## Abstract

This paper presents a conformal mapping approach to the study of planar Stokes flows near a wall with either one or two gaps. The technique reproduces, in a unified fashion, analytical solutions for various Stokes flows past a single gap in a wall found by previous authors using different methods. These include purely pressure-driven flows, shear flows and stagnation point flows near a gap in a wall. It is then shown how the conformal mapping method can be generalized to find new analytical solutions for the analogous flows when there are two gaps in the wall. Features of the new two-gap solutions are studied in detail.

## 1. Introduction

The study of slow viscous Stokes flows in complicated geometries is of recurring interest in fluid dynamics. The presence of boundaries renders both analytical and numerical methods for determining the flows more difficult. The monograph of Kim & Karrila (2005) provides an overview of the theoretical and numerical techniques available for the study of bounded Stokes flows. There has been a resurgence of interest in Stokes flows near boundaries owing to their relevance to microfluidics where it is required to control fluid motion in small-scale microchannels (for a review, see Squires & Quake 2005). Other applications are to swimming organisms at low Reynolds numbers (Lauga *et al.* 2006; Berke *et al.* 2008), studies of small robotic swimmers for biomedical applications (Kosa *et al.* 2007), the design of complex microstructured surfaces (Davis & Lauga 2009) and in a variety of electrophoresis and electro-osmosis applications where wall effects play a determining role (Zhao & Bau 2007). In the context of low Reynolds number swimmers, much recent effort has been made to understand their dynamics near an infinite plane wall (see Lauga *et al.* 2006; Berke *et al.* 2008 and references therein). How these swimmers evolve in geometrically more complicated domains is also of interest but does not appear to have been studied in any detail. In this paper, we focus on Stokes flows in a class of domains that is not as simple as the half plane above an infinite straight wall but are slightly more complicated: here, we suppose the fluid occupies the unbounded region exterior to an infinite plane wall of negligible thickness, in two dimensions, with either one or two finite-length gaps in it. In this case, we find that a broad range of analytical results are available. These paradigmatic flow solutions are expected to be useful in the various applications described above.

Hasimoto (1958) has studied low Reynolds number flows in precisely these two geometries (among others). His results involve flows driven purely by a pressure difference between the two sides of the wall. In that case, certain symmetry assumptions on the field variables across the wall allow an analogy to be made with flows of perfect fluids. Hasimoto builds his work on prior observations by Roscoe (1949). The flows of interest in this paper are more general: in addition to a possible pressure gradient across the wall, we also allow for flows which are not symmetric with respect to the two sides of the wall. Indeed, we assume that some background flow is imposed above the wall but that the flow far beneath the wall vanishes. Two specific background flows provide the focus of this study: uniform shear flows and stagnation point flows. For the case of a single gap in a wall, both flow situations have already been treated previously. Smith (1987) finds analytical solutions for a uniform shear flow in the region above a wall with a gap (with no flow far beneath the wall) with a view to understanding how the effects of flow along the wall can be transmitted across the gap. Ko & Jeong (1994) find solutions in the same geometry but driven by a stagnation point flow in the region above the wall (again with the assumption that there is no flow far beneath it). Ko and Jeong also allow for a possible pressure difference between the two sides of the wall. Concerning the analogous shear and stagnation point flows near a wall with two gaps, we have been unable to find any prior studies in the literature although Hasimoto (1958) has studied purely pressure-driven flow in the two-gap case. The present paper generalizes Hasimoto’s solution to more general flows in the same two-gap geometry.

Although we focus here on purely two-dimensional flows, it should be pointed out that Davis (1991) has found an exact solution, by solving a pair of dual integral equations, for the axisymmetric analogue of the problem considered by Smith (1987) in which there is a shear flow above an infinite wall with a single circular orifice (rather than a rectilinear slit as in the Smith problem). Davis’ work is a generalization of the classical Sampson (1891) flow solution for Stokes flow driven through a circular orifice in a wall.

