## Lets Look At How A Single Particle Behaves In A

During the motion of viscous flow over a stationary body or particle, certain resistances arise. To overcome these resistances or drag and to provide more uniform fluid motion, a certain amount of energy must be expended. The developed drag force and, consequently, the energy required to overcome it, depend largely on the flow regime and the geometry of the solid body. Laminar flow conditions prevail when the fluid medium flows at low velocities over small bodies or when the fluid has a relatively high viscosity. Flow around a single body is illustrated in Figure 1. As shown in Figure 1(A), when the flow is laminar a well-defined boundary layer forms around the body and the fluid conforms to a streamline motion. The loss of energy in this situation is due primarily to fiction drag. If the fluid's average velocity is increased sufficiently, the influence of inertia forces becomes more pronounced and the flow becomes turbulent. Under the influence of inertia forces, the fluid adheres to the particle surface, forming only a very thin boundary layer and generating a turbulent wake, as shown in Figure 1(B). The pressure in the wake is significantly lower than that at the stagnation point on the leeward side of the particle. Hence, a net force, referred to as the pressure drag, acts in a direction opposite to that of the fluid's motion. Above a certain value of the Reynolds number, the role of pressure drag becomes

Figure 1. Flow around a single particle.

Figure 1. Flow around a single particle.

We shall begin discussions by analyzing a dilute system that can be described as a low concentration of noninteracting solid particles carried along by a water stream. In this system, the solid particles are far enough removed from one another to be treated as individual entities. That is, each particle individually contributes to the overall character of the flow. Let's consider the dynamics of motion of a solid spherical particle immersed in water independent of the nature of the forces responsible for its displacement. A moving particle immersed in water experiences forces caused by the action of the fluid. These forces are the same regardless whether the particle is moving through the fluid or whether the water is moving over the particle's surface. For our purposes, assume the water to be in motion with respect to a stationary sphere. The fluid shock acting against the sphere's surface produces an additional pressure, P. This pressure is responsible for a force, R (called the drag force) acting in the direction of fluid motion. Now consider an infinitesimal element of the sphere's surface, dF, having a slope, a, with respect to the normal of the direction of motion (Figure 2). The pressure resulting from the shock of the fluid against the element produces a force, dF, in the normal direction. This force is equal to the product of the surface area and the additional pressure, PdF0. The component acting in the direction of flow, dR, is equal to dxcosa. Hence, the force, R, acting over the entire surface of the sphere will be:

where dF is the projection of dF0 on the plane normal to the flow. The term F refers to a characteristic area of the particle, either the surface area or the maximum cross-sectional area perpendicular to the direction of flow. The pressure P represents the ratio of resistance force to unit surface area (R/F), and it depends on several factors, namely the diameter of the sphere (d), its velocity (u), the fluid density (p) and the fluid viscosity (p,).: i.e., P = f(d, u, p, ¡x). Applying

dimensional analysis, the following dimensionless groups are identified:

where Eu is the dimensionless Euler number, defined as P/u2 p, and Re is the Reynolds number (Re = du p//i). By substituting for density using the ratio of specific gravity to the gravitational acceleration, an expression similar to the well-known Darcy-Weisbach expression is obtained:

Figure 2. Infinitesimal element of a sphere's surface inclined at angle a from the direction of flow.

where CD is the drag coefficient, which is a dimensionless parameter that is related to the Reynolds number. The relationship between CD and Re for flow around a smooth sphere is given by the plot shown in Figure 3. As shown in this plot, there are three regions that can be approximated by expressions for straight lines. These three regions are the Stokes law region, Newton's law region, and the intermediate region. Refer to the sidebar discussion for these expressions. By substituting the expressions for the drag coefficient into equation (3), we obtain a convenient set of expressions that will enable us to calculate settling velocities. The details are left to you. There are plenty of empirical correlations in the literature for the drag

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