Free surface shapes in rigid body rotation: Exact solutions, asymptotics and approximants
FFree surface shapes in rigid body rotation:Exact solutions, asymptotics and approximants.
Enrique Ram´e
Madrid, 28034 Madrid, Spain
Steven J. Weinstein
Department of Chemical EngineeringSchool of Mathematical SciencesRochester Institute of Technology, Rochester, NY 14623, USA
Nathaniel S. Barlow
School of Mathematical SciencesRochester Institute of Technology, Rochester, NY 14623, USA
Abstract
We analyze steady interface shapes in zero gravity in rotating right circular cylindrical con-tainers under rigid body rotation. Predictions are made near criticality, in which the interface,or part thereof, becomes straight and parallel to the axis of rotation. We examine geometrieswhere the container is axially infinite and derive properties of their solutions. We then exam-ine in detail two special cases of menisci in a cylindrical container: a meniscus spanning thecross section; and a meniscus forming a bubble. In each case we develop exact solutions forthe meniscus height and the bubble length as infinite series in powers of appropriate rotationparameters; and we find the respective asymptotic behaviors as the shapes approach their criti-cal configuration. Finally we apply the method of asymptotic approximants to yield analyticalexpressions for the height of the meniscus and the length of a spinning bubble over the wholerange of rotation speeds. As the spinning bubble method is commonly used to measure surfacetension, the latter result has practical relevance. a r X i v : . [ m a t h . NA ] F e b Introduction
The shapes of fluid interfaces in rigid body rotation have been well studied, with the spinning bub-ble tensiometer being a notable example, see [1, 2]. Since about the mid 1950s, interest in suchproblems has grown with the need to engineer fluid containers in zero gravity for spacecraft, whereguaranteeing a known location for the liquid phase is crucial for rockets to fire properly. Seebold[3] performed a stability analysis of menisci in circular cylindrical containers in rigid-body rotationwith arbitrary axial gravity and contact angle. He derived stability limits by a variational analysisof the Hamilton principle. Later Preziosi & Joseph [4] analyzed the stability of periodic interfaceshapes in rigid body rotation by minimization of a suitable energy potential under conditions ofnegligible gravity, obtaining results that are wholly consistent with those of Seebold [3].Similar to a static meniscus in a gravitational field, the shape of the interface between two im-miscible fluids in rigid body rotation with angular velocity ω , density difference ∆ ρ ≡ ρ − ρ ( > ) and surface tension σ , depends on the rotational Bond number λ ≡ ∆ ρ ω d /σ , where d issome appropriate length that depends on the geometry being considered. The studies cited aboveshow that a critical value λ c exists such that, when λ → λ c from below, the interface undergoesa critical transition, with an outcome that depends on the configuration of the fluid body and theparticular container. Figure 1 illustrates some typical configurations.More specifically, Seebold [3] showed that menisci having finite axial length and spanning theentire cross section of a cylinder with contact angle α at the cylinder wall (such as shown in fig.1a), exist in zero gravity under rigid body rotation for λ < f ( α ) , where f ( α ) is some functionof the contact angle and f (0) = 1 . When λ approaches f ( α ) the meniscus reorganizes with adivergent axial length into the shape of a straight cylinder (fig. 1b). Preziosi & Joseph [4] showedthat non-straight periodic interfaces that develop in rigid body motions for λ < become straightcylinders when when λ ≥ . This result is entirely consistent with the analysis of Seebold [3],who collected extensive experimental evidence (and also intuited without proof) that shapes be-come straight cylinders when λ ≥ λ c . Ross [5] studied the shapes of rotating drops and bubbles (asshown in fig. 1a for α = 0 and fig. 1c), and obtained several important results and interpretationsthat coincide with those of Seebold [3] and Preziosi & Joseph [4] in the case of bubbles, and withthose of Chandrasekhar [6] in the case of drops. 2 π /2 π /4 CL r=RGas Liquid H H Hz αθ r (a) CL r=Rr c GasLiquid (b)
