Asymptotic approximation equations

This relation for the thickness of the boundary layer has been obtained on the assumption that the velocity profile can be described by a polynomial of the form of equation 11.10 and that the main stream velocity is reached at a distance 8 from the surface, whereas, in fact, the stream velocity is approached asymptotically. Although equation 11.11 gives the velocity ux accurately as a function of v, it does not provide a means of calculating accurately the distance from the surface at which ux has a particular value when ux is near us, because 3ux/dy is then small. The thickness of the boundary layer as calculated is therefore a function of the particular approximate relation which is taken to represent the velocity profile. This difficulty cat be overcome by introducing a new concept, the displacement thickness 8. ... 

Figure 4.10 shows that two approximations, equations (4.3.22) and (4.3.23), become close in the middle of the intermediate region and both are approaching asymptotically equation (4.3.26). Note that the case of multipole... 

Figure 2. Equation 15 (circles) (a) asymptotic approximation for small values of a (eq 16), continuous line (b) asymptotic approximation for large values of a (eq 20), continuous line.

The error introduced by use of the Wien equation is less than 1 percent when XT < 3000 pm K. The Wien equation has significant practical value in optical pyrometry for T < 4600 K when a red filter (X = 0.65 pm) is employed. The long-wavelength asymptotic approximation for Eq. (5-102) is known as the Rayleigh-Jeans formula, which is accurate to within 1 percent for XT > 778,000 pm-K. The Raleigh-Jeans formula is of limited engineering utility since a blackbody emits over 99.9 percent of its total energy below the value of XT = 53,000 pm-K. 

It turns out, very fortunately, that this asymptotic approximation is also an exact solution of the Schrddinger equation Eq (7.29) with = 0, Just what happened for the harmonic-oscillator problem in Chapter 5. The solutions are designated Rnlit), where the label n is known as the principal quantum number, as well as by the angular momentum i, which is a parameter in the radial equation. The solution (7.32) corresponds to Rioir)- This should be normalized according to the condition... 

Introductory note Most transport and/or fluids problems are not amenable to analysis by classical methods for linear differential equations, either because the equations are nonlinear (or simply too comphcated in the case of the thermal energy equation, which is linear in temperature if natural convection effects can be neglected), or because the solution domain is complicated in shape (or in the case of problems involving a fluid interface having a shape that is a priori unknown). Analytic results can then be achieved only by means of approximations. One approach is to simply discretize the equations in some way and turn on the computer. Another is to use the family of approximations methods known as asymptotic approximations that lead to useful concepts such as boundary layers, etc. This course is about the latter approach. However, it is not just a... 

We saw in the previous example lor R, <approximate solution, (4-4), which we obtained by taking the limit R, —> 0 in the exact equation, (4-1), was just the first term in an asymptotic solution for Rr approximate equation and solution (4-17) that we obtain by taking the limit Rm -> oo in (4-16) will also be the first term in a formal asymptotic expansion of H for Rm y>> 1. Assuming this expansion is regular, it will take the form... 

Let us now return to the solution of our problem for Rr 1. Although the arguments leading to (4-25) were complex, the resulting equation itself is simple compared with the original Bessel equation. Our objective here is an asymptotic approximation of the solution for the boundary-layer region. In general, we may expect an asymptotic expansion of the form... 

With a first-order reaction, the governing equation is linear and could thus be solved without any use of scaling or asymptotic methods. However, we could just as easily assume that the reaction rate is second order in c or add other complications that do not so easily allow an exact analytic solution. The point here is to illustrate the idea of the asymptotic approximation technique, which is easily generalizable to all of these problems. 

Now, the dynamics of changes in bubble radius with time, starting from some initial radius that differs slightly from an equilibrium value, is a problem that is ideally suited to solution by means of a regular asymptotic approximation. Of course, the governing equation is still the Rayleigh Plesset equation. Before beginning our analysis, we follow... 

The reader may again be curious whether there are consequences of making the wrong choice for c, Uc, or Tc when there is more than one possibility available We will need to discuss this point in some detail, once we see how we intend to use the nondimensional-ized versions of our governing equations (and boundary conditions) in the development of asymptotic approximations. For now, we simply assume that the appropriate choices have been made. Then, for each additional dimensional scale that appears in a particular problem, we get one more dimensionless parameter, in addition to the two that will appear based on... 

Let us conclude by briefly summarizing the material of this section in the following manner. If we wish to determine 9 everywhere in the domain, we would have to solve the differential equation (11-6) - either exactly or by means of an asymptotic approximation-subject to either of the boundary conditions (11-108) or (11-110) at the body surface. If, on the other hand, we wish to determine only 6S (x), then it is advantageous to solve the boundary integral (11-109) or (11-111). This converts the original 2D problem into a ID problem and allows 9S (x) to be calculated directly without the necessity of determining 9 everywhere in the domain. 

A model due to Eager et estimates the strain energy in the ceramic in well-bonded ceramic-metal joints. For a small CTE mismatch between the ceramic (C) and the metal substrate (M), but with a large CTE mismatch between the interlayer (I) and the base materials, the elastic strain energy, Uec. >i> the ceramic for a disc-shaped joint is calculated in terms of the yield strength (oyi) of the braze, the radial distance from the center of the joint, and the elastic moduli of the ceramic (Ec) and braze (Ei). Eager et al proposed analytical expressions to calculate Uec as asymptotic approximations (to 1% accuracy) to their finite element calculations these analytical expressions (equations [l]-[3] in ref ) are used here to estimate the strain energy. 

A comparison of this equation and Equation (6.11) shows that the Ockham factor is approximately equal to the ratio p 9 Cj)/p 9 VXj) which is always less than unity if the data provide any information about the model parameters in the model class Cj. Indeed, for large N, the negative logarithm of this ratio is an asymptotic approximation of the information about 0 provided by data V [147]. Therefore, the log-Ockham factor In Oj removes the amount of information about 9 provided by T> from the log-likelihood In p T> 9, Cj) to give the log-evidence, In p(V Cj). 

The linear stability theory is exceptional in the sense that it can be fuUy based on the three-dimensional equations of fluid dynamics. All the additional effects lead to either direct numerical simulations or the asymptotic approximations. One of the most natural ways of the asymptotic description of the dynamics of jets is the quasi-one-dimensional approach. In the quasi-one-dimensional approximation. 

In the narrow resonance approximation (NR), the assumption is made that the absorption is large only in an interval small compared to aoE, the maximum energy loss in an absorber collision. In this case, the quantities inside the integral in Equation (3.10) can be replaced by their asymptotic values (Equation (3.11) and os = and one obtains... 

---