Fluid drift and spheroid rotation in viscous density stratified fluids

Abstract

This thesis broadly deals with the motion of passive particles in density stratified fluids inthe viscous limit and can be divided into two parts. The first part is concerned with theevaluation of the drift volume, as defined by Darwin [1953], due to slow settling of a spherein a weakly stratified fluid in the convection dominant limit. This problem is motivated by theintent to examine the validity of the rather provocative proposal of a biogenic contributionto ocean mixing [Katija and Dabiri, 2009; Leshansky and Pismen, 2010; Subramanian,2010]. The main intention is therefore to evaluate the drift volume due to a passive spheresettling, in the regime mentioned above, which in turn requires the velocity field generatedby sphere translation. Thus, the problem considered in the first part is concerned withthe evaluation of the velocity and density fields due to translation of a sphere in a stablystratified ambient in the limit of small Reynolds (Re≪1) and viscous Richardson (Riv≪1)numbers; here,Re=ρU aμandRiv=γa3gμUwithabeing the sphere radius,Uthe translationspeed,ρandμthe density and viscosity of the stratified ambient,gthe acceleration dueto gravity, andγ(>0)the density gradient (assumed constant) characterizing the stableambient stratification. In contrast to most earlier efforts, our study primarily considers theconvection dominant limit corresponding toPe=U aD≫1,Dbeing the diffusivity of thestratifying agent. However, the diffusion dominant (Pe→0) regime [Ardekani and Stocker,2010; List, 1971] is also considered for purposes of completeness, with a few new results,with regard to the structure of the velocity and density fields being presented. Using acombination of numerical computations and far-field asymptotics, we have characterizedin detail the velocity and density fields in what we term the Stokes stratification regime,defined byRe≪Ri1/3v≪1, and corresponding to the dominance of buoyancy over inertialforces. We have used a Fourier transform approach to write down the velocity and densityfields as Fourier integrals, and a far-field analysis of these integrals leads us to distinguishbetween different regions of the flow field. On the whole, buoyancy forces associated withthe perturbed stratification fundamentally alter the viscously dominated fluid motion at largedistances. At distances of order the stratification screening length, that scales asaRi−1/3vinthe large-Pelimit, the fluid motion transforms from the familiar fore-aft symmetric Stokesianform to a fore-aft asymmetric pattern of recirculating cells with primarily horizontal motion within; except in the vicinity of the rear stagnation streamline. At larger distances, the motionis vanishingly small except within (a) an axisymmetric horizontal wake whose vertical extentgrows asO(Ri−1/5vr2/5t),rtbeing the distance in the plane perpendicular to translation and (b)a buoyant reverse jet behind the particle that narrows as the inverse square root of distancedownstream. ForPe=∞, the motion close to the rear stagnation streamline starts off pointingin the direction of translation, in the inner Stokesian region, and decaying as the inverse ofthe downstream distance; the motion reverses beyond a distance of1.15aRi−1/3v, with theeventual reverse flow in the far-field buoyant jet again decaying as the inverse of the distancedownstream. For large but finitePe, the narrowing jet is smeared out beyond a distance ofO(aRi−1/2vPe1/2), leading to an exponential decay in the aforementioned reverse flow. The velocity field obtained above in the convection dominant limit, is used to calculatethe associated fluid pathlines, drift displacements, drift surfaces, and finally, the drift volumefor different viscous Richardson (Riv) and Peclet (Pe) numbers. In order to account for thefinite size of the sphere, a uniformly valid expression for the velocity field is found by meansof an additive composite approximation involving the summation of the inner and outerregion expressions and subtraction of the overlapping contribution. Numerical integration ofthe ODE’s governing the positions of the fluid elements is carried out using the compositevelocity field above (both axial and transverse components), at each time instant, with theouter-region contribution being given as Fourier integrals. At the initial timet=0, fluidelements are assumed to occupy a horizontal planez=0. The calculations of the upstreamand downstream drift components are then carried out by integrating to suitably large negativeand positive times, respectively. Our results clearly show that the pathlines and the driftdisplacements thus obtained do not show the characteristics associated with a homogeneousStokes flow. In particular, the drift displacements do not exhibit a logarithmic-in-timedivergence characteristic of a homogeneous Stokesian regime. Further, the downstreamcalculations show a reversal of fluid pathlines due to the effects of the aforementionedexistence rearward jet for timest>O(Ri−1/3v). The drift surfaces, corresponding to timesapproaching negative and positive infinity, are shown to asymptote to finite limiting forms,implying that the drift volume converges to a finite value in a density stratified fluid for anyfinite Pe. For sufficiently large Pe, the upstream and downstream components of the totaldrift volumes are shown to scale asRi−2/3v, although the convergence to this limiting large-Peplateau is much faster for the upstream component. The final drift volume has the characterof a reflux owing to the aforementioned reversal of fluid pathlines, and the cancellationbetween the upstream and downstream components appears to result in a net drift volumethat is smaller than that expected from naive scaling arguments, with the scaling obtainedfrom the numerics being closer toRi−1/3v. Further, the fact that the total drift volume has the character of a reflux clearly indicates the singular role of stratification, for sufficiently longtimes, on the drift induced by particles in density stratified fluids.