CL Gas bubbleLiquid R b (c) Figure 1:
Schematics of surface configurations with fluids in rigid body rotation about the axis of acylindrical container. Liquid is below the interface. (a): typical shapes at λ < λ c , for wall contactangles, α = 0 , π/ and π/ , as marked. θ is the slope angle of the interface. z = 0 is chosen so that thevolumes of liquid above and below are equal. H is the meniscus length. (b): Shape with λ very closeto λ c , for 90-degree contact angle. Most of the surface is a straight circular cylinder of radius r c . Thevortex height diverges as λ → λ c . (c): Spinning bubble. R b is the maximum radius. The half-bubble oneither side of the equator is mathematically identical to the zero-contact angle case in fig. 1a. In this work we derive new results about properties of menisci in rigid body rotation. Specifically,our goal is to develop analytical tools to describe the axial length of menisci in configurations ofpractical relevance. We analyze two configurations, depicted schematically in fig. 1: 1) a meniscusspanning the cylinder cross section, as in figs. 1a and 1b; and 2) a meniscus forming a bubble whoseaxis coincides with the axis of rotation, as in fig. 1c. The latter is the same geometry of the spinningbubble tensiometer [1]. For each geometry, we examine the respective shapes and axial lengthsas a function of λ . In particular, we identify the asymptotic behaviors of the divergence in axiallength (denoted H in fig. 1) as λ → λ c and find that the divergence law depends on the meniscusconfiguration. We then develop exact solutions for these lengths as infinite series in powers of λ .Since the infinite series solutions converge very poorly near λ c , we apply the method of asymptoticapproximants [7] to describe axial length uniformly over the whole range ≤ λ < λ c for bothconfigurations. 3 Analysis
Consider a liquid of density ρ in contact with a gas of negligible density in a cylindrical containerof radius R , rotating with angular velocity ω about its axis in rigid body rotation. The gas-liquidinterface has surface tension σ , its location is z = h ( r ) , and obeys the normal component of thedynamic boundary condition with a pressure whose gradient arises from the centrifugal accelerationonly.Neglecting the dynamics of the gas and using d as a characteristic length scale, the dimnesionlessgoverning equation is given as r ddr (cid:18) rh (cid:48) √ h (cid:48) (cid:19) = − P − λ r , (1) λ ≡ ρω d σ , (2)where h (cid:48) ( r ) = tan θ is the slope relative to the r -axis, P is the pressure difference across the inter-face at r = 0 made dimensionless with σ/d and λ is the rotational Bond number. (If the meniscuscontacts an interior surface that prevents it from reaching the center line, then P is the pressuredifference at r = 0 of the static extrapolation of the meniscus back to r = 0 .) In general, the char-acteristic length d depends on the configuration considered, as sketched in fig. 1. If the interfacespans the entire radius of the container, then d = R ; in the case of a bubble wholly surrounded bythe liquid, then d is the maximum radius of the bubble.Since h (cid:48) ( r ) = tan θ , the left-hand side of eq. (1) may be written as d ( r sin θ ) / ( r dr ) . This equa-tion may be integrated once to obtain an expression for sin θ vs r using two appropriate boundaryconditions. One boundary condition accounts for the axial symmetry and is imposed to all the casesstudied here: θ (0) = 0 . (3)The other boundary condition is applied at the outer end of the interface. In the case of a meniscusspanning the container, the interface obeys a contact angle condition: θ (1) = π − α, (4)4here α is the contact angle, as depicted in fig. 1a. In the case of a bubble wholly surrounded byliquid, the second boundary condition is: θ (1) = π . (5)Physically, this condition reflects the equatorial symmetry of the bubble; but mathematically it canbe seen to be identical to the case of a meniscus with contact angle α = 0 expressed in eq. (4). Theshape for arbitrary contact angle α is therefore: h (cid:48) √ h (cid:48) = sin θ = r cos α + λ r (1 − r ) . (6)where, from the preceding discussion, α must be set to zero to describe a bubble. A volume condi-tion, specific to each configuration determines h ( r ) after one additional integration. λ c r ( θ ) Figure 2:
Sine of the slope angle, vs. r at various λ for α = π/ , i.e., θ (1) = 0 . When λ < λ c , themaximum slope corresponds to an inflection point with slope < θ < π/ . Having shown that eq. (6) may be used both for a meniscus spanning the container with arbitrarycontact angle α and for a wholly immersed bubble by setting α = 0 , in this section we examine thecase of α = π/ , i.e., normal contact at the container wall. No generality is lost by focusing on thisspecial case; on the contrary, we will show below that, far from being special, critical shapes forarbitrary α may be obtained from the shape at any other α by suitable scaling manipulations. Fig. 2shows sin θ vs r for various λ . When λ < λ c , the slope attains a local maximum smaller than π/ ,hence the maximum slope coincides with an inflection point. However, at λ c the local maximum5s θ max = π/ , and the axial length of the meniscus diverges with infinite slope and with a straightcylindrical shape of radius equal to the location where θ max = π/ –in agreement with previousstudies, [3, 4]. To find this location, we solve (with α arbitrary) sin θ = 1 , d sin θdr = 0 , (7)to find λ c = 4 r c , r c = 12 cos (cid:0) ( π − α ) (cid:1) . (8)For α = π/ , λ c = 12 √ and r c = 1 / √ . Eqs. (8) provide exact relations in support of thenumerical results of Seebold [3] for zero gravity. In particular, if one views the vertical slope loca-tion, r c , as the radius of a straight circular cylinder, the rotational Bond number can be written as λ c ≡ r c λ c = 4 ; this result agrees with previous work [3, 4].An important property of critical shapes is that α = π/ generates a master shape from which allother critical shapes with ≤ α ≤ π may be constructed. The method is as follows: Starting withthe critical shape for α = π/ , and denoting the independent variable as r ∗ , we set λ = λ ∗ c = 12 √ and, using eq. (6), determine r ∗ w such that the slope is π/ − α , i.e., r ∗ w is the location where the crit-ical master shape has the same slope as the wall contact slope of interest: cos α = ( λ ∗ c / r ∗ w − r ∗ w ) ,i.e., r ∗ w is the location where the critical master shape has the same slope as the wall contact slopeof interest: r ∗ w = 2 √ (cid:18)
13 ( π − α ) (cid:19) . (9)We then rescale r ∗ , r = r ∗ r ∗ w . (10)Substituting for r ∗ and manipulating to expose the binomial ( r − r ) , we obtain the critical shapewith θ (1) = π/ − α : sin θ = r cos α + λ c r − r ) , (11)where λ c satisfies eqs. (8). This is the same equation that is solved in eq. (6) with arbitrary α for λ = λ c . 6 .3 Meniscus spanning the container radius When the meniscus spans the cylinder radius R , we identify d = R . One integration of eq. (6) witha volume condition fixes the absolute height of the interface. In a reference frame where the liquidvolumes above and below z = 0 are equal: (cid:90) h rdr = 0 . (12)The integration to compute h ( r ) must be performed numerically. We focus on the case of normalcontact ( α = π/ ). This case is special only because the maximum slope location is r c = 1 / √ for all λ . Apart from this distinction, interface shapes are qualitatively similar when contact is notnormal; and, as stated in sec. 2.2, interface shapes at criticality are easily scaled across differentcontact angles. r - - - z Figure 3:
Shapes with normal contact at r = 1 (contact angle α = π/ ) for values of λ noted in legend.The shape nearest z = 0 is for λ = 5 . Consecutive values of λ apply to shapes further away from z = 0 ,showing the divergence of the depth as λ → λ c = 12 √ ≈ . ... . λ ≈ √ Fig. 3 shows interface shapes z = h ( r ) when α = π/ , obtained numerically by integrating h (cid:48) given by eq. (6). It is clear that the axial length of the meniscus, H = (cid:82) h (cid:48) dr , diverges as λ → λ c .Identifying the leading asymptotic behavior of this divergence is of considerable theoretical andpractical interest because it helps control devices such as rotating reactors where two immisciblefluids of differing densities are present. Since h (cid:48) ( r c ) → ∞ as λ → λ c , it follows that h (cid:48) develops7 narrowing peak around the maximum slope location, r c ; and that the area under the peak –thoughdivergent– depends to leading order on the shape of this peak only, i.e., it is independent of thedetails outside the peak. To begin, we identify the radial scale around the peak. Let (cid:15) ≡ λ c − λ, η ≡ r − r c (cid:15) p , (13)where (cid:15) (cid:28) , η is a stretched radial distance centered at the peak, and p needs to be determined.Approximating h (cid:48) ∼ √ − sin θ near r c , and substituting r and λ from eq. (13), we find that − sin θ ∼ (cid:15) √
