Body forces, gravity

Body forces gravity force, centrifugal force, Coriolis force electromagnetic force. 

The change of momentum for a particle in the disperse phase is typically due to body forces and fluid-particle interaction forces. Among body forces, gravity is probably the most important. However, because body forces act on each phase individually, they do not result in momentum transfer between phases. In contrast, fluid-particle forces result in momentum transfer between the continuous phase and the disperse phase. The most important of these are the buoyancy and drag forces, which, for reasons that will become clearer below, must be defined in a consistent manner. However, as detailed in the work of Maxey Riley (1983), additional forces affect the motion of a particle in the disperse phase, such as the added-mass or virtual-mass force (Auton et al., 1988), the Saffman lift force (Saffman, 1965), the Basset history term, and the Brownian and thermophoretic forces. All these forces will be discussed in the following sections, and the equations for their quantification will be presented and discussed. 

The total force vector F[n] consists of surface forces (pressure, normal and shear stresses) acting on the fluid element surface and body forces (gravity and electromagnetic force) acting on the volume of the same fluid element. By replacing the x component of the force vector, we get the following notation ... 

The rate of work C done on the moving fluid element due to body forces (gravity, electromagnetic force) and surface forces (pressure, normal stresses, shear stresses) is mathematically defined as... 

Component of electric field along z Body force Gravity vector... 

To illustrate the use of the momentum balance, consider the situation shown in Figure 21c in which the control volume is bounded by the pipe wall and the cross sections 1 and 2. The forces acting on the fluid in the x-direction are the pressure forces acting on cross sections 1 and 2, the shear forces acting along the walls, and the body force arising from gravity. The overall momentum balance is... 

Momentum Balance Since momentum is a vector quantity, the momentum balance is a vector equation. Where gravity is the only body force acting on the fluid, the hnear momentum principle, apphed to the arbitraiy control volume of Fig. 6-3, results in the following expression (Whitaker, ibid.). 

The effects of the surface stresses are accounted for explicitly, and source terms Sj y, and include contributions due to the body process only. For example, the body force due to gravity would be modeled by S x = 0, S y = 0, and S z = -pg-... 

Thermally-Driven Buoyancy Flow. Thermal gradients can Induce appreciable flow velocities in fluids, as cool material is pulled downward by gravity while warmer fluid rises. This effect is Important in the solidification of crystals being grown for semiconductor applications, and might arise in some polymeric applications as well. To illustrate how easily such an effect can be added to the flow code, a body force term of pa(T-T ) has been added to the y-coraponent of the momentum equation, where here a is a coefficient of volumetric thermal expansion. 

Here, u, v, and w are the components of the velocity vector in the x, y, and z directions, respectively. Note that velocity is treated as a vector quantity, so that the vector sum ui + vj + wk (where i, j, and k are the unit vectors in the x, y, and z directions) represents both the direction and magnitude of the fluid velocity at a particular position and time. The symbol P represents fluid pressure, p is the fluid viscosity, p is the fluid density, and the F parameters are the components of a body force acting on the fluid in the x, y, and z directions. (A body force is a force that acts on the fluid as a result of its mass rather than its surface area gravity is the most common body force.)... 

It will be assumed that the only body force is that due to gravity ... 

Application of Newton s second law of motion to an infinitesimal element of an incompressible Newtonian fluid of density p and constant viscosity p, acted upon by gravity as the only body force, leads to the Navier-Stokes equation of motion ... 

If we assume that the body force is only due to gravity and use the definition r = —prL +1 where pr is the isotropic resin pressure, / is the unit tensor, and r is the deviatoric stress as well as assuming a constant density and define a new pressure Pr —pr + prgh, Equation 5.21 simplifies to,... 

Summation over repeated subscripts is again implied, in this case over the subscript j. Fi are body forces (such as gravity), which will be neglected. 

It is useful to think of two different kinds of forces, one that acts over the volume of a fluid element and the other that acts on the element s surface. The most common body force is exerted by the effect of gravity. If an element of fluid is less dense than its surroundings (e.g., because it is warmer), then a volumetric force tends to accelerate it upward—hot air rises. Other fields (e.g., electric and magnetic) can exert volumetric body forces on certain fluids (e.g., ionized gases) that are susceptible to such fields. Here we are concerned mostly with the effect of gravity on variable-density flows,... 

To illustrate, consider that f is the body force due to gravity and the r-6 problem is in a long horizontal tube. In this case... 

In equations 5-8, the variables and symbols are defined as follows p0 is reference mass density, v is dimensional velocity field vector, p is dimensional pressure field vector, x is Newtonian viscosity of the melt, g is acceleration due to gravity, T is dimensional temperature, tT is the reference temperature, c is dimensional concentration, c0 is far-field level of concentration, e, is a unit vector in the direction of the z axis, Fb is a dimensional applied body force field, V is the gradient operator, v(x, t) is the velocity vector field, p(x, t) is the pressure field, jl is the fluid viscosity, am is the thermal diffiisivity of the melt, and D is the solute diffiisivity in the melt. The vector Fb is a body force imposed on the melt in addition to gravity. The body force caused by an imposed magnetic field B(x, t) is the Lorentz force, Fb = ac(v X v X B). The effect of this field on convection and segregation is discussed in a later section. 

Some quantity which will act as a measure of the magnitude of the body force acting on the flow. Consideration here will only be given to buoyancy forces, i.e., to forces arising dueio gravity. However, the method used here to derive a measure of this force can easily be applied when other force fields exist. 

In some low-speed combustion problems natural convection is of importance, and consequently f, is not negligible. Since the acceleration due to gravity is the same for all species, f, is the same for all i in these problems. Body forces disappear from equations (3) and (7) and remain only in the momentum equation. 

The force per unit mass exerted on a particle of kind j by the surrounding gas will be denoted by F. Among the effects that may contribute to are (1) skin friction and separation drag, (2) gravity and other body forces. 

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