The rotation of a sedimenting anisotropic particle in a linearly stratified ambient

The atmosphere and oceans being, on average, in a stably stratified state, the motion of particles as well as living organisms in a stratified ambient is of obvious importance in natural settings. A large fraction of the research on the vertical motion of particles through stratified fluids, including cases of both sharp (Camassa et al. (2009)A.N. et al. (1999)) and continuous (Hanazaki (1988),Hanazaki et al. (2009),Yick et al. (2009),Doostmohammadi et al. (2012),Doostmohammadi et al. (2014)) stratification profiles, has, however, focused on spherical particles. Although this research has shed light on the non-trivial effects of stratification on the structure of the disturbance flow field induced by a sedimenting sphere, for instance, its sensitive dependence on the diffusivity of the stratifying agent via the Peclet number ($Pe$) (for instance, see List (1971), Ardekani & Stocker (2010), Doostmohammadi et al. (2012) and Varanasi & Subramanian (2021)), the vast majority of particles and living (micro)organisms in natural scenarios depart from the idealized spherical shape. Indeed, both marine phytoplankton and zooplankton come in an astonishing variety of shapes (Lab (2018),Kiorboe (2011)), and there are provocative questions to be addressed with regard to the large-scale effects of zooplankton migration across the oceanic pycnocline (Kunze et al. (2006),Visser (2007),Katija & Dabiri (2009),Subramanian (2010),Varanasi & Subramanian (2021)). Other classes of organic particles including marine snow aggregates (J.C. et al. (2015)), phytodetritus and faecal pellets, which make up the so-called biological pump (Turner (2015)), and undesired microplastics (Andrew & Luke (2011),Cole et al. (2011)), also depart significantly from the canonical spherical geometry. Extensive research over a long time has now led to a fairly mature understanding of the dynamics of anisotropic particles sedimenting in a homogeneous ambient (Guillaume & Magnaudet (2002),Franck et al. (2013)). While the non-trivial effects of unsteady wake dynamics come into play at higher Reynolds numbers ($Re$), as manifest by the onset of path instabilities of sedimenting spheroids (Ern et al. (2012)), the simplest scenario which prevails for low to moderate Reynolds numbers, when the wake has a quasi-steady character, involves inertial effects acting to turn sedimenting anisotropic particles broadside-on. For small $Re$, and in the case where the anisotropic particle is a prolate or an oblate spheroid, the inertial torque acting to turn the spheroid broadside-on has been determined analytically as a function of the spheroid aspect ratio (Dabade et al. (2015)), and has recently been shown to exert a crucial influence on the orientation distribution of such particles settling through a turbulent velocity field (Anand et al. (2020)).

The present effort is specifically motivated by very recent experiments involving cylindrical and disk-shaped particles (Mercier et al. (2020),Mrokowska (2018),Mrokowska (2020b),Mrokowska (2020a)) that are among the first to systematically explore the role of shape anisotropy for sedimenting particles in a heterogeneous stably stratified ambient. The experiments reported by Mercier et al. (2020) pertain to a linearly stratified ambient, while those reported in Mrokowska (2018),Mrokowska (2020b) and Mrokowska (2020a) pertain to a non-linearly stratified fluid layer sandwiched between homogeneous upper and lower layers. While the detailed results obtained for the two sets of experiments differ on account of the differing nature of the ambient stratification, one of the most important findings, common to both sets of experiments, pertains to the ability of the torque due to buoyancy forces to oppose, and even overwhelm the aforementioned inertial torque that acts in a homogeneous setting, thereby turning the particle longside-on. A recent theoretical effort (Dandekar et al. (2020)) has attempted to calculate the stratification-induced corrections to the force and torque acting on a non-spherical particle settling in a viscous linearly stratified ambient. While a correction to the force was determined in terms of the viscous Richardson number ($Ri_{v}$ defined below in section 2) for both chiral and achiral particles, a hydrodynamic torque was found to arise from buoyancy forces only for chiral particles, the origin of this torque being the translation-rotation coupling that already exists for such particles in a homogeneous ambient. Thus, the analysis does not explain the principal observation in the aforementioned experiments involving the stratification-induced transition of a sedimenting anisotropic but achiral particle from a broadside-on to an edgewise configuration. In the present effort, we show that buoyancy forces associated with the ambient stratification do lead to a torque for achiral particles modeled as prolate and oblate spheroids of an arbitrary aspect ratio. This stratification-induced torque consists of both hydrostatic and hydrodynamic components, with only the former contribution having been calculated in Dandekar et al. (2020) for a few restricted particle geometries. We show that the hydrostatic contribution always turns a spheroid broadside-on, regardless of aspect ratio, in agreement with Dandekar et al. (2020). More importantly, the hydrodynamic component of the stratification-induced torque is shown to be asymptotically larger than the hydrostatic one for large $Pe$, and orients spheroids edgewise, thereby offering the first explanation of the experimental observations above, of edgewise settling of an anisotropic particle in a stratified fluid.

