Potential flow of a second-order fluid over a tri-axial ellipsoid

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1 Potential flow of a second-order fluid over a tri-axial ellipsoid F. Viana 1, T. Funada, D. D. Joseph 1, N. Tashiro & Y. Sonoda 1 Department of Aerospace Engineering and Mechanics, Universit of Minnesota, 110 Union St. SE, Minneapolis, MN 55455, USA Department of Digital Engineering, Numau College of Technolog, 3600 Ooka, Numau, Shiouka, , Japan Revised 6 Januar, 005) The problem of potential flow of a second-order fluid around an ellipsoid is solved, following general expressions in Lamb 1993), and the flow and stress fields are computed. The flow fields are determined b the harmonic potential but the stress fields depend on viscosit and the parameters of the second-order fluid. The stress fields on the surface of a tri-axial ellipsoid depend strongl on the ratios of principal axes and are such as to suggest the formation of gas bubble with a round flat nose and two-dimensional cusped trailing edge. A thin flat trailing edge gives rise to a large stress which makes the thin trailing edge thinner. Kewords: Potential flow, Second-order fluid, Cusp, Normal extensional stress 1 Introduction Wang and Joseph 004) studied the potential flow of a second-order fluid over a sphere or an ellipse. The potential for the ellipse is a classical solution given as a complex function of a complex variable. The stress for a second-order fluid was evaluated on this irrotational flow. An important result of this stud is that the normal stress at a point of stagnation changes from compression to tension strongl under even mild conditions on the viscoelastic parameters. Here, we extend the three dimensional stud of Wang and Joseph 004) to the case of flow over an ellipsoid whose three principal axes ma be unequal. The solution of Laplace s equation φ = 0) bounded internall b an ellipsoid x a + b + =1, 1) c moving with constant velocit U in the direction x is given b Lamb 1993) p. 15) and Milne-Thomson 1996) p ). Since a is arbitrar their solution is readil adapted to the case of a translating ellipsoid in an of the three principal directions. To be definite we adopt the convention that a>b>c. ) The motion of an ellipsoid in an arbitrar direction ma be formed from superposing of motions in three principal directions. Here we compute solutions relative to a stationar ellipsoid in a uniform stream. Since our goal is the calculation of irrotational viscous and non-newtonian second-order) stresses we must compute working formulas, not in the literature, for velocities, pressure and the derivatives of velocit required to calculate stresses. We first use these formulas to compute the velocit field and pressure for the classical problem of irrotational flow of an inviscid fluid. Then we appl these same formulas to the case of a viscous, second-order, non-newtonian fluid. The main goal of our calculations for the second-order fluid model is to identif mechanisms which lead to two-dimensional cusps at the trailing edge of a gas bubble rising in an unbounded liquid where axismmetric solutions might be expected. We calculate the effects of viscosit, second-order viscoelasticit and inertia. The effects of viscoelasticit are opposite to the effects of inertia; under modest and realiable assumptions about the values of the second-order fluid parameters, the normal stresses at points of stagnation change from compression to tension. The effect of inertia and elasticit are essentiall smmetric in that the depend on squares of velocit and velocit gradients but the effects of viscosit are asmmetric. 1

2 a) b) Figure 1: Two orthogonal views showing the cusped a) and broad b) shape of the trailing edge of an air bubble cm 3 ), rising in a viscoelastic liquid S1). The two photographs are from Liu, Liao & Joseph 1995). For the rising gas bubbles, the effects of the second-order and viscous terms on the normal stress are such as to extend and flatten the trailing edge. These calculations suggest that two-dimensional cusping can be viewed as an instabilit in which a thin flat trailing edge gives rise to a large stress which makes the thin trailing edge even thinner. Fluid mechanics of two dimensional cusping at the trailing edge of gas bubbles rising in viscoelastic liquids. An air bubble rising freel in a non-newtonian liquid tends to be prolate and can develop a cusp at the trailing edge as shown in figure 1a. This cuspidal tale occurs onl in gas bubbles rising freel in non-newtonian liquids. Joseph et al. 1991) defined the cuspidal tails as point singularities of curvature. The also stated that the build-up of extensional stresses near stagnation points ma favor the formation of cusps. In its analsis of cusped interfaces, Joseph 199a) suggests that the strong tendenc for cusping in non- Newtonian fluids is a mechanism for eliminating stagnation points for the relaxation of elongational stresses. Hassager 1979) was the first to show that the cusp was not rotationall smmetric but two-dimensional, with a broad shape in one view and so flat that exhibits a point cusp when observed orthogonall. Liu, Liao & Joseph 1995) presented new experimental evidence of the two-dimensional characteristic of cusped bubbles see figure 1). The also reported the different shapes of the broad edge observed in experiments. A comprehensive review of the literature concerning bubbles rising in non-newtonian fluids and the analsis of two-dimensional cusps at the trailing edge of the bubbles can be found in Liu, Liao & Joseph 1995). From their experimental results for air bubbles of different sies rising in various viscoelastic liquids and in columns of different configurations, the concluded that the formation of cusps is independent of the sie and shape of the column. Liu, Liao & Joseph 1995) reported that the cusping tails occur at the trailing edge of a bubble rising in a non-newtonian liquid for capillar numbers Ca) of 1 or higher. Pillapakkam & Singh 001) developed a code to simulate the deformation of a gas bubble rising in an Oldrod-B liquid. The found that the shape of the bubble depends on both the Capillar Ca) and Deborah De) numbers. The observed that in general, the gas bubble assumes an elongated shape with the frontal part round and when both Ca and De numbers are of the order of 1 a two-dimensional cusp is developed at

