Shaking a container full of perfect liquid; a tractable case, a torus shell, exhibits a virtual wall

Manipulation (‘shaking’) of a rigid container filled with incompressible liquid starting from stationary generally results in some displacement, or mixing, of the liquid within it. If the liquid also has zero viscosity, a ‘perfect’ or ‘Euler’ liquid, Kelvin’s theorems dramatically simplify the flow analysis. Response is instantaneous; stop the container and all liquid motion stops. In fact an arbitrary manipulation can be considered as alternating infinitesimal translations and rotations of the container. Relative to the container, the liquid is stationary during every translation. Infinitesimal rotations (an infinitesimal vector along the rotation axis) resolve into three orthogonal components in the container frame. Each generates its own infinitesimal liquid displacement vector field. Their combined consequences are usually obscure. Rather than a volume flow, a surface flow in 3D is considerably easier, the liquid slipping freely in a shell, sandwiched between two nested closed surfaces with constant infinitesimal gap. The closedness avoids extra boundaries. The two dimensionality admits a scalar streamfunction determined by the container angular velocity vector. Manipulation of the container angular velocity at will, usually leads to an infinitely rich variety of area preserving re-configurations of the liquid. In particular, any chosen point of the liquid can be moved, in the shell container frame, to any other chosen point. However, for a torus (shell) with a small enough hole (diameter < 0.195 torus diameter), there exists a virtual wall, a hypothetical axial cylinder intersecting the torus. No matter how the torus is manipulated, liquid inside the cylinder stays inside; outside stays outside. The analysis, solving for the streamfunction, is based on the relative vorticity, and the conformal mapping of a torus to a (periodic) rectangle, which lead to a fairly simple convolution integral formula for the flow.

There is a virtual cylinder construcJon associated with surface flow on a rigid torus (driven by torus manipulaJon) provided the torus is fat enough.The cylinder is somewhat wider than the hole in the torus and therefore intersects it in two parallel circles or 'twin rings' as shown.This hypotheJcal cylinder consJtutes a virtual wall: no manipulaJon whatever of the torus can cause any liquid to cross it.Liquid starJng outside stays outside, inside stays inside.

