Velocity component

Figure Bl.14.10. Flow tlirough an KENICS mixer, (a) A schematic drawing of the KENICS mixer in which the slices selected for the experiment are marked. The arrows indicate the flow direction. Maps of the z-component of the velocity at position 1 and position 2 are displayed in (b) and (c), respectively, (d) and (e) Maps of the v- and the y-velocity component at position 1. The FOV (field of view) is 10 nnn. (From [31].)...

The application of a small external electric field A to a semiconductor results in a net average velocity component of the carriers (electrons or holes) called the drift velocity, v. The coefficient of proportionality between E and is known as the carrier mobility p. At higher fields, where the drift velocity becomes comparable to the thennal... 

The average random force over the time step is taken from a Gaussian with a varianc 2mk T y(St). Xj is one of the 3N coordinates at time step i E and R are the relevan components of the frictional and random forces at that time n, is the velocity component. 

In this element the velocity and pressure fields are approximated using biquadratic and bi-linear shape functions, respectively, this corresponds to a total of 22 degrees of freedom consisting of 18 nodal velocity components (corner, mid-side and centre nodes) and four nodal pressures (corner nodes). 

U-V-P schemes belong to the general category of mixed finite element techniques (Zienkiewicz and Taylor, 1994). In these techniques both velocity and pressure in the governing equations of incompressible flow are regarded as primitive variables and are discretized as unknowns. The method is named after its most commonly used two-dimensional Cartesian version in which U, V and P represent velocity components and pressure, respectively. To describe this scheme we consider the governing equations of incompressible non-Newtonian flow (Equations (1.1) and (1.4), Chapter 1) expressed as... 

Using different types of time-stepping techniques Zienkiewicz and Wu (1991) showed that equation set (3.5) generates naturally stable schemes for incompressible flows. This resolves the problem of mixed interpolation in the U-V-P formulations and schemes that utilise equal order shape functions for pressure and velocity components can be developed. Steady-state solutions are also obtainable from this scheme using iteration cycles. This may, however, increase computational cost of the solutions in comparison to direct simulation of steady-state problems. 

Typically velocity components along the inlet are given as essential (also called Dirichlet)-type boundary conditions. For example, for a flow entering the domain shown in Figure 3.3 they can be given as... 

On no-slip walls zero velocity components can be readily imposed as the required boundary conditions (v = v, = 0 on F3 in the domain shown in Figure 3.3). Details of the imposition of slip-wall boundary conditions are explained later in Section 4.2. 

Equations (3.59) and (3.60) are recast in terms of their components and solved together. After algebraic manipulations and making use of relations (3.61) slip-wall velocity components are found as... 

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... 

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

Integration of llie velocity components, given by liquation (5.43), with re.spect to gap-wise direction (i.e. z) yields the volumetric Dow rates per unit width m the. V and V directions as... 

Step 2 - use the calculated pressures and find the velocity components by variational recovery. 

Let H and L be two characteristic lengths associated with the channel height and the lateral dimensions of the flow domain, respectively. To obtain a uniformly valid approximation for the flow equations, in the limit of small channel thickness, the ratio of characteristic height to lateral dimensions is defined as e = (H/L) 0. Coordinate scale factors h, as well as dynamic variables are represented by a power series in e. It is expected that the scale factor h-, in the direction normal to the layer, is 0(e) while hi and /12, are 0(L). It is also anticipated that the leading terms in the expansion of h, are independent of the coordinate x. Similai ly, the physical velocity components, vi and V2, ai e 0(11), whei e U is a characteristic layer wise velocity, while V3, the component perpendicular to the layer, is 0(eU). Therefore we have... 

Assuming symmetry of vj and V2 with respect to mid-surface at = 0), the velocity components, given by Equation (5.70), are integrated to obtain flow rates in the lateral directions within the limits of the thin layer as... 

SETPRM Rearranges numbers of nodal degrees of freedom to make them compatible with the velocity components at each node. For example, in a niiie-noded element allocated degree of freedom numbers for v i and vj at node n are X and X +9, respectively. 

Molecules such as 3,4 and 5 in Figure 2.6, which have a zero velocity component away from the source, behave uniquely in that they absorb radiation of the same frequency Vj-es whether the radiation is travelling towards or away from R, and this may result in saturation (see Section 2.3.4). If saturation occurs for the set of molecules 3, 4 and 5 while the radiation is travelling towards R, no further absorption takes place as it travels back from R. The result is that a dip in the absorbance curve is observed at Vj-es, as indicated in Figure 2.5. This is known as a Lamb dip, an effect which was predicted by Lamb in 1964. The width of the dip is the natural line width, and observation of the dip results in much greater accuracy of measurement of v es. 

The functions v,aij,Sij v) represent the velocity, components of the stress tensor and components of the rate strain tensor. The dot denotes the derivative with respect to t. The convex and continuous function describes the plasticity yield condition. It is assumed that the set... 

Equation 7 should not be constmed to mean that the constant is the same for all stieamlines. Once the velocities are known, this immediately gives the piessuie. For a constant density, itiotational flow, the velocity components themselves are detivable from a velocity potential, ( ) ... 

The physics and modeling of turbulent flows are affected by combustion through the production of density variations, buoyancy effects, dilation due to heat release, molecular transport, and instabiUty (1,2,3,5,8). Consequently, the conservation equations need to be modified to take these effects into account. This modification is achieved by the use of statistical quantities in the conservation equations. For example, because of the variations and fluctuations in the density that occur in turbulent combustion flows, density weighted mean values, or Favre mean values, are used for velocity components, mass fractions, enthalpy, and temperature. The turbulent diffusion flame can also be treated in terms of a probabiUty distribution function (pdf), the shape of which is assumed to be known a priori (1). 

Example The differential equation of heat conduction in a moving fluid with velocity components is... 

An example of a linear hyperbolic equation is the adveclion equation for flow of contaminants when the x and y velocity components areii and i , respectively. 

One-dimensional Flow Many flows of great practical importance, such as those in pipes and channels, are treated as onedimensional flows. There is a single direction called the flow direction velocity components perpendicmar to this direction are either zero or considered unimportant. Variations of quantities such as velocity, pressure, density, and temperature are considered only in the flow direction. The fundamental consei vation equations of fluid mechanics are greatly simphfied for one-dimensional flows. A broader categoiy of one-dimensional flow is one where there is only one nonzero velocity component, which depends on only one coordinate direction, and this coordinate direction may or may not be the same as the flow direction. 

Rate of Deformation Tensor For general three-dimensional flows, where all three velocity components may be important and may vaiy in all three coordinate directions, the concept of deformation previously introduced must be generahzed. The rate of deformation tensor Dy has nine components. In Cartesian coordinates. 

Laminar and Turbulent Flow, Reynolds Number These terms refer to two distinct types of flow. In laminar flow, there are smooth streamlines and the fuiid velocity components vary smoothly with position, and with time if the flow is unsteady. The flow described in reference to Fig. 6-1 is laminar. In turbulent flow, there are no smooth streamlines, and the velocity shows chaotic fluctuations in time and space. Velocities in turbulent flow may be reported as the sum of a time-averaged velocity and a velocity fluctuation from the average. For any given flow geometry, a dimensionless Reynolds number may be defined for a Newtonian fluid as Re = LU p/ I where L is a characteristic length. Below a critical value of Re the flow is laminar, while above the critical value a transition to turbulent flow occurs. The geometry-dependent critical Reynolds number is determined experimentally. 

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-... 

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