3 the trailing edge. The claim to show that the pull out effects of sufficientl large viscoelastic stresses near the trailing edge of the bubble cause the formation of a cuspidal tail. An interesting aspect of the rise of gas bubbles in a viscoelastic liquid is that the fluid in the region behind the bubble moves in the opposite direction of the bubble. This phenomenon was reported for the first time b Hassager 1979) and was termed negative wake. Pillapakkam & Singh 003) presented some numerical results that indicate a negative wake in the region behind gas bubbles rising in viscoelastic liquids. The associated the presence of a negative wake with a certain range of two viscoelastic parameters, the Deborah number and the polmer concentration. The found that for a polmer concentration of, the shape of the fore part of the bubble is round and no negative wake is observed. For polmer concentrations higher than, the reported the existence of a negative wake in the region behind the bubble. Here we show that the normal stress distribution at the surface of the bubble ma cause the formation of a cusp at the trailing edge. We anale the effect of the normal stress on the shape of a gas bubble b computing the normal stress at the surface of a tri-axial ellipsoid immersed in a uniform irrotational flow of a second-order fluid. 3 Irrotational flow of an incompressible and inviscid fluid over a stationar ellipsoid The flows for which the vorticit vector vanishes ω = u) everwhere in the flow field are said to be irrotational. Since for an scalar function φ it is satisfied that φ = 0, the condition of irrotationalit is redefined b choosing u = φ. In which case the function φ is called the velocit potential. B using the continuit equation of an incompressible fluid u = 0) gives the Laplace s equation φ = 0). In this section we present the velocit potential for the flow induced b an ellipsoid that translates along the x-axis given b Lamb 1993) and Milne-Thomson 1996). In addition, we compute the velocit components and the inviscid pressure for the irrotational flow around a stationar ellipsoid. The harmonic function presented b Lamb 1993) represents the solution to the Laplace s equation expressed in terms of a special sstem of orthogonal curvilinear coordinates known as ellipsoidal coordinates. The equation x a + θ + b + θ + =1 a>b>c 3) c + θ where a, b, c are fixed and θ is a parameter, represents for an constant value of θ a central quadric of a confocal sstem. In particular, when θ = 0, we have the ellipsoid given b equation 1). Equation 3) leads to the expression f θ) =x b + θ ) c + θ ) + c + θ ) a + θ ) + a + θ ) b + θ ) a + θ ) b + θ ) c + θ ) 4) =0 which is a cubic equation in θ and has three roots, sa λ, µ and ν, that are distributed as follows see Kellogg 199): a ν b µ c λ. 5) The values of x,, can be expressed as functions of λ, µ and ν b the following equations, a x + λ ) a + µ ) a + ν ) =, a b )a c ) b + λ ) b + µ ) b + ν ) =, 6) b c )b a ) c + λ ) c + µ ) c + ν ) = c a )c b. ) 3

