## Abstract

We study the rapid dissolution of a dense rigid body (a cylinder or sphere; density *ρ*_{p}), fixed in a uniform flow of speed *U* (density *ρ*_{f}), within an inviscid framework. The body consists of *N* shells, and dissolves through a series of discrete steps, where after each shell dissolves, it is pushed out by a radial flow. The flow through the surface of the body pushes bound vorticity into the ambient flow, creating free vorticity and satisfying, at the same time, Kelvin's circulation theorem. The circulation of each shell is described by a Lagrangian equation describing the position of each shell and an integral relationship between the circulation of the shells dissolved and the bound vorticity. The thermal component is neglected from this analysis.

The impulse of the exterior fluid increases to compensate for the decrease in the momentum of the body. The bodies are fixed as they dissolve so that the sum of the flow impulse and body momentum is only approximately conserved. The impulse of the vorticity field created after the body has disappeared, *I*_{T}, is approximately equal to the initial momentum of the body, *ρ*_{p}*UV* (where *V* is the initial volume of the body). The total circulation in the upper half-plane is *Γ*≈*Γ*_{0}+*C*_{Γ}(*ρ*_{p}/*ρ*_{f})^{1/N}*a*_{0}*U*, where *N*=2, 3 for a cylinder, sphere, respectively, and *C*_{Γ}∼*O*(1) a constant. The total kinetic energy associated with the vorticity field after the body has disappeared is *T*∼(1/2)*ρ*_{p}*C*_{T}*VU*^{2}, where *C*_{T}<1. The sum of the kinetic energy of the flow and body is not conserved.

After the cylinder/sphere has disappeared, the vorticity distribution rolls up to produce a dipolar vortex/vortex ring. The properties of these dipolar structures are estimated assuming a final form and the conservation of *T*, *Γ* and *I*_{T}. A dense cylinder dissolves to create a vortex that has a radius 1.3*a*_{0}(*ρ*_{p}/*ρ*_{f})^{1/2} and moves with a speed 0.28*U*, while a dense sphere dissolves to create a vortex of radius 0.65*a*_{0}(*ρ*_{p}/*ρ*_{f})^{1/3}, which moves with a speed 0.50*U*.

## 1. Introduction

A number of studies have examined dissolving two-dimensional infinitesimally thin plates (e.g. Klein 1910; Betz 1950). Moore (1975) analysed the roll-up of a vortex sheet created by a rigid plate with bound vorticity suddenly removed (or dissolved). His calculations (and others) have provided general insight into the roll-up of vortex sheets shed from a three-dimensional lifting surface. The roll-up of the vortex sheet ultimately produces a dipolar vortex. Taylor (1953) studied the dissolution of a circular disc moving perpendicular to its flat surface and used integral invariants, such as impulse, circulation and kinetic energy, to estimate the ultimate radius, core size and the speed of propagation of the vortex ring. Although the final state of the vortex ring was assumed to be a hollow-cored vortex ring, Taylor's calculations provide a leading-order description of the eventual form of the vortex. Saffman (1992, ch. 6) summarized these physical processes. More recent numerical calculations have compared the long time evolution of a vortex created by a dissolving cylinder and sphere (with the same density as the ambient fluid). While a Taylor-type argument provides a leading-order description of the vortex dynamics for a few eddy turnover times, ultimately a fraction of vorticity is detrained from a spherical vortex sheet rolling up (Nitsche 2001).

Rottman *et al*. (1987) studied theoretically and experimentally the development and final form of a cylinder of fluid released in a uniform flow. This work was motivated by environmental concerns relating to the spread of a release of finite release of neutrally buoyant contaminant (from rest) in a crossflow. The main component of the problem was studied by considering the inviscid problem of a cylinder that dissolves instantaneously and they drew on Taylor's (1953) original model. An energy argument was used to estimate the ultimate size and speed of propagation of the dipolar vortex using the work of Pierrehumbert (1980). The generation of a dipolar vortex by translating and removing an open ended cylinder—identical to the Rottman's experiments—has been variously applied in geophysical flow experiments. Eames & Flór (1998) used this method in their study of the transport of dipolar vortices in rotating systems. While the separation of the cylinder wake contributes vorticity to the final dipolar vortex, provided the cylinder is displaced only a fraction of its radius before it is removed, most of the vorticity is generated by the shear layer created by the forward motion of the fluid within the cylinder and the external fluid. Rottman *et al*. (1987) also studied the vortex generated by a cylinder of fluid whose density was different from the ambient fluid, but did not study dense bodies that dissolve to produce fluid of the same density as the ambient fluid.