Mo+va+on and results, pictorially
ManipulaJon ('shaking') of a rigid container filled with incompressible liquid starJng from staJonary generally results in some displacement, or mixing, of the liquid within it.This applies even if the liquid has zero viscosity -a perfect liquid, treated here.One might quesJon this statement -think of a liquid filled spherical container -a simple rotaJon of the container about its centre obviously does nothing at all to the liquid which remains staJonary, slipping fricJonlessly.But, in a frame of reference fixed in the container, the liquid has moved -by a rigid counter rotaJon.This same tendency is present for a general container shape.Kelvin's theorems (recalled at the end of this secJon) dramaJcally reduce the flow analysis to geometry (rather than dynamics).From his KineJc energy theorem, response is instantaneous; stop the container and all liquid moJon stops.In fact an arbitrary manipulaJon can be considered as alternaJng infinitesimal translaJons and rotaJons of the container.RelaJve to the container, the liquid is staJonary during every translaJon.For infinitesimal rotaJons, however (except around a symmetry axis), the infinitesimal liquid displacement fields are much less trivial.There is a separate such field for each of three rotaJon axis direcJons in the container frame and the consequences of a sequence of them are obscure.
Striving for yet more simplicity, reduced dimensionality is ocen a^racJve; here realized by a rigid container that 'sandwiches' a liquid in a shell of uniform infinitesimal thickness so that the flow is confined to a surface.The analyJcal benefit of 2D is the existence of a scalar streamfuncJon and Kelvin's Constant circulaJon theorem relates this funcJon in the next secJon, to the container angular velocity.A boundaryless 'closed' surface is simplest.Except for a sphere (spherical shell) for which any sequence of rotaJons leads to a 'rigid' relaJve displacement of the liquid, the freedom to manipulate the container angular velocity at will, leads to an infinitely rich variety of area preserving re-configuraJons of the liquid.It is natural to expect that at least, in common with the sphere, any chosen point of the liquid can be moved to any chosen point in the shell container.This will prove not to be the case (fig 1), a fat torus providing a striking counterexample.Less symmetric shell shapes than the torus, however, seem very likely to admit, like the sphere, the arbitrary point to arbitrary point displacement that is lacking in the fat torus.
Specializing, from now on, to the torus, natural axes in the torus frame of reference are the symmetry axis a, and two orthogonal 'diametral' ones.The relaJve flow consequence of turning the torus about its symmetry axis is obvious.The liquid merely counter rotates rigidly with fricJonless slipping.RotaJon about a diametral axis at angular velocity w^ causes a more elaborate flow pa^ern that needs calculaJng in the remaining secJons.(The streamline pa^ern is independent of |w^|).Some features are forced by symmetry and have an important consequence later.The two circles in the mirror symmetry plane normal to the turning axis (figs 2 and 4) are necessarily streamlines, though with uneven flow speed.
There is thus no flow across either of them, and, by incompressibility, there can be no net flux across any longitude circle f=const (fig 2).The same zero net flux also applies to every laJtude circle q=const.Again, zero flux for a single circle suffices, say the largest q =p, which has zero net flux by the p rotaJonal symmetry of the flow about the diametral turning axis.Fig 2 .VerJcal cross secJon bisecJng a horizontal torus, defining its shape and size.The raJo r/R describes the torus shape, from thinnest to fa^est: 0<r/R<1.The raJo can be neatly expressed in solid angle terms: it is the proporJon of all 4p solid angle taken by the torus when viewed from its centroid C. Natural coordinates on the torus are: a laJtude-like one q (but with -p<q<p) as shown, and a longitude-like one f (with -p<f< p) not shown, measuring azimuth angle around the symmetry axis.
In case a qualitaJve descripJon of the outcome suffices, or perhaps moJvates the proper calculaJon, it can be supplied now, pictorially.There is a difference in the topology of the diametral turning flow for a thin torus, r/R<0.674…and a fat torus, r/R>0.674.The flow streamlines on the torus have the respecJve topologies of contours of w^.r and of w^.n, where n is the unit surface normal vector at r.These separate types of contours are shown in fig( 3).Fig 3 .Turning a torus about a diametral axis (horizontal, at any angular velocity w^) produces flow of a perfect liquid confined to its surface.For visual clarity and analyJcal assimilaJon, the lines shown here correctly capture the flow topology but have the oversimplified analyJcal forms stated.Evidently the two pictures differ.Lec: Thin torus (r/R=0.3<0.674..) with contours of w ^.r drawn.They are somewhat unrealisJc geometrically since the true streamline loops are not typically planar.Right: Fat torus (r/R=0.8>0.674..) with some contours of w ^.n drawn, where n is the unit surface normal vector.The important two horizontal circle streamlines, the 'twin rings', are however, unrealisJcally located at the top and bo^om -they should be closer together as in fig 1, the intersecJons of the torus with a fairly narrow axial cylinder.The cylinder radius for thinner tori approaches the hole radius, the twin rings coalescing and annihilaJng for r/R=0.674.The twin rings are shown correctly located in fig 4 .A fat torus has twin horizontal circular streamlines, 'twin rings', intersecJons of the torus with an axial cylinder as in fig 1. Crucially, and obviously by axial symmetry, this same twin ring pair, applies for any diametral axis of turning of the torus.And for axial turning of the torus, the twin pair is sJll also obviously present, included among the enJrety of horizontal circular streamlines.That means: no liquid can be made to cross the twin pair under any manipulaJon whatever of the torus; liquid inside the intersecJng cylinder stays inside, outside stays outside.the raJos r/R being 0.6, 0.7, 0.8, 0.9, 1.0 (r is the black circle radius, and 2R the centrecentre distance).Also shown for each torus, in red, is the edge-on projecJon (not cross-secJon) of the twin horizontal virtual wall rings lying in the torus surface.The excepJon is the thinnest torus shown with raJo r/R=0.6<0.674, which has no virtual wall rings.The fa^est possible torus r/R=1 (closed hole) has twin rings with verJcal separaJon 2r sin(0.352),computed from (12).The separaJon is graphed in fig 5. Concluding this secJon, Kelvin's two theorems [Lamb 1932] for a perfect liquid (incompressible and having zero viscosity) should be specified.They apply when boundary moJon alone drives the liquid moJon, so this covers the 'shaking' of a full container of perfect liquid.If started from staJonary his Constant convected circulaJon theorem states that the circulaJon (the line integral of velocity around a hypotheJcal test circuit within the liquid) remains zero for any test circuit.This will relate the (Laplacian of the) streamfuncJon, in the next secJon, to the local normal component of container (shell) angular velocity.His KineJc energy theorem dictates that the instantaneous moJon of the liquid is that with the least total kineJc energy consistent with the prescribed normal component of moJon of the boundary.This proves the instantaneous response menJoned earlier; there is no history dependence.