4 It follows that, and hence, x λ = 1 x a + λ, λ = 1 b + λ, λ = 1 c + λ, 7) ) h 1 = 1 x 4 a + λ) + b + λ) + c + λ). 8) The square of the scale factors h 1, h, h 3 in ellipsoidal coordinates are given b, h 1 = 1 λ µ)λ ν) 4 a + λ)b + λ)c + λ) h = 1 µ ν)µ λ) 9) 4 a + µ)b + µ)c + µ) h 3 = 1 ν λ)ν µ) 4 a + ν)b + ν)c + ν). The direction-cosines of the outward normal to the three surfaces which pass through x,, ) will be 1 x h 1 λ, 1 h 1 λ, 1 ) 1 x, h 1 λ h µ, 1 ) h µ, 1 1 x, h µ h 3 ν, 1 h 3 ν, 1 ). 10) h 3 ν We ma note that if λ, µ, ν be regarded as functions of x,, the direction-cosines of the three line-elements above considered can also be expressed in the forms ) λ λ λ µ h 1, h x,h 1,h 1 x,h µ,h µ ) ν, h 3 x,h ν 3,h ν 3 ), 11) from which, and from 10), various interesting relations can be inferred. For our present purpose the following relations would be useful, λ x = 1 x h 1 λ = x h 1 a + λ) λ = 1 h 1 λ = h 1) 1 b + λ) λ = 1 h 1 λ = h 1 c + λ). The Laplacian ) of the scalar function φ in ellipsoidal coordinates can be written in the form 1 φ = h 1 h h 3 { λ h h 3 h 1 ) φ + λ µ h3 h 1 h ) φ + µ ν h1 h h 3 )} φ. 13) ν Equating this to ero, we obtain the Laplace s equation that is the general expression of continuit given in ellipsoidal coordinates. Solutions to this equation are called ellipsoidal harmonics. From Milne-Thompson 1996) and Lamb 1993), the corresponding ellipsoidal harmonics are given b, dλ φ x = Cx a + λ) a + λ)b + λ)c + λ), 14) λ dλ φ = C b + λ)c + λ) a + λ)b + λ)c + λ), 15) λ 4

5 where C is an arbitrar constant, and x,, are supposed expressed in terms of λ, µ, ν b means of equations 6). For a full account of the solution of Laplace s equation in ellipsoidal coordinates we must refer to Lamb 1993) and Milne-Thomson 1996). For the ellipsoid given b equation 1), which corresponds to λ = 0, moving in the direction of the x-axis with velocit U, the boundar condition is φ n = U cos θ φ x x or = U, λ =0. 16) λ λ Thus when λ =0,φ = Ux, and when λ, φ 0. These conditions are satisfied b the function φ x of equation 14). Appling the boundar conditions to equation 14) gives, C = abcu, α 0 where, dλ α 0 = abc 0 a + λ) a + λ)b + λ)c + λ). 17) The constant α 0 depends solel on the semi-axes a, b, c of the ellipsoid. Its numerical evaluation requires the use of elliptic integrals. Thus, finall, φ = abcux dλ, 18) α 0 λ a + λ) 3/ b + λ) 1/ c 1/ + λ) and on the surface of the ellipsoid we have, from equation 14), with λ =0, φ = xα 0 U. 19) α 0 Equation 18) represents the potential for the space external to the ellipsoid 1) that moves with velocit U in a liquid at rest at infinit. This result corresponds to an origin moving with the ellipsoid. B superposing a uniform flow with velocit U, in the positive direction of the x-axis, gives, φ = xu abc dλ α 0 a + λ) a + λ)b + λ)c + λ) +1 0) λ and defining, Γ=U abc α 0 λ dλ a + λ) a + λ)b + λ)c + λ) +1, 1) it follows that, φ = x Γ ) where, Γ is a function of λ onl. Equation ) represents the velocit potential for the flow around a stationar ellipsoid. The function Γ given b the equation 1) involves the elliptic integral, dλ I = a + λ) a + λ)b + λ)c + λ). 3) λ 5

6 The solution to this integral, obtained from the handbook of elliptic integrals written b Brd & Friedman 1971) p. 5), is given b where I = [F ϕ, k) E ϕ, k)] k a c ) a c, 4) a c ϕ = arcsin 5) λ + a a k = b 6) a c and the functions F ϕ, k) and Eϕ, k) represent the incomplete elliptic integral of the first kind and the Legendre s incomplete elliptic integral of the second kind respectivel. The values of the functions F ϕ, k) and Eϕ, k) are tabulated in Brd & Friedman 1971) for given values of ϕ and k. The differentiation of the elliptic functions F ϕ, k) and Eϕ, k) with respect to ϕ ields see Brd & Friedman 1971), p. 84), d dϕ F ϕ, k) = 1 1 k sin ϕ, 7) d dϕ E ϕ, k) = 1 k sin ϕ. 8) With the expression for the elliptic integral given b equation 4), the equation 1) becomes, [ ] abc [F ϕ, k) E ϕ, k)] Γ=U α 0 a b ) +1. 9) a c For an irrotational flow the velocit components are given b, u = φ = φ. 30) x i Appling equation 30) to the scalar function given b equation ) gives, u = xγ) x =Γ+x Γ λ λ x =Γ+ x Γ h 1 a + λ) λ, v = xγ) w = xγ) = x Γ λ λ = x Γ h 1 b + λ) λ, 31) = x Γ λ λ = x Γ h 1 c + λ) λ, which represent the velocit components of the irrotational flow of an inviscid fluid around an ellipsoid. It is pertinent to introduce at this point, the first, second and third derivatives of Γ that would be used in determining the velocit components and their derivatives. Γ λ = abcu, 3) α 0 )a + λ) 3/ b + λ) 1/ c + λ) 1/ Γ = Γ [ ] 3 λ λ a + λ) + 1 b + λ) + 1, 33) c + λ) 3 Γ = Γ [ ] 3 λ 3 λ a + λ) + 1 b + λ) + 1 c + λ) + Γ λ [ 3 a + λ) + 1 b + λ) + 1 c + λ) ]. 34) 6