The purpose of this paper is to examine the permanent flow signature that remains after a body loses mass and disappears rapidly in the absence of a gravitational force. Regardless of the thermal effect of a phase change due to the loss of mass, momentum must be globally conserved so that the consequence of any body disappearing is to generate a permanent vortex signature whose momentum/impulse is equal to the sum of the initial momentum of the ambient fluid and body. Most studies of evaporating droplets have always been led with treatments of the viscously dominated flows, either numerically or asymptotically (see Del Alamo & Williams 2007). But in this paper, we try to understand the inviscid limit corresponding physically to droplets moving and evaporating over a short distance compared with their overall size and where the evaporative time scale is much shorter than the viscous time scale. Despite the overwhelming simplification adopted in this paper, where a body disappears through a series of discrete steps, the shell model introduced is successful in explaining how the momentum of the body is communicated to the ambient fluid as it loses mass.

To understand some of the fundamental issues relating to a body rapidly dissolving in an inviscid fluid, we consider the examples of a cylinder (§2) and sphere (§3). We consider the case when a body of density *ρ*_{p} and volume *V* dissolves to generate fluid with the same density as the ambient fluid, and restrict our attention to bodies that are denser than the ambient fluid (*ρ*_{p}/*ρ*_{f}≥1). The effect of the dissolution process is to generate a flow through the surface of the boundaries in order that mass is conserved. The formulation that we describe is based on an analysis of the circulation of fluid elements. After the bodies have dissolved, the vortex sheets will ultimately roll-up to produce dipolar vortical structures. We estimate in §4 the characteristics of the dipolar vortical structures using the conservation of impulse, kinetic energy and circulation, assuming a prescribed form of the vortex. Our results are discussed more generally in §5, where we put the results into a physical context.

## 2. Dissolving circular cylinder

We analyse a cylinder of initial radius *a*_{0} and density *ρ*_{p}(≥*ρ*_{f}) fixed in a uniform stream −*U* of inviscid fluid and which dissolves in an ambient fluid of density *ρ*_{f}. The analysis presented is identical to when the cylinder is projected with a velocity *U* into a fluid at rest. To simplify our discussion, we assume that the cylinder dissolves instantaneously and symmetrically about its centre through a series of discrete steps and the dissolution creates fluid of density *ρ*_{f}. The pressure gradient generated by the mass loss is radial and perpendicular to the symmetric circular shells, so that the baroclinic torque on the dissolved boundaries is necessarily zero. The vorticity created by the dissolution process is due to the conversion of bound vorticity to free vorticity.

### (a) Fundamental concepts and model formulation

Consider a cylinder of density *ρ*_{p}/*ρ*_{f}=1 whose boundary dissolves at a rate |d*a*/d*t*| much faster than *U*, the speed of the ambient flow (figure 1*a*). At *t*=0, the tangential slip velocity at the surface of the cylinder is 2*U* sin *θ* (Batchelor 1967) and corresponds to bound vorticity *ω*=2*U* sin *θδ*(*r*−*a*_{0}). As the cylinder dissolves, bound vorticity is converted to free vorticity and a circular vortex sheet of radius *a*_{0} is created. Since *ρ*_{p}/*ρ*_{f}=1, the flow within the circular vortex sheet encapsulating the cylinder (*a*<*r*<*a*_{0}) is stationary due to the combination of free and image vorticity and the incident flow (figure 1*b*). The bound vorticity at the cylinder surface is zero because the tangential velocity at the surface of the cylinder is zero and so no further vorticity is swept into the ambient fluid. The momentum associated with the bound, image and free vorticity components are *I*_{b}=*ρ*_{f}*πa*^{2}*U*, *I*_{i}=−2*ρ*_{f}*πa*^{2}*U* and , respectively. Thus, the sum of the momentum of the cylinder and exterior flow is conserved, i.e. (since *ρ*_{p}=*ρ*_{f}). When the cylinder has disappeared (*a*=0), the total circulation associated with the positive vorticity is *Γ*=4*a*_{0}*U* and impulse . These results are identical to Rottman *et al*. (1987; except for a typographical error in (2.28)), though our interpretation is different.

When *ρ*_{p}/*ρ*_{f}≠1, a radial source flow is generated in the region *r*>*a* by the dissolving cylinder, where(2.1)For *ρ*_{p}/*ρ*_{f}>1, a positive volumetric source is generated, which pushes material radially outwards; for *ρ*_{p}/*ρ*_{f}<1, a sink flow is generated. Since *ρ*_{p}/*ρ*_{f}>0, the flow relative to the surface of the cylinder is outwards, since *u*_{r}−d*a*/d*t*>0, which means that there is always a through surface flow.

