2.1. Kelvin's minimum energy theorem

We begin by stating the minimum energy theorem of Kelvin (Reference Kelvin1849), which will be generalized thereafter. The statement is as follows.

Theorem 2.1 Consider an ideal fluid domain $D\subseteq \mathbb {R}^2$, with velocity field $\boldsymbol {v}(\boldsymbol {x})=\boldsymbol {\nabla } \phi (\boldsymbol {x})$ satisfying $\nabla ^2\phi =0$ with no-flux boundary conditions on $\partial D$, where $\phi$ is single valued. Further assume the volume of each boundary does not change in time and the velocity vanishes at infinity. Then any other incompressible flow $\boldsymbol {q}(\boldsymbol {x})$, satisfying the boundary conditions on $\partial D$, will possess kinetic energy greater than or equal to that of $\boldsymbol {v}$,

(2.1)\begin{equation} \tfrac{1}{2} \rho \int_{D}\|\boldsymbol{v}\|^2\, {\rm d}V \leq \tfrac{1}{2} \rho \int_{D}\|\boldsymbol{q}\|^2\,{\rm d}V, \end{equation}

when $\|\boldsymbol {v}\|^2$ contains no non-integrable singularities.

When there exist source singularities in $D$, the integrals in (2.1) are not well defined. This point is important since our model in § 4 involves point sources. We reserve a discussion of this technicality for § 4.1. In these analyses we show that all results of § 2 still hold if the flow possesses a source singularity, as long as the energy is defined appropriately. Until § 4, we will continue to assume that there are no non-integrable singularities in the flow. We now present the proof of Kelvin's minimum energy theorem as stated in Theorem 2.1, when the flow possesses no singularities.

Proof. We now prove Theorem 2.1. We choose to decompose $\boldsymbol {q}$ according to $\boldsymbol {q}=\boldsymbol {v}+\boldsymbol {w}$, where $\boldsymbol {w}$ is finite in $D$. Since $\boldsymbol {v}$ already satisfies the appropriate boundary conditions, $\boldsymbol {w}$ must satisfy the no-flux condition $\boldsymbol {w}\boldsymbol {\cdot }\boldsymbol {n}=0$ on the boundary, $\boldsymbol {x}\in \partial D$. Noting that $(\|\boldsymbol {q}\|^2-\|\boldsymbol {v}\|^2)=2\boldsymbol {\nabla } \phi \boldsymbol {\cdot } \boldsymbol {w}+\|\boldsymbol {w}\|^2$, we find that

(2.2)\begin{equation} \tfrac{1}{2} \rho \int_{D}\left(\|\boldsymbol{q}\|^2-\|\boldsymbol{v}\|^2\right)\, {\rm d}V = \rho\left(\int_{C}\phi \boldsymbol{w}\boldsymbol{\cdot} \boldsymbol{n}\,{\rm d}A-\int_{D} \phi \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{w}\, {\rm d}V\right)+\tfrac{1}{2} \rho \int_{D} \|\boldsymbol{w}\|^2\,{\rm d}V, \end{equation}

where $C$ is equal to $\partial D$, since we took $\phi$ to be single valued. Then the first integral on the right-hand side vanishes by the no-flux condition on $\boldsymbol {w}$. The second integral on the right-hand side vanishes by the incompressibility of $\boldsymbol {q}$. Meanwhile, the last term is non-negative.

In general, there is no unique solution to the Laplace problem with Neumann boundary conditions: solutions are only determined up to an unspecified amount of circulation around each boundary. In the above proof, assuming that $\phi$ was a single-valued function specified a particular solution to the Laplace problem. This solution is defined by the property that the flow $\boldsymbol {\nabla } \phi$ possesses no circulation in the sense that $\int _{\gamma }\boldsymbol {\nabla } \phi \boldsymbol {\cdot } \boldsymbol {dl}=0$ around any closed loop $\gamma$. Hence, Theorem 2.1 is applicable to the solution that possesses no circulation. As was noted by Gonzalez & Taha (Reference Gonzalez and Taha2022), the correct conclusion is that the minimal kinetic energy solution is the one possessing no circulation; in their paper, it is argued that the true solution is the one minimizing curvature. However, in the present paper, we focus on the zero circulation solution.