This paper is structured as follows. First, we reappraise the work of Smith (1987) and Ko & Jeong (1994) and retrieve their solutions by introducing a new method based on conformal mapping (Hasimoto’s result is a subcase of the Ko–Jeong solution). The method is then generalized to produce new analytical solutions for the analogous Stokes flows past a wall with two gaps. The latter results are of special interest because the fluid domain is then doubly connected (i.e. there are two disconnected walls or boundaries to the flow) and to write down the solutions requires some special results from function theory. Exact solutions for Stokes flows in doubly connected regions are rare and only a few are known, mostly for flows involving circular cylinders. Jeffery (1922) used a formulation in terms of bipolar coordinates to identify analytical expressions for the Stokes flow in the bounded annular region between one cylinder contained in another (and where one or both of the cylinders are in motion). Using reflection ideas, Frazer (1926) considered the problem in which the two circular cylinders are in motion in an unbounded viscous fluid. Much later, and again using bipolar coordinates, Jeffrey & Onishi (1981) gave formulae for the flow due to a circular cylinder in translation and rotation above an infinite plane wall (and calculated the forces and torques associated therewith). These classical solutions continue to find use in a variety of applications ranging from studies of mixing in slow viscous flows (Finn & Cox 2001) to investigations of wall effects in electrophoresis (Zhao & Bau 2007). Finn & Cox (2001) give a helpful unification of the many disparate analytical results spread throughout the literature on Stokes flows between two cylinders. Hasimoto’s (1958) solution for purely pressure-driven flow across a wall with two gaps is, to the best of our knowledge, the only analytical solution in a doubly connected fluid region that does not involve circular cylinders.

## 2. Complex variable formulation

A complex variable formulation of the problem will be used. In two dimensions, the streamfunction *ψ* describing an incompressible Stokes flow is known to satisfy the biharmonic equation
2.1
and, once *ψ* is determined, the velocity field (*u*,*v*) follows from
2.2
A general representation of the solution *ψ* is given by
2.3
where *f*(*z*) and *g*(*z*) are two analytic functions of the complex variable *z*=*x*+*iy*. *f*(*z*) and *g*(*z*) are allowed to have singularities at infinity associated with various background flows of interest. It is possible to express all physical quantities in terms of *f*(*z*) and *g*(*z*) (Langlois 1964)
2.4
where *μ* is the fluid viscosity, *p* is the pressure, *ω* is the vorticity and *e*_{ij} denotes the fluid rate-of-strain tensor. It is clear from equation (2.4) that if
2.5
where *c* is an arbitrary constant, then all physical quantities associated with the flow remain the same. This means there is an additive degree of freedom in the definition of *f*(*z*). Here, the no-slip condition will be applied on any solid boundaries so that *u*+*iv*=0 there. This condition, together with the imposed singular behaviour in the far field, will determine *f*(*z*) and *g*(*z*) and, hence, the flow.

## 3. Conformal mapping: single gap

Our solution method involves the use of conformal mappings. For the single-gap case, the mapping is from the interior of the unit *ζ*-disc to the unbounded fluid region exterior to a wall with a gap between ±1. It is readily found to be
3.1
Equation (3.1) is obtained by considering successive composition of the two simpler conformal maps
3.2
The first map takes the interior of the unit *ζ*-disc to the unbounded region in an *η*-plane exterior to the slit [−1,1]; the second map takes this slit region to the required flow domain in a *z*-plane as it maps *η*=0 to . *ζ*=i maps to infinity in the upper half *z*-plane while *ζ*=−i maps to infinity in the lower half *z*-plane. Figure 1 shows a schematic of the mapped regions. Equation (3.1) can be explicitly inverted to give
3.3
This function has square root branch points at *z*=±1 (corresponding to the two edges of the walls). This fact will be important later.

## 4. Stokes flow near a 2*π*-corner

Dean & Montagnon (1949) and Moffatt (1964) have studied the local structure of Stokes flows in corners and around wedges. For the study here, we concern ourselves only with the singularities of *f*(*z*) and *g*(*z*) near a 2*π*-corner (the edge of a wall of negligible thickness). Near the wall edge at *z*=1, they are known to have the local behaviour
4.1
where *r*_{0},*r*_{1},*s*_{0} and *s*_{1} are constants. On differentiation, *g*′(*z*) will have unbounded behaviour of the form near the edge of the wall at *z*=1. The functions have analogous behaviour at *z*=−1. It is important to note that *ζ*(*z*) as given in equation (3.3) has the same square root branch point behaviour near *z*=±1 as required of *f*(*z*) and *g*(*z*). This is a key observation because it means that *ζ* can act as a uniformization variable for the problem if we introduce
4.2
These functions will then be single-valued functions of *ζ* in |*ζ*|≤1 and *G*(*ζ*) will have simple poles at *ζ*=±1.