The cylinder is envisaged to consist of *N* shells where the *i*th shell has radius *a*_{i}=(1−*i*/*N*)*a*_{0} and *a*_{N}=0. We consider a cylinder that dissolves rapidly through a series of discrete steps, where, after each shell dissolves, it is pushed out by the radial source (2.1). The rate of dissolution is assumed to be sufficiently high that the radial speed of the source flow is much larger than the ambient flow and the shells remain circular after dissolving. As the first shell is created and pushed radially outwards, the slip velocity on the cylinder increases (figure 2*b*), so at the next stage when the second ring dissolves, bound vorticity is created on the cylinder surface, which creates the circulation associated with the second shell (figure 2*c*). Image vorticity is required to ensure the kinematic condition on the surface of the cylinder is satisfied. A circular vortex sheet created at the *j*th step is stretched to a radius *R*_{v}(*j*; *i*) when the *i*th shell dissolves, where(2.2)Equation (2.2) is a discrete form of (2.1).

For a rapidly dissolving cylinder, we track the circulation of fluid elements that form part of the individual shells. Because the initial slip velocity is proportional to sin *θ*, bound and free vorticity associated with the circular vortex sheets will also have a sin *θ* dependence. If the *j*th shell has local circulation *κ*_{j} sin *θ* (where *κ*_{j} is constant on the shell) and is pushed out to a radial distance *R*_{v}(*j*; *i*), the free vorticity consists of a circular vortex sheet that generates a horizontal velocity *κ*_{j}/2*R*_{v}(*j*; *i*) within the shell. The radial and azimuthal components of flow inside a circular vortex sheet are *u*_{r}=*κ*_{j}/2*R*_{v} cos *θ* and *u*_{θ}=−*κ*_{j}/2*R*_{v} sin *θ*, respectively. The image vorticity required to satisfy the kinematic boundary condition on the cylinder consists of a circular vortex sheet of radius and circulation −*κ*_{j} sin *θ* (see Saffman 1992). The circular image vortex sheet has components and (for ). In combination, the image and free vorticity satisfy the kinematic constraint on the cylinder (since *u*_{r}=0 at *r*=*a*_{i}) and create a tangential velocity *u*_{θ}=−*κ*_{j} sin *θ*/*R*_{v}(*j*; *i*) on the surface of the cylinder. The tangential velocity at the surface of the cylinder when the *i*th shell has dissolved is equal to the sum of the contribution from the flow around the cylinder (2*U* sin *θ*) and previous stretched vortex sheets encapsulating the cylinder . The local circulation of the *i*th circular vortex sheet is *κ*_{i} sin *θ*, where(2.3)and *κ*_{0}=2*a*_{0}*U*. The properties of all the shells can be calculated from (2.2) and (2.3). The shell model is consistent with the results for *ρ*_{p}/*ρ*_{f}=1, since *κ*_{0}=2*a*_{0}*U* and *κ*_{i}=0 for *i*≥1.

### (b) Diagnostics

To interpret the results of the model, we calculate the impulse as the cylinder disappears, the final impulse, circulation and the kinetic energy after the cylinder has dissolved. For *ρ*_{p}/*ρ*_{f}=1, the force on the cylinder is zero as it disappears and the impulse and kinetic energy are conserved. The cylinder is constrained at the origin, and for *ρ*_{p}/*ρ*_{f}≠1 a source flow does work by displacing the vorticity field outside the cylinder. Kinetic energy is not conserved during this process because the rate of decay of total kinetic energy is equal to , which is negative. And while constraining the cylinder at the origin means that momentum is not globally conserved, only a fraction is lost due to this assumption.

The total impulse of the vortex shells after the *i*th shell (1≤*i*≤*N*) has dissolved is (see Batchelor 1967; Saffman 1992)(2.4)The second term in the brackets of (2.4) is the impulse of the image vorticity, which is zero after the cylinder has dissolved when *a*_{N}=0. The total circulation *Γ* associated with the vorticity in the region *y*≥0 is (since , where *A* is the region *y*≥0)(2.5)For *ρ*_{p}/*ρ*_{f}≥1, the circulation is positive in the top half of the cylinder and negative in the bottom half, with the net circulation being zero.

