Time derivative of velocity

Following the motion of a fluid, the increment of velocity dv in the Eulerian specification results not only from a local increase at point rbut also from the displacement of the elementary volume of the fluid during the interval of time At. Taking into 

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You have possibly heard about a hot new Porsche that can accelerate from 0 to 60 mph in 3.8 seconds. Just as velocity is the time derivative of distance, acceleration is the time derivative of velocity ... 

Acceleration is the time derivative of velocity and the second derivative with respect to time of displacement. Thus, sensors of displacement and velocity can be used to determine acceleration when their signals are appropriately processed. In addition, there are direct sensors of acceleration based on Newton s second law and Hooke s law. The fundamental structure of an accelerometer is shown in Figure 2.3. A known seismic mass is attached to the housing by an elastic element. As the structure is accelerated in the sensitive direction of the elastic element, a force is applied to that element according to Newton s... 

Accurate calculation of the time derivative of velocity potential is very crucial in obtaining correct pressure and force on the body sm-face at each time step. There are several ways to obtain this velocity potential. Backward difference is the simplest way using the potential values of previous time steps. In case of a stationary structure, more accurate finite-difference formulae can also be used. The wave force on the body surface is calculated by integrating Bernoulli s pressure over the instantaneous wetted surface from the nonlinear wave. While the above development is shown for a wave force computation, the method can be easily extended to include moving structures as well. ... 

Lagrangian-Eulerian (ALE) method. In the ALE technique the finite element mesh used in the simulation is moved, in each time step, according to a predetermined pattern. In this procedure the element and node numbers and nodal connectivity remain constant but the shape and/or position of the elements change from one time step to the next. Therefore the solution mesh appears to move with a velocity which is different from the flow velocity. Components of the mesh velocity are time derivatives of nodal coordinate displacements expressed in a two-dimensional Cartesian system as... 

We have used a common notation from mechanics in Eq. (5-4) by denoting velocity, the first time derivative of a , x, and acceleration, the second time derivative, x. In a conservative system (one having no frictional loss), potential energy is dependent only on the location and the force on a particle = —f, hence, by differentiating Eq. (5-3),... 

If the position (r), velocity (v), acceleration (a) and time derivative of the acceleration (b) are known at time t, then these quantities can be obtained dX. t + 8t by a Taylor expansion ... 

By definition, acceleration is the second derivative of displacement (i.e., the first derivative of velocity) with respect to time ... 

A phase modulation can also be expressed as frequency modulation. The corresponding frequency deviation is the time derivative of the modulated phase angle (Pm t). According to the basic relationships Afrequency deviation Af(f) with respect to the carrier frequency fg, commonly known as the Doppler frequency shift... 

The pair correlation function of the velocities and the pair correlation functions of some time derivatives of the velocity are sometimes taken into account.75 However, the validity of this description in the nonadiabaticity regions also has to be proved. The dynamic description or the description using the differentiable random process is more rigorous in this region.76... 

Thus, for incompressible flow the net rate of expansion is zero. Note that the flow need not be steady for equation A.8 to hold the time derivative of p disappears because p is constant but the velocity components in equation A.8 may change with time. 

Because acceleration a is the partial derivative of velocity on time, through some substitutions, and posing... 

As an example of the application this work, Kapral [285] and Pagistas and Kapral [37] have considered the reaction rate between iodine atoms (or some other similar species) effectively distributed uniformly in solution. They compared their calculations with those of the diffusion equation analysis and with the molecular pair approach rather than compare rate coefficients, Kapral [285] compared the rate kernels (which are approximately the time derivatives of rate coefficients). Over long times, these kinetic theory and molecular pair rate kernels both reduce to the typical form of the Smoluckowski rate kernel. However, with parameters such as R — 0.43 nm and D = 6 x 10 9m2s 1, the time beyond which the rate kernels of kinetic theory and the Smoluchowski theory are in reasonably close agreement is 20 ps, a time much longer than the velocity... 

The time derivative of the displacement vector r is the velocity V, which, of course, assumes that the fluid system is moving with the fluid velocity. The left-hand side of the energy equation now represents the convective transport, and it remains to develop the heat-transfer and work terms on the right-hand side... 

This equation uses a vector notation v to emphasize that there are three components to the velocity, which is the time derivative of the vector specifying the molecule s position x. 

Since j-c-v, the electrical current density autocorrelation function and the velocity autocorrelation function are proportional to each other. The latter function, however, can be expressed with the help of the time derivative of the decaying pro-... 

The other arises from the rotation of the unit vectors i,j, and k with angular velocity m. Since the time derivatives of the unit vectors are the vector products of [Pg.9]

In mechanism b, under given conditions of heat transfer, each elementary quantity of combustion products is cooled at a certain rale. For a given time derivative of the temperature, the derivative with respect to the coordinate is inversely proportional to the propagation velocity, and the same is true of the heat flux in the direction of the flame which carries heat out of the... 

In this equation the angular velocity vector co is referred to the center of mass frame coordinate system and the s are the time derivatives of the internal coordinates. Moreover, the kinetic energy matrix coefficients may be expressed as... 

Philo, J. S. (2000). A method for direcdy fitting the time derivative of sedimentation velocity data and an alternative algorithm for calculating sedimentation coefficient distribution functions. Anal. Biochem. 279(2), 151—163. 

Equation 3.3 uses the fact that velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity with respect to time. Equation 3.3 is actually a vector equation. If the vectors are expressed in Cartesian coordinates (Section 1.3) it is identical to the three equations... 

In general, this Ck)ulomb yield criterion can be used to determine what stress will be required to cause a ceramic powder to flow or deform. All that is needed are the two characteristics of the ceramic powder the angle of friction, 8, and the cohesion stress, c, for each particular void fraction. With these data, the effective yield locus can be determined, from which the force required to deform the powder to a particular void fraction (or density) can be determined. This Coulomb yield criterion, however, gives no information on how fast the deformation will take place. To determine the velocity that occurs durii flow or deformation of a dry ceramic powder, we need to solve the equation of motion. The equation of motion requires a constitutive equation for the powder. The constitutive equation gives the shear and normal states of stress in terms of the time derivative of the displacement of the material. This information is unavailable for ceramic powders, and the measurements are particularly difficult [76, p. 93]. 

The total time derivatives of the generalized coordinates dqa/dt are called the generalized velocities. They are denoted by qa. Again, q specifies the set qa, oc = 1,2,. ..,n. 

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... 

The flow velocities in flame systems are such that transport processes (diffusion and thermal conduction) make appreciable contributions to the overall flows, and must be considered in the analysis of the measured profiles. Indeed, these processes are responsible for the propagation of the flame into the fresh gas supporting it, and the exponential growth zone of the shock tube experiments is replaced by an initial stage of the reaction where active centres are supplied by diffusion from more reacted mixture sightly further downstream. The measured profiles are related to the kinetic reaction rates by means of the continuity equations governing the one-dimensional flowing system. Let Wi represent the concentration (g. cm" ) of any quantity i at distance y and time t, and let F,- represent the overall flux of the quantity (g. cm". sec ). Then continuity considerations require that the sum of the first distance derivative of the flux term and the first time derivative of the concentration term be equal to the mass chemical rate of formation q,- of the quantity, i.e. 

In the strict kinematic limit in which no forces act, one takes the time derivative of Eq. (3) to conclude that the transformation Eq. (3) also relates the initial and final velocities. Often however there is a strong repulsion between the products and in the sudden limit this force gives rise to an impulsive change of the velocities. It follows that the velocities transform as... 

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