7 A two dimensional representation of the flow field at the centerline of a tri-axial ellipsoid is shown in figure. This representation corresponds to a tri-axial ellipsoid with semi-axes a/b = b/c = and a Renolds number of x Figure : Velocit field of an irrotational, inviscid flow around an ellipsoid with semi-axes a = 3, b = 1.5, c = 0.75, and a Renolds number of 5. Two dimensional representation of the velocit field at the centerline of the ellipsoid = 0). Integration of the Euler s equation ields the Bernoulli equation. Thus, for an incompressible, irrotational and stead flow the inviscid pressure equation can be written as p I = ρ U φ ) + p, 35) where U and p are the velocit and the pressure far awa from the flow field. Introducing the expression for the magnitude of the velocit vector in equation 35) ields, p I = ρ ) ] [U Γ x Γ a + λ) + dγ,λ dγ,λ + p. 36) h 1 At the surface of the ellipsoid λ = 0) the pressure distribution can be obtained b evaluating the following expression, ρu [ 4x b 4 c 4 ] p Is = α 0 ) α 0 α 0 4) + x b 4 c 4 + a 4 c 4 + a 4 b 4 + p. 37) ) The non-dimensional pressure on the surface of the ellipsoid is obtained b substituting equation 37) into the pressure coefficient defined as, C p p p 1/ ρu. 38) A plot of this function in the x-) and x-) planes is given in figure 3. It shows a smmetric pressure distribution over the ellipsoid. At the fore and rear stagnation points the pressure force is maximum and C p = 1. As we move around the ellipsoid, the fluid accelerates and the pressure drops accordingl. At θ = π/ 7

8 x,) plane x,) plane Cp θ Figure 3: Pressure coefficient distribution on the surface of an ellipsoid for two cross sections in the x-) and x-) planes with = 0 and = 0 respectivel. the pressure has dropped to C p = The pressure drops faster in the x-) plane where the cross section is flatter than in the x-) plane due to the difference in curvature. The velocit gradient can be decomposed into its smmetric and anti-smmetric parts. The smmetric part is associated with the straining motions while the anti-smmetric part indicates the rotational motion of a fluid element. Thus, the velocit gradient can be written as, L = u = D + Ω, 39) where D = 1 / u + u T ) and Ω = 1 / u - u T ), represent the strain or deformation and the rotation tensors respectivel. Since the rotation tensor is directl proportional to the vorticit vector, for an irrotational flow field, the anti-smmetric part of the velocit gradient is identicall ero. Thus, the strain tensor is equal to the velocit gradient tensor D = L). For the flow field given b equation 31) the components of the strain tensor are: L 11 = u x = λ x + λ x x L 1 = v x = u λ = L 13 = w x = u = L = v = x λ Γ λ + x L 3 = w = v = x λ Γ L 33 = w = x λ Γ λ + x D ij =L ij = u j x i, 40) ) ) Γ λ λ + x Γ x λ, ) ) + x λ Γ λ x λ + x λ Γ x λ, ) ) λ + x λ Γ λ x λ + x λ Γ x λ, ) λ Γ λ, ) λ λ + x λ Γ λ, ) λ Γ λ. 8