The kinetic energy of the perturbation flow generated by the vortex shells is calculated from the radial and azimuthal components of the velocity field *u*_{r} and *u*_{θ} at step *N*. In the region *a*_{i+1}(*ρ*_{p}/*ρ*_{f})^{1/2}<*r*<*a*_{i}(*ρ*_{p}/*ρ*_{f})^{1/2} (within the outer shell), the flow is(2.6)Beyond the outer shell (*r*≥*a*_{0}(*ρ*_{p}/*ρ*_{f})^{1/2}), the velocity components are(2.7)For homogeneous fluids with compact regions of vorticity and no net circulation, the far-field flow is dipolar and of the form(2.8)where *Φ*_{S}=(log *r*)/2*π* is the velocity potential for a unit source in a planar flow and ** D** is the dipole moment. The dipole moment

**is proportional to the impulse of the flow,(2.9)where is the unit vector parallel to the ambient flow. Equation (2.8) describes the flow in the region**

*D**r*≥(

*ρ*

_{p}/

*ρ*

_{f})

^{1/2}

*a*

_{0}, which can be seen by substituting (2.4) into (2.7) (for

*i*=

*N*).

The kinetic energy of the disturbance to the ambient flow after the cylinder has dissolved is defined by(2.10)The kinetic energy contribution from the flow outside the vortex is . The contribution to the total kinetic energy within a distance *a*_{0}(*ρ*_{p}/*ρ*_{f})^{1/2} of the origin is calculated numerically using (2.6).

### (c) Analysis and numerical results

A sensitivity study of the results to the number of shells *N* is described in appendix A. Providing *N*>1000, the total impulse, circulation and kinetic energy are within 1% of the limiting value of *N*→∞, for density ranges 1<*ρ*_{p}/*ρ*_{f}<1000.

Figure 3 shows the variation of the total impulse for *ρ*_{p}/*ρ*_{f}=1–1000, compared against the initial momentum of the cylinder (*ρ*_{p}*VU*). The difference between *I*_{T}(*N*)=2*ρ*_{f}*VU* and the estimate *ρ*_{p}*VU* is large for *ρ*_{p}/*ρ*_{f}=1, but decreases for large *ρ*_{p}/*ρ*_{f}. Because, in this calculation, the cylinder is constrained to be at the origin, we would not expect momentum to be conserved exactly, the fractional difference for large *ρ*_{p}/*ρ*_{f}>100 is less than 10% and decreases with increasing *ρ*_{p}/*ρ*_{f}.

For *ρ*_{p}/*ρ*_{f}≫1, and the radius of the vortex shells (after the cylinder has disappeared) scale as (*ρ*_{p}/*ρ*_{f})^{1/2}*a*_{0}, we would anticipate *Γ*∼*I*_{f}(*N*)/(*ρ*_{p}/*ρ*_{f})^{1/2}*a*_{0} or *Γ*∼*C*_{Γ}(*ρ*_{p}/*ρ*_{f})^{1/2}*a*_{0}*U*, where *C*_{Γ}∼*O*(*π*). Also, *Γ*=*Γ*_{0} where *Γ*_{0}=4*a*_{0}*U* for *ρ*_{p}/*ρ*_{f}=1. This scaling is borne out in figure 4 where *Γ*=*Γ*_{0} is plotted as a function of (*ρ*_{p}/*ρ*_{f})^{1/N}−1 (where *N*=2). The numerical calculation appears to support the scaling *Γ*−*Γ*_{0}∼*C*_{Γ}(*ρ*_{p}/*ρ*_{f})^{1/2}*a*_{0}*U*, where *C*_{Γ}≈2.5.

Figure 5*a* shows the variation of *κ*_{i} with the shell radius *a*_{i}, for *i*=1, …, *N* for large *ρ*_{p}/*ρ*_{f}.1 The circulation is made dimensionless with *κ*_{i}/(*ρ*_{p}/*ρ*_{f})^{1/2}Δ*aU*, where Δ*a*=*a*_{0}−*a*_{1} is the initial separation of the shells. For large *ρ*_{p}/*ρ*_{f}, the numerical results collapse on to a single curve and support the scaling *κ*_{i}∼(*ρ*_{p}/*ρ*_{f})^{1/2}*a*_{i}Δ*aU*/*a*_{0}. For (*a*_{0}−*a*_{i})/*a*_{0}≪1 and *ρ*_{p}/*ρ*_{f}≫1, from (2.3) the outer shells are approximately described by(2.11)For 1≪*i*, *j*≪*N*, the summation can be approximated as a convolution integral,(2.12)and solved using Laplace transform theory, to give(2.13)where *Γ*_{f} is the gamma function. Figure 5*b* shows a good comparison between (2.13) and the numerical results close to the outer shell for 1≪*i*≪*N*. As *i*→1, the circulation is not singular since it is readily demonstrated from (2.3) that *κ*_{1}/(*ρ*_{p}/*ρ*_{f})^{1/2}Δ*aU*∼(*ρ*_{p}/*ρ*_{f})^{1/2}. Equation (2.13) provides a leading-order description of *κ*_{i} within the outer 10% of the shells and integrating (2.13) over *i*, we obtain *Γ*≈*Γ*_{0}+4.2(*ρ*_{p}/*ρ*_{f})^{1/2}*Ua*_{0}. Although this estimate of the leading coefficient is higher than *C*_{Γ}, it lends support to the results presented in figure 4.