318 + (cid:15) p η , (cid:15) → . (14)This suggests that, for h (cid:48) to be integrable at η = 0 we must have p = 1 / so that h (cid:48) ∼ h (cid:48) asy = 13 √ (cid:15) (cid:113) √ + η . (15)Thus, the leading behavior of H may be found in closed form from H ∼ (cid:90) (1 − √ ) √ (cid:15) − √ (cid:15) dη (cid:113) √ + η , (16)and then expanding for (cid:15) → : ●●●●●●●●●●●●●●●●●●●●●● - - - - - ( λ * - λ ) H Figure 4:
Comparison of numerical and asymptotic evaluations of depth H . λ c − λ = (cid:15) and λ c = 12 √ .Black dots: numerical. Dashed line: eq. (17). H ∼ H asy = −
13 ln (cid:15) + H . (17)8 is an O (1) constant and is found by numerical integration, H = lim (cid:15) → (cid:90) ( h (cid:48) − h (cid:48) asy ) dr ≈ . , (18)Fig. 4 shows the agreement between the numerical and asymptotic evaluations of the meniscusdepth. H ( λ ) In order to have a description of H ( λ ) over the rest of the λ -domain, we now seek a series solutionfor H ( λ ) in powers of λ about λ = 0 . Since sin θ ∼ λ for α = π/ , it follows that h (cid:48) ( r, λ ) ,obtained from eq. (6), h (cid:48) ( r, λ ) = sin θ (cid:112) − sin θ , (19)may be expanded in a Taylor series about λ = 0 as h (cid:48) ( r, λ ) = ∞ (cid:88) n =0 a n +1 (sin θ ) n +1 = ∞ (cid:88) n =0 a n +1 (cid:18) r − r (cid:19) n +1 λ n +1 , (20)where a = 1 , a n = a n − (cid:18) n − n − (cid:19) , n = 3 , , , ... (21)As seen in fig. 2, sin θ remains below 1 for λ < λ c . Thus, the series in eq. (20) converges for λ < λ c and may be integrated term by term to obtain the meniscus length: H ( λ ) = ∞ (cid:88) n =0 a n +1 b n +1 λ n +1 , (22)where b n = 18 n (cid:90) ( r − r ) n dr, n odd . (23)It is easy to show that b = 1 / and b n +2 = 164 ( n + 2)( n + 1) (cid:0) n +12 (cid:1)(cid:0) n +72 (cid:1) (cid:0) n +52 (cid:1) (cid:0) n +32 (cid:1) b n . (24)Even though this series solution was motivated by a desire to examine the small- λ behavior of H , theability to compute all the terms of the infinite series permits evaluation of the radius of convergenceof the series given in eq. (22). The ratio criterion guarantees convergence iff lim n →∞ a n +1 a n − b n +1 b n − λ < . (25)9valuation of this criterion using eqs. (21) and (24) shows that the series does converge for λ < √ λ c as was stated above. Thus, eq. (22) is an exact solution. Unfortunately, though, theconvergence is poor and nonuniform with increasing λ beyond λ ≈ due to the influence of thelogarthmic singularity, see sec. 2.3.1. In sec. 2.5 we use asymptotic approximants to generate arapidly converging and uniform representation of H ( λ ) over the entire range ≤ λ < √ . If the gas volume in a finite container is small enough, or the axial dimension of the container islong enough, the bubble can become critical (i.e., it can adopt a straight circular cylindrical shape)before the interface touches the end plates of the container. This is the basis for the well-knownspinning bubble method to determine surface tension [1, 2]. In this geometry (see fig. 1c) and inzero-gravity, λ < λ c = 4 , and the characteristic length, d , is the maximum bubble radius, R b . Forextensive detail on the challenges of interpreting and operating the spinning drop tensiometer in agravitational field, see Manning & Scriven [8] and references therein.In this section we derive the asymptotic behavior of the bubble shape as λ → , which can be usedto better inform the bubble length being measured; and develop a solution for the bubble length asa series in powers of λ . Using the bubble maximum radius, R b , as the characteristic length, theslope angle of the interface relative to the r -axis is found by integrating eq. (6) subject to θ (0) = 0 , θ (1) = π/ : sin θ = r + λ r (1 − r ) . (26)Since sin θ = h (cid:48) / √ h (cid:48) , numerical integration with a volume condition yields the interfaceshape, z = h ( r ) . The bubble has infinite slope ( θ max = π/ ) for all λ at r = 1 , in contrast withthe vortex analyzed in sec. 2.3 where the maximum slope is at an inflection point with θ max < π/ and < r < , and θ max → π/ only as λ → λ c . To probe the character of the spinning bubbleshape, we note that, when λ < , dr/dz = 0 at ( r, z ) = (1 , but d r/dz < there. Thissuggests that − r ∼ z , so that sin θ ∼ − A (1 − r ) as r → for some constant A . In contrast,when the shape is critical at λ = 4 , the end-cap shape approaches a straight cylinder asymptoticallyat a distance from the bubble tip that is large compared with the radius; therefore, in the criticalcondition, − r ∼ exp[ B ( H − z )] for some constant B , where ( r, z ) = (0 , H ) is the tip location.This implies that sin θ ∼ − C (1 − r ) as r → for λ = 4 and some constant C . We conclude,10 r ( θ ) Figure 5:
Shapes of spinning bubbles at various λ = 1 , , , . Only λ = 4 has zero slope d (sin θ ) /dr at r = 1 . ● ● ● ● ● ● ● ● ● ● ● ● ● - - - - - ( - λ ) H Figure 6:
Length of spinning bubble vs. λ . Dots: numerical, eq. (26). Dashed line: Asymptotics, eq.(31). therefore, that the critical shape requires d (sin θ ) dr = 0 at r = 1 , λ = 4 (27)whereas subcritical shapes satisfy d (sin θ ) dr > at r = 1 , λ < . (28)Not surprisingly, the shapes of eq. (26), a few of which are shown in fig. 5, display these properties.In contrast to the meniscus spanning the cylinder radius, the distinct character of the spinning bubbleconfiguration is that sin θ = 1 always at r = 1 ; but d (sin θ ) /dr (cid:54) = 0 at r = 1 unless λ = λ c = 4 .11 .4.1 Asymptotics for λ ≈ We derive the asymptotic behavior of the shape as λ → by the same method of sec. 2.3.1. Let (cid:15) ≡ − λ and η ≡ ( r − /(cid:15) p , where p > is to be determined. It may be shown that, when (cid:15) issmall, sin θ ∼ − (cid:15) p η + (cid:15) p η ... (29)The only choice that ensures integrability at η = 0 is p = 1 , yielding h (cid:48) ∼ h (cid:48) asy = 1 (cid:15) (cid:113) η − η . (30)The half-length of the near-critical bubble is then H asy = (cid:90) − (cid:15) (cid:113) η − η dη ∼ − √ − λ ) + H , λ → . (31)where the constant H ≈ . is found using the method of eq. (18). Fig. 6 shows a comparisonof the numerical and the asymptotic interface shapes.The (dimensionless) volume depends on λ : V ( λ ) = 2 π (cid:90) h ( r, λ ) rdr (32)and provides a relation between the dimensional volume and maximum radius. The closer thebubble is to the critical configuration the closer its shape is to a straight cylinder of radius 1, hencethe volume grows progressively more linearly with πH as λ → . A fit of the numerical volumeevaluation shows that V ∼ πH − . , H (cid:29) , (33)in good agreement with the asymptotic behavior of Ross’ exact expression for the volume (eqn. 15in ref. [5]) as λ → . H ( λ ) In order to construct a Taylor series representation of the bubble half-length, we note that, as inSec. 2.3, h (cid:48) = sin θ/ (cid:112) − sin θ . However, in contrast to that analysis, here sin θ (cid:54) = O ( λ ) , which12 r1 H Figure 7:
Schematic of a half-bubble spinning about the z − axis. The blue area represents the liquidand is equal to the volume deficit of the actual bubble relative to a straight circular cylinder where thebubble is inscribed. The spinning container is larger than the bubble size and is not shown. H V Figure 8:
Volume vs bubble half-length H . Dashed line: V asy = πH − . . complicates evaluation of a Taylor series in powers of λ for H ( λ ) = (cid:82) h (cid:48) dr . One way to obtainthis series is to first generate the series h (cid:48) ( r, λ ) = ∞ (cid:88) n =0 c n ( r ) λ n . (34)Let sin θ = A ( r ) + λB ( r ) (35)where A = r and B = r − r . Now the denominator in the expression for h (cid:48) is (cid:112) − A − λAB − λ B = (cid:88) j =0 a j λ j / , (36)13here a = 1 − A , a = − AB and a = − B . Using J.C.P. Miller’s formula for the seriesexpansion of a series raised to any power [9], we evaluate the series for the inverse of (36): (cid:88) j =0 a j λ j − / = ∞ (cid:88) j =0 b j λ j , (37)to find the following recursion for the coefficients, c = A b c n> = A b n + B b n − , (38)where b = 1 √ − A , b = AB √ − A ; b n> = − n (1 − A ) (cid:20)(cid:18) − n (cid:19) A B b n − + (1 − n ) B b n − (cid:21) . (39)Since c n ’s are linear combinations of b n ’s, convergence properties of the series in eq. (34) can