Non-continuum lubrication flows between particles colliding in a gas
Non-continuum lubrication flows between particles colliding in a gas
复制标题
气体中碰撞的颗粒之间的非连续润滑流
作者：
R. R. Sundararajakumar;D. Koch
如果连续不可压缩流体的润滑力在所有颗粒分离时保持不变，气体中光滑颗粒之间的固体碰撞就不会发生。当粒子之间的间隙是气体的平均自由程λ0的数量级时，气体的离散分子性质变得重要。对于半径a小于约50μm的粒子，以与其终端速度相当的相对速度在空气中碰撞，缝隙中气体的可压缩性的影响并不重要。缝隙中流动的性质取决于最小缝隙厚度H0≡aε、平均自由路径λ0和曲率影响变得重要的距离aε1/2的相对大小。Hocking(1973年)在颗粒表面用麦克斯韦滑移边界条件分析了滑移流区a[GT]λ0。为了求出在过渡区域(aε∼O(λ0))的润滑力，我们使用Cercignani&Daneri(1963)的结果作为泊松叶槽流中压力梯度的函数。当ε[Lt]λ0[Lt]aε1/2时，人们可能认为缝隙中的局部流受努森扩散的控制。然而，试图计算平行板之间的克努森扩散系数会导致对数发散，分子间碰撞会切断对数发散，因此通量与h0clog(λ0/h0)成正比，其中c是平均分子速度。当颗粒分离趋于零时，非连续润滑力具有微弱的对数-对数发散。因此，碰撞过程中消耗的能量是有限的。在大惯性极限下，耗散的能量为6πμU0a2(logH0/λ0-1.28)，其中2U0为粒子的相对速度。当λ0[gT]是ε1/2时，我们在能隙中有自由分子流。在这种情况下，由于颗粒的曲率，流量与压力梯度的关系是非局部的。我们分析了两个圆柱体之间的自由分子流动，得到了润滑力的标度。
Solid-body collisions between smooth particles in a gas would not occur if the lubrication force for a continuum incompressible fluid were to hold at all particle separations. When the gap between the particles is of the order of the mean free path λ0 of the gas, the discrete molecular nature of the gas becomes important. For particles of radii a smaller than about 50 μm colliding in air at a relative velocity comparable to their terminal velocity, the effects of compressibility of the gas in the gap are not important. The nature of the flow in the gap depends on the relative magnitudes of the minimum gap thickness h0 ≡ aε, the mean-free path λ0, and the distance aε1/2 over which the effects of curvature become important. The slip-flow regime, a[Gt ]λ0, was analysed by Hocking (1973) using the Maxwell slip boundary condition at the particle surface. To find the lubrication force in the transition regime (aε ∼ O(λ0)), we use the results of Cercignani & Daneri (1963) for the flux as a function of the pressure gradient in a Poiseuille channel flow. When aε[Lt ]λ0[Lt ]aε1/2, one might expect the local flow in the gap to be governed by Knudsen diffusion. However, an attempt to calculate the Knudsen diffusivity between parallel plates leads to a logarithmic divergence, which is cut off by intermolecular collisions, and the flux is therefore proportional to h0c log(λ0/h0), where c is the mean molecular speed. The non-continuum lubrication force is shown to have a weak, log - log divergence as the particle separation goes to zero. As a result, the energy dissipated in the collision is finite. In the limit of large particle inertia, the energy dissipated is 6πμU0a2(log h0/λ0 – 1.28), where 2U0 is the relative velocity of the particles. When λ0[Gt ]aε1/2, we have a free molecular flow in the gap. In this case, owing to the curvature of the particles, the flux versus pressure gradient relation is non-local. We analyse the free molecular flow between two cylinders and obtain scalings for the lubrication force.