The layout of the paper as follows. In section 2, we describe the reciprocal theorem formulation which yields the torque acting on a spheroid sedimenting in a linearly stratified viscous ambient in terms of distinct contributions arisng from the effects of fluid inertia and the buoyancy forces associated with the ambient stratification. The fluid inertial torque and the hydrodystatic component of the stratification torque are readily evaluated on account of their regular character, and this calculation is given in section 3. The calculation of the hydrodynamic component of the stratification torque is more involved, being dependent on $Pe$, and is carried out in sections 4.1 and 4.2 in the limits $Pe\ll 1$ and $Pe\gg 1$, respectively. Finally, section 5 discusses the transition from broadside-on to edgewise settling that arises due to the completing influences of the inertial and hydrodynamic components of the stratification torque, at large $Pe$, and ends with a comparison with recent experiments. In section 6, we briefly indicate possible lines of investigation for the future.

The torque acting on a spheroid, sedimenting in a stably stratified ambient, is derived below using the generalized reciprocal theorem (see Kim & Karrila (1991);Dabade et al. (2015, 2016)). The theorem relates two pairs of stress and velocity fields, and may be stated in the form:

where $S_{p}$ denotes the surface of the spheroid, and with the neglect of the surface integral at infinity, the volume integral on the right hand side of (1) is over the unbounded fluid domain external to the spheroid. In (1), the pair $({\boldsymbol{\sigma}}^{(1d)},{\boldsymbol{u}}^{(1)})$ denotes the dynamic stress and velocity fields associated with the problem of interest viz.​ a torque-free spheroid sedimenting under gravity in an ambient linearly stratified medium for small Reynolds ($Re$) and viscous Richardson ($Ri_{v}$) numbers. These dimensionless parameters measure the relative importance of inertial and buoyancy forces relative to viscous forces, respectively, and the aforementioned limit corresponds to the case where inertia and stratification act as weak perturbing influences about a leading order Stokesian approximation. The Reynolds and Richardson numbers are defined later in this section when writing down the non-dimensional system of governing equations; the precise definition of the dynamic stress field, ${\boldsymbol{\sigma}}^{(1d)}$, is also provided at the same place. The pair $({\boldsymbol{\sigma}}^{(2)},{\boldsymbol{u}}^{(2)})$, that defines the test problem in (1), corresponds to the stress and velocity fields associated with the Stokesian rotation of the same spheroid, about an axis orthogonal to its axis of symmetry, in a homogeneous and otherwise quiescent ambient with the same (assumed constant) viscosity as the medium in the actual problem. The equations governing the test problem may be written as:

with the boundary condition ${\boldsymbol{u}}^{(2)}={\boldsymbol{\Omega}}^{(2)}\wedge{\boldsymbol{x}}$ on $S_{p}$, ${\boldsymbol{\Omega}}^{(2)}$ being the angular velocity of the spheroid in the test problem, and the far-field decay condition for both ${\boldsymbol{u}}^{(2)}$ and $p^{(2)}$. Use of this surface boundary condition in (1) leads to

where the second integral on the left hand side in (4) now denotes the torque due to the dynamic stress field ${\boldsymbol{\sigma}}^{(1d)}$. We postpone further simplification of (4) until after we define the pair $({\boldsymbol{\sigma}}^{(1d)},{\boldsymbol{u}}^{(1)})$ below.

