Infinitesimal elements

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

The foregoing are volume integrals evaluated over the entire volume of the rigid body and dw is an infinitesimal element of weight. If the body is of uniform density, then the center of gravity is also called the centroid. Centroids of common lines, areas, and volumes are shown in Tables 2-1, 2-2, and 2-3. For a composite body made up of elementary shapes with known centroids and known weights the center of gravity can be found from... 

The momentum flow rate over the cross sectional area of the pipe is easily determined by writing an equation for the momentum flow through an infinitesimal element of area and integrating the equation over the whole cross section. The element of area is an annular strip having inner and outer radii r and r+ Sr, the area of which is 2-nrSr to the first order in 8r. The momentum flow rate through this area is 2irr8r.pv2 so the momentum flow rate through ihe whole cross section of the pipe is equal to... 

The volumetric flow rate is determined by writing the equation for the volumetric flow rate across an infinitesimal element of the flow area then integrating the equation over the whole flow area, ie the cross-sectional area of the pipe. It is necessary to use an infinitesimal element of the flow area because the velocity varies over the cross section. Over the infinitesimal area, the velocity may be taken as uniform, and the variation with r is accommodated in the integration. 

Vector force per unit area on an infinitesimal element of area that has a given normal and is at a given point in a body. 

The fundamental physical laws governing motion of and transfer to particles immersed in fluids are Newton s second law, the principle of conservation of mass, and the first law of thermodynamics. Application of these laws to an infinitesimal element of material or to an infinitesimal control volume leads to the Navier-Stokes, continuity, and energy equations. Exact analytical solutions to these equations have been derived only under restricted conditions. More usually, it is necessary to solve the equations numerically or to resort to approximate techniques where certain terms are omitted or modified in favor of those which are known to be more important. In other cases, the governing equations can do no more than suggest relevant dimensionless groups with which to correlate experimental data. Boundary conditions must also be specified carefully to solve the equations and these conditions are discussed below together with the equations themselves. 

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 ... 

The dual pseudotensor of any antisymmetric tensor in 4-space arises from the integral over a two-dimensional surface in 4-space [101], in which the infinitesimal element of surface is given by the antisymmetric tensor ... 

Note the mean residence time for this distribution is the same as for the pure plug flow. However, in this case, only an infinitesimal element of the flow has this precise residence time. Much of the fluid passes through the vessel by mixing rather than convective forces so that it spends too short a time in the vessel. This is offset by elements of fluid which spend too long a time in the vessel, and thus reduce Its effective capacity. 

A Riemann surface is a 2-dimensional compact differentiable surface, together with an infinitesimal element of length (see textbooks on differential and Riemannian geometry, for example, [Nak90]). The curvature K(x) at a point x is the coefficient a in the expansion ... 

Notice that Eq. (6) is not evaluated directly because the metallic states are a continuum, here taken care of by the infinitesimal element 0+. The principal part, Eq. (6), corresponds to the real part of the denominator in Eq. (6). 

The infinitesimal element of scatter signal recorded in a detector subtending a solid angle of A/2det at a small sample having volume AV = Adi on irradiation by N0 photons is ... 

The standard hemispherical monochromatic gas emissivity is defined as the direct volume-to-surface exchange area for a hemispherical gas volume to an infinitesimal area element located at the center of the planar base. Consider monochromatic transfer in a black hemispherical enclosure of radius ft that confines an isothermal volume of gas at temperature Tg. The temperature of the bounding surfaces is T. Let A2 denote the area of the finite hemispherical surface and dAi denote an infinitesimal element of area located at the center of the planar base. The (dimensionless) monochromatic direct exchange area for exchange between the finite hemispherical surface A2 and d then follows from direct integration of Eq. (5-116a) as... 

Irradiance (E) The radiant flux or radiant power, P, of all wavelengths incident on an infinitesimal element of surface containing the point under consideration divided by the area of the element (dP/dS, simplified expression E = P/S when the radiant power is constant over the surface area considered). The SI unit is Wm. Note that E = ExdX, where E is the spectral irradiance at wavelength X. For a parallel and perpendicularly incident beam not scattered or reflected by the target or its surroundings Jluence rate (Eo) is an equivalent term. 

A general definition of flame stretch for planar flames is the time derivative of the logarithm of an area of the flame sheet [15], [93], the boundary of the area being considered to move with the local transverse component of the fluid velocity at the sheet. This definition is applied to an infinitesimal element of surface area at each point on the flame sheet to provide the distribution of stretch over the sheet. Thus at any given point on... 

Figure 4.3 Components of the stress tensor in rectangular coordinates on an infinitesimal element.

Now consider the stresses arising inside a deformed elastic body. The forces acting on an elastic body fall into two types volume and surface forces. The volume forces act on the various elements of the body volume. For example, we assume that the force on an infinitesimal element of the volume dv is equal to Fdv, where F is the density of the volume force. 

Now a differential balance around an infinitesimal element of a for component c gives... 

A collection of mono-atomic gas molecules are characterized by their position r in space and their velocity c at time t. An infinitesimal spatial space containing the point r is denoted by dr (e.g., in Cartesian coordinates = dxdydz). In a similar manner, an infinitesimal element in a hypothetical velocity space containing the velocity c is denoted by dc (e.g., in Cartesian coordinates = dcx dcy dcz). The imaginary or hypothetical space containing both dr and dc constitutes the six-dimensional phase space. Therefore, by a macroscopic point (r, c, t) in phase space is meant an infinitesimal volume, dr dc, centered at the point (r, c,t), having an extension sufficient to contain a large number of molecules as required for a statistical description to be valid, but still small compared with the scale of the natural changes in the macroscopic quantities like pressure, gas velocity, temperature and density of mass. 

In the gas we consider an infinitesimal element of surface area dA as sketched in Fig. 2.1. The orientation of the surface area is defined by a unit vector n normal to the surface, and u> is the angle between C and n. Imagine further that the element of surface area moves along with the fluid having the velocity v(r, t). The collection of molecules will then move back and forth across this element area with their peculiar velocities C about the mean velocity V, in accordance with (2.59). 

In this theorem A/ is the area of the interface between phase k and the other phase, Ilk is the outward unit normal of the infinitesimal element of area a of phase k, and v/ is the velocity of the local interface. The theorem, which was originally derived by [236], represents a special form of the Leibnitz rule which is necessary for the particular case when the time derivative is discontinuous and reflects a Dirac delta function like character [90, 239, 58]. 

This equation is written on the basis of the observation that the infinitesimal element of area swept out by the dislocation is characterized by the vector product dy X d. Note that dy is the local excursion of the segment, while d is a vector along the line at the point of interest. If we now recall that the configurational force is given as SEm = —Em m, then we may evidently write the configurational force in the present context as... 

Figure 1.1 An infinitesimal element of volume dR—dxdydz at the point R.

The infinitesimal element of the configuration of a single molecule is denoted by... 

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