless momentum of the jet (equivalently, the Reynolds number Re = ν −1 √ M , or the parameter A) affects the exponent β. The same happens in the wedge where the dimensionless strength of the source (the Péclet number) affects the decay exponent β.

VIII. CONCLUSION
We studied the first-passage characteristics of a particle diffusing in a wedge with absorbing boundaries, and advected by the flow generated by a source at the apex of the wedge. The survival probability decays algebraically with time. The decay exponent is easy to compute in the case of an ideal incompressible fluid. For the viscous fluid, we have reduced the determination of the exponent to finding the ground state energy of the quantum particle in an infinitely deep potential well. The shape of the well is determined by an exact solution of the Navier-Stokes equations for the incompressible viscous flow inside the wedge.
We employed perturbation techniques and deduced analytical predictions for the exponent β describing the decay of the survival probability and for the mean exit time which is finite when β > 1. The calculation of the mean exit time in planar domains is an active research subject [23,[61][62][63][64][65][66]. Due to the scale invariance of the wedge, the dependence of T (r, θ) on the distance r from the apex is fixed by dimensional arguments, Eq. (16). The angular dependence is simple for ideal flows, Eq. (18). In the viscous case, we have established the angular dependence in the Stokes limit, Eqs. (70)-(71).
Amongst three-dimensional analogs of advection in the wedge, the closest is the jet flow (Sec. VII). The analysis of the first passage characteristics is parallel to the analysis in the case of the wedge.
I am grateful to Alexei Skvortsov for correspondence.

Appendix A: Circular Cones
In contrast to the wedge case, one cannot define a global Reynolds number in the case of the flow in a cone. Far away from the apex v ∼ Q/r 2 , so the local Reynolds number rv/ν ∼ (Q/ν)r −1 asymptotically vanishes irrespectively on the strength of the source. Thus, at least sufficiently far from the apex, one can employ the low Reynolds number approximation. In this r Q/ν region, the velocity is purely radial: v = (v, 0, 0) in the spherical (r, θ, φ) coordinates. The continuity equation and axial symmetry fixes the radial dependence, v = r −2 u(θ), which is integrated to yield p = 2ν r 3 u(θ) + F (r) (A4) Substituting (A1) and (A4) into (A2) we obtain ν −1 r 4 dF dr = u + u cot θ + 6u (A5) The left-hand side of (A5) depends on r, while the righthand side depends on θ. Hence both sides must be equal to the same constant. In particular u + u cot θ + 6u = const Note that it becomes singular at α * = 2π/3, so the solution (A10) is applicable only when α < 2π/3.
For wide cones, α ≥ 2π/3, the leading behavior is different from (A1) and (A10). As in the case of the wedge (see Sec. VI D) one can take into account the finiteness of the input region.