arXiv subjects

H. K. Moffatt

Publications and source records attributed to H. K. Moffatt.

4 recordsLinked to original sources

arXiv2024

Flow induced by the rotation of two circular cylinders in a viscous fluid

The low-Reynolds-number Stokes flow driven by rotation of two parallel cylinders of equal unit radius is investigated by both analytical and numerical techniques. In Part I, the case of counter-rotating cylinders is considered. A numerical (finite-element) solution is obtained by enclosing the system in an outer cylinder of radius $R_{0}\!\gg\!1$, on which the no-slip condition is imposed. A model problem with the same symmetries is first solved exactly, and the limit of validity of the Stokes approximation is determined; this model has some relevance for ciliary propulsion. For the two-cylinder problem, attention is focused on the small-gap situation $\varepsilon \ll 1$. An exact analytic solution is obtained in the contact limit $\varepsilon=0$, and a net force $F_{c}$ acting on the pair of cylinders in this contact limit is identified; this contributes to the torque that each cylinder experiences about its axis. The far-field torque doublet (`torquelet') is also identified. Part II treats the case of co-rotating cylinders, for which again a finite-element numerical solution is obtained for $R_{0}\!\gg \!1$. The theory of Watson (1995) is elucidated and shown to agree well with the numerical solution. In contrast to the counter-rotating case, inertia effects are negligible throughout the fluid domain, however large, provided Re $\ll 1$. In the concluding section, the main results for both cases are summarised, and the situation when the fluid is unbounded ($R_{0}=\infty$) is discussed. (...)

physics.flu-dyn↗

arXiv2019

Singularities in Fluid Mechanics

Singularities of the Navier-Stokes equations occur when some derivative of the velocity field is infinite at any point of a field of flow (or, in an evolving flow, becomes infinite at any point within a finite time). Such singularities can be mathematical (as e.g. in two-dimensional flow near a sharp corner, or the collapse of a Mobius-strip soap film onto a wire boundary) in which case they can be resolved by refining the geometrical description; or they can be physical (as e.g. in the case of cusp singularities at a fluid/fluid interface) in which case resolution of the singularity involves incorporation of additional physical effects; these examples will be briefly reviewed. The finite-time singularity problem for the Navier-Stokes equations will then be discussed and a recently developed analytical approach will be presented; here it will be shown that, even when viscous vortex reconnection is taken into account, there is indeed a physical singularity, in that, at sufficiently high Reynolds number, vorticity can be amplified by an arbitrarily large factor in an extremely small point-neighbourhood within a finite time, and this behaviour is not resolved by viscosity. Similarities with the soap-film-collapse and free-surface--cusping problems are noted in the concluding section, and the implications for turbulence are considered.

physics.flu-dyn↗

arXiv2019

Chiral transfer of angular momentum

Suppose that viscous fluid is contained in the space between a fixed sphere $S_2$ and an interior sphere $S_1$ which moves with time-periodic velocity ${\bf U}(t)$ and angular velocity ${\bf Ω}(t)$, with $ \left<{\bf U}(t)\right> = \left<{\bf Ω}(t)\right> = 0$. It is shown that, provided this motion is chiral in character, it can drive a flow that exerts a non-zero torque on $S_2$. Thus angular momentum can be transferred through this mechanism.

physics.flu-dyn↗

arXiv2019

Towards a finite-time singularity of the Navier-Stokes equations. Part 2. Vortex reconnection and singularity evasion

In Part 1 of this work, we have derived a dynamical system describing the approach to a finite-time singularity of the Navier-Stokes equations. We now supplement this system with an equation describing the process of vortex reconnection at the apex of a pyramid, neglecting core deformation during the reconnection process. On this basis, we compute the maximum vorticity $ω_{max}$ as a function of vortex Reynolds number $R_Γ$ in the range $2000\le R_Γ\le 3400$, and deduce a compatible behaviour $ω_{max}\sim ω_{0}\exp{\left[1 + 220 \left(\log\left[R_Γ/2000\right]\right)^{2}\right]}$ as $R_Γ\rightarrow \infty$. This may be described as a physical (although not strictly mathematical) singularity, for all $R_Γ\gtrsim 4000$.

physics.flu-dyn↗