Flow past a sphere undergoing unsteady rectilinear motion and unsteady drag at small Reynolds number

• Published in: Journal of fluid mechanics Vol. 446; pp. 95 - 119
• Main Author: ASMOLOV, EVGENY S.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 10.11.2001
• Subjects: Flow; Fourier transforms; Kinematic viscosity; Reynolds number
• DOI: 10.1017/S0022112001005675
• ISSN: 0022-1120

The flow induced by a sphere which undergoes unsteady motion in a Newtonian fluid at small Reynolds number is considered at distances large compared with sphere radius a. Previous solutions of the unsteady Oseen equations (Ockendon 1968; Lovalenti & Brady 1993b) for rectilinear motion are refined. Three-dimensional Fourier transforms of the disturbance field are integrated over Fourier space to derive new concise equations for the velocity field and history force in terms of single history integrals. Various slip-velocity profiles are classified by the ratio A of the particle relative displacement, z′p(t′) − z′p(τ′), to the diffusion length, l′D = 2[v(t′ − τ′)]1/2, where v is the kinematic viscosity of the fluid. Most previous studies are concerned with large-displacement motions for which the ratio is large in the long-time limit. It is shown using asymptotic calculations that the flow at any point at large distance z past a sphere for arbitrary large-displacement and non-reversing motion is the same as the steady-state laminar wake if z is expressed in terms of the time elapsed since the particle was at that point in the laboratory frame. The point source solution for the remainder of the far flow is also valid for the unsteady case. A start-up motion with slip velocity V′p = γ′(t′)−1/2, t′ > 0, is investigated for which A is finite. A self-similar solution for the flow field is obtained in terms of space coordinates scaled by the diffusion length, u′ = auss(η)/t′ where η = r′/2(vt′)1/2. The unsteady Oseen correction to the drag is inversely proportional to time. When A is small in the long-time limit (a small-displacement motion) the flow field also depends on the space coordinates in terms of η. The distribution of the streamwise velocity uz is symmetrical in z.