Equation of motion dimensionless

For convenience, we focus our discussion initially on streaming flow past a sphere. The configuration is sketched in Fig. 10-10, where we also indicate the direction of motion relative to the spherical coordinates (r, 9). The dimensionless equations of motion for the outer region, in which lc = a (the sphere radius), are identical to (7-50), and it is clear that the leading-order approximation for Re >> I is thus... 

Note that the Stokes number is the ratio of the particle stop distance sp to the characteristic length of the flow L. As particle mass decreases, the Stokes number also decreases. A small Stokes number implies that the particle is able to adopt the fluid velocity very quickly. Since the dimensionless equation of motion depends only on the Stokes number, equality between two geometrically similar flows indicates similarity of the particle trajectories. 

The Reynolds number (i.e.. Re) represents an order-of-magnitude ratio of convective forces to viscous forces, and it appears as the most important dimensionless number on the left-hand side of the dimensionless equation of motion ... 

Why doesn t the Froude number appear in the dimensionless equation of motion, given by equation (8-42) Use one or two sentences to answer this question. 

The primary focus of this chapter is to analyze the dimensionless equation of motion in the laminar flow regime and predict the Reynolds number dependence of the tangential velocity gradient at a spherical fluid-solid interface. This information is required to obtain the complete dependence of the dimensionless mass transfer coefficient (i.e., Sherwood number) on the Reynolds and Schmidt numbers. For easy reference, the appropriate correlation for mass transfer around a solid sphere in the laminar flow regime, given by equation (11-120), is included here ... 

As illustrated in Chapter 8 via equation (8-42), if dynamic pressure 5 is dimen-sionalized using a characteristic viscous momentum flux (i.e., iV/L), then = lYand one obtains the following form of the dimensionless equation of motion for laminar flow ... 

The tangential component of the dimensionless equation of motion is written explicitly for steady-state two-dimensional flow in rectangular coordinates. This locally flat description is valid for laminar flow around a solid sphere because it is only necessary to consider momentum transport within a thin mass transfer boundary layer at sufficiently large Schmidt numbers. The polar velocity component Vo is written as Vx parallel to the solid-liquid interface, and the x direction accounts for arc length (i.e., x = R9). The radial velocity component Vr is written... 

In the primary flow direction, parallel to the interface, within the mass and momentum boundary layers, molecular transport of x momentum in the x direction (i.e., ixd vxldx ) is neglected relative to convective transport of x momentum in the x direction (i.e., pvxdvx/dx). Hence, when convective, viscous, and dynamic pressure forces are equally important, the x component of the dimensionless equation of motion is... 

The latter approach is adopted because dynamic pressure gradients are calculated in the potential flow regime, outside the momentum boundary layer and far from the solid-liquid interface, where fi 0 and viscous forces are negligible. Then, 3P /3x is imposed across the boundary layer. This is standard practice for momentum boundary layer problems. Hence, if v represents the dimensionless velocity vector in the potential flow regime, then the steady-state dimensionless equation of motion is... 

Based on the dimensionless equation of continuity [i.e., eq. (12-18dimensionless equation of motion [i.e., eq. (12-18e)] in the laminar flow regime, one concludes that both v and v are functions of dimensionless spatial coordinates x and y, as well as the Reynolds number and the geometiy of the flow configuration. Unfortunately, the previous set of equations does not reveal the specific dependence of v and v on Re. 

However, the Reynolds number does not appear explicitly in the renormalized form of the dimensionless equation of motion ... 

Three basic approaches have been used to solve the equations of motion. For relatively simple configurations, direct solution is possible. For complex configurations, numerical methods can be employed. For many practical situations, particularly three-dimensional or one-of-a-kind configurations, scale modeling is employed and the results are interpreted in terms of dimensionless groups. This section outlines the procedures employed and the limitations of these approaches (see Computer-aided engineering (CAE)). 