As mentioned above, problem 1 corresponds to an arbitrarily oriented spheroid, sedimenting under the action of a gravitational force $F\hat{\boldsymbol{g}}$, in an ambient medium that is linearly stratified (along $\hat{\boldsymbol{g}}$) in the absence of the fluid motion induced by the spheroid. The unit vector $\hat{\boldsymbol{g}}$ is aligned along gravity, with $g$ denoting the magnitude of the gravitational acceleration, and $F=\frac{4\pi}{3}Lb^{2}\Delta\rho g\,(\frac{4\pi}{3}L^{2}b\Delta\rho g)$ denoting the buoyant weight for a prolate (oblate) spheroid. Here, $L$ and $b$ are the semi-major and semi-minor axes of the spheroid, with $\kappa=L/b$ being the aspect ratio of a prolate spheroid, and $\kappa=b/L$ that of an oblate spheroid; thus, $\kappa>1$ and $<1$ for prolate and oblate spheroids, respectively. The density difference that enters the buoyant weight above is $\Delta\rho=\rho_{s}-\rho^{(1)}_{\infty}({\boldsymbol{x}}_{c})$, with $\rho_{s}$ being the density of the spheroid (assumed homogeneous), and $\rho^{(1)}_{\infty}({\boldsymbol{x}}_{c})=\rho_{0}$ being the ambient fluid density at the center of the spheroid. The latter simplification arises because of the linear stratification and the fore-aft symmetry of the spheroid, both of which imply that the weight of the equivalent stratified spheroidal fluid blob that gives the buoyant force is the same as the weight of a homogeneous fluid blob with density equal to the ambient value at the spheroid center. In a lab-fixed reference frame, the ambient density field in problem 1 may be written in the form:

where ${\boldsymbol{x}}^{L}$ denotes the position vector in laboratory coordinates with the spheroid center as the origin, and $\gamma>0$ is the constant density gradient that characterizes the stable ambient stratification. The calculations for the torque are, however, best done in a reference frame translating with the spheroid where a quasi-steady state is assumed to prevail at leading order. The latter assumption is motivated by the asymptotically weak rotation of the sedimenting spheroid in the limit $Re,Ri_{v}\ll 1$. The precise condition for the quasi-steady state assumption to hold depends on $Pe$, being more restrictive for large $Pe$, and is stated later alongside the results for the spheroid angular velocity for small and large $Pe$, obtained below.

The ambient density in the particle-fixed reference frame takes the form:

${\boldsymbol{x}}$ being the position vector in the new reference frame. In (6), ${\boldsymbol{U}}$ is the spheroid settling velocity, and related to the force ($F\hat{\boldsymbol{g}}$) via a mobility tensor that is a known function of the spheroid aspect ratio $\kappa$. In terms of the spheroid orientation vector ${\boldsymbol{p}}$, one may write ${\boldsymbol{U}}=\frac{1}{\mu L}[X_{A}^{-1}{\boldsymbol{p}}{\boldsymbol{p}}+Y_{A}^{-1}({\boldsymbol{I}}-{\boldsymbol{p}}{\boldsymbol{p}})]\cdot(F\hat{\boldsymbol{g}})$, $X_{A}(\kappa)$ and $Y_{A}(\kappa)$ being the axial and transverse translational resistance functions. The aspect ratio dependence of these functions is well known (see Kim & Karrila (1991)), and is given in Appendix A for convenient reference. Note that the ambient density at the center of the spheroid (${\boldsymbol{x}}=0$) is given by $\rho_{0}+\gamma(U_{i}\hat{g}_{i})t$, the time dependence arising from the spheroid translation. The equations of motion for problem 1, within a Boussinesq framework where the fluid density multiplying the inertial terms is taken as a constant $\rho_{0}$ (say), may be written as:

One now defines the perturbation density via $\rho^{(1)}=\rho_{0}+\gamma(x_{i}+U_{i}t)\hat{g}_{i}+\rho^{\prime(1)}$. Next, using the scales $U=F/(\mu LX_{A})$ for the velocity, $L$ for the length, $\mu U/L$ for the pressure and $\gamma L$ for $\rho^{\prime(1)}$, one obtains the following system of non-dimensional equations:

where $Re=\rho_{0}UL/\mu$ and $Ri_{v}=\gamma L^{3}g/(\mu U)$ are the Reynolds and viscous Richardson numbers; we continue to use the same notation for the dimensionless fields for simplicity, with $\hat{\boldsymbol{U}}$ now being a dimensionless vector along the direction of settling; note that $\hat{\boldsymbol{U}}$ is not a unit vector for an arbitrarily oriented spheroid, and reduces to one only for a spheroid aligned with gravity. In the particle-fixed frame, ${\boldsymbol{u}}^{(1)}\rightarrow-\hat{\boldsymbol{U}}$ at large distances, corresponding to the ambient uniform flow far away from the particle, and thus, the combination $(\hat{\boldsymbol{U}}+{\boldsymbol{u}}^{(1)})\cdot\hat{\boldsymbol{g}}$ in (12) denotes the convection of the (constant) base-state density gradient by the component of the disturbance velocity field along gravity. Finally, the time dependence of ${\boldsymbol{u}}^{(1)}$ in (11), and that of $\rho^{\prime(1)}$ in (12) in particular, that arise from the (slow) rotation of the spheroid, have been neglected owing to the quasi-steady state assumption made in (10-12); the time dependence of the density multiplying the inertial terms, on account of spheroid translation, has already been neglected within the Boussinesq approximation.

One now defines a disturbance pressure field via $p^{(1)}=p_{0}^{(1)}+p^{\prime(1)}$ with

so that $p^{(1)}_{0}$ defines the baseline hydrodystatic contribution arising from the ambient linear stratification. Having incorporated the baseline hydrostatic variation in $p^{(1)}_{0}$, and defining the disturbance velocity field as ${\boldsymbol{u}}^{(1)}=-\hat{\boldsymbol{U}}+{\boldsymbol{u}}^{\prime(1)}$, one may write the governing equations above in terms of the disturbance velocity, pressure and density fields as follows:

where the left hand side of (11) has been written in terms of the dynamic stress field ${\boldsymbol{\sigma}}^{(1d)}$ defined by ${\boldsymbol{\sigma}}^{(1d)}=-p^{\prime(1)}{\boldsymbol{I}}+({\boldsymbol{\nabla}}{\boldsymbol{u}}^{\prime(1)}+{\boldsymbol{\nabla}}{{\boldsymbol{u}}^{\prime(1)}}^{\dagger})$. Thus, one has the relation ${\boldsymbol{\sigma}}^{(1)}=-p_{0}^{(1)}{\boldsymbol{I}}+{\boldsymbol{\sigma}}^{(1d)}$ between the total and the dynamic stress fields of problem 1.

Assuming the spheroid in problem 1 to rotate with an angular velocity ${\boldsymbol{\Omega}}^{(1)}$, one has ${\boldsymbol{u}}^{(1)}={\boldsymbol{\Omega}}^{(1)}\wedge{\boldsymbol{x}}$ on $S_{p}$. Using this in the first surface integral in (1), and substituting the divergence of the dynamic stress from (15) in the volume integral in (1), one obtains:

where $\boldsymbol{\mathcal{L}}^{\sigma(1)d}$ now denotes the torque contribution due to the dynamic stress ${\boldsymbol{\sigma}}^{(1d)}$. Now, the particle in problem 1 is torque-free. In light of the above relation between ${\boldsymbol{\sigma}}^{(1)}$ and ${\boldsymbol{\sigma}}^{(1d)}$, the total torque may be written as $\boldsymbol{\mathcal{L}}^{(1)}=\boldsymbol{\mathcal{L}}^{\sigma(1)d}+\boldsymbol{\mathcal{L}}^{\sigma(1)s}$, where the dynamic torque component $\boldsymbol{\mathcal{L}}^{\sigma(1)d}$ includes both inertia and stratification-induced contributions, while $\boldsymbol{\mathcal{L}}^{\sigma(1)s}$ is the hydrostatic contribution due to the pressure field $p^{(1)}_{0}$ associated with the linearly varying density field of the stably stratified ambient, and defined by (13). Thus, $\boldsymbol{\mathcal{L}}^{(1)}=0\Rightarrow\boldsymbol{\mathcal{L}}^{\sigma(1)d}=-\boldsymbol{\mathcal{L}}^{\sigma(1)s}$, and the relation involving the spheroid angular velocity in problem 1 takes the following form:

where

with $p_{0}^{(1)}$ being defined in (13). Since the buoyancy force in a homogenous ambient acts through the centre of the spheroid, only the linearly varying term in (13) contributes to the hydrostatic torque, which may therefore be written as:

the contribution above remaining the same regardless of the choice of reference frame (${\boldsymbol{x}}$ or ${\boldsymbol{x}}^{L}$). On substitution of the above expression for ${\mathcal{L}}_{k}^{\sigma(1)s}$, and using the relation ${\boldsymbol{u}}^{(2)}_{i}={\boldsymbol{U}}^{(2)}_{ij}{\boldsymbol{\Omega}}^{(2)}_{j}$ (on account of the linearity of the Stokes equations), the second order tensor ${\boldsymbol{U}}^{(2)}_{ij}$ being known in closed form (see Dabade et al. (2015), and section 3 below), (18) takes the form:

Again, on account of linearity, one may write the torque on the rotating spheroid, in the test problem, in the form $\boldsymbol{\mathcal{L}}^{(2)}=[X_{C}{\boldsymbol{p}}{\boldsymbol{p}}+Y_{C}({\boldsymbol{I}}-{\boldsymbol{p}}{\boldsymbol{p}})]\cdot\boldsymbol{\Omega}^{(2)}$, where $X_{C}$ and $Y_{C}$ are the axial and transverse rotational resistance functions, and are known functions of $\kappa$ (Kim & Karrila (1991)), whose expressions are given in Appendix A. By symmetry, the sedimenting spheroid cannot spin about its axis regardless of its orientation, and therefore without loss of generality, the test problem 2 can be taken as that of a tranverse Stokesian rotation (${\boldsymbol{\Omega}}^{(2)}\cdot{\boldsymbol{p}}=0$), in which case the test-torque-angular-velocity relation takes the simpler form $\boldsymbol{\mathcal{L}}^{(2)}=Y_{C}\boldsymbol{\Omega}^{(2)}$. Finally, accounting for the fact that the test angular velocity ${\boldsymbol{\Omega}}^{(2)}$ can point in an arbitrary direction in a plane perpendicular to ${\boldsymbol{p}}$, one obtains the following relation for the spheroid angular velocity in problem 1:

Since a settling spheroid in a homogeneous ambient must retain its initial orientation in the Stokes limit on account of reversibility, expectedly, the rotation of the spheroid, as given by (22), arises due to the combined (weak) effects of fluid inertia and the ambient stratification. The first term in (22) corresponds to the inertial torque, while the second and third terms which have been grouped together correspond to the hydrostatic and hydrodynamic components of the stratification torque, respectively. The hydrostatic torque only involves knowledge of the ambient density field, and is easily evaluated. The inertial torque has a regular character in that the dominant contributions to the $O(Re)$ volume integral in (22) arise from a volume of $O(L^{3})$ around the sedimenting spheroid, and therefore, the integral may again readily be determined at leading order using the Stokesian approximations for the velocity fields involved, as has been done in Dabade et al. (2015). The evaluation of these two simpler contributions is detailed in the next section. The nature of the hydrodynamic torque arising from the perturbed stratification depends crucially on $Pe$, and this more complicated calculation is given in sections 4.1 and 4.2, respectively.