9 4 Second-order fluid model For an incompressible fluid, the stress tensor can be written as, T = pi + S, 41) where p is pressure and S is the extra stress which is modeled b a constitutive equation. There is not a single constitutive equation for all flow motions. For ver slow flows, all the models collapse into a single form, the second-order fluid. A second-order fluid is an asmptotic approximation to the stress for nearl stead and ver slow flow. It is quadratic in the shear rate and represents the recent memor of the fluid b a time derivative Joseph 1996). The approximation to S for a second-order fluid is given b see Joseph 199b), S = ηa + α 1 B + α A, 4) where A = L + L T, is twice the smmetric part of the velocit gradient L, and B = A t +u )A + AL+ LT A. 43) Thus, for a second-order fluid the stress tensor can be written as, T = pi + ηa + α 1 B + α A 44) where A and B are known as the first and second Rivlin-Ericksen tensors see Joseph & Feng 1996). The parameter η is the ero-shear viscosit and the parameters α 1 = n 1 / and α = n 1 + n, the quadratic constants, are related b ˆβ =3α 1 +α 0; where n 1 and n are constants obtained from the first and second normal stress differences and ˆβ is the climbing constant. After Joseph 199b), the Bernoulli equation for potential flow of a second-order fluid and in particular for stead flow can be written as, p = ρ U φ ) + ˆβ 4 tra + p. 45) B introducing the scalar function for the pressure given b equation 45), and the stead form of equation 43), into equation 44) and rearranging we get, [ ρ T = U φ ) + ˆβχ ] + p I + ηa + α 1 u ) A +α 1 + α )A. 46) In index notation we have, T ij = [ ρ φ A ij = = u j, x i x j x i χ = 1 4 tra = φ φ = u ) k u i uk =, x i x k x k x i x i x k x i U φ ) ] x i + ˆβχ φ + p δ ij + ηa ij + α 1 x i x k A ij +α 1 + α ) A ik A kj. 9

10 5 Irrotational flow of a second-order fluid over a stationar ellipsoid The set of equations that full define the irrotational and stead flow of a second-order fluid around an ellipsoid is given next, φ = xγλ), u = φ, u =0, T = pi + ηa + α 1 u ) A +α 1 + α ) A, ρ u ) u = p + η u + [α 1 u ) A +α 1 + α ) A ], p = ρ U φ ) + ˆβ 4 tra + p, A = L + L T = φ, x i x j 1 4 tra = φ x i x k u ) A = φ x k φ, x k x i A ij. x k The normal component of the stress is computed as, T nn = n T n, 47) where n is the unit vector normal to the surface of the ellipsoid 1) and is given b, x n = a i + b j + ) / x c k a + 4 b + 4 c. 48) 4 The normal component of the stress tensor in index notation, in terms of the velocit components, is given b, [ ρ ) ] T nn = n i n j T ij = U u j u i u i + ˆβχ + p +ηn i n j x i Expanding equation 49) ields, T nn = ρ [ U u + v + w )] p [ ) ) u v 3α 1 +α ) + + x [ +η n x +α 1 [n x u x + v n + n u u x + v v x + w w +n x n u v x + v u { [ u +4 α 1 + α ) +n n x [ w ) + x +α 1 n i n j u k u j x k x i +4α 1 + α ) n i n j u k x i u j x k. 49) ) ) ) w v w x x ] w +n v xn x +n w xn x +n w n ) ) x + n u u + v v + w w + n ) + w w +n x n u w x x + v w x + w u ) ) ) ] [ ) ) v w v v n + + x x x x ) ] [ w u v +n x n x x + v v x + w ] w x ]} ) w + [ u w +n x n x x + v w x + w x ] [ w v w +n n x x + v w + w 10 ) ] w u u ) + v v + w w ) +n n u w w v x + v w ) ] )] + w v

11 where, u x = v x = w x = u = ) ) 3 λ x + λ Γ λ x 3 x 3 λ +3 λ x x + λ ) 3 Γ λ x x λ + x 3 Γ x λ, 3 ) λ x + x 3 λ Γ x λ + λ λ x + λ λ x x +x λ ) λ ) Γ λ x x λ + x λ x ) λ x + x 3 λ Γ x λ + λ λ x + λ λ x x +x λ ) λ ) Γ λ x x λ + x λ x ) ) ) λ + x 3 λ Γ λ x λ + +x λ λ x + x λ λ Γ λ x λ + x λ x 3 Γ λ 3, 3 Γ λ 3, ) 3 Γ λ 3, v = x 3 λ 3 Γ w = x 3 λ u = λ λ +3x λ λ ) 3 Γ λ + x 3 Γ λ, 3 Γ λ + x λ λ +x λ ) λ Γ λ + x ) ) λ + x 3 λ Γ λ x λ + +x λ x x λ λ + x λ ) λ Γ λ +3x λ λ ) 3 Γ λ λ + x 3 Γ λ 3, ) λ + x 3 λ Γ λ x λ + λ + x λ v = x 3 λ Γ λ + w = x 3 λ 3 Γ w x = +x λ λ λ 3 Γ x λ 3. ) λ λ λ + x λ λ + x λ λ x + x λ x 3 Γ λ, ) 3 λ x ) λ 3 Γ λ, 3 Γ λ + x λ x λ + x λ λ x ) λ 3 Γ λ, 3 ) Γ λ The higher order derivatives of the ellipsoidal parameter λ, are obtained from the first derivatives given b equation 1) with λ = λx,, ). With u, v and w given b equation 31), the resulting normal stress for a second-order fluid around an ellipsoid is of the form T nn = T nn x,,,λ). On the surface of the ellipsoid λ = 0, so that the normal component of the stress on the surface of the ellipsoid is a function of the Cartesian coordinates onl. The dimensionless form of the normal stress at the surface is expressed as, 6 Normal stress distribution on the ellipsoid T nn = T nn + p ρu /. 50) Here we present the results of the normal stress evaluated at the surface of an ellipsoid, immersed in a uniform flow of a second-order fluid, for a Renolds number Re = ρua/η) of 5. Three different cases are shown to illustrate the effects of the semi-axes ratios in the stress distribution. First, the normal stress is evaluated on a tri-axial ellipsoid with a/b = b/c =. The second case corresponds to a flatter ellipsoid with a/b = and b/c = 5. Finall, we show the distribution of the stress at the surface of a prolate spheroid with a/b = and b = c. As an example of second-order fluid we used the liquid M1 with the following properties Hu et al. 1990): [ρ =0.895 g/cm 3, η =30P,α 1 = 3 and α =5.34 g/cm]. The distribution and sign of the normal stress at the surface is depicted b arrows around ellipses that represent cross sections of the ellipsoid see figures 4-13, 16 and 17). If the normal stress is negative it gives rise to compression and if positive, it induces tensions that ma be responsible for the deformation of gas bubbles in viscoelastic liquids. Inward arrows represent the negative values and outward arrows represent the positive 11