The streamlines of the perturbation flow created by the vortex shells are calculated from (2.6) and (2.7) and plotted in figure 6*a*. For *ρ*_{p}/*ρ*_{f}=1, the flow inside the circular vortex sheet is uniform and has speed *U* (figure 6*a*(i)). For *ρ*_{p}/*ρ*_{f}>1, the circulation of the inner shells is positive, which causes the streamlines within the vortex to converge/diverge (figure 6*a*(ii)). The streamlines have a finite change of direction at the edge of the outer vortex shell because the outer cylindrical vortex sheet has a finite circulation (per unit length).

Figure 7 shows the variation of the final kinetic energy (calculated from (2.7), (2.10) and (2.6)) with *ρ*_{p}/*ρ*_{f}. The fitted straight line to the results gives *T*∼(1/2)*C*_{T}*ρ*_{p}*UV* for *ρ*_{p}/*ρ*_{f}≫1, where _{T}≈0.62. This is less than the sum of the initial kinetic energy of the cylinder and outer flow, 1/2(*ρ*_{p}+*ρ*_{f})*U*^{2}*V*, indicating that a fraction is lost during the dissolution process.

Figure 8*a* shows the fractional reduction of the impulse of the free and image vorticity compared with the final value, *I*_{T}(*i*)/*I*_{T}(*N*), as a function of the fraction of the shells *i*/*N* that have disappeared. Since the impulse in the exterior flow is generated to conserve momentum, we would anticipate that the total impulse increases at the expense of the reduction of the momentum of the cylinder. For comparison, the fractional loss of cylinder momentum,(2.14)is plotted with symbols. For *ρ*_{p}/*ρ*_{f}=1, the numerical calculations and model are exactly identical. The prediction (2.14) is within a few per cent of the numerical results shown in figure 8*a*.

## 3. Dissolving sphere

Consider a sphere of density *ρ*_{p}, initial radius *a*_{0}, fixed in a uniform flow −*U* and which consists of *N* shells. The sphere is assumed to dissolve instantaneously and symmetrically. When expressed in terms of circulation, the model and diagnostics for studying a dissolving sphere are similar to a dissolving cylinder.

### (a) Modelling aspects

For *ρ*_{p}/*ρ*_{f}=1, the sphere dissolves with no source flow (figure 1). The tangential slip velocity at the edge of a sphere fixed in a uniform flow is (3/2)*U* sin *θ* (Batchelor 1967). The azimuthal vorticity field associated with the bound vorticity is *ω*=(3/2)*U* sin *θδ*(*r*−*a*_{0}). The impulse associated with the bound, image and free vorticity is *I*_{b}=2*ρ*_{f}*πa*^{2}*U*/3, *I*_{i}=−2*ρ*_{f}*πa*^{3}*U* and , respectively, so that the sum of the momentum of the body and vorticity field is . The total impulse of flow inside and outside the sphere is initially (*ρ*_{p}+*C*_{m}*ρ*_{f})*UV* and is conserved while the sphere dissolves.

For *ρ*_{p}/*ρ*_{f}≠1, the dissolution of the sphere generates a source flow(3.1)The radius of a spherical vortex sheet, which was created at the *j*th step increases to *R*_{v}(*j*; *i*) when the *i*th shell dissolves, where(3.2)