In Appendix B we present a generalization of Theorem 2.1 to the case where arbitrary circulations may exist around each flow boundary. When circulation is present, the above proof is modified because the contour $C$ cannot be taken equal to $\partial D$; instead, $C$ must be chosen not to cross branch cuts associated with a multivalued $\phi$. It turns out that Theorem 2.1 is still applicable even when there is circulation, provided that a particular flux condition is satisfied across branch cuts (see Appendix B).

We emphasize again that in the present paper we restrict our attention to the behaviour of the zero circulation solutions, to which Theorem 2.1 applies. We proceed by showing that an added stationary boundary never decreases the energy. This corollary will be important in establishing the result of § 2.2, regarding energy non-monotonicity.

Corollary 2.2 Suppose a stationary impenetrable body $C$, with boundary $\partial C\subset D$, is added to the previous flow. Then a potential flow solution exists with velocity $\boldsymbol {\nabla } \phi _C$ defined over $D \backslash C$, where $\nabla ^2 \phi _C=0$. Furthermore, the kinetic energy of that flow is greater than the boundaryless potential flow solution,

(2.3)\begin{equation} \tfrac{1}{2} \rho \int_{D\backslash C} \|\boldsymbol{\nabla} \phi_C\|^2\, {\rm d}V \geq \tfrac{1}{2} \rho \int_{D} \|\boldsymbol{v}\|^2\,{\rm d}V. \end{equation}

Proof. The ideal flow, in the region $D \backslash C$, has a solution by standard existence theorems (Thomson Reference Thomson1848). That solution has no flux into the region $C$ and a perfectly valid choice of an incompressible flow $\boldsymbol {q}$ defined over all $D$, as defined in Theorem 2.1, is

(2.4)\begin{equation} \boldsymbol{q}(\boldsymbol{x}) = \left\{ \begin{array}{@{}ll@{}} \boldsymbol{\nabla} \phi_C, & \boldsymbol{x}\in D\backslash C, \\ \boldsymbol{0}, & \boldsymbol{x}\in C. \end{array} \right. \end{equation}

This flow has precisely the kinetic energy of the potential flow solution in the presence of the boundary. However, by Theorem 2.1, the flow possesses more energy than the boundaryless flow. Note that $\partial C$ has measure zero, $\boldsymbol {v}$ is defined over $D$ and $\boldsymbol {\nabla } \phi _C$ over $D\backslash C$.

In the proof of Theorem 2.1, it may have appeared that the reason $\boldsymbol {q}$ has more energy than $\boldsymbol {v}$ is either that the former is rotational or possesses non-zero circulation. However, we have demonstrated a third possibility via corollary 2.2. Namely, if $\boldsymbol {q}$ is irrotational and possesses zero circulation, but is subject to additional boundary conditions, it will also possess an energy greater or equal to that of $\boldsymbol {v}$. This statement is understood most simply by appealing to a variational argument; see, for example, Whitham (Reference Whitham1974, p. 434) for a deeper discussion of the variational formulation of potential flow. The general argument is as follows. A solution to the Laplace equation $\phi$ can be viewed as a function that extremizes the energy functional $S[\phi (\boldsymbol {x})]=(\rho /2)\int {(\boldsymbol {\nabla }\phi \boldsymbol {\cdot }\boldsymbol {\nabla } \phi )}\,{\rm d}V$, so that $\delta S=0$, subject to a given set of constraints. In other words, $\phi (\boldsymbol {x})$ can be thought of as the result of a constrained optimization of $S[\phi (\boldsymbol {x})]$. In potential flow, the extremization occurs at a minimum of $S[\phi (\boldsymbol {x})]$ and the constraints are given by the boundary conditions of the flow. It follows that imposing additional boundary conditions (constraints) on the constrained optimization problem can never result in a better minimization of $S[\phi (\boldsymbol {x})]$.

This is in contrast to the electrostatic problem with a neutral conducting boundary. In the electrostatic problem, surface charge is allowed to distribute itself on the surface of the conductor, as long as the net charge is zero. The presence of surface charge affects the relevant energy functional. Hence, the addition of a conducting boundary presents an additional degree of freedom in the optimization of the electrostatic energy functional, and so neutral conducting boundaries decrease (or keep constant) the electrostatic energy.