12 values of the normal stress. We also present the results of the dimensionless normal stress T nn, as a function of the polar angle θ see figures 14 and 15). The normal stress distribution on the surface of a tri-axial ellipsoid with a =3,b =1.5 and c =0.75 cm, shown in figures 4-15, suggest the formation of a gas bubble with a prolate shape in the front and a flat two-dimensional shape in the back with an elongated trailing edge. We made the former ellipsoid even flatter b making c =0.3 cm. The computed normal stress distribution is depicted in figure 16. The results indicate that as the rear part of the bubble gets flatter, the pulling out effect at the right and left poles increase giving rise to a thinner tail. Once the bubble acquires the prolate shape in the front, the stresses around the ellipsoid become constant for each cross section. This is shown in figure 17, that represent the stresses around the ellipsoid with a =3.0, b =1.5 and c =1.5 cm. The normal stress for each cross section located at: x =.9,.6,.0, 1.0,, 1.0,.0,.6 and.9, is constant and the are all compression in the front and tension in the back. For the three semi-axes ratios covered in this stud, there is a strong tension near the rear stagnation point that could be responsible for the build up of a two-dimensional cusp at the trailing edge of a bubble. Figure 14 shows a maximum compression near the leading edge. The maximum tension, that is almost twice the maximum compression, is located ver close to the trailing edge see figure 15). 1

13 .0 a b c d e f g h i x Figure 4: Distribution of the normal stress at the surface of a tri-axial ellipsoid, immersed in a uniform stream that moves from left to right. Two-dimensional representation in the x, )-plane of an ellipsoid with semi-axes a =3,b =1.5, c =0.75 cm). The arrows represent the normal stress T nn n x + T nn n ) /150 dn/cm evaluated for different cross sections along the ellipsoid. The cross sections are located at: x =.9 a),.6 b),.0 c), 1.0 d), e), 1.0 f),.0 g),.6 h) and.9 i) cm. A perpendicular view of the different cross sections is shown in figures There is compression in the fore part of the ellipsoid and a strong tension in the rear, near the trailing stagnation point. 13

14 Figure 5: Normal stress T nn n + T nn n ) /600 dn/cm for an ellipse /b + /c =1.9 /a at x =.9 cm in the,)-plane. Perpendicular view of the cross section a) in figure 4. The normal stress at the front of the ellipsoid gives rise to compression near the leading edge. This ma explain the flat top observed in gas bubbles rising in viscoelastic fluids see figure 1) Figure 6: Normal stress T nn n + T nn n ) /300 dn/cm for an ellipse /b + /c =1.6 /a at x =.6 cm in the,)-plane. Perpendicular view of the cross section b) in figure 4. There is a bigger compression indicated b bigger inwards arrows) at the right and left poles of the ellipse. This ma cause that the cross section of the ellipsoid becomes more round and in consequence the ellipsoid would tend to be prolate. 14

