Velocity derivation

IlyperChem can either use initial velocilies gen eraled in a previous simulation or assign a Gaussian distribution of initial velocities derived from a random n iim her generator. Random numbers avoid introducing correlated motion at the beginn ing of a sim illation. ... 

If Restart is ch ecked ih eti the vckicitics arc th c existing assign cd velocities derived from a previous m olecti lar dyn am ics simulation or included in the HI.N file when It was first read In. You can thus restart a trajectory at exactly the poln t it was term in ated by iisiri g the Restart check bo.x. 

In Equation (4.12) the discretization of velocity and pressure is based on different shape functions (i.e. NjJ = l,n and Mil= l,m where, in general, mweight function used in the continuity equation is selected as -Mi to retain the symmetry of the discretized equations. After application of Green s theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and the pressure terms (to maintain the consistency of the formulation) and algebraic manipulations the working equations of the U-V-P scheme are obtained as... 

After application of Green s theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and algebraic manipulations the working equations of the continuous penalty scheme are obtained as... 

The extra stress is proportional to the derivatives of velocity components and consequently the order of velocity derivatives in terms arising from... 

Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure bound-aiy condition and two velocity boundaiy conditions (for each velocity component) to completely specify the solution. The no sBp condition, whicn requires that the fluid velocity equal the velocity or any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-... 

Assume that the flow is fully developed, that is, all velocity derivatives van-... 

Some researchers (e.g., Abramovich,Baturin,Rajaratnam,- and Nielsen and Moller ) consider x to be the distance from a point located at some distance Xq upstream from the diffuser face. Equations for the jet boundaries and velocity profile used in the centerline velocity derivation assume that the jet is supplied from the point source. Addition of the distance Xq to the distance from the outlet corrects for the influence of the outlet size on the jet geometry. For practical reasons some researchers neglect Xq. 

Liquid entrainment mass velocity, lb entrainment/ (min) (ft ), based on net tray area of tower minus twice downcomer area Assumed allowable liquid entrainment mass velocity derived from assumed allow able loss mols liq-uid/mol vapor, Ib/hr (ft ), based on net tray areas same as for We... 

The expected value on the left-hand side is taken with respect to the entire ensemble of random fields. However, as shown for the velocity derivative starting from (2.82) on p. 45, only two-point information is required to estimate a derivative.14 The first equality then follows from the fact that the expected value and derivative operators commute. In the two integrals after the second equality, only /u,[Pg.264]

Based on the form of Eq. 4.63, we may anticipate a general result in which the product of the friction factor and the Reynolds number is a constant, /Re = constant. We seek a nondimensional analysis that leads to a general friction-factor result. We choose a length scale based on the long dimension of the channel, a. The velocity scale is based on a mean velocity derived from the mass flow rate, U = m/pAc ... 

In the finite-gap case, however, the radial velocity gradient must also be determined from the solution. Using the definition of the scaled velocity V = v/r, we write the radial-velocity derivative as... 

Because of the incredible precision of interferometric techniques, this measured velocity is altogether one percent of the earth s circumference velocity derived from the orbital motion. Very-long-baseline interferometry (VLBI)— which is an exhaustively improved Pogany experiment—can detect Ago 10-9 in the earth s rotation. 

Earlier studies (ref. 440-442) with ordinary air microbubbles (without any synthetic surfactant coating) have already shown that echocardiographic contrast produced by microbubbles is useful in the qualitative analysis of blood flow and valvular regurgitation. In addition, quantitative studies (ref. 440) have shown a correlation between individual contrast trajectories on M-mode echocardiography and invasive velocity measurements in human beings. Meltzer et al. (ref. 441) have shown that velocities derived from the slopes of contrast trajectories seen on M-mode echocardiography correlate with simultaneous velocities obtained by Doppler techniques. (This correlation is expected because both measures represent the same projection of the microbubble velocity vector, that is, in the direction of the sound beam.) More detailed studies (ref. 442) confirmed that microbubble velocity obtained from either Doppler echocardiography or M-mode contrast trajectory slope analysis correlates well with actual (Doppler-measured) red blood cell velocity. Thus, these early studies have shown that microbubbles travel with intracardiac velocities similar to those of red blood cells. 

Assume that the flow is fully developed, that is, all velocity derivatives vanish in the x direction. Since the flow field is infinite in the z direction, all velocity derivatives should be zero in the z direction. Therefore, velocity components are a function of y alone. It is also assumed that there is no flow in the z direction, so vz = 0. The continuity equation Eq. (6-21), with vz = 0 and dvx/dx = 0, reduces to... 

In Section 12.3 we found that when the two mobile phase terms A and Cmv are equal, the flow velocity is approximately equal to the fundamental value ve = DJdpy which is used as a scale factor to get reduced velocity. Derive expressions for the velocities at which A = Blv and Cmv = B/v, showing that these velocities are also of the same order of magnitude as fundamental velocity vc. 

Once the dimensionless mixing length distribution has been found using the above equations, the dimensionless eddy viscosity distribution, in terms of which the numerical method has been presented, can be found using the finite-difference approximation for the velocity derivative that was introduced earlier, i.e., using ... 

To obtain a theoretical basis, Chapman et al. [13] compared a pickup velocity derived from the force balance ... 

Fig. 15. Oxygen dependence of iron uptake in yeast. (A) Progress curves for Fe uptake by wild-type yeast strain DEY1457 are shown. The individual uptake curves are obtained at different concentrations of O2 ranging from 1.2 to 240 (JlM. (B) The uptake velocities derived from the traces in (A) are plotted versus the [O2] the line is the theoretical fit of the data to the Michaelis-Menten equation using the htted constants given in Table IV.

Mach number M is the ratio of the speed of fluid in the duct to the speed of sound in the fluid. The derivatives in these equations are rates of change with length as the fluid passes through a duct. Equation (4-160) relates the pressure derivative, and Eq. (4-161), the velocity derivative, to the entropy and area derivatives. According to... 

This velocity derivative (or gradient) is also called the strain rate, shear, time rate of shear, or rate of deformation. 

It is important to note that the value effective migration velocity, derived from Deutsch, is not equal to the theoretical value uith. derived from Eq. (14). 

To velocity derivatives are approximated by use of the central difference scheme ... 

However, we should always expect the deposition velocity derived from boundary layer theory to be somewhat larger than observed in practice (neglecting measurement errors). Since the physical chemistry limitations should be independent of size and shape of the object in question, we may use boundary layer calculations to indicate the relative characteristics of different situations. 

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