Acceleration particle fluid

A particle falling freely in vacuum is subjected to a constant acceleration, and its velocity increases continuously. The velocity at any point depends only on the distance from the starting point, and is independent of the size and the density of the particle. Thus a heavy stone and a feather fall at exactly the same rate in an evacuated system. However, in the event of a particle falling in a fluid medium, there is resistance to this fall or movement. The resistance increases as the velocity of the particle increases, and this continues until the forces tending to accelerate the particle and the fluid resistance forces become equal. The particle is then said to have attained its terminal velocity it continues to fall, but with a uniform velocity. 

This expression shows that, unlike the terminal velocity of the particle, its initial acceleration is independent of the particle size and depends only on the densities of the solid particle and the fluid. This type of acceleration, known as differential acceleration, may be exploited by designing equipment which provides frequent opportunities for accelerating the particles from rest. If a particle is allowed to accelerate from rest for a brief period of time and then arrested and subsequently allowed to fall once more, the total distance travelled by it will be influenced more by the differential acceleration and, therefore, by the specific gravities of the particle and of the liquid, than by its terminal velocity, or in other words, by its size. In this way, as the preferential movement of dense particles to the bottom of a bed... 

The Basset force may be negligible when the fluid-particle density ratio is small, e.g., in most gas-solid suspensions, and the time change is much longer than the Stokes relaxation time or the acceleration rate is low. 

The particle mobility B is defined as B = U. Generally, the particle velocity is given in terms of the product of the mobility and a force F acting externally on the particle, such as a force generated by an electrical field. Under such conditions, the particle motion is called quasi-stationary. That is, the fluid particle interactions are slow enough that the particle behaves as if it were in steady motion even if it is accelerated by external forces. Mobility is an important basic particle parameter its variation with particle size is shown in Table II along with other important parameters described later. 

Equation (11) is written in the form of Newton s second law and states that the mass times acceleration of a fluid particle is equal to the sum of the forces causing that acceleration. In flow problems that are accelerationless (Dx/Dt = 0) it is sometimes possible to solve Eq. (11) for the stress distribution independently of any knowledge of the velocity field in the system. One special case where this useful feature of these equations occurs is the case of rectilinear pipe flow. In this special case the solution of complex fluid flow problems is greatly simplified because the stress distribution can be discovered before the constitutive relation must be introduced. This means that only a first-order differential equation must be solved rather than a second-order (and often nonlinear) one. The following are the components of Eq. (11) in rectangular Cartesian, cylindrical polar, and spherical polar coordinates ... 

C Is the acceleration of a fluid particle necessarily zero in sffady flow Explain. 

In fully developed laminar flow, each fluid particle moves at a constant axial velocity along a streamline and the velocity profile i/(r) remains unchanged in the flow direction. There is no motion in the radial direction, and thus the velocity component in the direction normal to flow is everywhere zero. There is no acceleration since the flow is steady and fully developed. 

The terminal velocity of a particle is a constant value of velocity reached when all forces (gravity, drag, buoyancy, etc.) acting on the particle are balanced. The sum of all the forces is then equal to zero (no acceleration). To calculate this velocity, a dimensionless constant K determines the appropriate range of the fluid-particle dynamic laws that apply ... 

Fluid flow may be steady or unsteady, uniform or nonuniform, and it can also be laminar or turbulent, as well as one-, two-, or three-dimensional, and rotational or irrotational. One-dimensional flow of incompressible fluid in food systems occurs when the direction and magnitude of the velocity at all points are identical. In this case, flow analysis is based on the single dimension taken along the central streamline of the flow, and velocities and accelerations normal to the streamline are negligible. In such cases, average values of velocity, pressure, and elevation are considered to represent the flow as a whole. Two-dimensional flow occurs when the fluid particles of food systems move in planes or parallel planes and the streamline patterns are identical in each plane. For an ideal fluid there is no shear stress and no torque additionally, no rotational motion of fluid particles about their own mass centers exists. 

The basis for any derivation of the momentum equation is the relation commonly known as Newton s second law of motion which in the material Lagrangian form (see Fig. I.IB) expresses a proportionality between the applied forces and the resulting acceleration of a fluid particle with momentum density, P, (e.g., [89]) ... 

In order to complete our discussion on momentum transfer, we must consider the final forms of the mesoscale acceleration models in the presence of all the fluid-particle forces. When the virtual-mass force is included, the mesoscale acceleration models must be derived starting from the force balance on a single particle ... 

Summarizing the results for Couette flow, we have seen that the fluid moves in circular paths and thus the fluid particles are being accelerated. As a consequence, the inertial terms in the equations of motion are nonzero clearly the flow is not unidirectional. However, this centripetal acceleration does not produce a secondary flow because the nonlinear... 

The term in brackets is positive and independent of r. Hence the velocity on the free surface is constant but smaller than the velocity of the solid plate. The speed of a fluid particle that travels along the interface must therefore increase discontinuously as it reaches the plate and turns the corner - that is, it must undergo an infinite acceleration. This infinite acceleration is produced by an infinite stress and pressure, 0(r 1), on the plate in the limit r —> 0. Again, we conclude that the solution breaks down in the limit r —r 0. 

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