15 Figure 7: Normal stress T nn n + T nn n ) /150 dn/cm for an ellipse /b + /c =1.0 /a at x =.0 cm in the,)-plane. Perpendicular view of the cross section c) in figure 4. Strong compression at the right and left poles of the ellipse and mild compression towards the center Figure 8: Normal stress T nn n + T nn n ) /50 dn/cm for an ellipse /b + /c =1 1.0 /a at x = 1.0 cm in the,)-plane. Perpendicular view of the cross section d) in figure 4. Small compression at the center and high compression at the right and left poles of the ellipse indicated b small and big inwards arrows). This suggests that the upper part of a gas bubble rising in a viscoelastic fluid would tend to be prolate. 15

16 Figure 9: Normal stress T nn n + T nn n ) /.5 dn/cm for an ellipse /b + /c =1atx = cminthe,)-plane. Perpendicular view of the cross section e) in figure 4. There is still compression right in the middle of the ellipsoid. 16

17 Figure 10: Normal stress T nn n + T nn n ) /50 dn/cm for an ellipse /b + /c =1 1.0 /a at x =1.0 cm in the,)-plane. Perpendicular view of the cross section f) in figure 4. There is a change of sign of the normal stress right after the front half of the ellipsoid. It changes from negative compression) to positive tension). Strong tension near the right and left poles and mild tension towards the center. This ma induce the formation of a flat tail for a gas bubble. The same effect is observed in the other cross sections of the rear part of the ellipsoid see figures 11-13). 17

18 Figure 11: Normal stress T nn n + T nn n ) /150 dn/cm for an ellipse /b + /c =1.0 /a at x =.0 cm in the,)-plane. Perpendicular view of the cross section g) in figure 4. Strong tension at the right and left poles, causes the ellipsoid to get flatter Figure 1: Normal stress T nn n + T nn n ) /300 dn/cm for an ellipse /b + /c =1.6 /a at x =.6 cm in the,)-plane. Perpendicular view of the cross section h) in figure 4. Strong tension at the right and left poles, causes the ellipsoid to get flatter. 18

19 Figure 13: Normal stress T nn n + T nn n ) /600 dn/cm for an ellipse /b + /c =1.9 /a at x =.9 cm in the,)-plane. Perpendicular view of the cross section i) in figure 4. The strong tension observed near the trailing stagnation point ma explain the formation of a two-dimensional cusping tail in a gas bubble rising in viscoelastic fluids. 19

20 x=-.9 x=-.6 x=-.0 x=-1.0 x= Tnn* θ Figure 14: Dimensionless normal stress as a function of the polar angle θ for the different cross sections of the frontal part of the ellipsoid shown in figures 5 to 9 x 0). The normal stress is compression in the fore part of the ellipsoid. The maximum compression is observed near the leading edge, and decreases as we move towards the rear of the ellipsoid. 0

21 000 x= x=1.0 x=.0 x=.6 x= Tnn* θ Figure 15: Dimensionless normal stress as a function of the polar angle θ for the different cross sections of the rear part of the ellipsoid shown in figures 9 to 13 x 0). The normal stress is tension in the rear of the ellipsoid, with a strong tension near the trailing edge. 1

22 0.3.0 c d e b f a) T nn n x + T nn n ) /100 dn/cm for an ellipsoid x /a + /b =1 /c with =0, in the x, )-plane. x b) T nn n + T nn n ) /7500 dn/cm for an ellipsoid /b + /c =1 x /a with x =.9 cm, in the,)-plane c) T nn n + T nn n ) /800 dn/cm for an ellipsoid /b + /c =1 x /a with x =.0 cm, in the,)-plane d) T nn n + T nn n ) /10 dn/cm for an ellipsoid /b + /c =1 x /a with x = cm, in the,)-plane e) T nn n + T nn n ) /800 dn/cm for an ellipsoid /b + /c =1 x /a with x =.0 cm, in the,)-plane f) T nn n + T nn n ) /7500 dn/cm for an ellipsoid /b + /c =1 x /a with x =.9 cm, in the,)-plane. Figure 16: Distribution of the normal stress T nn on various cross sections on the,)-planes of an ellipsoid with semi-axes a =3,b =1.5, c =0.3 cm) immerse in a flow of fluid M1 moving from left to right), with a Renolds number of 5. The properties of the liquid M1 that are used in the calculations are: ρ =0.895 g/cm 3, η =30P,α 1 = 3 and α =5.34 g/cm. The normal stress is represented b vectors at the surface of the ellipsoid on the,)-planes corresponding to the cross sections of the ellipsoid at x =.9 b),.0 c), d),.0 e), and.9 f) cm. The orientation of the vectors indicate compression inwards) or tension outwards) exerted b the normal stress at the surface of the ellipsoid. In the first half of the ellipsoid the normal stress is mainl compression and, it changes to tension as we approach the trailing edge.