## 5. Shear flow past a wall with a gap

Smith (1987) has studied shear flow over a wall with a single gap in the Stokes regime. He imposed a uniform shear (aligned along the wall) in the far field in the upper half plane above the wall and no flow far beneath the wall. His solution method involves a separation of the solution into two components: one has identical shear flow both above and below the wall (and is easy to find) while the second solution has shear flow in opposite senses above and below the wall and is found by solving a pair of dual integral equations and making use of transform methods. A sum of these two solutions produces the required solution (the shear flows far below the wall cancel out producing quiescent flow there, as required). Smith’s analysis is a special case of more general work by O’Neill (1977) who used bipolar coordinates to study shear flow over a cylindrical trough in a plane.

The approach to the problem here is different to that of both Smith (1987) and O’Neill (1977). Smith supposed that, as , *ψ*∼*y*^{2}. He assumed there is no flow as . From equation (2.4), it can be deduced that the far-field boundary conditions for *f*(*z*) and *g*′(*z*) are
5.1
where denotes infinity in the upper half plane (and is infinity in the lower half plane) and
5.2
where and are constants. From the required singularities of *f*(*z*) and *g*′(*z*) at infinity and at the edges of the wall, it follows that we can write *F*(*ζ*) and *G*(*ζ*) in the form
5.3
where and are to be determined: they are analytic and single-valued inside and on the boundary of the unit disc. They will therefore have Taylor series expansions valid in |*ζ*|≤1.

The no-slip condition on the wall is
5.4
which, in the *ζ*-plane, takes the form
5.5
where we have used the fact that on |*ζ*|=1. If the representations (5.3) are substituted into equation (5.5) the result, after multiplication by (1−*ζ*^{2})(*ζ*−*i*)^{2}, is
5.6
This relation between analytic functions is valid everywhere including inside the unit disc. Careful inspection of this equation shows that the Taylor expansion of about *ζ*=0 can only have a finite number of terms. Indeed, it must be a linear function of *ζ*. From equation (5.3), this means we can write
5.7
The additive degree of freedom in *f*(*z*), and hence in *F*(*ζ*), means we can set *A*=0. *B* is determined by insisting that the far-field condition (5.1) is satisfied. From equation (3.1), it follows that
5.8
as *ζ*→*i*. It follows that *B*=*i*/2 so
5.9
which is a simple rational function. On substitution of equation (5.9) into
5.10
which follows from equation (5.5) an explicit representation of *G*(*ζ*) is also obtained. It is also a rational function. After some algebraic manipulation, it can be shown explicitly (details are omitted here but can be found in Samson (2010)) that the solution just derived is exactly the same as that found by Smith (1987) using transform methods.

## 6. Stagnation flow past a wall with a gap

Ko & Jeong (1994) studied the problem of stagnation point flow in exactly the same geometry as Smith (1987). Although their work appeared later, Ko and Jeong do not seem to be aware of Smith’s paper and adopt a different approach to him. They solve the problem using Riemann–Hilbert methods.

We now retrieve their solution by adapting the conformal mapping method just described. In this case, the far-field conditions on *f*(*z*) and *g*′(*z*) are
6.1
and
6.2
where and are some constants, *k* is the imposed strain rate in the far-field and we have also allowed for a pressure difference of 2Δ*P* across the gap. The problem involving a wall with a single gap considered by Hasimoto (1958) corresponds to the special case with *k*=0. Now, it can be argued that
6.3
where and are to be determined: they are analytic and single-valued for |*ζ*|≤1. If the representations (6.3) are substituted into equation (5.5) and the equation then multiplied by (*ζ*−*i*)^{2}(*ζ*^{2}−1)(*ζ*+*i*), the result is
6.4
The structure of this equation reveals that the Taylor expansion of about *ζ*=0 can only have a finite number of terms: indeed, it must be a cubic polynomial. This means we can write *F*(*ζ*) in the partial fraction form
6.5
for some constants *A*,*B*,*C* and *D*.