2.2. Non-monotonic kinetic energy

We now demonstrate that the fluid kinetic energy need not vary monotonically with the distance between a fluid system and a newly introduced boundary $\partial C$, a result that follows from corollary 2.2. We define a fluid system by a collection of boundary conditions on $\partial D$ as in Theorem 2.1. A stationary boundary is introduced mathematically as an additional no-flux boundary $\partial C$, as in corollary 2.2. As usual, slip is allowed in an ideal fluid. The boundary-separation distance is defined as the distance between a chosen point attached to $\partial C$ and one in $\partial D$. We now demonstrate by example, referring to figure 2, as follows.

Figure 2.

A cylinder of unit radius and unit speed is pictured with its boundaryless streamlines. Configurations of a streamline-approximating boundary are superimposed and labelled $A,B,C$.

Consider $D$ to be the exterior to a unit circle that translates vertically at unit velocity. We choose a boundary $\partial C$ comprising a pair of infinitely thin arcs resembling streamlines, which are drawn in three distance configurations, labelled $A,B,C$. The boundary-separation distance, labelled $x$, is taken as the vertical distance from the centre of the circle to the bottom of the boundary. We call the kinetic energy of the system $T(x)$. When $x\rightarrow \infty$, the boundaryless energy is recovered. At other distances, $T(x)\geq T(\infty )$ by corollary 2.2. However, when $x=x_B$, the boundary does not perturb the cylinder flow, as it coincides precisely with streamlines, and $T(x_B)=T(\infty )$. Therefore, $T(x_B)$ corresponds to a local minima of the kinetic energy as a function of the boundary-separation distance. Hence, the kinetic energy dependence need not be monotonic.

Similarly, one can conclude that the kinetic energy of the configuration in figure 1(d) is non-monotonic in the boundary-separation distance, $y$. Since the perfect slits do not perturb the source flow when $y=0$, the kinetic energy $T(y)$ satisfies $T(0)=T(\infty )$. At other distances, $T(y)\geq T(\infty )$. One can then deduce the schematic plot of the kinetic energy given in figure 1(b). A main goal of our investigation will be to analyse the physical manifestation of the change in sign of the slope of $T(y)$, in the highlighted region $y\gtrsim 0$.

The electrostatic energy of the configuration considered by Levin & Johnson (Reference Levin and Johnson2011) (see figure 1c) is plotted schematically in figure 1(a). An energy theorem of electrostatics states that a neutral conducting boundary must decrease (or keep constant) the potential energy. The curve shape then follows from the fact that $V(0)=V(\infty )$ and $V(x)\leq V(\infty )$. Thus, the electrostatic energy theorem predicts the non-monotonic shape of the curve in figure 1(a), where the highlighted window manifests as a repulsive electrostatic force, $F_{{e}}=-{\rm d}V/{{\rm d}\kern0.7pt x}$.

Although these discussions considered infinitely thin boundaries, we show in § 4.2 that the corresponding effects on fluid forces persist for finite thicknesses.

3.1. Added mass cannot decrease

Consider a single translating body in an unbounded ideal fluid. The Buckingham–Pi theorem (Buckingham Reference Buckingham1914) implies that the total fluid kinetic energy can be written as

(3.1)\begin{equation} T_f=\frac{C_a}{2}\rho V U^2, \end{equation}

where $U$ is the body velocity, $C_a$ is a dimensionless constant called the added mass coefficient, and $\rho$ and $V$ are the fluid density and body volume, respectively. Considering acceleration at a constant rate $a$, (3.1) gives the necessary work to move a distance $\delta x$, $\delta W=C_a \rho V a \delta x$. The resulting force is therefore $F=-\delta W/\delta x=-C_a \rho V a$; the reaction is equivalent to a mass augmentation of amount $C_a \rho V$. Corollary 2.2 showed that boundaries may not decrease the total kinetic energy $T_f$. Therefore, when the presence of a boundary is modelled as an effective change in the added mass, increasing $T_f$ implies an increased $C_a$ in (3.1).

Hence, boundaries cannot be used to decrease the added mass and attain higher accelerations. The connection between boundary-induced added mass increase and Kelvin's theorem was mentioned by Lamb (Reference Lamb1975, § 93). Although boundaries cannot reduce the added mass, geometries such as those described in § 2.2 can lead to interesting fluid force effects, to be analysed in the remainder of this paper. The effects presented in the remainder of the paper rely on energy non-monotonicity as discussed in § 2.2, and are thus in analogy with the electrostatic results of Levin & Johnson (Reference Levin and Johnson2011).