23 0.6.0 c d e b f a) T nn n x + T nn n ) /100 dn/cm for an ellipsoid x /a + /b =1 /c with =0, in the x, )-plane. x b) T nn n + T nn n ) /50 dn/cm for an ellipsoid /b + /c =1 x /a with x =.9 cm, in the,)-plane c) T nn n + T nn n ) /50 dn/cm for an ellipsoid /b + /c =1 x /a with x =.0 cm, in the,)-plane d) T nn n + T nn n ) dn/cm for an ellipsoid /b + /c =1 x /a with x = cm, in the,)-plane e) T nn n + T nn n ) /50 dn/cm for an ellipsoid /b + /c =1 x /a with x =.0 cm, in the,)-plane f) T nn n + T nn n ) /50 dn/cm for an ellipsoid /b + /c =1 x /a with x =.9 cm, in the,)-plane. Figure 17: Distribution of the normal stress T nn on various cross sections on the,)-planes of an ellipsoid with semi-axes a =3,b =1.5, c =1.5 cm) immerse in a flow of fluid M1 with a Renolds number of 5. The properties of the liquid M1 that are used in the calculations are: ρ =0.895 g/cm 3, η =30P,α 1 = 3 and α =5.34 g/cm. The normal stress is represented b vectors at the surface of the ellipsoid on the,)-planes corresponding to the cross sections of the ellipsoid at x =.9 b),.0 c), d),.0 e), and.9 f) cm. The orientation of the vectors indicate compression inwards) or tension outwards) exerted b the normal stress at the surface of the ellipsoid. The normal stress is constant at each cross section, and it is compression in the front and tension in the back. 3

24 7 Conclusions We developed the analsis of viscoelastic potential flow of a second-order fluid around an ellipsoid. To carr out these calculations we formed expressions for the velocit, the inviscid pressure, the velocit derivatives and the composition of these derivatives to compute stresses. The calculations give rise to normal stress distribution compatible with experimental observations of gas bubbles rising in viscoelastic liquids. In particular, the normal stress at the top of a rising bubble is compression and the side stress tends to round the elliptic shape. The trailing edge of the bubble is stretched into a cusp and the side of the ellipse tends to flatten into the remarkable two-dimensional cusps observed in experiments. Since the velocit is obtained from a harmonic potential, the velocit field does not depend on viscous or viscoelastic parameters. This suggests that the cusping effect is associated primaril with normal stresses rather than with secondar effects due to changes in velocit not computed here. References [1] Brd P.F & Friedman M. D Handbook of Elliptic Integrals, nd Edition, Springer-Verlag, New Springer-Verlag, New York, Heidelberg, Berlin. [] Hu H. H., Riccius O., Chen K. P., Arne M. & Joseph D. D Climbing constant, second-order correction of trouton s viscosit, wave speed and delaed die swell for M1. J. of Non-Newtonian Fluid Mech., [3] Joseph D. D Two-dimensional cusped interfaces. J. Fluid Mech. 3, [4] Joseph D. D. 199a Understanding cusped interfaces. J. of Non-Newtonian Fluid Mech., [5] Joseph D. D. 199b Bernoulli equation and the competition of elastic and inertial pressures in the potential flow of a second-order fluid. J. of Non-Newtonian Fluid Mech., [6] Joseph D. D Flow induced microstructure in Newtonian and Viscoelastic Fluids. Proceedings of the 5 th World Congress of Chemical Engineering, Particle Technolog Track. Second Particle Technolog, San Diego, AIChE New York, [7] Joseph D. D. & Feng J A note on the forces that move particles in a second-order fluid. J. of Non-Newtonian Fluid Mech., [8] Kellogg, O. D. 199 Foundations of Potential Theor, Dover Publications Inc., New York. [9] Lamb, S. H Hdrodnamics, First Cambridge Universit Press Edition. [10] Liu Y. J., Liao T. Y. & Joseph D. D A two-dimensional cusp at the trailing edge of an air bubble rising in a viscoelastic liquid. J. Fluid Mech. 304, [11] Milne-Thomson L. M Theoretical Hdrodnamics, Fifth Edition, Dover Publications Inc., New York. [1] Pillapakkam S. B. & Singh P. 001 A level-set method for computing solutions to viscoelastic two-phase flow. J. of Comp. Phs. 174, [13] Pillapakkam S. B. & Singh P. 003 Negative wake and velocit of a bubble rising in a viscoelastic fluid. Proceedings of 003 ASME International Mechanical Engineering Congress FED-59, [14] Wang J. & Joseph D. D. 004 Potential flow of a second order fluid over a sphere or an ellipse. J. Fluid Mech. 511,