If the *j*th shell has local circulation *κ*_{j} sin *θ* (where *κ*_{j} is constant on the shell) and is pushed out to a radial distance *R*_{v}(*j*; *i*), the free vorticity consists of a spherical vortex sheet that generates a horizontal velocity 2*κ*_{j}/3*R*_{v}(*j*; *i*) within the shell. The radial and azimuthal components of flow inside a spherical vortex sheet are *u*_{r}=(2*κ*_{j}/3*R*_{v})cos *θ* and *u*_{θ}=−(2*κ*_{j}/3*R*_{v})sin *θ*. The image vorticity required to satisfy the kinematic boundary condition on the sphere consists of a spherical vortex sheet of radius and circulation (see Saffman 1992). The spherical image vortex sheet has components and (for ). In combination, the image and free vorticity satisfy the kinematic constraint on the sphere (since *u*_{r}=0 at *r*=*a*_{i}) and create a tangential velocity *u*_{θ}=−*κ*_{j} sin *θ*/*R*_{v}(*j*; *i*). The tangential velocity at the surface of the sphere when the *i*th shell has dissolved is equal to the sum of the contribution from the flow around the sphere ((3*U*/2)sin *θ*) and previous stretched vortex sheets encapsulating the sphere . The local circulation of the *i*th spherical vortex sheet is *κ*_{i} sin *θ*, where(3.3)The circulation associated with the first shell is *κ*_{0}=(3/2)*Ua*_{0} sin *θ*.

### (b) Diagnostics

At each stage of the calculation (e.g. step *i*, 0≤*i*≤*N*), the total impulse associated with the spherical free and image vortex sheets, defined by(3.4)is calculated. The second term on the r.h.s. of (3.4) is the impulse associated with the image vorticity field. After the sphere has disappeared, the total circulation *Γ*, defined by(3.5)is calculated (since where *A* is the region *y*≥0). For axisymmetric flows, the definition of total circulation is identical to that for two-dimensional flows and is conserved even when the vorticity is stretched.

The radial and azimuthal components of the flow, *u*_{r} and *u*_{θ}, are calculated from the individual contributions of the vortex shells. For *r*<*a*_{0}(*ρ*_{p}/*ρ*_{f})^{1/3}, the velocity components at a distance *r* from the origin, where *a*_{i+1}(*ρ*_{p}/*ρ*_{f})^{1/3}<*r*<*a*_{i}(*ρ*_{p}/*ρ*_{f})^{1/3}, are(3.6)Beyond the outer shell (*r*>*a*_{0}(*ρ*_{p}/*ρ*_{f})^{1/3}), the velocity components are(3.7)The flow (3.7) is described by (2.8), where *Φ*_{S}=1/4*πr*^{2} is the velocity potential for a unit source flow in a three-dimensional flow and the dipole moment is (2.9).

The kinetic energy of the flow disturbance created by the vorticity field after the sphere has dissolved is defined by(3.8)The contribution to the total kinetic energy from flow within a distance *a*_{0}(*ρ*_{p}/*ρ*_{f})^{1/3} of the origin is calculated numerically using (3.6). The contribution to the total kinetic energy from flow outside a distance *a*_{0}(*ρ*_{p}/*ρ*_{f})^{1/3} is .

### (c) Analysis and numerical results

The sensitivity of *I*_{T}, *T* and *Γ* to the number of shells *N* is similar to those of a cylinder, requiring *N*>1000 for convergence (see appendix A).

Figure 3 shows the variation of the final impulse *I*_{T}(*N*) with the density of the sphere. For *ρ*_{p}/*ρ*_{f}=1, the total impulse of the final vortex system is and includes both the initial momentum of the sphere and its added-mass contribution, as confirmed in these calculations. As *ρ*_{p}/*ρ*_{f} increases, the fractional difference between the total impulse and the initial momentum of the sphere decreases and is much less than 1% for *ρ*_{p}/*ρ*_{f}>10.

The total circulation that remains after the sphere has disappeared is plotted in figure 4*b* as a function of its initial density. Because (for *ρ*_{p}/*ρ*_{f}≫1) and the radius of the shells (after the sphere has disappeared) scale with (*ρ*_{p}/*ρ*_{f})^{1/3}*a*_{0}, we would anticipate the total circulation scales as *Γ*∼(*ρ*_{p}/*ρ*_{f})^{1/3}*a*_{0}*U* for *ρ*_{p}/*ρ*_{f}≫1. Figure 4*c* shows a total circulation plotted as *Γ*−*Γ*_{0} against (*ρ*_{p}/*ρ*_{f})^{1/N}−1 (where *N*=3). An approximate line through the data gives a coefficient *C*_{Γ}≈2.4.

For (*a*_{0}−*a*_{i})/*a*_{0}≪1 and *ρ*_{p}/*ρ*_{f}≫1, from (3.3) the circulation of the outer shells are approximately described by(3.9)For 1≪*i*, *j*≪*N*, the summation can be approximated as a convolution integral,(3.10)and solved using Laplace transform theory, to give(3.11)where *Γ*_{f} is the gamma function. Figure 5*c*,*d* shows the variation of the final circulation distribution for dense spheres. In figure 5*c*, the circulation close to the centre varies as . Close to the edge of the outer shell, the circulation tends to take the asymptotic form described by (3.11), as indicated in figure 5*d*.