Rela+ve flow streamfunc+on
Since the incompressible liquid shell has uniform infinitesimal thickness the relaJve surface flow is divless.Such a flow has the property that if, at every point, the relaJve velocity tangent vector is hypotheJcally twisted by a right angle about the local surface normal vector (all in the same sense), that new vector field is curlless.It is therefore the gradient of a scalar funcJon s (possibly globally mulJvalued but always locally meaningful -more correctly the gradient is a closed one-form).The div of this right angle-twisted vector field Ñ.(Ñs)ºÑ 2 s equals the curl of the relaJve velocity field which lies along the surface normal direcJon, that is, its relaJve vorJcity.The important points are that this relaJve vorJcity is not zero (the flow is not irrotaJonal in a rotaJng frame), but that it is explicitly known.It is simply expressed in terms of the angular velocity vector w of the rigid shell and the local normal unit vector n of the shell.At all Jmes and at all posiJons The jusJficaJon is as follows.In general for a perfect liquid the vorJcity, the curl of liquid velocity in space (as opposed to relaJve vorJcity), is well known to remain permanently zero everywhere if it starts so.But its interpretaJon is subtle for a flow confined to a moving surface, and it is safest to use the equivalent integral version, Kelvin's Constant convected circulaJon theorem: any chosen closed circuit carried, or convected, by the liquid moJon in 3D, has unchanging line integral of velocity around it, in this case zero, since the liquid starts staJonary.In parJcular, infinitesimal circuits have permanently zero circulaJon.This circulaJon is twice the infinitesimal circuit area Jmes the dot product of n with the local angular velocity vector of the liquid in space.But this angular velocity in space equals the angular velocity w of the shell plus the angular velocity of the liquid relaJve to the shell frame of reference, ln, say, since it must point along the local surface normal.So one has 2(w+ln).n=0,yielding l=-2(w.n)as the relaJve vorJcity of the liquid on the right hand side of (1).
As indicated before, since the flows produced by components of the torus angular velocity simply superpose, it is convenient to treat them separately, resolving w into axial and diametral vectors: w = w a + w ^ where, in terms of the unit axial vector a, w a=a( w.a) and w ^=w-a(w.a).The streamfuncJon corresponding to w a is obvious without reference to the equaJon (1).The rigid counter rotaJon means r ¶s/ ¶q = -(w.a)(R-rcosq).The streamfuncJon corresponding to w ^ is to derived in the next secJons.It requires the relaJve vorJcity expressed in coordinates with the point  =  = 0 defined as lying on the 'hole waist' circle of radius  −  at the azimuth, say, where w^ and n are anJparallel.From here on, the torus frame of reference will be understood, so the qualificaJon 'relaJve' will be dropped.
This paragraph is a pre-empJve dismissal of a potenJal complicaJon.The velocity field that derives from a given vorJcity field is generally not unique.On a torus there is a two parameter family of undetermined div-less, curl-less flows, briefly described now, but dismissed as zero by the symmetry of the flow from the diametral turning of the torus.One div-less, curl-less, flow has equal circulaJon around all laJtude circles (so speed inversely proporJonal to circle radius).At posiJon vector r, the velocity is v(r) µ (r Ù a)/ (r Ù a) 2 .Another such flow is along longitude circles.It is obtained by rotaJng each previous vector by a right angle (in the same sense) about the local normal vector n.These two flows [Klein 1932], and therefore also any linear combinaJon of them, exhaust all the div-free, curl-free flows allowed by torus topology.The two div-less, curl-less flows described have non zero flux respecJvely across longitude circles, and across laJtude circles.But as noted in the previous secJon diametral turning of the torus causes zero net flux across all longitude and laJtude circles.The desired flow, constructed from a vortex lauce below, has zero fluxes built in.

