It is argued that the appropriate generalization of Darcy' s law when inertia effects are included takes the form Vp = -(Mk) V -(pc/kl/2)lv[v, div V = 0, where k is the permeability of the medium and the 'form drag constant' c is a coefficient which is independent of the pressure p, the seepage velocity V, and the density p and viscosity g of the fluid but which is dependent on the geometry of the medium. We formulate a nonlinear extension of Brinkman's self-consistent theory for the flow of a viscous fluid through a swarm of spherical particles. We equate the drag per unit volume given by the right hand side of the first of the above equations to the total drag ND on the N particles contained within that unit volume, in an infinite region 11, where D is the drag on a single particle placed in a velocity field v subject to p(v. V)v + grad p = •2¾ _ /.dk v -(cp/kl/2)lvlv, div v = 0, vlon is a prescribed constant, where g is the viscosity. Without solving these equations, we obtain an estimate for c from the known experimental drag law for a solid sphere placed in a uniform stream.

Summary.The value of R = 180 which has been given by Orr [1] as a limit for sure stability of Hagen-Poiseuille flow is incorrect. A lower value, R = 82.88, can be associated with an eigenfunction possessing a first mode azimuthal variation (N = 1) and no streamwise variation. This eigenfunction is obtained as an exact solution of the appropriate Euler equation. A yet lower value, R = 81.49, is associated with a spiral mode with N = 1 and wave number a ft! 1. Corresponding results are obtained for Poiseuille flow between cylinders. For all but the very smallest radius ratios the smallest eigenvalue of Euler's equation is assumed for the purely azimuthal disturbance. The mode shape for the pipe flow is consistent with the experimental situation as it is now understood [2], though the stability limit is much smaller than the experimental value (R 2100). For the annulus, the variation of the energy limit with the radius ratio (from pipe flow to channel flow) is consistent with the experimentally observed stability limit. All the results of the energy analysis hold equally if the pipe is in rigid rotation about its axis. If the rotation is "fast" linear and energy results nearly coincide.Consider the arbitrary motion of a viscous fluid governed by the Navier-Stokes equations. It is known that if the Reynolds number (Re) of this motion satisfies the inequality Re < R,whereii v r / Vv : Vv dxthen the motion is asymptotically stable in the mean. Here 13 is a bounded region andwhere m is a constant, v is the kinematic viscosity, d is the unit of length, V is the velocity of the basic motion, and v is the difference in the velocity of the basic and disturbed motions. The region 13 may also be taken as unbounded in directions for which v maŷ

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