As with the case of a cylinder, the streamline pattern of the vortex shells created immediately after the sphere has disappeared (figure 6) show straight and divergent streamlines, depending on whether *ρ*_{p}/*ρ*_{f}=1 or >1. The streamlines have a finite change in direction at the edge of the outer shell because the circulation is finite, even in the limit of *N*→∞.

The kinetic energy associated with the vortex shells is calculated and plotted in figure 7 for *ρ*_{p}/*ρ*_{f} varying from 1 to 1000. A straight line of the form *T*∼(1/2)*ρ*_{p}*C*_{T}*VU*^{2} is fitted to the data (where *C*_{T}≈0.73). Because *C*_{T}<1, a fraction of the kinetic energy is lost during the dissolution process.

Figure 8 shows the comparison of the impulse ratio *I*_{T}(*i*)/*I*_{T}(*N*) with the fraction of the shells *i*/*N* that have dissolved. Because the impulse in the exterior flow is generated to conserve momentum, we would anticipate that the total impulse increases at the expense of the reduction of the momentum of the sphere. For comparison, the fractional loss of sphere momentum,(3.12)is plotted with symbols. For *ρ*_{p}/*ρ*_{f}=1, the numerical calculations and predictions (3.12) are identical. The agreement between (3.12) and the numerical results are quite good, indicating physically that the decrease in the momentum of the sphere is compensated by an increase of the impulse of the exterior flow.

## 4. Characteristics of the dipole vortex signature for dense bodies

After the dissolution of the dense cylinder and sphere, the flow consists of a compact region of vorticity that has a net zero circulation. Following this, the vorticity goes through an inviscid adjustment, during which kinetic energy, impulse and circulation (of the positive/negative components of vorticity) are conserved, to finally produce a dipolar/ring vortex. As with Klein (1910), Taylor (1953), Rottman *et al*. (1987) and Saffman (1992, ch. 6), the final state of the vortex can be estimated using the integral constraints of kinetic energy, impulse and circulation. An inherent weakness of this approach, commented on by Saffman (1992), is that a form of the vorticity distribution within the vortex cores must be assumed. Nevertheless, it is instructive to estimate the characteristic size and speed of the resulting vortex and compare them with the initial size of the body and the free-stream velocity.

### (a) Dense cylinder

The mean distance of the centroid of vorticity from the vortex centre,(4.1)is a scalar invariant. The integrals are taken over the region *y*≥0, denoted by *A*. By dimensional analysis, the dipolar vortex created by the inviscid roll-up of the vorticity field, has a radius *R*_{ω} and moves with a speed *U*_{ω}, where(4.2)The coefficients *α*_{2} and *β*_{2} depend on the assumed form of the final vortex. If the final state is assumed to consist of cores of uniform vorticity, the coefficients *α*_{2} and *β*_{2} are determined by interpolating table 1 of Pierrehumbert (1980), using the dimensionless kinetic energy . Figure 9 shows the variation of the vortex radius and speed of propagation as a function of *ρ*_{p}/*ρ*_{f}>1.

These results can be contrasted against those obtained by assuming that the final state is a Lamb's dipolar vortex. The impulse of a Lamb's dipolar vortex is , where the speed and circulation are related through *U*_{ω}=−*CkJ*_{0}(*kR*_{ω}) and , where *kR*_{ω}=3.83. In this case, the coefficients(4.3)are constants. Figure 9 shows a comparison between the results assuming a uniform core and a circular dipolar vortex. The difference is small, suggesting that the vortex core is fat and close to a Lamb's dipolar vortex. The vortex finally produced is estimated to move with speed 0.28*U* and have a radius 1.3*a*_{0}(*ρ*_{p}/*ρ*_{f})^{1/2}.

### (b) Dense sphere

For axisymmetric flows, the mean squared distance of the vorticity from its centre is a scalar invariant, i.e.(4.4)where the region *y*≥0 is denoted by *A*. By dimensional analysis, the final radius of the vortex ring and speed of propagation scale is(4.5)The coefficients *α*_{3} and *β*_{3} depend on the assumed form of the final vortex ring.