3.2. Effect of non-monotonic energy on unsteady force

When a body translates toward a boundary, according to the added mass formulation, the energy can be expressed as in (3.1) with $C_a=C_a(x)$, where $x$ is the separation distance to the boundary. Motion can only be taken as one dimensional after either assuming a constraint or by exploiting a symmetry in the $y$ direction, as is present in figure 2. This assumption is made to simplify the calculation and illustrate our main argument. Note again that we will ignore the intrinsic mass of the cylinder. If the cylinder had mass $m_c$, then one should replace every $C_a$ with $C_a +m_c/(\rho V)$ in the following analysis.

For one-dimensional free motion toward the boundary, the Euler–Lagrange equation is

(3.2)\begin{equation} 2\frac{\partial C_a(x)}{\partial x}\dot{x}^2+2C_a(x)\ddot{x}-\frac{\partial C_a(x)}{\partial x}\dot{x}^2=\frac{1}{\dot{x}}\frac{{\rm d}}{{\rm d}t}\left( C_a(x)\dot{x}^2 \right)=0, \end{equation}

or $C_a(x)\dot {x}^2=\mathrm {const.}$ since $\dot {x}\neq 0$. Lamb (Reference Lamb1975, § 136) justified the Lagrangian formulation, and the validity of (3.1), in the context of a body approaching a boundary with such a symmetry. In analysing the motion of a sphere toward a plane wall, Lamb (Reference Lamb1975, § 137) deduced from (3.2) a repulsive force since $\partial C_a/\partial x$ was always negative there. However, this need not always be the case as discussed in § 2.2: if the boundary coincides, at some finite $x$, with the streamlines of a travelling body then there is a local minima of $C_a(x)$. The equation of motion (3.2) implies, setting the constant to 1, that $\partial \dot {x}/\partial x=-C_a^{-{3}/{2}}(\partial C_a/\partial x)/2$ which switches sign at the local minima.

Consider the geometry of figure 2. A circle translates vertically and its boundaryless streamlines, which are a function of velocity, are drawn for reference. The circle approaches a boundary that approximates flow streamlines. If, as the circle evolves according to (3.2), at some moment the free streamlines coincide with the boundary (as pictured for distance $x_B$), then the force will change sign when the boundary slits get aligned with the streamlines, $x=x_B$. In a different scenario where the circle is pushed at constant velocity, the necessary external force to maintain the velocity reverses sign at $x_B$.

4.1. On the validity of Kelvin's theorem with source singularities

In two dimensions the kinetic energy of a source singularity, in an otherwise unbounded fluid, diverges logarithmically. A source at the origin has velocity $\|\boldsymbol {v}\| \propto 1/r$, so that the absolute kinetic energy scales as $\int _0^{2{\rm \pi} }\int _{\epsilon }^L \|\boldsymbol {v}\|^2 r \,{\rm d}r \,{\rm d}\theta \propto \log {(L/\epsilon )}$. The integral is divergent both at the source location, as $\epsilon \rightarrow 0$, and at infinity, as $L\rightarrow \infty$. Therefore, the absolute kinetic energy of a source is not a meaningful quantity. However, forces and physical observables are related to differences, and in particular gradients, of energy. Therefore, the energy may be redefined with respect to any reference system, without altering physical predictions. In what follows, we will demonstrate a manner in which a finite energy can be defined even in the presence of sources. It is with respect to this definition of energy that the arguments of § 2 will hold.

We note that a similar issue of energy divergence arises in the theory of electrostatics. The electromagnetic field energy of a point charge is non-integrable at the location of a charge in three dimensions. The divergence at the location of the source is eliminated (regularized) by choosing to measure the energy with respect to that of an isolated point source. Similarly, by choosing to measure the fluid kinetic energy with respect to that of an isolated source singularity, the divergence at the location of a fluid source can be regularized. What is left is to deal with the issue of the integral's convergence at infinity. The divergence at infinity is related to the fact that we are working in two dimensions. Sources of the Laplace equation are integrable at infinity in dimensions $n \geq 3$.

