Effects of inertia and viscoelasticity on sedimenting anisotropic particles

Abstract

An axisymmetric particle sedimenting in an otherwise quiescent Newtonian fluid, in the Stokes regime, retains its initial orientation. For the special case of a spheroidal geometry, we examine analytically the effects of weak inertia and viscoelasticity in driving the particle towards an eventual steady orientation independent of initial conditions. The generalized reciprocal theorem, together with a novel vector spheroidal harmonics formalism, is used to find closed-form analytical expressions for the O(Re) inertial torque and the O(De) viscoelastic torque acting on a sedimenting spheroid of an arbitrary aspect ratio. Here, Re = UL/ν is the Reynolds number, with U being the sedimentation velocity, L the semi-major axis and ν the fluid kinematic viscosity, and is a measure of the inertial forces acting at the particle scale. The Deborah number, De = (λU)/L, is a dimensionless measure of the fluid viscoelasticity, with λ being the intrinsic relaxation time of the underlying microstructure. The analysis is valid in the limit Re, De 1, and the effects of viscoelasticity are therefore modelled using the constitutive equation of a second-order fluid. The inertial torque always acts to turn the spheroid broadside-on, while the final orientation due to the viscoelastic torque depends on the ratio of the magnitude of the first (N 1 ) to the second normal stress difference (N 2 ), and the sign (tensile or compressive) of N 1 . For the usual case of near-equilibrium complex fluids -a positive and dominant N 1 (N 1 > 0, N 2 < 0 and |N 1 /N 2 | > 1) -both prolate and oblate spheroids adopt a longside-on orientation. The viscoelastic torque is found to be remarkably sensitive to variations in κ in the slender-fibre limit (κ 1), where κ = L/b is the aspect ratio, b being the radius of the spheroid (semi-minor axis). The angular dependence of the inertial and viscoelastic torques turn out to be identical, and one may then characterize the long-time orientation of the sedimenting spheroid based solely on a critical value (El c ) of the elasticity number, El = De/Re. For El < El c (> El c ), inertia (viscoelasticity) prevails with the spheroid settling broadside-on (longside-on). The analysis shows that El c ∼ O[(1/ln κ)] for κ 1, and the viscoelastic torque thus dominates for a slender rigid fibre. For a slender fibre alone, we also briefly analyse the effects of elasticity on fibre orientation outside the second-order fluid regime.

The effect of inertia on the orientation dynamics of anisotropic particles in simple shear flow

“…That this approximation leads to a convergent integral may be seen by noting that, for a linear flow at Re = 0, we have u (1) s ∼ O(1/r 2 ) for r L, and from (2.11), it is then seen that (Du (1) s )/Dt ∼ O(1/r 2 ) for large r. Since u (2) ∼ O(1/r 2 ) for r L, the O(Re) integrand based on the Stokes approximation is O(1/r 4 ) for r L, implying convergence. As for the case of sedimentation in a quiescent fluid (Dabade et al 2015), this points to the regular nature of the O(Re) correction, with the dominant contribution to the O(Re) torque arising due to fluid inertial forces acting within a volume of order the size of the particle itself. It may be shown that the next correction to the angular velocity is O(Re 3/2 ), and is singular in character, arising from the effects of inertia acting on length scales of O(Re −1/2 ).…”

“…Since the structure of the formalism, and a comparison with a similar expansion of the velocity field in terms of spherical harmonics, originally given by Lamb (for instance, see Kim & Karrila (1991, chap. 4)), has already been explained in some detail in Dabade et al (2015), we will be brief here. The formalism is based on expressing the general solution of the Stokes equations, around an arbitrary number of spheroidal particles, as a superposition of growing and decaying vector harmonics in local spheroidal coordinates defined with respect to a Cartesian system centred at each particle, and aligned with the particle axis of symmetry.…”

“…Further, noting that the velocity field in the test problem is linear in Ω 2 , one may write u (2) = U (2) · Ω 2 , L 2 = L (2) · Ω 2 and σ (2) = Σ (2) · Ω 2 , where U (2) and L (2) are second-order tensors, and Σ (2) is a third-order tensor, dependent only on the geometry of the spheroidal particle, and are known in closed form as a function of the aspect ratio (see § 3; also see Dabade et al (2015)). Accounting for Ω 2 being arbitrary, (2.19) takes the form: valid for arbitrary Re and St.…”

“…While the expressions for these component Stokesian velocity fields may be obtained using earlier results based on the method of singularities (Chwang & Wu 1974, herein we use the vector spheroidal harmonics formalism developed by Kushch and co-workers (Kushch 1997(Kushch , 1998. The reasons for this choice have been outlined in Dabade et al (2015), where the formalism was used for a single sedimenting particle in an otherwise quiescent fluid. Since the structure of the formalism, and a comparison with a similar expansion of the velocity field in terms of spherical harmonics, originally given by Lamb (for instance, see Kim & Karrila (1991, chap.…”

“…The shape-selective nature of inertial lift forces relies crucially on the coupled positional and orientation dynamics of anisotropic particles in the bounded channel geometry. Other applications of interest, pertaining to atmospheric and geophysical scenarios, have been mentioned in Dabade, Marath & Subramanian (2015).…”