If the final state is a steady vortex with a non-circular core of vorticity whose strength is proportional to distance from the centreline, we can interpolate table 2 of Norbury (1973), using the dimensionless kinetic energy to determine *α*, a dimensionless measure of vortex core size, by matching against (where , , and are, respectively, the dimensionless kinetic energy, circulation, impulse and velocity used by Norbury 1973). Once *α* is determined, the coefficients and are calculated by interpolating over table 2 of Norbury (1973). If the final state is a Hill's spherical vortex, where and *U*_{ω}=*Γ*/5*R*_{ω}, then(4.6)

Figure 9 shows a comparison between the results assuming a non-circular core and a spherical vortex. The difference is small, suggesting that the vortex core is fat and close to a Hill's spherical vortex. The vortex finally produced is estimated to move with speed 0.50*U* and have a size 0.65*a*_{0}(*ρ*_{p}/*ρ*_{f})^{1/3}.

## 5. Discussion and conclusions

The generation of vorticity in an inviscid fluid by rigid bodies dissolving is well known and is interpreted as bound vorticity being converted to free vorticity (Saffman 1992, ch. 6). In this paper, we have studied dense bodies that dissolve using a shell model of the dissolution process, where each shell dissolves and then pushes outwards. Despite the simplicity of this shell model that is described by (2.3) and (2.4) or (3.3) and (3.4), it has some important features. First, it explains how the impulse of the exterior flow is increased by the source flow that stretches or separates the opposite-signed free vorticity. The increase in the impulse of the free vorticity compensates for the decrease in the momentum of the body. Second, it shows that the final impulse of the exterior flow is approximately equal to the initial momentum of the body.

In general, the sum of momentum of the body and the impulse of the free, image and bound vorticity is conserved (see Eames *et al*. 2007). In our model, the cylinder and the sphere were fixed at the origin and this is why the final impulse of the vortex is not exactly equal to the initial sum of the momentum of the body (*ρ*_{p}*UV*) and impulse of the flow *ρ*_{f}*C*_{m}*UV* (where *C*_{m} is the added-mass coefficient), except when *ρ*_{p}/*ρ*_{f}=1. But, as we see from figure 3, the fractional difference between *I*_{T}(*N*) and *ρ*_{p}*UV* decreases as *ρ*_{p}/*ρ*_{f} increases. Indeed, this difference is small for a dissolving sphere. Kinetic energy is conserved for *ρ*_{p}/*ρ*_{f}=1, but in general this quantity is not conserved during the dissolution process because the source flow does work by stretching or displacing the vorticity field.

The dissolution process generates an initial outer shell that has a circulation 4*a*_{0}*U* or 3*a*_{0}*U* (for a cylinder and sphere, respectively) and is independent of *N*. Physically, this corresponds to the vortex sheet studied by many authors, but here a source flow is present which increases the impulse of the sheet. The inner shells certainly depend on the shell thickness Δ*a*=*a*_{0}/*N*. But the main point is that the vorticity distribution, expressed as circulation per unit area, converges for large *N*. Physically, it appears that within an inviscid context, the flow consists of a singular vortex sheet (the first shell) and a continuous distribution of vorticity within this shell. As shown in appendix A, the total impulse, circulation and kinetic energy converge for large *N*.

This work was motivated by a physical problem of droplet evaporation, where viscous effects could be important, even for rapid evaporation. The flow within these droplets tends to be initially small, relative to the droplet velocity, and increases by the action of the weak tangential shear stress acting on their surface. In many cases, an adequate description of the flow around such droplets is to apply a non-slip condition on the surface of the droplets (see Del Alamo & Williams (2007) for a brief discussion). In this paper, we have analysed the problem within an inviscid context. The radial blow velocity sweeps the bound vorticity away from the surface of the body and reduces the slip velocity to zero. The main effect of viscosity is to diffuse the vorticity field and this effect is small when the evaporative time scales are much shorter than the viscous time scale. Although it is usual to discount these types of idealized inviscid calculations, in this case the inviscid analysis is of relevance to the high Reynolds number flow. For the case of a fuel droplet rapidly evaporating, *ρ*_{p}/*ρ*_{f}≈700 and we would expect the vortex ring that is finally generated to move with speed 0.5*U* and have a radius 5.8*a*_{0}. To our knowledge, only Frohn & Roth (2000) have reported the generation of a vortex by a moving droplet explosively evaporating. Certainly from the short sequence they present, the vortex after the droplet has disappeared is visible, but the final image only shows the vortex roll-up and not its final state. These observations are encouraging, but further experimental work is required to make a comparison with our predictions.

## Acknowledgments

I.E. gratefully acknowledges support from the Philip Leverhulme Prize (2005), EPSRC overseas travel grant (EP/E029302/1), Royal Society overseas travel grant and a Royal Academy of Engineering Global Research Award (2007). Prof. George Bergeles kindly provided support at the National Technical University of Athens, where this work was completed.