It remains to determine *A*,*B*,*C* and *D*. As *F*(*ζ*) is only determined up to an arbitrary additive constant, we set *A*=0. From equations (5.8), it follows that
6.6
thus, as *ζ*→*i*,
6.7
A comparison with equation (6.1) gives
6.8
Similarly, as *ζ*→−*i*,
6.9
A comparison with equation (6.1) results in
6.10
It follows, after simplification, that
6.11
This is the final expression for *F*(*ζ*); again, it is a simple rational function. An explicit expression for *G*(*ζ*) follows from substitution of equation (6.11) into equation (5.10).

By inverting equation (3.1) to find *ζ* as a function of *z*, it can be confirmed (Samson 2010) that this solution is the same as that obtained in the paper by Ko & Jeong (1994). A convenient feature of the present approach is that *F*(*ζ*) and *G*(*ζ*) are both single-valued rational functions.

## 7. Stokes flow past a wall with two gaps

We now consider the case where the wall has two gaps. Attention will be focused on the case of two symmetric gaps given by the intervals [−1,−*r*] and [*r*,1] where 0≤*r*<1 although the analysis is readily generalized to two arbitrary gaps. In the symmetric gap case, there is a finite section of no-slip wall in the interval [−*r*,*r*] and it must be supposed that, for a general Stokes flow in this configuration, there will be a non-zero total force on this finite section of the wall. We refer to this finite section of wall as the central plate. It can be shown from equation (2.4) that the complex form of the fluid stress −*pn*_{i}+2*μe*_{ij}*n*_{j} (where *n*_{i} denotes the components of the unit normal vector to the wall) is 2*μi* *dH*/*ds* where *s* denotes arclength along the boundary and
7.1
The total force on the central plate is given by
7.2
where *C* is a contour taken anticlockwise around the finite section of the central plate and the square brackets denote the total change in the quantity contained inside them as a single circuit of *C* is traversed.

## 8. Conformal mapping: two gaps

With two gaps in the wall, the flow domain is doubly connected. For any doubly connected region, it is known (Nehari 1952) that there must exist a conformal mapping to it from an annulus *ρ*<|*ζ*|<1 in a complex *ζ*-plane for some 0<*ρ*<1. For the region exterior to an unbounded wall with two gaps, the relevant mapping has been found by Crowdy & Marshall (2006) to have the explicit form
8.1
where *P*(*ζ*,*ρ*) is defined by
8.2
In practice, this function is readily computed by truncation of the infinite product. This function has simple zeros at *ζ*=*ρ*^{2k} for . The latter function can be shown, directly from its definition (8.2), to satisfy the two functional relations
8.3
Equation (8.1) maps the unit circle |*ζ*|=1 to the two portions of the wall extending off to infinity while |*ζ*|=*ρ* maps to the central plate. Equation (8.1) depends on a single real parameter *ρ*. Different choices of *ρ* give mappings to regions with gaps of different lengths (i.e. different values of *r*). Figure 2 shows a graph of the central plate length 2*r* against *ρ*.

By making use of the relations (8.3), the map (8.1) can be shown to satisfy the functional relation
8.4
This fact is crucial in the analysis to follow. It is also useful to note that
8.5
so that
8.6
as *ζ*→*i*, where
8.7
and
8.8
where *P*_{ζ}(*ζ*,*ρ*)≡∂*P*(*ζ*,*ρ*)/∂*ζ* and *P*_{ζζ}(*ζ*,*ρ*)≡∂^{2}*P*(*ζ*,*ρ*)/∂*ζ*^{2}. We will also need to use the fact that
8.9
as *ζ*→−*i* .

It is not easy to explicitly invert the map (8.1) to find the inverse function *ζ*(*z*) as in the simply connected case (3.3). Nevertheless, from general arguments, it is known that the map (8.1) has square root branch point behaviour at the points *ζ*=±1,±*ρ* corresponding to the preimages of the four endpoints of the wall. Once again, *ζ* will serve as a uniformization variable for this problem so that *F*(*ζ*) and *G*(*ζ*) will be expressible as single-valued functions of *ζ* near these points.

