Slow steady fall of rigid bodies in a second-order fluid
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 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 in the Stokes approximation. If Ψ1 > 0, the force per unit area acting on is compressive. For special geometries of , we can explicitly evaluate ωS. We do this in the case of a prolate spheroid and find that 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 under the action of gravity coincides with the velocity that 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 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, 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 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 , under the action of gravity g, performing a steady translational fall in a fluid . This means that the motion of from a fixed inertial frame is a steady translation, while the motion of as seen from a frame attached to is steady. We shall assume throughout that is homogeneous (constant density). Moreover, the constitutive equation for the fluid is given by the relationwhere 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 is a prolate spheroid. In such a case, the expression for vS is well known (see, e.g. Ref. [20]) and we havewhere the major axis of symmetry of is directed along e1 and (without loss) we assume U = (U1,U2,0). Moreover,
References (20)
- et al.
A note on the forces that move particles in a second-order fluid
J. Non-Newtonian Fluid Mech.
(1996) - et al.
Uniqueness and drag for fluids of second grade in steady motion
Int. J. Non-Linear Mechanics
(1978) - et al.
Slow motion of a body in a fluid of second grade
Int. J. Engng Sci.
(1997) - et al.
Orientation of long bodies falling in a viscoelastic liquid
J. Rheol.
(1993) - et al.
Sedimentation of particles in polymer solutions
J. Fluid Mech.
(1993) - et al.
Direct simulation of the sedimentation of elliptic particles in Oldroyd-B fluids
J. Fluid Mech.
(1998) - et al.
A three-dimensional computation of the force and torque on an ellipsoid settling slowly through a viscoelastic fluid
J. Fluid Mech.
(1995) The stokes resistance of an arbitrary particle II
Chem. Engng. Sci
(1964)- V. Happel, H. Brenner, 1965, Low Reynolds Number Hydrodynamics, Prentice...
- G.P. Galdi, 1998, Slow motion of a body in a viscous incompressible fluid with application to particle sedimentation,...
Cited by (9)
Numerical simulations on the settling dynamics of an ellipsoidal particle in a viscoelastic fluid
2022, Journal of Non-Newtonian Fluid MechanicsCitation Excerpt :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].
Effects of inertia and viscoelasticity on sedimenting anisotropic particles
2015, Journal of Fluid MechanicsSedimentation of spheroids in Newtonian fluids with spatially varying viscosity
2024, Journal of Fluid MechanicsOrientation dependent elastic stress concentration at tips of slender objects translating in viscoelastic fluids
2019, Physical Review Fluids