The divergence at infinity may be resolved by two different approaches. The first approach involves treating the two-dimensional problem as the limit of a three-dimensional problem. The two-dimensional flow due to a point source at $x=y=0$ is equivalent to the flow produced by a uniform continuum (line) of three-dimensional sources along the entire $z$ axis. If one considers a line of sources with density $m$, in three dimensions, extending from $z=-L$ to $z=L$, the radial velocity profile in the $z=0$ plane is given by $v_r(r)=m/(2{\rm \pi} r \sqrt {1+(r/L)^2})$; the integrated kinetic energy in the $z=0$ plane of such a source is then convergent at infinity for any finite $L$. In the presence of sources, the energy can be made finite at infinity by regulating each source according to the prescription $m/(2{\rm \pi} r)\rightarrow m/(2{\rm \pi} r \sqrt {1+(r/L)^2})$ for some finite regularizing parameter, $L$.

A second approach, which does not require invoking three dimensions, is described by Saffman (Reference Saffman1993, p. 125). In this approach, a small circle of radius $\epsilon > 0$ around each singularity is excluded from the domain when computing the energy. The boundary at infinity is also replaced by a circle of large radius $R$. In the limit of $\epsilon \rightarrow 0$ and $R\rightarrow \infty$, the energy can be expressed as the sum of a finite part and a divergent constant $C \propto \log {(R/\epsilon )}$. Since constant shifts in energy are not observable, only the finite part of the energy needs to be retained. This procedure is reminiscent of the procedure of dimensional regularization used in quantum field theory, a connection nicely illustrated by Olness & Scalise (Reference Olness and Scalise2011).

The key takeaways from this section are as follows. The absolute kinetic energy of a source flow is not finite in two dimensions, relative to a static configuration, since the kinetic energy integral diverges both at the source location and at infinity. Measuring the energy relative to that of an isolated source eliminates the divergence at the source location. The divergence at infinity can be regulated in two ways, and a finite regularized energy can be defined. In terms of the regularized energy, all of the discussions of § 2 apply, along with the interesting force behaviours predicted when boundaries approximate streamlines, even when there exists a source singularity in the flow. In what follows, we analyse the source flows of figure 3, and examine the force behaviour when boundaries approximate the source streamlines.

4.2. Single slit and source

Consider an ellipse (which may degenerate to a slit) centred at the origin in the complex plane, with some orientation $\alpha$, as shown in figure 3(a). When $x=0$, the source is located directly above the plate and Bernoulli suction attracts the ellipse to the source, as shown in figure 4(a). However, when $x \neq 0$ is fixed, there is a value of $y=x\tan {\alpha }$ where the ellipse falls on a natural source streamline. For a perfect slit, $y=x\tan {\alpha }$ corresponds to an energy minima as a function $y$, as follows from the discussion of § 2.2. We thus expect the vertical steady force to flip sign near this point for sufficiently slender ellipses.

Figure 4.

(a) Plot of the vertical steady force on the slit in the geometry shown in figure 3(a) when $x=0$, $s=1$, for various $\alpha$. The force is positive for $y>0$, indicating attraction as in the standard Bernoulli gripper. (b) Same plot when $\alpha ={\rm \pi} /4$, $x=2$, for various slenderness parameters $s$. The slit approximates a streamline when $y/D=1$, leading to an energy minima and local sign reversal of the force only for slender ellipses ($s\approx 1$).

We employ a conformal mapping approach. The simpler problem of a source exterior to a unit disk is solved and mapped to the problem of interest (figure 3a) by a conformal map. We denote the physical domain coordinates by $z$ and the pre-image coordinates by $\zeta$. The complex potential for a source of strength $m=2{\rm \pi}$ at location $\zeta _s$, where $|\zeta _s|>1$, is $\log {(\zeta -\zeta _s)}$. The complex potential exterior to a stationary unit disc is then given by the Milne–Thomson theorem (Milne-Thomson Reference Milne-Thomson1962, § 6.21),

(4.1)\begin{equation} W=\log{\left(\zeta-\zeta_s\right)}+\log{\left((1-\overline{\zeta_s}\zeta)/\zeta\right)}. \end{equation}

The unit disk exterior maps to the tilted ellipse exterior by a Joukowsky-type map,

(4.2)\begin{equation} z(\zeta)=\left(\zeta + \frac{s}{\zeta}\right){\rm e}^{\mathrm{i}\alpha}, \end{equation}

where $s\in [0,1]$ for univalency (Llewellyn Smith, Michelin & Crowdy Reference Llewellyn Smith, Michelin and Crowdy2008); $s$ defines a homotopy between the circle ($s=0$) and the slit of length $D=2$ ($s=1$). Intermediate shapes are ellipses. To achieve the configuration in figure 3(a), $\alpha$ in (4.2) corresponds directly to $\alpha$ in the figure. However, the source pre-image, $\zeta _s$, must be chosen to ensure that $z(\zeta _s)\equiv z_s=x+\mathrm {i}y$. After some algebra, one finds the pre-image location, $\zeta _s(z_s)=(z_s {\rm e}^{-\mathrm {i}\alpha }+\sqrt {(z_s {\rm e}^{-\mathrm {i}\alpha })^2-4s})/2$.

