Instytut Podstawowych Problemów Techniki

Streszczenie:

The virial corrections to short-time self- and collective diffusion coefficients as well as the effective viscosity are calculated for suspensions of hard spheres with the same radii and constant (blocked within the particle) magnetization modelled by a point dipole. Analytic, integral formulae derived from basic principles of statistical mechanics are provided for both cases – in the absence and in the presence of an external magnetic field. In the former case the diffusion and viscosity coefficients are evaluated numerically as a function of the strength of magnetic interactions between the particles and it is reported that the translational collective diffusion coefficient is significantly greater than all the other coefficients. In the presence of an external magnetic field the coefficients become anisotropic and are evaluated in the asymptotic regime of weak interparticle magnetic interactions.

Słowa kluczowe:

colloids, low-Reynolds-number flows, magnetic fluids

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Streszczenie:

The Rotne–Prager–Yamakawa (RPY) approximation is a commonly used approach to model the hydrodynamic interactions between small spherical particles suspended in a viscous fluid at a low Reynolds number. However, when the particles overlap, the RPY tensors lose their positive definiteness, which leads to numerical problems in the Brownian dynamics simulations as well as errors in calculations of the hydrodynamic properties of rigid macromolecules using bead modelling. These problems can be avoided by using regularizing corrections to the RPY tensors; so far, however, these corrections have only been derived for equal-sized particles. Here we show how to generalize the RPY approach to the case of overlapping spherical particles of different radii and present the complete set of mobility matrices for such a system. In contrast to previous ad hoc approaches, our method relies on the direct integration of force densities over the sphere surfaces and thus automatically provides the correct limiting behaviour of the mobilities for the touching spheres and for a complete overlap, with one sphere immersed in the other one. This approach can then be used to calculate hydrodynamic properties of complex macromolecules using bead models with overlapping, different-sized beads, which we illustrate with an example.

Słowa kluczowe:

complex fluids, computational methods, low-Reynolds-number flows, mathematical foundations, suspensions

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Streszczenie:

Rotne-Prager-Yamakawa approximation is a commonly used approach to model hydrodynamic interactions between particles suspended in fluid. It takes into account all the long-range contributions to the hydrodynamic tensors, with the corrections decaying at least as fast as the inverse fourth power of the interparticle distances, and results in a positive definite mobility matrix, which is fundamental in Brownian dynamics simulations. In this communication, we show how to construct the Rotne-Prager-Yamakawa approximation for the bulk system under shear flow, which is modeled using the Lees-Edwards boundary conditions.

Słowa kluczowe:

Tensor methods, Hydrodynamics, Shear flows, Brownian dynamics, Boundary value problems

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Streszczenie:

The Rotne–Prager–Yamakawa approximation is one of the most commonly used methods of including hydrodynamic interactions in modelling of colloidal suspensions and polymer solutions. The two main merits of this approximation are that it includes all long-range terms (i.e. decaying as R−3 or slower in interparticle distances) and that the diffusion matrix is positive definite, which is essential for Brownian dynamics modelling. Here, we extend the Rotne–Prager–Yamakawa approach to include both translational and rotational degrees of freedom, and derive the regularizing corrections to account for overlapping particles. Additionally, we show how the Rotne–Prager–Yamakawa approximation can be generalized for other geometries and boundary conditions.

Słowa kluczowe:

computational methods, low-Reynolds-number flows, suspensions

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Streszczenie:

We study the effect of stratification and compressibility on the threshold of convection and the heat transfer by developed convection in the nonlinear regime in the presence of strong background rotation. We consider fluids both with constant thermal conductivity and constant thermal diffusivity. The fluid is confined between two horizontal planes with both boundaries being impermeable and stress-free. An asymptotic analysis is performed in the limits of weak compressibility of the medium and rapid rotation (τ−1/12 ≪ |θ| ≪ 1, where τ is the Taylor number and θ is the dimensionless temperature jump across the fluid layer). We find that the properties of compressible convection differ significantly in the two cases considered. Analytically, the correction to the characteristic Rayleigh number resulting from small compressibility of the medium is positive in the case of constant thermal conductivity of the fluid and negative for constant thermal diffusivity. These results are compared with numerical solutions for arbitrary stratification. Furthermore, by generalizing the nonlinear theory of Julien and Knobloch [Fully nonlinear three-dimensional convection in a rapidly rotating layer. Phys. Fluids 1999, 11, 1469–1483] to include the effects of compressibility, we study the Nusselt number in both cases. In the weakly nonlinear regime we report an increase of efficiency of the heat transfer with the compressibility for fluids with constant thermal diffusivity, whereas if the conductivity is constant, the heat transfer by a compressible medium is more efficient than in the Boussinesq case only if the specific heat ratio γ is larger than two.

Słowa kluczowe:

Anelastic convection, Heat transfer, Compressibility

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Streszczenie:

We revisit the problem introduced by Gilman (1970) and Acheson (1979) of linear stability of a plane layer of compressible fluid permeated by a horizontal magnetic field of magnitude decreasing with height with respect to short-wavelength two-dimensional perturbations varying in the directions perpendicular to the applied field. We show, that in the limit of large horizontal wave numbers the perturbations become strongly localised in the vertical direction. The motiavtion for this study is of astrophysical nature and comes from the common belief, that the magnetic buoyancy effects produce short-wavelength instabilities in the solar tachocline. We analyse the solar tachocline parameter regime to speculate about the strength of the magnetic field at the base of the solar convective zone and the time scales of the field variations induced by the magnetic buoyancy instability on the Sun.

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