Consider the simple elastic system of Fig. 13. Equations of motion under the applied (simplified) force pulse can be easily written and solved (see Refs. 15 and 21), and a dimensionless form of the maximum response Xjnax can be plotted versus another dimensionless ratio which relates loading time T to structural natural period (Figure 14). In these two figures, the various symbols represent ... 

When considering boundary conditions, a useful dimensionless hydrodynamic number is the Knudsen number, Kn = X/L, the ratio of the mean free path length to the characteristic dimension of the flow. In the case of a small Knudsen number, continuum mechanics will apply, and the no-slip boundary condition assumption is valid. In this formulation of classical fluid dynamics, the fluid velocity vanishes at the wall, so fluid particles directly adjacent to the wall are stationary, with respect to the wall. This also ensures that there is a continuity of stress across the boundary (i.e., the stress at the lower surface—the wall—is equal to the stress in the surface-adjacent liquid). Although this is an approximation, it is valid in many cases, and greatly simplifies the solution of the equations of motion. Additionally, it eliminates the need to include an extra parameter, which must be determined on a theoretical or experimental basis. 

By appropriate rescaling of the variables x t and h to X, T and H, the parameters in the equation of motion can be scaled out, leading to a dimensionless equation. Considering the present form (4) of the current, without the ) term, the equation of motion of the surface can be written in form of a functional derivative ... 

Equations (11-30) to (11-32) give a good description of the motion of particles released from rest (C8, Ol, 02). In dimensionless form, the equation of motion is (C8) ... 

For a power law solution the equation, of motion is non-dimensionalized in a similar manner as with the Newtonian solution, except that dimensionless pressure is defined as... 

Wave Propagation in Reduced FKN Mechanisms. The full FKN mechanism has many species. However, there are a number of rather fast reactions. Using scalings on concentrations and rate coefficients for concentrations of interest to wave propagation, it has been shown that the FKN mechanism may be reduced to a three variable problem (12). With this procedure it is found that the three reduced concentrations obey the following equations of motion for a wave propagating at constant dimensionless velocity c ... 

A thermostat that rigorously corresponds to a canonical ensemble has been developed by Nose [79]. This significant advance also adds a friction term to the equation of motion, but one that maintains the rigorously correct distribution of vibrational modes. It achieves this by adding a new dimensionless variable to the... 

The relationship (7.4) can also be derived, if the equation of motion (Navier-Stokes differential equations) are drawn up and dimensionlessly formulated under given boundary conditions (here the continuity and energy equations). W. Nusselt followed this path (1909/1915). The thus derived pi-numbers were later named by... 

Start with the full 2D equations of motion in the Cartesian coordinate system (x, y ). Assume that t2 is the appropriate characteristic length scale in both thex and y directions. Hence, in this case, the dimensionless variables are... 

As is shown in the second part of the review, in spite of the smallness of b, the equation of motion, governing the b(t) time dependence, is nonlinear. In terms of the dimensionless variables s and (p, it takes the form... 

In this section the full equations of motion for the external problem sketched in Fig. 4.1a are simplified by using approximations appropriate to natural convection, and the resulting equations are nondimensionalized to bring to light the important dimensionless groups. Although... 

Important Dimensionless Groups. The average Nusselt number for steady-state heat transfer from the body shown in Fig. 4.1a depends on the dimensionless groups that arise in the nondimensionalized equations of motion and their boundary conditions [78]. With Tw and T constant, the only dimensionless groups that appear in the boundary conditions are those associated with the body shape. Provided the simplified equations (Eqs. 4.5 1.7) are valid, the only other dimensionless groups are the Rayleigh and Prandtl numbers. Thus, for a given body shape,... 

These also result even if the motion is unsteady, providing that ajxV and the other dimensionless terms remain finite in the limit R = 0 Equations (7) and (8) are then referred to as the quasi-static or quasi-steady Stokes equations. In this case the time variable enters the equations of motion only in an implicit form. The precise relationship between the solutions of Eqs. (7) and (8) and the asymptotic solutions of the Navier-Stokes equations at small Reynolds numbers is discussed in Section III. 

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