Conformal map of a torus
The task is to invert the Laplacian to obtain s in terms of the vorJcity -2(w^.n).The difficulty is that the torus is not flat, but it can be mapped in a standard way to a flat rectangle with periodic boundary condiJons on which the Laplace inversion process is more familiar.Once found, the stream funcJon s can be carried back directly onto the torus by the inverse mapping.The laJtude and longitude circles on the torus (fig 2) map to horizontal and verJcal straight lines on the rectangle, but simply taking (f,q ) as cartesian coordinates forming the rectangle is not adequate.
Only for a conformal mapping (angle preserving, thus locally shape preserving) is the Laplacian of a funcJon proporJonal (via the local area raJo) to the Laplacian of the mapped funcJon.For this local shape preservaJon, the rectangle coordinates need to be (f,q ') where q '(q ) is a specific funcJon (4) below, a result apparently [1] going back to Kirchhoff.
Since all the coordinate meshes on the rectangle have the same shape, the funcJon must be such that laJtude lines drawn on the torus at equal intervals of values of q ' (rather than q) form a mesh pa^ern on the torus in which all the mesh shapes are the same (with varying size), fig( 6).The torus laJtude lines have differing circumferences proporJonal to (1-scosq) where s =r/R.For mesh shape matching one therefore needs where the denominator ensures that, when integrated, ′ runs from − to  like .The common aspect raJo of the mesh shapes df ´ dq ' can be read off from the case q = /2 where dq/dq '=1/√1 −  !, so df ´ dq '= df ´ dq / √1 −  ! which on the torus has aspect raJo r/R/ √1 −  != s/√1 −  ! .This is the aspect raJo of the rectangle conformally mapped from the torus with s =r/R.The rectangle is a square at s = 1/Ö2; it is taller than wide for fa^er tori, or wider than tall for thinner tori.The overall size of the rectangle is arbitrary; for definiteness, that with horizontal width 2pR is chosen, the verJcal height then being 2pRs/√1 −  ! .

Vortex pair laAce flow
The vorJcity distribuJon (5) on the periodic rectangle is to be built up as the weight factor in a distribuJon of shiced point vortex lauces, idenJcal rows each having alternaJng ±d-funcJons.The key point of this convoluJon, or 'Green funcJon', approach is that an individual lauce of point vorJces has a known associated flow field (fig 7).This flow involves a special funcJon, the EllipJc funcJon sn.However it is only invoked temporarily since when convolved with the vorJcity distribuJon, the outcome is expressed in terms of elementary funcJons, so in the end no reference to the intricacies of EllipJc funcJons is required.where sn is the EllipJc funcJon in a periodic rectangle (here pictured for the 'modulus' parameter k for which the rectangle is square) that has zero div everywhere, and zero curl except at the two point vorJces of opposite signs (±d-funcJons of curl).Right: The corresponding streamlines are contours of the real part of the integral of sn, explicitly Re∫ sn(, ) For the present periodic rectangle (deriving from the torus) with coordinates (f ,q¢ ), one takes z=(2Kf +iK'q¢ )/p, with K(k) and K'(k) being the standard complete EllipJc integrals with k such that K'/2K= s/√1 −  !, the rectangle aspect raJo where s =r/R.The appropriate streamfuncJon sdd is then simply proporJonal to Re(òsn) with a prefactor such that Ñ 2 sdd =±d/2 at the vortex points.The 2 is needed because there is a pair of opposite vorJces per rectangle and each will contribute equally to the convoluJon because of the anJsymmetry of the vorJcity funcJon.Since the residue of the poles of sn are ±1/k, the circulaJon around each is ±2p/k.Thus the desired streamfuncJon with the point vorJces solving the above Laplacian equaJon is sdd =(k/4p) Re(òsn), wri^en out more fully in ( 6), ( 7) and ( 8) below.This is both a funcJon in the complex z plane (periodic with dimensions 4K´2K') and, via the link above of z to (f ,q¢ ), in the periodic rectangle (dimensions 2pR´2pR s/√1 −  ! ).These rectangles differ only in scale (not aspect raJo).The corresponding velocity fields are related by the reciprocal of the scaling factor, and thus they have idenJcal circulaJon values around corresponding arbitrary test circuits, equal to half the count of vorJces enclosed.