bedetermined from those of (cid:80) n b n λ n . Dividing through by b n − in eq. (39) we form two ratios ofconsecutive b n . Assuming that this ratio has a limit, R ∞ ( r ) , say, as n → ∞ , solving a quadraticequation yields R ∞ ( r ) = ( r + r ) / . Based on the ratio criterion, convergence is guaranteed iffMax r [ R ∞ ( r )] λ < , i.e., λ < . As in the problem of sec. 2.3, the series in eq. (34) converges inthe entire range of λ where shapes exist, i.e., ≤ λ < , and is therefore an exact solution. It cantherefore be integrated term-by-term to produce another convergent exact solution for H ( λ ) , e.g., H ( λ ) = ∞ (cid:88) n =0 (cid:18)(cid:90) c n ( r ) dr (cid:19) λ n = ∞ (cid:88) n =0 C n λ n (40)In the appendix we show an evaluation of C n without recursion. Because convergence of series40 is poor as λ increases beyond λ ≈ . , we show in Sec. 2.5 how to implement the methodof asymptotic approximants to obtain an analytical expression for H ( λ ) that is uniform across theentire range ≤ λ ≤ . Asymptotic approximants provide uniformly convergent approximations to H ( λ ) defined in eqs.(22) (the height of a rotating meniscus) and (40) (the length of a spinning bubble) over their entirerespective intervals ≤ λ ≤ λ c . The method has most recently been revisited and reexaminedby Barlow et al. [7], and interested readers may consult their article and references therein for an14xtensive presentation of the method applied to problems covering a broad range of physics.Briefly, asymptotic approximants go beyond Pad´e in that they incorporate asymptotic behaviorsthat are often singular in ways other than just poles [10], thus dramatically improving the approxi-mant’s power to extend the region of convergence. Both power series for H ( λ ) in the present work(eqs. (22) and (40)) have logarithmic divergence at their respective λ c . Because we have the powerseries expanded about λ = 0 , as well as the logarithmic divergence behavior as λ approaches λ c , anasymptotic approximant may be used to join these behaviors. In the two problems considered here,we propose the following approximant for H ( λ ) defined in eqs. (22) and (40): H A ( λ, N ) = N (cid:88) n =0 A n ( λ c − λ ) n + A L + B L log( λ c − λ ) , (41)where A L and B L have been computed from the respective asymptotic analyses in eqs. (17) and(31). The coefficients A i are determined from the condition that the N -term Taylor series of H A ( λ, N ) about λ = 0 is equal to the N -term Taylor series of H ( λ ) (eqs. (22) and (40)). Theform in eq. (41) imposes the fully stripped-off asymptotic logarithmic divergence as λ → λ c . ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●● λ H Figure 9:
The height of the rotating meniscus versus λ . Dots: numerical calculation. Line: 20-term H A ; A L = 1 . , B L = 1 / . At λ = 0 the interface is flat, therefore H (0) = 0 . The form of an asymptotic approximant to a given function is not uniquely determined but ex-perience allows one to pose forms that exhibit superior convergence. In this work we have notattempted to optimize the form of the approximant that minimizes the number of terms N requiredto produce a given error. Figures 9-10 show the numerically calculated H ( λ ) and H A ( λ, N ) for the15 ●●●●●●●●●●●●●● λ H Figure 10:
The half-length of the spining bubblen versus λ . Dots: numerical calculation. Two solidlines, the 5-term and 10-term H A , are indistinguishable; A L = 1 . , B L = 1 / √ . At λ = 0 and inzero-gravity, the bubble is spherical, i.e., H (0) = 1 . rotating meniscus and the spinning bubble, respectively.We define the error of the N -term approximant, Err N ( λ ) , as the pointwise absolute error between H A ( λ, N ) and the numerical values of H . Both approximants seem to converge to the numer-ical calculation as N increases; but, as shown in figs. 11 and 12, the convergence has a smallnon-uniformity near λ c . This is a well-known phenomenon due to remaining singularities that ap-proach zero as λ → λ c [11]. The largest errors of the most accurate approximants calculated areErr ≈ × − for the rotating meniscus and Err ≈ . × − for the spinning bubble. λ Δ Figure 11:
Pointwise error, | H ( λ ) − H A ( λ ) | , at numerically calculated points for the rotating meniscusfor approximants with number of terms N as indicated.
