Movements of a sphere rolling down an inclined plane

Solution of Unsteady Rolling Motion of Spheres Equation in Inclined Tubes Filled with Incompressible Newtonian Fluids by Differential Transformation Method

“…It is reasonable to postulate that since smooth walled tubes have been studied, F R is expected to be negligible (or alternately F D + F R can be regarded as the total resistance to sphere motion, which is included in the drag coefficient). The all of detail of problem was explained at [15].The equation of motion is gained as follow from [15]: …”

“…1 illustrates a schematic view of the present problem. Figure.1 a schematic figure of current problem [15] The forces acting on the spheres are the fluid-drag, F D , fluid lift force, F L , buoyancy force, F B , gravitational force, F g , solid-solid resistance force due to rolling, F R , virtual mass force, F VM due to relative acceleration of the fluid around the particle. It is reasonable to postulate that since smooth walled tubes have been studied, F R is expected to be negligible (or alternately F D + F R can be regarded as the total resistance to sphere motion, which is included in the drag coefficient).…”

“…(6) is in very good agreement with results of the reference [12] and correlates the data with average error of 8.4883%.The maximum difference between values of Eqs. (5) and (6) .Aside from mentioned work of Chhabra et al [12], all other surveys of the rolling motion of the particles are related to open channels [12][13][14][15]. In reality, when a sphere is rolling in a tube, the wall exerts an extra retardation effect on it due to upward motion of the fluid through the eccentric annular gap between the particle and the wall this issue distinguishes the mechanism of rolling in tubes from open channels.…”

“…classification, centrifugal and gravity collection or separation, where it is often necessary to determine the trajectories of particle accelerating in a fluid for proposes of design or improved operation [15,16]. Unfortunately, there are few studies in the literature in the filled of rolling particles and the major part of the available investigations are related to the use of a rolling ball viscometer to measure viscosity of liquids [5,6].…”

Efficient Analytical Approaches For Motion Of A Spherical Solid Particle In Plane Couette Fluid Flow Using Nonlinear Methods

“…A combined experimentalnumerical study was performed for a drag coefficient-Reynolds correlation for a single spherical particle rolling down a smooth plane for three classified regimes of Re from low up to 10 5 , in an incompressible Newtonian media by Jan and Chen [2]. Explicit equations of correlation between Reynolds and drag coefficients proposed by Yow et al for different shapes of particles including spheres, cube octahedrons, octahedrons, cubes, tetrahedrons, discs, cylinders and rectangular parallelepipeds [3].…”