Acceleration body force

Basically, Newtonian mechanics worked well for problems involving terrestrial and even celestial bodies, providing rational and quantifiable relationships between mass, velocity, acceleration, and force. However, in the realm of optics and electricity, numerous observations seemed to defy Newtonian laws. Phenomena such as diffraction and interference could only be explained if light had both particle and wave properties. Indeed, particles such as electrons and x-rays appeared to have both discrete energy states and momentum, properties similar to those of light. None of the classical, or Newtonian, laws could account for such behavior, and such inadequacies led scientists to search for new concepts in the consideration of the nature of reahty. 

Centripetal and Centrifugal Acceleration A centripetal body force is required to sustain a body of mass moving along a curve tra-jec tory. The force acts perpendicular to the direction of motion and is directed radially inward. The centripetal acceleration, which follows the same direction as the force, is given by the kinematic relationship ... 

Several additional studies [Winitzer, Sep. ScL, 8(1), 45 (1973) ibid., 8(6), 647 (1973) Maru, Wasan, and Kintner, Chem. Eng. Set., 26, 1615 (1971) and Rapacchietta and Neumann, J. Colloid Inteiface ScL, 59(3), 555 (1977)] which include body forces such as gravitational acceleration and buoyancy have been made. A typical example of a force balance describing suen a system (Fig. 22-39) is summarized in Eq. (22-41). 

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,... 

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. 

If the propellant is to be accelerated by electric body forces a primary requirement is that the propellant be a charged particle. While interest has centered on positively charged atomic ions, the use of both negatively and positively charged colloids has been considered. 

The characterization and control of electrostatic forces are of particular interest. Electrostatic forces depend on the electric charge and potential at the particle surfaces. When subjected to a uniform, unidirectional electric field E. charged colloidal particles accelerate until the electric body force balances the hydrodynamic drag force, so that the particles move at a constant average velocity v. This motion is known as electrophoresis, and v is the electrophoretic velocity. 

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. 

Acceleration of a flame sheet produces effects of the same type as those of body forces. In a noninertial coordinate system that moves with the flame (or in which the flame moves at a constant velocity), an effective body force appears as a consequence of flame acceleration. This may be seen from equation (89) by bringing Rd f/dx to the right-hand side of the equation the effective nondimensional body force per unit volume becomes R(Fj[ — d f/dx ). Thus a flame acceleration in the direction of its motion (a positive rate of increase of — df/dx) is equivalent to a body force directed from the fresh mixture to the burnt gas. A body-force instability is therefore associated with flame acceleration. By the preceding reasoning, the wavelength... 

This is the relation for the momentum balance in the x-direction, and is known as the x-momentnm equation. Note that we would obtain the. same result if we used momentum flow rates for the left-hand side of this equation instead of mass limes acceleration. If there is a body force acting in the x-direction, it can be added to the right side of the equation provided that it is expressed per unit volume of the fluid. 

In these equations, tt is the molecular flux of momentum and g and F are gravitational acceleration and external body forces, respectively. The physical interpretation of the various terms appearing in these equations again follows similar lines the first term is the rate of increase in momentum per unit volume the second term represents... 

The body forces have an effect on all the particles in the body. They are far ranging forces and are caused by force fields. An example is the earth s gravitational field. The acceleration due to gravity g acts on each molecule, so that the force of gravity on a fluid element of mass AM is... 

In the calculation of the velocity profile of a hydrodynamic, fully developed, laminar flow we will presume the flow to be incompressible and all properties to be constant. The velocity profile of a fully developed, tubular flow is only dependent on the radius r, wx = wx(r) and wy = 0. Therefore the acceleration term gdw /dt in the Navier-Stokes equation (3.59) disappears body forces are not present, so fc = 0. An equilibrium develops between the pressure and friction forces. Balancing the forces on an annular fluid element, Fig. 3.32, gives... 

General physical laws often state that quantities like mass, energy, and momentum are conserved. In computational mechanics, the most important of these balance laws pertains to linear momentum (when reckoned per unit volume, linear momentum may be expressed as the material density p times velocity v). The balance equation for linear momentum may be considered as a generalization of Newton s second law, which states that mass times acceleration equals total force. As we saw in the previous section, stresses in a material produce tractions, which may be considered as internal forces. In addition, external forces such as gravity may contribute to the total force. These are commonly reckoned per unit mass and are usually referred to as body forces to distinguish them from tractions, which may be considered as surface forces. For a one-dimensional motion, balance of linear momentum requires that (37,38)... 

Han J, Tryggvason G (1999) Secondary breakup of axisymmetric liquid drops. I. Acceleration by a constant body force. Physics of Fluids ll(12) 3650-3667... 

A comparison of equations (7.103) and (7.104) shows that the Newton s second law of motion in the inertial frame O is identical in form to that in O except that the latter formulation contains several additional fictitious body forces. The term —2mil x v is the Coriolis force, and —mQ x (f2 x r) designates the centrifugal force. No name is in general use for the term — x r. The acceleration —ao compensates for the translational acceleration of the frame. 

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