In what follows, we analyse the vertical component of force as $y$ is varied for a given $x$. The force is computed in the $\zeta$ plane since the complex potential is known there according to (4.1). The steady force is given by the expression of Tchieu et al. (Reference Tchieu, Crowdy and Leonard2010, equation (18)),

(4.3)\begin{equation} F=\overline{\frac{\mathrm{i}}{2}\oint_{|\zeta|=1}{\left(\frac{{\rm d}W}{{\rm d}\zeta}\right)^2 \left(\frac{{\rm d}z}{{\rm d}\zeta}\right)^{{-}1}\,{\rm d}\zeta}}, \end{equation}

after setting the fluid density to be equal to 1. The vertical component is $F_s\equiv \mathrm {Im}\{ F \}$. Using (4.1) and (4.2), the residue theorem yields the vertical force on the ellipse,

(4.4)\begin{equation} F_s=\mathrm{Im}\left\{-\frac{2{\rm \pi}\zeta_s^2(\zeta_s-s\overline{\zeta_s})}{(\zeta_s^2-s)^2(|\zeta_s|^2-1)}{\rm e}^{-\mathrm{i}\alpha}\right\}, \end{equation}

which is plotted in figure 4(a) for the case of $x=0$ and $s=1$. The usual Bernoulli suction effect is captured there; a low pressure on the top of the plate leads to a net vertical (attractive) force. The suction persists regardless of the angle $\alpha <{\rm \pi} /2$ when $x=0$. Note that taking $s<1$ does not qualitatively affect the shape of the curve.

We now examine the case where the slit may fall on a streamline of the source, $x\neq 0$. Figure 4(b) plots the case of $x=2$ and $\alpha ={\rm \pi} /4$ for various slenderness parameters, $s$. When $s=1$, the slit coincides with a natural (radial) streamline of the source at $y/D=1$. As per the discussion of § 2.2, this value of $y$ corresponds to a local energy minimum as a function of $y$, and the vertical force should thus change sign when $y/D=1$. The plot in figure 4(b) shows the force changes sign at precisely this point for a perfect slit, $s=1$. There is a local region of repulsion for $y/D>1$. For large $y/D$, the force becomes attractive again. Meanwhile, the onset of sign reversal of the force is shifted for an ellipse of finite slenderness ($s<1$) until the effect vanishes for $s=0.79$ where there is no repulsive region. For large $y/D$, all force plots are positive, indicating attraction.

4.3. Two slits and source

The vertical steady force is computed by a similar procedure in the two-body geometry shown in figure 3(b). The canonical domain for the doubly connected problem is the annulus, where the problem can be solved simply and then mapped to the physical domain by a conformal map. The conformal map from the annulus, $|\zeta |\in [\rho,1]$, to the two-slit geometry shown in figure 3(b) is

(4.5)\begin{equation} z(\zeta)={-}\mathrm{i}A{\rm e}^{3\mathrm{i}\alpha}\frac{\omega(\zeta,\sqrt{\rho}{\rm e}^{2\mathrm{i}\alpha})\omega(\zeta,\sqrt{\rho}^{{-}1}{\rm e}^{2\mathrm{i}\alpha})}{\omega(\zeta,\sqrt{\rho})\omega(\zeta,\sqrt{\rho}^{{-}1})}, \end{equation}

where $\omega$ is the Schottky–Klein prime function defined by Crowdy (Reference Crowdy2020, pp. 64,75); a similar map was used by Crowdy (Reference Crowdy2009) to analyse the Weis-Fogh mechanism. Note that we consider figure 3(b) with zero-thickness slits. To accommodate a source, a sink of equal strength is included in the annulus for mass conservation, with the sink located at the pre-image of infinity, $\zeta _{\infty }=\sqrt {\rho }$. The complex potential in the $\zeta$ plane is (Crowdy Reference Crowdy2013)