## 9. Shear flow past a wall with two gaps

From equation (7.2), it is clear that if there is a non-zero force on the finite section of wall in the gap *F*(*ζ*) and *G*(*ζ*) are not single-valued in the annulus *ρ*<|*ζ*|<1. We now let
9.1
where *F*_{l} is some constant to be determined and and are analytic and single-valued in *ρ*<|*ζ*|<1. This choice of logarithmic singularities is made in order that the velocity field is single-valued while rendering the force on the central plate equal to
9.2
The well-known Stokes paradox (Batchelor 1967) associated with certain planar flows is not problematic here even though there is a non-zero force on the central plate. This is because there is an equal and opposite force on the unbounded portion of the wall leading to non-divergent velocities in the far field.

The no-slip condition on |*ζ*|=1 takes the usual form
9.3
while the condition on |*ζ*|=*ρ* is
9.4
On substitution of equation (9.1) into equations (9.3) and (9.4), we find
9.5
and
9.6
Equation (9.6) is a relation between analytic functions and holds off the circle |*ζ*|=*ρ* so we can let *ζ*↦*ρ*^{2}*ζ* in equation (9.6) to find
9.7
Subtraction of equation (9.7) from equation (9.5) gives
9.8
But, on use of the special property (8.4) of the mapping, this functional equation simplifies considerably to
9.9
It is this circumstance that enables us to find analytical solutions to this problem.

Now introduce the new function
9.10
then, from the functional relation (8.2), it can be shown that
9.11
By inspection of the second of these, it is clear that *K*(*ζ*,*ρ*) satisfies a functional relation very similar to that satisfied by *F*(*ζ*) in equation (9.9). *K*(*ζ*,*ρ*) has simple pole singularities at the points *ζ*=*ρ*^{2k} where . In particular, it has the behaviour
9.12
near *ζ*=1.

The boundary conditions on *f*(*z*) and *g*′(*z*) for this problem are given again by equations (5.1) and (5.2). As *f*(*z*) has a simple pole at infinity in the upper half plane but tends to a constant in the lower half plane, we let
9.13
for some constant *A*. On use of equations (9.9) and (9.11), we must pick
9.14
It remains to determine *F*_{l}. Given equations (8.6) and (9.12), it follows that
9.15
as *ζ*→*i*. But we require *f*(*z*)∼*iz*/2 so we arrive at
9.16
*G*(*ζ*) then follows from equation (9.1) with given by equation (9.5) and this completes the determination of the flow. As a check on the analysis, in the limit *ρ*→0 the conformal mapping (8.1) can be seen to reduce to equation (3.1) while the solution (9.16) reduces to the expression (5.9).

The total force on the central plate calculated from equation (9.2) is found to be directed in the *x*-direction and is graphed in figure 3. The zero vertical component is perhaps surprising given that the flow has no up–down symmetry across the plate. It is, however, consistent with the observation by Smith (1987), who, in the case of shear flow past a single gap, found that while the shear flow above the wall induces a flow in the region below the wall there is no net mass flux across the gap and no mixing of the fluids on the two sides of the wall. Instead, the flow below the wall is generated purely by the shear stress across the gap. The horizontal force on the central plate just calculated is a result of the differential shear stress across it and the absence of mixing across the gap explains the absence of any vertical force on the plate.

Typical streamlines for central plates of lengths 0.3,1 and 1.9 are shown in figure 4. When the central plate length is 1.9, so the gaps in the wall are narrow, the streamlines closely resemble those sketched in fig. 2 of the paper by Smith (1987), who considered the solution for shear flow past an infinite set of equally spaced narrow gaps in the wall. This gives a qualitative verification of our analysis.

## 10. Stagnation flow past a wall with two gaps

In the case of stagnation point flow past a wall with two gaps, the far-field conditions on *f*(*z*) and *g*′(*z*) are given by equations (6.1) and (6.2). Many of the details of the shear flow example carry over to this case, in particular, *F*(*ζ*) must satisfy condition (9.9). The problem involving a wall with two gaps considered by Hasimoto (1958) corresponds to the special case with *k*=0.

To solve for this flow, it is useful to introduce another special function *L*(*ζ*,*ρ*) related to *K*(*ζ*,*ρ*) by
10.1
where *K*_{ζ}(*ζ*,*ρ*)=∂*K*(*ζ*,*ρ*)/∂*ζ*. From the functional relations (9.11), it can be shown that
10.2
*L*(*ζ*,*ρ*) clearly has second-order poles at the points *ζ*=*ρ*^{2k}, where . In particular, it has the behaviour
10.3
near the point *ζ*=1.

