The drag on sedimenting discs in broadside motion in tubes

Thin discs, sedimenting in broadside motion, under gravity, experience an enhanced drag due to wall effects. In cylindrical tubes, when the disc just blocks the tube, the drag becomes infinite according to the inverse of the gap distance squared. This singular behaviour of the drag coefficient is computed by summing the perturbation series in radius ratio ε of the disc to the cylinder. An extension of the regular perturbation series for the thin disc in broadside motion is computed, permitting the nature of the singular behaviour as the disc scrapes the tube to be characterized and explicitly accounted for, yielding a prediction of the drag force which is a valid approximation for the entire range of the blockage factor parameter space. The next closest set of poles in the complex plane of blockage factor are complex conjugates which are accompanied by a sign pattern of seven, rarely found in long perturbation series.     

Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall

The motion of a spherical particle in infinite linear flow and near a plane wall, subject to the slip boundary condition on both the particle surface and the wall, is studied in the limit of zero Reynolds number. In the case of infinite flow, an exact solution is derived using the singularity representation, and analytical expressions for the force, torque, and stresslet are derived in terms of slip coefficients generalizing the Stokes–Basset–Einstein law. The slip velocity reduces the drag force, torque, and the effective viscosity of a dilute suspension. In the case of wall-bounded flow, advantage is taken of the axial symmetry of the boundaries of the flow with respect to the axis that is normal to the wall and passes through the particle center to formulate the problem in terms of a system of one-dimensional integral equations for the first sine and cosine Fourier coefficients of the unknown traction and velocity along the boundary contour in a meridional plane. Numerical solutions furnish accurate predictions for (a) the force and torque exerted on a particle translating parallel to the wall in a quiescent fluid, (b) the force and torque exerted on a particle rotating about an axis that is parallel to the wall in a quiescent fluid, and (c) the translational and angular velocities of a freely suspended particle in simple shear flow parallel to the wall. For certain combinations of the wall and particle slip coefficients, a particle moving under the influence of a tangential force translates parallel to the wall without rotation, and a particle moving under the influence of a tangential torque rotates about an axis that is parallel to the wall without translation. For a particle convected in simple shear flow, minimum translational velocity is observed for no-slip surfaces. However, allowing for slip may either increase or decrease the particle angular velocity, and the dependence on the wall and particle slip coefficients is not necessarily monotonic.     

The use of stokeslets to describe the arbitrary translation of a disk near a plane wall

A previously presented method is extended to describe the fully three-dimensional Stokes flow generated by the translation in any direction of an arbitrarily oriented disk in fluid bounded by a plane wall. The velocity field is represented solely in terms of stokeslet distributions on the disk, modified to take account of the bounding wall according to the century-old idea of Lorentz. Sets of integral equations of the second kind, not all disjoint, are obtained for the Abel transforms in each Fourier mode of the density functions. However, only a few modes need be considered in determining the flow field to order D ^-3, where D is the distance of the disk axis from the wall. Less detail is required to evaluate the drag force and torque experienced by the disk.     

Effect of a protruding rod-supported disk on the drag of a nose-controlled cylinder

The drag C _x of a cylinder of diameter D with a front protruding disk supported on a rod of length l has been studied as a function of the relative distance l/D under the conditions of high (supersonic) flight velocities. It is established that the optimum (minimum) drug C _x exists, the value of which agrees with the results of numerical simulations.     

Shear flow over a protuberance on a plane wall

Simple shear flow over a protuberance with an axisymmetric shape projecting from a plane wall with its axis normal to the wall is studied by means of a boundary-integral method that is suitable for computing three-dimensional Stokes flow in axisymmetric domains. The problem is formulated in terms of a system of three scalar Fredholm integral equations of the first kind for the distribution of traction over the surface of the protuberance and the wall, and is solved by means of a boundary-element method. Numerical computations are performed for a family of protuberances whose exposed surface is a section of a sphere or of an oblate spheroid with its minor axis normal to the wall, and the results are in agreement with those of previous analytical computations for hemi-spherical and spherical shapes. The numerical computations provide accurate information on the hydrodynamic force and torque exerted on the protuberances due to the shear flow, and the distribution of shear stresses, and illustrate the kinematical structure of the flow with reference to the development of stagnation points and flow reversal.     

