Brownian motion (diffusion) The ratio between these two mobilities is independent of the viscosity:Rotational and translational diffusion of protein and lipid molecules in biological membranes has recently become accessible to experimental study (1-4). The problem can be studied theoretically by applying the classical analysis of Brownian motion to a hydrodynamic model. A simple model is one in which .the membrane is taken as an infinite plane sheet of viscous fluid (lipid) separating infinite regions of less viscous liquid (water). The protein molecule is regarded as a cylinder, with axis perpendicular to the plane of the sheet, moving about in the sheet under the action of Brownian motion ( Fig. 1).Diffusion of a particle due to Brownian motion is described by diffusion coefficients, DT and DR, for translational and rotational displacements. For motion in a plane and rotation about a perpendicular axis, r2=4DTt, =2DRt, [1] where ' and B are the mean square displacement and angular rotation in time t, respectively.The diffusion coefficients are related to the mobilities of the particle by the Einstein relations DT = kBTbT; DR= kBTbR [2] where kB is Boltzmann's constant, T is the absolute temperature, and b is the mobility (independent of force or torque) defined as the velocity (or angular velocity) produced by steady unit force (or torque) (5).Iipid, qi -1water, al FIG. 1. The hydrodynamic model. A cylindrical particle embedded in a lipid bilayer membrane bounded by aqueous phases on both sides. The particle is permitted to move laterally in the x-y plane, and to rotate around the z-axis.bT/bR = 3 a2 [4] where a denotes the particle radius, it being assumed that the Reynolds number is small so that the equations of slow viscous motion (inertial terms neglected) apply.For our model of the protein in the membrane, matters are not so simple. We denote by 1A the viscosity of the fluid representing the membrane and by u' the viscosity of the exterior liquid. It is supposed that ,ut' << A, If At' is neglected completely, there is no viscous stress transmitted across the surfaces of the sheet and the hydrodynamical problem is that of the motion of a cylinder through a viscous fluid in directions perpendicular to its generators. Finding the rotational mobility is a trivial calculation, giving 1 bR = 4rpta2h ' [5] where h denotes the thickness of the sheet and a now stands for the radius of the cylindrical particle. However, the translational mobility does not exist, for there is no solution of theslow viscous flow equations for steady translational motion in two dimensions (the so-called Stokes paradox) (6).A finite translational mobility bT can be obtained by taking account of the inertia of the viscous fluid, replacing the slow viscous flow equations by the Oseen equations (6). It is then found that bT = 4(log + 2 ) [6] where p denotes the density of the fluid, U is the (steady) translational velocity, and y is Euler's constant (0.5772). But the mobility is now not independent of force and the argument le...

It is shown that a sphere moving through a very viscous liquid with velocity V relative to a uniform simple shear, the translation velocity being parallel to the streamlines and measured relative to the streamline through the centre, experiences a lift force 81·2μVa2k½/v½ + smaller terms perpendicular to the flow direction, which acts to deflect the particle towards the streamlines moving in the direction opposite to V. Here, a denotes the radius of the sphere, κ the magnitude of the velocity gradient, and μ and v the viscosity and kinematic viscosity, respectively. The relevance of the result to the observations by Segrée & Silberberg (1962) of small spheres in Poiseuille flow is discussed briefly. Comments are also made about the problem of a sphere in a parabolic velocity profile and the functional dependence of the lift upon the parameters is obtained.

This paper proposes a theory of collisions between small drops in a turbulent fluid which takes into account collisions between equal drops. The drops considered are much smaller than the small eddies of the turbulence and so the collision rates depend only on the dimensions of the drops, the rate of energy dissipation E and the kinematic viscosity u. Reasons are given for believing that the collision efficiency for nearly equal drops is unity, and the collision rate due to the spatial variations of turbulent velocity is shown to be N = 1.30(r1 + r2)3nln2(~/~)1/2, valid for r1/y2 between one and two. A numerical integration has been performed using this expression to show how an initially uniform distribution will change because of collisions. An approximate calculation is then made to take account also of collisions which occur between drops of different inertia because of the action of gravity and the turbulent accelerations. The results are applied to the case of small drops in atmospheric clouds to test the importance of turbulence in initiating rainfall. Estimates of E are made for typical conditions and these are used to calculate the initial rates of collision, the change in mean properties and the rate of production of large drops. It is concluded that the effects of turbulence in clouds of the layer type should be small, but that moderate amounts of turbulence in cumulus clouds could be effective in broadening the drop size distribution in nearly uniform clouds where only the spatial variations of velocity are important. In heterogeneous clouds the collision rates are increased, and the effects due to the inertia of the drop soon become predominant. The effect of turbulence in causing collisions between unequal drops becomes comparable with that of gravity when E is about 2000 cm2 s~c-~.

The structure of the intense-vorticity regions is studied in numerically simulated homogeneous, isotropic, equilibrium turbulent flow fields at four different Reynolds numbers, in the range Re, = 35-170. In accordance with previous investigators this vorticity is found to be organized in coherent, cylindrical or ribbon-like, vortices ('worms'). A statistical study suggests that they are simply especially intense features of the background, O(o'), vorticity. Their radii scale with the Kolmogorov microscale and their lengths with the integral scale of the flow. An interesting observation is that the Reynolds number y/v, based on the circulation of the intense vortices, increases monotonically with ReA, raising the question of the stability of the structures in the limit of Re,-z co. Conversely, the average rate of stretching of these vortices increases only slowly with their peak vorticity, suggesting that self-stretching is not important in their evolution. One-and two-dimensional statistics of vorticity and strain are presented; they are non-Gaussian and the behaviour of their tails depends strongly on the Reynolds number. There is no evidence of convergence to a limiting distribution in this range of Re,, even though the energy spectra and the energy dissipation rate show good asymptotic properties in the higher-Reynolds-number cases. Evidence is presented to show that worms are natural features of the flow and that they do not depend on the particular forcing scheme.

A theoretical justification is given for an empirical boundary condition proposed by Beavers and Joseph [1]. The method consists of first using a statistical approach to extend Darcy's law to non homogeneous porous medium. The limiting case of a step function distribution of permeability and porosity is then examined by boundary layer techniques, and shown to give the required boundary condition. In an Appendix, the statistical approach is checked by using it to derive Einstein's law for the viscosity of dilute suspensions.