The $O(Re)$ inertial torque in (22) has recently been calculated for spheroids, both prolate and oblate, of an arbitrary aspect ratio (see Dabade et al. (2015)). Although the analysis in Dabade et al. (2015) pertains to the limit $Re\ll 1$, the results have been shown to remain qualitatively valid even for $Re$’s of order unity (see Jiang et al. (2020)). As mentioned above, this torque has a regular character, and the regularity may be seen from the convergence of the inertial volume integral in (18) based on a leading order Stokesian estimate for the integrand. As argued in Dabade et al. (2015), the inertial acceleration ${\boldsymbol{u}}^{(1)}\cdot\nabla{\boldsymbol{u}}^{(1)}\sim\hat{\boldsymbol{U}}\cdot\nabla{\boldsymbol{u}}^{(1)}\sim O(1/r^{2})$ for distances large compared to $L$, or $r\gg 1$ in dimensionless terms, on using ${\boldsymbol{u}}^{(1)}\sim O(1/r)$ for the Stokeslet field due to the translating spheroid. The test velocity field ${\boldsymbol{u}}^{(2)}$ due to the rotating spheroid has the character of a rotlet (and stresslet) in the far-field, and is therefore $O(1/r^{2})$. This leads to an integrand that decays as $\hat{\boldsymbol{U}}\cdot\nabla{\boldsymbol{u}}^{(1)}\cdot{\boldsymbol{u}}^{(2)}\sim O(1/r^{4})$ for $r\gg 1$, implying a convergent volume integral. This volume integral has been evaluated in closed form using spheroidal coordinates in Dabade et al. (2015). For the prolate case, the spheroidal coordinates $(\xi,\eta,\phi)$ are defined by the relations: $x_{1}+\mathrm{i}x_{2}=d\bar{\xi}\bar{\eta}\exp(i\phi)$, $x_{3}=d\xi\eta$, with the $3$-axis of the Cartesian system aligned with the spheroid axis of symmetry. Here, $1\leq\xi<\infty$, $|\eta|\leq 1$ and $0\leq\phi<2\pi$, with $\bar{\xi}=(\xi^{2}-1)^{\frac{1}{2}}$ and $\bar{\eta}=(1-\eta^{2})^{\frac{1}{2}}$. The constant-$\xi$ surfaces correspond to confocal prolate spheroids and the constant-$\eta$ surfaces to confocal two-sheeted hyperboloids, both with the interfoci distance $2d$, and the constant-$\phi$ surfaces are planes passing through the axis of symmetry. The corresponding expressions for the oblate case may be obtained by the substitutions $d\leftrightarrow-\mathrm{i}d$, $\xi\leftrightarrow\mathrm{i}\bar{\xi}$. In either case, the spheroid is the surface $\xi=\xi_{0}$, its aspect ratio being given by $\kappa=\frac{\xi_{0}}{\bar{\xi_{0}}}$ and $\frac{\bar{\xi}_{0}}{\xi_{0}}$ for the prolate and oblate cases; thus, the near-spherical limit ($\kappa\rightarrow 1$) for either prolate or oblate spheroids corresponds to $\xi_{0}\rightarrow\infty$, while the slender fiber ($\kappa\rightarrow\infty$) and flat disk ($\kappa\rightarrow 0$) limits correspond to $\xi_{0}\rightarrow 1$. The fluid domain in the volume integrals in (22) corresponding to $\xi\geq\xi_{0}$.

For a prolate spheroid, the actual velocity field ${\boldsymbol{u}}^{(1)}$ and the test velocity field tensor ${\boldsymbol{U}}^{(2)}$ in (22), may be written in terms of the appropriate decaying vector spheroidal harmonics (see Dabade et al. (2015);Dabade et al. (2016)) as:

The $\boldsymbol{S}^{(3)}_{t,s}$’s and $\boldsymbol{S}^{(2)}_{t,s}$’s in (23) and (24) denote the decaying (biharmonic and harmonic) vectorial solutions of the Stokes equations in spheroidal coordinates, and are given by the following expressions:

with the $P_{t}^{s}(\eta)$ and $Q_{t}^{s}(\xi)$ being the associated Legendre functions of the first and second kind, respectively (for $s=0$, the $P_{t}^{s}(\eta)$ correspond to the usual Legendre polynomials). The resulting volume integration may then be carried out analytically, and the inertial angular velocity ($\boldsymbol{\Omega}^{(1)I}$) is given by:

for prolate spheroids. The corresponding expression for the oblate case may be obtained by the aforementioned substitutions viz. $d\leftrightarrow-\mathrm{i}d$, $\xi_{0}\leftrightarrow\mathrm{i}\bar{\xi}_{0}\,$in the dimensional angular velocity, and is given by:

The expressions for $F^{p}_{I}(\xi_{0})$ and $F^{o}_{I}(\xi_{0})$, as functions of the spheroid eccentricity ($e=1/\xi_{0}$), were first obtained by Dabade et al. (2015), and are given in Appendix A. The inertial angular velocity given by (29) and (30) orients sedimenting spheroids broadside-on regardless of $\kappa$. The combination of the aspect-ratio-dependent functions, $F^{p/o}_{I}(\xi_{0})X_{A}/(Y_{C}Y_{A})$, that multiplies $Re(\hat{\boldsymbol{g}}\cdot{\boldsymbol{p}})(\hat{\boldsymbol{g}}\wedge{\boldsymbol{p}})$, and that determines the $\kappa$-dependence of the inertial angular velocities above, are plotted as a function of the eccentricity in Figure 1, for both the prolate and oblate cases. One obtains the expected $O(1/\xi_{0}^{2})$ scaling in the near-sphere limit ($\xi_{0}\rightarrow\infty$); at the other extreme($\xi_{0}\rightarrow 1$), the inertial angular velocity approaches zero as $O[\ln(\xi_{0}-1)]^{-1}$ in the slender fiber limit, consistent with viscous slender body theory (Khayat & Cox (1989),Subramanian & Koch (2005)), while remaining finite in the limit of a flat disk.