Convolu+on
The convoluJon (between the vorJcity distribuJon and the vortex lauce flow on the rectangle) will eliminate the EllipJc funcJon.This comes about using an idenJty that can be proved by using the Fourier transform product rule for convoluJons, and the known Fourier series [Lawden 1989] of the funcJon sn.In this series only the first term gives a non-zero contribuJon: sn() = !+√ -./($%-) sin(/2) + ⋯ , where  = expM−2/√1 −  !N, and as always s =r/R.The idenJty, with z=(2Kj +iK' " )/p where − <  " <  , and the curly forms ,  " of f, q ' used as integraJon variables henceforth, then reads: The line in braces ( 8) is not essenJal because sn can be replaced by its first Fourier term, as menJoned, leading directly to the next line, but it is included for explicitness.A shic now needs to be taken into account.The point vorJces of sdd (fig 6), are offset verJcally, none is at the origin.But for the purposes of the convoluJon a vortex needs to be located at the origin.The required shic is  " →  " −  so that  22 (, ′) becomes  22 (,  " − ).It would be incorrect, however, simply to make that subsJtuJon in (9), because that has assumed that − <  " < .Outside this range the cosh is truncated and repeated periodically.However the  shic can be accommodated by transfer to the convolving partner (5).
The 2D convoluJon of sdd =(k/4p) Re(òsn) with the vorJcity distribuJon (5) supplying the stream funcJon s(f,q¢) on the periodic rectangle for the diametral turning can finally be wri^en down using ( 9) and ( 5).There is a Jacobian factor R 2 s/√1 −  !converJng between this 2pR´2pRs/√1 −  !rectangle and the 2p´2p range of (f, q ¢).A parJcular case s =r/R=0.7 is chosen here for the contour map (of the flow streamfuncJon).The streamfuncJon is the product of a simple cosine horizontally (f axis), and the convoluJon of the two red funcJons verJcally (q ¢ axis).The other colours correspond to different values s = 0.1, 0.3, 0.5, and 0.9.All of the vorJcity graphs (the right hand set) have two zeros, but when convolved with the chopped cosh graphs (lec hand set) only the sharper pointed ones (0.9 and 0.7) retain two zeros, yielding the twin horizontal streamlines shown for the 0.7 case.The less sharp ones smear away the zeros and would yield a product contour map with two diagonal crosses (separatrices) like those in fig 7 (and, on the torus in fig 3).

