Slow steady fall of rigid bodies in a second-order fluid

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Abstract

We consider the steady, slow translational fall of a rigid body B in a second-order fluid, under the action of the force of gravity g. We find a general expression for the total force and torque acting on B. In particular, the force per unit area is always compressive, if the first normal stress coefficient Ψ1 is positive. We then specialize these formulas to the case when B is a prolate spheroid of eccentricity e, and show that, when 0 < e < 1, there are only two orientations of fall allowed, namely, when the major axis of B is either perpendicular or parallel to g. However, we show that if Ψ1 >0, only this latter orientation is stable to small disorientations, in agreement with the recent experimental results of Joseph and coworkers.

Introduction

This paper deals with the slow steady fall of a homogeneous rigid body in a second order fluid and is motivated by the recent work of D.D. Joseph and his coworkers on the orientation of ellipsoids settling through a quiescent viscoelastic fluid 1, 2, 3, 4. In particular, these authors have observed experimentally 1, 2, and computed numerically 4, 3, that such a body will eventually settle with its major axis parallel to the gravity, whenever the inertia of the fluid can be disregarded (creeping flow).1 This situation should be contrasted with the purely Newtonian case where, as is well known, in a creeping flow all possible orientations are allowed 5, 6, 7. The different behavior is due to the fact that a viscoelastic fluid generates a nonzero torque on the ellipsoid which is otherwise zero in the Newtonian case, if inertia is not taken into account.2 For creeping flow around slender rod-like particles in a second-order fluid, a nonzero torque was calculated by Leal [12], using a formal expansion of the solution at small Weissenberg numbers (see, also Ref. [9]). Very recently, Joseph and Feng [10] have furnished a qualitative analysis of the force acting on a particle moving slowly3 in a second-order fluid satisfying the condition Ψ1 + 2Ψ2 = 0 [11], where Ψ1 and Ψ2 are the first and second normal stress coefficients. This analysis shows that, if Ψ1 > 0, the normal stresses on a rigid particle are compressive and provides intuition on why and how particles with fore-and-aft symmetry may turn while settling.

One of the objectives of this paper is to furnish a quantitative analysis of the force F and the torque T acting on a rigid body B translating slowly and steadily with velocity U in a second-order fluid satisfying Ψ1 + 2Ψ2 = 0, under the action of gravity. For small Weissenberg numbers, we find that F and T have a simple expression and are related to the square of the vorticity ω of the fluid evaluated at the surface of the particle. Moreover, ω = ωS, where ωS is the vorticity calculated in the corresponding Newtonian flow around B in the Stokes approximation. If Ψ1 > 0, the force per unit area acting on B is compressive. For special geometries of B, we can explicitly evaluate ωS. We do this in the case of a prolate spheroid and find that F is identically zero, in agreement with the results of Leal [12], for a slender rod-like particle with fore-and-aft symmetry, and of Leslie [13], for a sphere. This result implies that there is no non-Newtonian contribution to drag and lift. Therefore, the velocity U of a steady fall of B under the action of gravity coincides with the velocity that B has, under the same circumstance, in a Newtonian fluid. As far as T is concerned, we show that it has only one non-identically zero component in a direction orthogonal to the plane containing U and the gravity g and that this component is a function of the eccentricity e and of the angle θ formed by U with the major axis of symmetry. For a fixed θ, the magnitude of the torque has its maximum at e = 0.95, that is, when B is sufficiently slender. In a steady fall, T must vanish. If e < 1, this happens if, and only if, U is directed along g and both vectors are parallel to an axis of symmetry. If e = 1, any value for θ is allowed, as in the Newtonian case. These results imply that for 0 < e < 1, B can only fall with its major axis either parallel or orthogonal to the gravity. However, if Ψ1 > 0, we show that T tends to rotate B toward the orientation where its broadside is parallel to the gravity, in agreement with the results of Joseph.

Section snippets

Conditions for slow steady translational fall of a body in a second-order fluid

We consider a rigid body B, under the action of gravity g, performing a steady translational fall in a fluid I. This means that the motion of B from a fixed inertial frame I is a steady translation, while the motion of I as seen from a frame S attached to B is steady. We shall assume throughout that B is homogeneous (constant density). Moreover, the constitutive equation for the fluid is given by the relationT=−pIA1(v)+STN(v,p)+Swhere T is the stress tensor, p the pressure (Lagrange

Slow steady translational fall of a prolate spheroid in a second-order fluid

We wish now to specialize the calculations performed in the previous section when B is a prolate spheroid. In such a case, the expression for vS is well known (see, e.g. Ref. [20]) and we havevS=U1e1+U2e2−(2A1e1+A2e2)B1−(A1rer+A2x2e1)(R−12−R−11)+(A1re1+A2x2er)rB2+gradΦwhere the major axis of symmetry of B is directed along e1 and (without loss) we assume U = (U1,U2,0). Moreover,A1=U1e2(1+e2)log1+e1−e−2e−1≡U1A1(e),A2=2U2e2(3e2−1)log1+e1−e+2e−1≡U2A2(e),R1=(x1+ea)2+r2, R2=(x1−ea)2+r2, er=x2e2+x3e3r,

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    Such final orientation is reached regardless of the initial orientation, particle aspect ratio, and fluid rheology. This behavior is due to a nonzero torque acting on the particle generated by fluid elasticity that has opposite sign with respect to the inertial one, as predicted by asymptotic analyses [20,52] and semi-analytical calculations [53–55] for a second order fluid, and numerical simulations [14,56]. Recently, the effect of inertia and viscoelasticity on the settling dynamics of a spheroidal particle is analyzed in the limit of small Reynolds and Deborah numbers [16].

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