The hydrostatic component of the stratification torque is also readily evaluated in spheroidal coordinates. For the prolate case, the dimensionless position vector that appears in (20) is given by $\boldsymbol{x}=\frac{\bar{\xi}_{0}}{\xi_{0}}\bar{\eta}(\cos\phi{\boldsymbol{1}}_{1}+\sin\phi{\boldsymbol{1}}_{2})+\eta{\boldsymbol{1}}_{3}$, and the unit normal is $\boldsymbol{n}={\boldsymbol{1}}_{\xi}=\frac{\xi_{0}\bar{\eta}}{\sqrt{\xi_{0}^{2}-\eta^{2}}}(\cos\phi{\boldsymbol{1}}_{1}+\sin\phi{\boldsymbol{1}}_{2})+\frac{\bar{\xi_{0}}\eta}{\sqrt{\xi_{0}^{2}-\eta^{2}}}{\boldsymbol{1}}_{3}$. Using these expressions, and the areal element $dS=h_{\eta}h_{\phi}d\eta d\phi$, with $h_{\eta}=\frac{\sqrt{\xi_{0}^{2}-\eta^{2}}}{\xi_{0}\bar{\eta}}$ and $h_{\phi}=\frac{\bar{\xi}_{0}\bar{\eta}}{\xi_{0}}$, one obtains the angular velocity due to the hydrostatic torque as

for the prolate case, and using the transformations mentioned above,

for the oblate case. The angular velocities given by (31) and (32) also orient the spheroid broadside-on like the inertial torque above. The aspect-ratio-dependent functions that multiply $Ri_{v}(\hat{\boldsymbol{g}}\cdot{\boldsymbol{p}})(\hat{\boldsymbol{g}}\wedge{\boldsymbol{p}})$ in (31)and (32) are plotted as functions of $\xi_{0}$ in Fig. 2. Since the hydrostatic torque is only a function of the particle geometry, these aspect ratio functions are algebraically small in both the near-sphere, and the slender fiber and flat-disk limits.

The inertial and hydrostatic angular velocities above have an identical angular dependence, of the form $(\hat{\boldsymbol{g}}\cdot{\boldsymbol{p}})(\hat{\boldsymbol{g}}\wedge{\boldsymbol{p}})$, one which is easily inferred based on the requirement that the angular velocity be a pseudovector quadratic in $\hat{\boldsymbol{g}}$ (Dabade et al. (2015)). The dependence implies that the maximum angular velocity occurs midway between the horizontal ($\hat{\boldsymbol{g}}\cdot{\boldsymbol{p}}=0$) and vertical  ($\hat{\boldsymbol{g}}\cdot{\boldsymbol{p}}=1$) orientations. The hydrostatic torque arises because the point of action of the upward buoyant force, the center of mass of the equivalent stratified fluid blob (in the Archimedean interpretation), lies below the geometric center through which the weight of the spheroid acts downward. The two forces therefore constitute a couple that turns the spheroid broadside-on. The broadside-on nature of the inertial torque is on account of ‘wake-shielding’ - the wake associated with the front portion of the spheroid shields the rear, which catches up with the front as a result. As pointed out in Dabade et al. (2015), this not literally true for small $Re$. A signature of the wake arises only on length scales greater than $O(LRe^{-1})$, in contrast to the scaling arguments above which show that the $O(Re)$ inertial torque arises from fluid inertial forces in a region of $O(L^{3})$ (the inner region) around the sedimenting spheroid. Nevertheless, the velocity field in the inner region reflects the asymmetry of the outer Oseen field, and the sense of rotation remains the same for small $Re$. Importantly, the broadside-on nature of the inertial and hydrostatic torques imply that the transition from broadside-on to edgewise settling, observed in the recent experiments (see Mercier et al. (2020)) discussed in the introduction, must depend entirely on the hydrodynamic component of the stratification torque, that is, the second term within square brackets on the right hand side in (22). While the calculation above shows the hydrostatic component to be $O(Ri_{v})$, consistent with the nominal order in (22), this is not true of the hydrodynamic component. As will be shown in section 4 below, the hydrodynamic component scales as $O(Ri_{v})$ only for $Pe\ll 1$ when the dominant contribution to the associated torque integral comes from length scales of $O(L)$ similar to the inertial torque above. In the opposite limit of $Pe\gg 1$, and for the so-called Stokes stratification regime corresponding to $Re\ll Ri_{v}^{\frac{1}{3}}$ (see Varanasi & Subramanian (2021)), the dominant contributions to the torque integral arise from much larger length scales of $O(LRi_{v}^{-\frac{1}{3}})$, and the hydrodynamic component scales as $O(Ri_{v}^{\frac{2}{3}})$, being much larger than the hydrostatic component above.