Remarks
(i) For the extremes of thin (s=0) and fat (s=1) tori, the convoluJon integral (12) admit analyJc evaluaJon.These evaluaJons will not be carried out, but the appropriate approximaJons are as follows.The thin limit comes from straighworward Taylor expansion in s in the integrand of (12).The fat limit comes from expanding the two funcJons being convolved in (12) about their peaks, namely the gradient disconJnuity for the cosh, and the p argument of the cos.Only the constant, and first non-constant, terms are required, and lead to integrals consisJng of exponenJal decays Jmes simple raJonal funcJons, and therefore resulJng in exponenJal integral special funcJons like Ei.The qualitaJve behaviour in this fat limit is interesJng.The peak widths of the two convolved funcJons are both of order Öe where e=1-s, so the convoluJon is the same order Öe in width.But these are small widths about q¢=p on the rectangle, not on the torus.To convert to the torus one needs the mapping (4) and this undoes the smallness since √1 − /√1 + ~√ (ii) The surprise that any surface shape possesses an intrinsic wall to flow induced by turning is perhaps emphasised by the fact that both more symmetric and less symmetric surfaces lack it.A sphere lacks it, any chosen iniJal point can be driven to any other chosen posiJon by a suitable sequence of rotaJons.Even a spheroid with its axial symmetry like the torus does not exhibit an intrinsic wall because it lacks the circles along which the surface normal is parallel to the axis.A torus is geometrically generated by swinging a circle about an axis in its plane, but the wall phenomenon would sJll be present with a deformed circle instead (though the conformal mapping to a rectangle will be less straighworward [Guenther 2020]).Also the surface of a thick bowl might posess an intrinsic wall.An ellipsoid with disJnct axis lengths would not because there would be no streamline common to all three rotaJon axes.The same would seem to apply to even less simple non symmetric surfaces.
(iii) An aspect of the reconfiguraJon of the liquid that has not been menJoned is angle holonomy.It is quite separate from the wall phenomenon except insofar as it also derives from the geometric nature (Jming irrelevance) of the reconfiguraJon.Angle holonomy in classical mechanics is restricted to an integrable system that is subjected to an adiabaJc excursion of its Hamiltonian [Hannay 1985].(Actually the arena of integrable systems is also tori, but in phase space, and not directly connected with the present real space torus).A simple example is a bead gliding fricJonlessly around a rigid wire ring or hoop.The hoop need not be planar, though the easiest example is a circular one.If the hoop is manipulated so that it ends up where it started, the bead will have acquired an extra shic beyond where it would have travelled to had the hoop not moved at all.If the manipulaJon is adiabaJc (slow compared to the bead's sliding circulaJon) the extra shic (in arc length) equals the solid angle swept out by the area vector, Jmes twice its magnitude, and divided by the hoop's perimeter.(The area vector is half the integral of posiJon vector cross unit tangent vector darclength).This same shic is experienced by a perfect liquid in a thin rigid hollow hoop.Now however there is no adiabaJcity condiJon, the liquid can start staJonary -it is a 1D flow version of the present 2D flow problem, and one should expect an analogous shic holonomy to apply to the torus.TreaJng the torus manipulaJon as infinitesimal alternaJng diametral and axial rotaJons, it is indeed clear that only the la^er ma^er, and the area averaged azimuthal liquid coordinate f shics by the solid angle swept out by the torus axis, in accordance with the 1D formula applied to a circular hoop.
(iv) For the more general problem of bulk 3D flow instead of the surface flow considered here, there is some connecJon with previous study of the external flow problem -'swimming' in a perfect liquid [Saffman 1967] [Hannay 2012].The 'swimmer' could be taken as rigid object (here the empty container) immersed in the liquid filling all the outside space.
The object is manipulated, perhaps by moving a mass around inside.Despite much similarity with the internal flow problem, an obvious difference is that zero container angular velocity vector does not mean that the liquid is staJonary in the container frame of reference frame, the container velocity vector ma^ers too.
(v) Returning to a contained perfect liquid in bulk 3D, there are similariJes and differences with the surface flow problem.Only rotaJons ma^er, and each component of container angular velocity has an associated basis flow, the three of them superposing linearly.Each flow has uniform relaJve vorJcity: minus twice the relevant resolved component of angular velocity.Each is divless but in 3D there is no streamfuncJon to simplify analysis.A virtual wall would be a surface to which all three basis flow velocity fields were tangent.

Fig
Fig 1.There is a virtual cylinder construcJon associated with surface flow on a rigid torus (driven by torus manipulaJon) provided the torus is fat enough.The cylinder is somewhat Fig 4 shows the computed wall circles for several tori of different raJos r/R.

Fig 4 .
Fig 4. Five torus pictures in one.Each shows the bisecJng cross secJon of a torus in black;

Fig 5 .
Fig 5. Graph showing how the separaJon of the twin virtual wall rings in fig 4 depends on the fatness s =r/R of the torus (taking R=1).

Fig 6 .
Fig 6.Lec: the torus with the horizontal, laJtude, circles drawn closer together on the inside proporJonally to the closeness of verJcal, longitude, circles.This creates meshes all having the same shape, though not size.Right: those same horizontal and verJcal circles mapped to straight lines with that same mesh shape, suitably resized.This defines the conformal (local shape preserving) map (4) of the torus (here with s =r/R =0.5) onto a periodic rectangle.It has a definite aspect raJo s/√1 −  ! .

Fig 7 .
Fig 7. Two equivalent pictures of the flow associated with a lauce of fixed point vorJces.Every unit cell contains a single posiJve vortex (one quarter vortex in each corner) and a single negaJve vortex (two halves).Lec: The flow velocity vector field (Im(sn), Re(sn)) where sn is the EllipJc funcJon in a periodic rectangle (here pictured for the 'modulus' parameter k for which the rectangle is square) that has zero div everywhere, and zero curl except at the two point vorJces of opposite signs (±d-funcJons of curl).Right: The corresponding streamlines are contours of the real part of the integral of sn, explicitly Re∫ sn(, )

5!
. (in the relaJvisJc aberraJon context this is called the headlight effect).The result is that the features of the flow, importantly the twin rings in fig 1and fig 4, are not in extreme locaJons on the torus.