We now propose that
10.4
where *A*,*B*,*C* and *D* are constants to be determined. Using up the additive degree of freedom in *f*(*z*), we set *D*=0 without loss of generality. Comparison with equation (9.9) and use of equation (9.11) implies that
10.5
Further conditions on these constants are determined by matching to the far-field conditions (6.1) and (6.2). As
10.6
it can be shown that
10.7
as in the upper half plane while
10.8
as in the lower half plane. Comparison with equation (6.1) means that
10.9
From equation (10.5), we also have
10.10
The final expression for *F*(*ζ*) is
10.11
*G*(*ζ*) then follows from equation (9.3) and completes the determination of the flow. A check on the analysis is provided by the limit *ρ*→0 in which the conformal mapping (8.1) reduces to equation (3.1) while the solution (10.11) reduces to the expression (6.11) derived for the single-gap case.

We computed the total force on the slit in the particular case when *k*=Δ*P*=*μ*=1 and found it to be parallel to the *y*-direction. It is given by equation (9.2) and is graphed in figure 5. The zero horizontal force is to be expected given the left–right symmetry (about the *y*-axis) of the forcing flow and the geometrical configuration. Hasimoto (1958) included a similar graph for the case of purely pressure-driven flow (*k*=0) and also noted that the ‘blocking effect’ of even a central plate of very small length (as *ρ*→0 in our analysis) is significant owing to the logarithmic term in the denominator of equation (10.10): from figure 2, it is clear that the central plate length 2*r* is an approximately linear function of *ρ* as *ρ*→0 hence the force on the plate falls off like as *r*→0 (which decays very slowly as *r*→0).

It is interesting to study the flow streamlines as the oncoming flow is altered and the geometry of the domain is changed. First, we fix the strength of the stagnation point flow and the viscosity by setting *k*=1 and *μ*=1, and vary the pressure difference across the wall. Following Ko & Jeong (1994), we define the number
10.12
When *N* is positive, the pressure gradient promotes flow downwards through the gap; when *N*=0 the flow is driven by the stagnation point flow only; finally, when *N* is negative, the pressure gradient promotes flow upwards through the gap.

First, we set *N*=1. When there is no central plate, the streamlines pass downwards through the single gap as in fig. 5*a* of the paper by Ko & Jeong (1994). When the central plate is introduced, but is small in length, small (Moffatt-type) recirculating eddies are found to form underneath it as shown in figure 6. The size of these eddies grows as the plate length increases and the width of the gaps decreases.

When *N*=0.07, there is only a weak pressure difference across the plate so the fluid motion is driven predominantly by the stagnation point flow. When there is no central plate and the gap width is 2 the streamlines are as shown in fig. 5*b* of the paper of Ko & Jeong (1994). When a central plate is introduced and varied in length, eddies form once again on its lower side although they appear to be oriented close to the vertical. Indeed, the flow beneath the wall in figure 7 for the case where the length of the central plate is 1.9 has streamlines strongly reminiscent of those depicted in fig. 2 of the paper by Smith (1987) for the case of shear flow past a periodic array of very narrow gaps. This is to be expected because the gaps are then narrow and are sufficiently far from the middle of the central plate that the oncoming flow just above the gaps locally resembles a shear flow.

When *N* is negative, as shown in figure 8, the pressure gradient is such that it directs the flow upwards through the gap from below and competes with the stagnation point flow tending to send the fluid downwards. In the absence of the central plate, the streamlines are as in fig. 5*c* of the paper of Ko & Jeong (1994). The flow creates eddies underneath the walls in the vicinity of the sharp edges. When a central plate is introduced, we see that the same eddies are present, but vary their position as the slit length is increased and the width of the gaps becomes smaller. In particular, the eddies do not tend to form beneath the central plate but beneath the sidewalls.

Finally, on comparison of the three cases *N*=1,0.07 and −0.5 shown in figures 6–8, it can be seen that the pressure gradient causes the viscous eddies formed beneath the gaps to be inclined at different angles: when the pressure gradient is almost absent (for *N*=0.07), the major axes of the eddies are oriented near vertically; when *N* is positive, these axes rotate as *N* increases so that the eddies form underneath the central plate while if *N* becomes negative the axes tilt outwards so that the eddies become inclined away from the central plate and form underneath the sidewalls.