Before proceeding with the calculation of the hydrodynamic component of the stratification torque, it is worth remarking on the nature of the coupling between the inertial and stratification torque contributions that is not obvious from the formal result (18) above, where they appear as separate additive contributions. On account of the convergent volume integral, the $O(Re)$ inertial angular velocity, as given by (29) and (30), only involves the Stokesian fields in a homogeneous ambient, and is evidently independent of the ambient stratification. The correction to this leading order estimate is dependent on the nature of the ambient stratification, however, even within the Boussinesq framework. To see this, we return to the inertial volume integral, and estimate the next correction. Recall that the angular velocities in (29) and (30) were based on the approximating the volume integral by Stokesian estimates, and the torque contribution at the next order requires one to examine the next term in the small-$Re$ expansion for the velocity field in problem 1. Writing ${\boldsymbol{u}}^{(1)}={\boldsymbol{u}}^{(10)}+Re\,{\boldsymbol{u}}^{(11)}$, ${\boldsymbol{u}}^{(10)}$ is the Stokesian approximation given by (23) above and is $O(1/r)$ for $r\gg 1$, while ${\boldsymbol{u}}^{(11)}$ remains $O(1)$ in the far-field. The latter, of course, implies that the above regular expansion breaks down at length scales of $O(LRe^{-1})$, a manifestation of the singular nature of inertia in an unbounded domain (the so-called Whitehead’s paradox; see Leal (1992)). Provided one assumes buoyancy forces to dominate the inertial ones on scales much smaller than the inertial screening length (of $O(LRe^{-1}))$, the above far-field estimate of ${\boldsymbol{u}}^{(11)}$ may still be used to estimate the correction to the leading $O(Re)$ contribution. The $O(Re^{2})$ inertial acceleration is now $\hat{\boldsymbol{U}}\cdot\nabla{\boldsymbol{u}}^{(11)}\sim O(1/r)$, and using ${\boldsymbol{u}}^{(2)}\sim O(1/r^{2})$ , the resulting volume integral, at $O(Re^{2})$, is logarithmically divergent. The divergence will be cut off at the stratification screening length that is $O(Ri_{v}Pe)^{-\frac{1}{4}}$ for $Pe\ll 1$ (Ardekani & Stocker (2010),List (1971)), and $O(Ri_{v}^{-\frac{1}{3}})$ for $Pe\gg 1$ (Varanasi & Subramanian (2021)), implying that the next correction to the inertial angular velocity is $O[Re^{2}\ln(Ri_{v}Pe)^{-\frac{1}{4}}]$ for $Pe\ll 1$ and $O[Re^{2}\ln Ri_{v}^{-\frac{1}{3}}]$ for $Pe\gg 1$, and is thereby a function of the ambient stratification. For self consistency, one requires that both of the aforementioned stratification screening lengths be less than $O(Re^{-1})$, which translates to the requirement $Ri_{v}\gg Pe^{-1}Re^{4}$ for small $Pe$, and for one to be in the aforementioned Stokes stratification regime ($Ri_{v}^{\frac{1}{3}}\gg Re$) for large $Pe$.

Owing to the differing character of the hydrodynamic component in the limits of small and large $Pe$, the calculations in these two asymptotic regimes are carried out in separate subsections below. Keeping in mind that $Pe=RePr$, $Pr$ being the Prandtl number, the small $Pe$ case doesn’t necessarily place a restriction on $Pr$ which may either be small or large (although, large $Pr$ imposes a greater restriction on the smallness of $Re$ since $Re$ must now be smaller than $O(Pe^{-1})$). However, the limit of small $Re$ that characterizes the analysis here implies that the large $Pe$ case necessarily requires a large $Pr$ which may be realized in experiments that use salt as a stratifying agent.