10 151 2 3 4 λ Δ Figure 12:
Pointwise error, | H ( λ ) − H A ( λ ) | , at numerically calculated points for the spinning bubblefor approximants with number of terms N as indicated. In configurations with arbitrary wall contact angle, the shape is described by eq. (6), rewritten here, sin θ = r cos α + λ r − r ) . (42)For a given α it is instructive to find the smallest λ for which sin θ has a maximum at r = 1 . Byrequiring that d (sin θ ) /dr = 0 at r = 1 this value is found to be λ min = 4 cos α . In order to achievecriticality in < r < , λ must be greater than λ min . The location of the maximum slope for λ > λ min is found to be r = (cid:20) (cid:18) λ cos α (cid:19)(cid:21) / . (43)Inserting this result into the expression for sin θ above, we find the maximum value of sin θ for agiven α as function of λ : sin θ | r = 13 √ λ (cid:18) λ cos α (cid:19) / . (44)This expression becomes 1 when λ = λ c as given in eq. (8). Fig. 13 illustrates this argument for α = π/ . Shapes do exist for λ < λ min but without an inflection point. As λ increases beyond λ min ,the inflection point moves down from r = 1 toward smaller r and the slope at the inflection becomesincreasingly vertical as λ approaches λ c . Fig. 14 shows the location of the inflection (where sin θ ismaximum) versus λ .The length of the rotating meniscus has two qualitatively different configuration types. The first17 min λ c r ( θ ) Figure 13:
Shapes for α = π/ (60-deg contact angle) at various rotation rates. λ min = 2 and λ c ≈ . . The location of the maximum at each λ is plotted below in fig. 14. type is associated to contact angles larger than zero. In these cases, the radial position of the maxi-mum slope, r , corresponds to an inflection point, e.g., h (cid:48) ( r ) > , h (cid:48)(cid:48) ( r ) = 0 (i.e., zero curvature)for all λ < λ c . The normal contact case we analyzed in sec. 2.3.1 is typical of this category, andin all these cases the coefficient of the logarithmic divergence is / . The second type of behaviorhas a single element in the zero-contact angle case. This case has always infinite slope at r = 1 ,but there is a non-zero curvature at the wall for all λ < λ c given by r (cid:48)(cid:48) ( z ) , at the contact point ( r, z ) = (1 , H ) , see fig. 7. The coefficient of the logarithmic divergence for the zero-contact anglecase is / √ . Perhaps more significantly, this case is mathematically identical to the spinning bub-ble.Our results for the spinning bubble have distinctly practical implications for the measurement ofsurface tension. The analysis of eq. (26) shows that when λ = 4 the bubble length diverges –while volume conservation requires that the radius of the straight-cylindrical region decrease. Thisis why the spinning bubble can only operate arbitrarily close to but not at λ = 4 . In practicalterms, however, this distinction is unimportant since, when an instrument spins to produce a bubblewith, say, H = 5 (i.e. the half length is 5 times the maximum radius), the theory indicates that λ ≈ − . × − . Hence, for all practical purposes a bubble with H = 5 can be consideredto be in critical configuration with λ = 4 . Joseph’s [2] argument that bubbles with H > canbe considered to be at λ = 4 is inaccurate, as the present calculation predicts that λ ≈ . when H = 2 . A bubble with H = 2 may thus not be long enough to be considered “critical”. As the plot18 λ r Figure 14: r vs. λ between λ min = 2 and λ c ≈ . , for α = π/ (60-deg contact angle). of fig. 8 shows, H = 2 is close to but not yet in the limiting long- H regime where volume increaseslinearly with H . The error in surface tension arising from assuming λ = 4 with H = 2 is 13% butdrops to 0.07% for H = 5 .In principle the experimenter need not assume λ = 4 , however, since we now have the ratio ofbubble half-length to maximum radius, H ( λ ) , described with a uniformly convergent asymptoticapproximant (see Sec. 2.5) over the entire range ≤ λ < . Thus, the approximant for the bubblelength allows one to extend surface tension measurement to arbitrary values of λ with just a simpleevaluation of H A ( λ ) which can be measured by the ratio of bubble half-length to maximum radius.In the absence of gravity, the approximant