## 11. Discussion

This paper has presented exact solutions for Stokes flows past a wall with one or two gaps. By developing a new mathematical approach based on conformal mapping, the results of Hasimoto (1958), Smith (1987) and Ko & Jeong (1994) for (respectively) pressure-driven, shear and stagnation point flows past a single gap in a wall have been rederived in a unified fashion. The analysis has then been extended, in a natural way, to derive analogous exact solutions for Stokes flows past a wall with two gaps. The two-gap solutions extend the purely pressure-driven flows found (in the same geometry) by Hasimoto (1958) to more general situations driven by imposed background shear and stagnation point flows. The latter results are believed to be new and constitute important (and rare) examples of analytical solutions for Stokes flows in doubly connected domains.

We have documented the solutions here as a contribution to the mathematical theory of Stokes flows. The solutions have been used to study the occlusion effects of a segment of wall (of differing lengths) being placed in a gap. The solutions are expected to be useful in many applications: indeed, the authors have already made use of them in the problem that motivated this investigation—a study of the dynamics of low Reynolds number swimmers near gaps in walls (Samson 2010). In work closely related to that presented here, Davis (1991) found an exact solution for Stokes flow past a circular orifice in an infinite plane wall and was motivated by a study of fluid skimming and particle entrainment performed by Yan *et al.* (1991). Such problems are other possible applications of our results.

Following Hasimoto (1958), we have focused on the symmetric two-gap situation. On the matter of generalization, we note that a conformal mapping from the annulus *ρ*<|*ζ*|<1 to the fluid region exterior to a wall with two generally non-symmetric gaps is
11.1
where *R* is a real parameter (controlling the relative lengths of the two gaps). *R*=1 corresponds to the symmetric gap case treated here. Equation (11.1) also satisfies the functional relation *z*(*ρ*^{2}*ζ*)=*z*(*ζ*) and this proved to be a key factor in the foregoing analysis. The solutions here can therefore be generalized, in a straightforward way, to the case *R*≠1 for non-symmetric gaps.

As the analysis of Hasimoto (1958) of purely pressure-driven flows past two gaps in a wall involved use of special results from elliptic function theory, it should be pointed out that the functions *P*(*ζ*,*ρ*),*K*(*ζ*,*ρ*) and *L*(*ζ*,*ρ*) used to express our solutions for more general flows in the same geometry have close connections with the Weierstrass sigma, zeta and functions (Whittaker & Watson 1996). Thus, the solutions can, in principle, be translated into the language of elliptic function theory. For both theoretical and computational purposes, however, we believe the function theoretical approach adopted here to be more transparent; in particular, the analysis here is self-contained and we have not needed to invoke any results from special function theory.

In the different problem of vortex motion in geometrically complicated geometries, explicit formulae for the conformal mappings (from an appropriate preimage domain) to fluid regions exterior to a wall with *any* finite number of gaps were presented and solutions to the vortex problem considered there were duly found by Crowdy & Marshall (2006) (that analysis made use of the so-called Schottky–Klein prime function of which the function *P*(*ζ*,*ρ*) used in this paper is a particular example). As this paper has also used a conformal mapping approach, it is natural to ask if the Stokes flow results here can be extended to analogous flows past any number of gaps (i.e. more than two) in the wall. We have investigated this and have found that, for three or more gaps, only the symmetric pressure-driven flows considered by Hasimoto (1958) appear to have solutions expressible in closed form (those extensions, not treated by Hasimoto himself, will be presented in a separate paper). For now, we have been unable to generalize the shear and stagnation point flow solutions to the case of walls with more than two gaps (such solutions can, of course, be computed by purely numerical means). The one- and two-gap cases therefore appear to be rather special.

## Acknowledgements

D.C. acknowledges the support of an EPSRC Advanced Research Fellowship. He also acknowledges the hospitality of the California Institute of Technology where he was a Visiting Professor during 2009 when some of this work was carried out. O.S. is supported by an EPSRC studentship.

## Footnotes

- Received January 27, 2010.
- Accepted March 5, 2010.

- © 2010 The Royal Society