Stagnation-point flow against a liquid film on a plane wall

Summary Stagnation-point flow of a semi-infinite viscous fluid against a liquid film resting on a plane wall is considered under conditions of Stokes flow and for arbitrary Reynolds numbers. Assuming that the interface remains flat and parallel to the wall at all times, the lateral spatial coordinate is scaled out to yield a simplified system of governing equations in the transverse coordinate normal to the wall, describing the evolution of the flow and displacement of the interface. In this inherently unsteady flow the film keeps thinning in time, while the rate of thinning decreases as the interface approaches the wall. Orthogonal two-dimensional, axisymmetric, three-dimensional, and oblique two-dimensional flow are individually considered. In each case, exact solutions of a similarity form are constructed, and an evolution equation describing the film thickness is formulated and solved by numerical methods. Quasi-steady solutions compare favourably with full calculations of the unsteady flow, suggesting that the unsteady terms have only a minor effect on the rate of film thinning. Dual solutions are uncovered in the case of three-dimensional flow.     

On the moment of a plane disk in a non-Newtonian fluid

Summary Exact analytic solution for the flow of non-Newtonian fluid of grade two generated by periodic oscillations of a plane disk is obtained. The velocity field and the moment of the frictional forces are calculated and the results are compared with those for Newtonian fluid. The moments caused by certain special oscillations are also discussed.     

Oscillatory free convection from a disk rotating in a vertical plane

Unsteady free convective flow of a viscous incompressible fluid from a heated disk rotating in a vertical plane induced by periodic variation in the mean temperature of the disk is analysed for low and high frequencies of oscillation. The oscillatory solutions of the energy equation are also obtained. The solutions are matched at a suitable frequency parameter which increases with the Prandtl number of the fluid.     

Fluid flow between a rotating disk and a fixed wall

The results of a theoretical investigation of the fluid flow in the gap between a rotating disk and a fixed wall are presented. The variations of the rate of twisting of the fluid and the statistical pressure along the radius are analyzed for the case of centripetal flow.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 26, No. 4, pp. 611–617, April, 1974.     

The movement of a metal disk toward pulsed laser radiation

It is shown in works [1–5] that the electron heat transfer mechanism yields the dominating value and duration of the tensile phase in pulses of the elastic stresses excited by pulse laser heating. This leads to a nonzero average mechanical momentum and the possibility of the movement of heat conducting objects toward the pulse heat source. In this work, we present the results of an experimental study of the movement of a pendulum—an aluminum disk 20 mm in diameter and 0.85 mm thick—toward pulse laser radiation.     

Low-Reynolds-number translation of a slender cylinder near a plane wall

Summary Low-Reynolds-number results are presented for the drag and induced torque on a slender circular cylinder translating near a single plane wall. Four representative situations are investigated, the principal feature of the analysis being that it is valid for all distances from the wall which are large compared with the radius of the cylinder. In particular, the results hold for distances from the wall of the same order of magnitude as the length of the cylinder. The direction and rate of rotation are given for those cases where it occurs.This paper was written while N. J. de Mestre was a visitor at the Department of Applied Mathematics and Theoretical Physics, University of Cambridge. W. B. Russel was supported by a NATO Postdoctoral Fellowship.     

Calculation of the nonstationary thermoelastic stresses in a plane wall

An approximate method is proposed for calculating the nonstationary thermoelastic stresses in a plane wall for an arbitrary law of variation of fluid temperature with time.     

On the motion of a spherical bubble deforming near a plane wall

Equations of motion are derived for an expanding spherical bubble in potential flow near a plane wall using the Lagrange-Thomson method and an extended Rayleigh dissipation function to account for the drag. This method is shown to yield the same acceleration of the bubble center as that obtained using the Lagally theorem. An extended Rayleigh-Plesset equation is derived to describe deformation in the vicinity of a plane wall, and expressions relating the drag force to the distance from the wall and the bubble growth rate are derived. The solution method for the velocity potential can also be applied to the case of non-spherical deformation.     

Optimum relationships in a plane multilayer wall with internal heat release

We consider the possibility of reducing the heat flows through a flat multilayer wall with internal heat release.     

Heat and mass transfer in a slit gap with sublimation of a rotating disk wall

A solution is obtained for the problem of the temperature distribution in a narrow slit gap between a motionless heated disk and a moving disk with a subliming coating. It is established that rotation leads to intensification of the heat-transfer process.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 48, No. 4, pp. 587–592, April, 1985.     

Effect of a moving wall on the characteristics of heat transfer in plane boundary layers

The results of modeling of the hydrodynamics and heat transfer in a boundary layer on a plane constantly moving surface are presented. Specific features of temperature fields are studied as functions of the boundary conditions and the Prandtl number. Detailed tables of numerical solutions are given. Polotsk State University, Novopolotsk. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 69, No. 5, pp 821–825, September–October, 1996.     

Determination of the velocity potential and forces on a body of rotation leaving the plane of a wall

The body of rotation is replaced by a system of sources and dipoles. The potential of the resulting velocity and the forces on the emergent body are found approximately.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 4, pp. 639–647, October, 1969.