(4.6)\begin{equation} W(\zeta)=\frac{m}{2{\rm \pi}}\log{\left(\frac{\omega(\zeta,\zeta_s) \omega(\zeta,\overline{\zeta_s}^{{-}1})}{\omega(\zeta,\zeta_{\infty}) \omega(\zeta,\overline{\zeta_{\infty}}^{{-}1})}\right)}, \end{equation}

up to a constant that does not enter our calculation. The source location in the annulus, $\zeta _s$, must be chosen such that its image lies on the imaginary axis as in figure 3(b). Conveniently, we notice the circle $|\zeta |=\sqrt {\rho }$ maps to the imaginary axis. We proceed by resolving the vertical component of force on the boundary. By symmetry, we can simply double the force on one slit; thus, the total force is twice that given in (4.3). Note that the map of (4.5) has zeros at the locations of the sharp slit edges. To avoid the corresponding poles while evaluating (4.3), the contour may be deformed slightly into the fluid domain, as was noted by Tchieu et al. (Reference Tchieu, Crowdy and Leonard2010).

The derivatives ${\rm d}W/{\rm d}\zeta$ and ${\rm d}z/{\rm d}\zeta$, and hence, the integrand of (4.3), can be written analytically in terms of the functions $K(\zeta, \alpha )=\zeta \partial \log \omega (\zeta,\alpha )/\partial \zeta$. The integrand may then be explicitly evaluated using the series representations given by Crowdy (Reference Crowdy2020, pp. 278,280), and easily integrated numerically. The result is plotted in figure 5 for the case of two slits near parallel to the real axis.

Figure 5.

Vertical force on the slits shown in figure 3(b), near $\alpha ={\rm \pi} /2$ (slits on real axis). For large $|y/D|$, there is attraction as in the usual Bernoulli gripper. When $y=0$, slits lie on streamlines and there is local repulsion. For tilted slits, there is only repulsion for $y/D\gtrsim 0$ as indicated by the shaded region.

For large positive $y$, the force on the plates is positive, indicating attraction between the source and slits. For large negative $y$, the force on the plates is negative, indicating attraction again. However, a small region of repulsion is encountered near the point $y=0$, when the source is aligned with the slits. At small distances above the point $y=0$, when $y\gtrsim 0$, the source is evidently repelled by the slits as the force on the slits is negative. Just below the point $y=0$, when $y\lesssim 0$, the force on the slits becomes positive, indicating attraction again since the source is still above the slits. When $y$ becomes sufficiently negative, so that it lies below the slits, the force on the slits again becomes negative, indicating attraction as we already noted.

All force plots vanish when $y=0$, and the force switches from attractive to repulsive. The location of this zero is precisely predicted by the discussion of § 2.2: when the natural source streamlines align with the slits, at $y=0$ as seen in figure 3(b), there exists a local energy minima as a function of $y$.

When the slits are parallel to the real axis, $\alpha ={\rm \pi} /2$, the force is an odd function of $y$. Interestingly, the odd symmetry is quickly disrupted for deviations from $\alpha ={\rm \pi} /2$. For $\alpha <{\rm \pi} /2$, the repulsive region is retained, but the repulsive force magnitude is diminished. Since decreasing $\alpha$ pushes the reference point $y=0$ further away from the plates (see figure 3b), it is to be expected that the magnitude of the repulsive force should decrease since the source flow decays with distance. Despite its diminishing magnitude, the repulsive region persists for $\alpha < {\rm \pi}/2$, as is seen in figure 5.

As was found in the single-body calculation of § 4.2, we expect the repulsion found in the two-body problem to be valid up to some finite slit widths. However, we do not present computations for finite slit widths in the two-body problem.

Additionally, we note that no Kutta condition was imposed in the calculation of § 4.3. The forces in figure 5 should thus be interpreted as those on slightly inflated slits with zero circulation and smooth corners. If the slits were truly perfect, a Kutta condition would be necessary, which would affect the vertical force. By slightly inflating the ellipse, the singularity in the conformal map is pushed off of the object boundary, into the exterior of the fluid domain; hence, this interpretation is consistent with our method of calculating (4.3) by pushing the contour slightly into the fluid domain to exclude the map singularity.