Nondimensionalization characteristic scales

The distinguishing characteristics of the stagnation-flow subcases depend on the domain and on the rotation. The characteristic scales are different for the subcases, but the equations themselves are the same. The boundary conditions also differ among the subcases. Table 6.1 shows the applicable scales and nondimensional groups that apply to each of the four subcases. 

B. Steady Unidirectional Flows - Nondimensionalization and Characteristic Scales... 

B. STEADY UNIDIRECTIONAL FLOWS - NONDIMENSIONALIZATION AND CHARACTERISTIC SCALES... 

Finally, we return briefly to the problem of choosing appropriate characteristic scales for nondimensionalization. In particular, it is important to recognize the role played by nondimensionalization so that the consequences of an incorrect (or inappropriate) choice of a characteristic scale is clear. The first and most important point is that the introduction of dimensionless variables does not change the problem at all, but simply renames variables. Suppose, for example, that we had used lc/uc = n/GR as a characteristic time scale in (3-93) instead of R2/v. Then, instead of... 

We begin, as in previous examples, by nondimensionalizing using characteristic scales. We need a characteristic length scale, and for this it seems reasonable to select the tube radius lc = a. The other quantity to scale in (3-194) is the temperature 9. However, there is no explicit temperature specified that provides a convenient (or appropriate) choice for scaling 9. Indeed, what is specified is the heat flux q, and if we think about the problem from a qualitative point of view, it is evident that the constant heat flux at the wall will cause the temperature to increase continuously as we move downstream in the tube. Because the increment in 9 from its initial value to its value at any fixed z will be proportional to q, we use the quantity 9C = (qa/k), which has the same units as 9, to nondimensionalize the temperature. Hence,... 

Now, assuming that the characteristic scales used to nondimensionalize (3-196) are correct, all of the terms in (3-196) involving dimensionless variables should be independent of the independent parameters of the problem. It follows from this that the relative magnitudes of the various terms should be determined completely by the magnitude of Pe. Thus, for small values of Pe <convection term should be small compared with either of the conduction terms in (3-196), and a first approximation is that the evolution of the temperature field in the heated portion of the tube is dominated by radial and axial conduction. 

The first step, as usual, is to introduce characteristic scales and nondimensionalize the problem. In this case, appropriate characteristic scales are... 

Of course, we can never change a physical (or mathematical) problem by simply nondi-mensionalizing variables, no matter what the choice of scale factors. It is only when we attempt to simplify a problem by neglecting some terms compared with others on the basis of nondimensionalization that the correct choice of characteristic scales becomes essential. Fortunately, as we shall see, an incorrect choice of characteristic scales resulting in incorrect approximations of the equations or boundary conditions will always become apparent by the appearance of some inconsistency in the asymptotic-solution scheme. The main cost of incorrect scaling is therefore lost labor (depending on how far we must go to expose the inconsistency for a particular problem), rather than errors in the solution. 

To be consistent with our preceding analyses, we begin by nondimensionalizing. For this purpose, we identify characteristic scales. Because the perturbation to the channel shape is assumed to be very small, it is apparent that the relevant velocity scale is... 

Following our usual custom, we now nondimensionalize. The physically obvious characteristic scales are the length scales for variations of the velocity and perturbation pressure in the x and v directions, and the characteristic magnitude of the velocity in the x direction,... 

The characteristic scales that have been used to produce these nondimensionalized equations are... 

Now, the drop shape will be non-spherical if this is necessary to satisfy the boundary conditions at its surface, and, specifically, to satisfy the normal-stress balance, (2 135). To determine the condition that leads to small deformations, we can nondimensionalize the boundary conditions using the same characteristic scales that were used for the governing equations, (7 198) and (7-199). The result for the normal-stress balance is... 

It is also convenient to express (12-125) and (12-126) in dimensional form because the characteristic scales that were used in nondimensionalizing these equations are not the most appropriate choice for the thin-film problem (for example we used lc = R, whereas the gap width is a much more appropriate choice for a characteristic length scale in the narrow gap problem). Hence, reversing the nondimensionalization (12-119) and introducing the approximation (12-142), we have... 

Now we nondimensionalize by using characteristic scales relevant to the narrow-gap limit,... 

The above results correspond to an important family of axisymmetric, convex geometries. The results presented in dimensional form or in nondimensional form as given above do not reveal an important property presented by this family of geometries and other geometries when the appropriate physical characteristic scale length is used for the nondimensionalization. 

Consider a fluidic system with surface tension effects. The characteristic scales for length and velocity are L and U. The physical parameters are density p, viscosity v, gravity g, and surface tension y. By using the Buckingham n theorem, one can obtain three independent nondimensional groups from these six variables. An option for a set of three independent nondimensional numbers is... 

Thus, the mathematical model (2), (6), (7), (8) describes the change in the velocity field in the formation of thrombus in the vessel. To simulate the obstacles of arbitrary shape (in this problem blood clot) is introduced by a discretetime artificial power. This force is applied only on the surface and within the constraints of the body. Force application point disposed in a spaced, similar velocity components defined on a staggered grid. When the point of application of force coincides with a virtual border, an artificial force is applied so as to satisfy the boundary conditions on the obstacle. The cell containing the virtual boundary, does not satisfy the equation of conservation of mass. Therefore, we introduce the source / drain weight to the cell that contains the virtual border. Discrete in time force is used to meet the conditions of adhesion on a virtual border, while the source / drain weight, to meet the conservation of mass for the cell that contains the virtual boundary. Procedure nondimensionalization this system involves choosing the characteristic scales the concentrations Oq and 4>q, lines size L, the characteristic scale of velocity V. In view of the above equations (1) - (2) takes the form ... 

The proper way to verify whether the nonlinear term (v V)v in the N-S equation can be neglected is to make the equation dimensionless. This means that we express all physical quantities, such as length and velocity, in units of the characteristic scales, for example, Lq for length and Vq for velocity. Reynolds number appears in the nondimensionalized N-S equation and the relative importance of different terms in the can be verified for small Reynolds number assumption. We explain this in the following sections using two examples. 

Equations (11.1)-(11.3) respectively express conservation of mass, momentum, and energy. To nondimensionalize these equations, the initial diameter do, the initial centreline velocity Uq, and a characteristic temperature difference To are used as scales. If the heat is deposited uniformly over a unit volume at a constant rate J, for a total time th into the flow. To may be defined as the resultant net temperature change that would result if this heat was deposited To = Jth/ P p ... 

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