allows measurements in the intermediate- λ region wherethe sensitivity to error in H is still moderate. But working at a lower than critical λ has the drawbackthat it would require precise measurements of both radius and length. We have reexamined the problem of interface shapes in fluid systems under rigid-body rotation witha focus on finding exact solutions and the asymptotics of singular behaviors near λ c . We studiedtwo configurations of practical importance, e.g., a meniscus spanning the rotating container radiuswith arbitrary contact angle at the container wall; and a spinning bubble where the meniscus doesnot contact the container wall. Finding the asymptotic behavior of the each meniscus configurationlength as the critical rotation is approached and the series solution about λ = 0 is important not19nly because of its intrinsic theoretical interest; it also has application in controlling such interfacesin processes of practical relevance, for example, a rotating reactor or a spinning bubble tensiometer.Knowing the form of the asymptotic divergence, one may construct efficient asymptotic approxi-mants to evaluate each meniscus length at any rotation velocity uniformly and without solving adifferential equation numerically.In conclusion, this work provides analyses that advance the interpretation of interface shapes offluids in rigid body rotation. The analyses are strictily valid in zero-gravity, but their validity maybe extended to normal gravity as long as the gravitational Bond number, ρgR /σ , is much smallerthan λ . For two canonical configurations (meniscus spanning container radius and spinning bubble)we have found exact solutions over the whole range of λ and asymptotic behaviors near critical rota-tion. To remedy the poor convergence of the infinite-sum exact solutions we constructed convergentasymptotic approximants that greatly improve the convergence efficiency of the exact solution. Ourresults provide proof of concept of useful analytical calculation tools for applications ranging fromcontrolling rotating reactors to measurement of surface tension with the spinning bubble method. Acknowledgement
One of us (ER) is grateful to R. Balasubramaniam for help formulating initial ideas and calculationsand for providing a critical sounding board during frequent discussions.
Appendix A Evaluation of C n in eq. (40) In this section we show an evaluation for C n that does not require the use of recursions. Because f ( r, λ ) ≡ sin( θ ) ≤ , we attempt expanding the denominator in h (cid:48) = f / (cid:112) − f as a series inpowers of f . This gives: h (cid:48) = f (cid:112) − f = ∞ (cid:88) m =0 f m +1 Γ( + m ) √ π Γ( m + 1) . (45)Note that this is just the product of f by the series of even powers for the denominator. Now expandthe powers of f = r + λ ( r − r ) using binomial theorem: f m +1 = m +1 (cid:88) i =0 (2 m + 1)! i ! (2 m + 1 − i )! (cid:18) λ (cid:19) m +1 − i r i ( r − r ) m +1 − i . (46)20 closed-form is available for the integration of the r-dependence in the above sum: (cid:90) f m +1 dr = m +1 (cid:88) i =0 (2 m + 1)! i ! (2 m + 1 − i )! (cid:18) λ (cid:19) m +1 − i Γ(1 + m )Γ(2 m + 2 − i )2 Γ(3 m + 3 − i ) . (47)We now use this to write the r -integral of eq. (45) as: H ( λ ) = ∞ (cid:88) m =0 Γ( + m )Γ(2 m + 2) √ π m +1 (cid:88) i =0 (cid:18) λ (cid:19) m +1 − i
12 Γ( i + 1)Γ(3 m + 3 − i )= ∞ (cid:88) p =0 C p λ p . (48)It remains to extract the coefficient of λ p , C p . Let m + 1 − i = p . To extract the p -th power fromthe double sum in eq. (48), for each m set i = 2 m + 1 − p in the finite i -index sum. It follows thatthe coefficient of λ p is the result of an infinite sum: C p = 12 √ π p ∞ (cid:88) m = m Γ( + m ) Γ(2 m + 2)Γ(2 m + 2 − p ) Γ(2 + m + p ) . (49)For a given p , the argument of Γ(2 m + 2 − p ) in the denominator cannot be less than 1, i.e., m + 1 ≥ p . This sets the lowest m in the sum, as m ≥ m ≥ ( p − / . When p is odd, thiscondition sets the lowest m directly; when p is even, the lowest m is the smallest integer that islarger than ( p − / . References [1] Bernard Vonnegut. Rotating bubble method for the determination of surface and interfacialtensions.
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