Similarity solutions velocity

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundaiy-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, separation of variables (not the classical concept) and the use of continuous transformation groups. The basic theoiy is available in Ames (see the references). 

It is only po.ssible to obtain similar solutions in situations where the governing equations (Eqs. (12.40) to (12.44)) are identical in the full scale and in the model. This tequirement will be met in situations where the same dimensionless numbers are used in the full scale and in the model and when the constants P(i, p, fj.Q,.. . have only a small variation within the applied temperature and velocity level. A practical problem when water is used as fluid in the model is the variation of p, which is very different in air and in water see Fig. 12.27. Therefore, it is necessary to restrict the temperature differences used in model experiments based on water. 

The similarity solution for a flow field in front of a steady piston is a special case from a much larger class of similarity solutions in which certain well-defined variations in piston speed are allowed (Guirguis et al. 1983). The similarity postulate for variable piston speed solutions, however, sets stringent conditions for the gas-dynamic state of the ambient medium. These conditions are unrealistic within the scope of these guidelines, so discussion is confined to constant-velocity solutions. 

Similarity solutions of the velocity profile functions for the Von Karman problem. (From Von Karman, Th, Z., Angew. Math. Mech., 1,231,1921.)... 

Boundary layer similarity solution treatments have been used extensively to develop analytical models for CVD processes (2fl.). These have been useful In correlating experimental observations (e.g. fi.). However, because of the oversimplified fiow description they cannot be used to extrapolate to new process conditions or for reactor design. Moreover, they cannot predict transverse variations In film thickness which may occur even In the absence of secondary fiows because of the presence of side walls. Two-dimensional fully parabolized transport equations have been used to predict velocity, concentration and temperature profiles along the length of horizontal reactors for SI CVD (17,30- 32). Although these models are detailed, they can neither capture the effect of buoyancy driven secondary fiows or transverse thickness variations caused by the side walls. Thus, large scale simulation of 3D models are needed to obtain a realistic picture of horizontal reactor performance. 

Lack of perfect specificity in carrier-solute recognition provides for the possibility that structurally similar solutes may compete for carrier availability. Analysis of competitive [Eq. (18)] and noncompetitive [Eq. (19)] inhibition as well as cooperativity effects (allosteric modulation by structurally dissimilar solutes) on carrier-mediated solute flux is equivalent to assessment of the velocity of enzyme reactions. 

It is important to note that the algebraic equation (131) probably holds even when the velocity of suction at the wall v0 x) assumes a form different from Eq. (114). The latter equation involves the existence of a similarity solution, but this is not required for applying the algebraic method of interpolation. 

This is a linear parabolic partial differential equation that can be readily solved as soon as boundary conditions are specified. There is a symmetry condition at the centerline, and it is presumed that the mass fraction Yk vanishes at the wall, Yk = 0. It is important to note that it has been implicitly assumed that the velocity profile has been fully developed, such that the similarity solution / is valid. This assumption is analogous to that used in the Graetz problem (Section 4.10). 

Consider the Jeffrey-Hamel flow in a planar wedge channel as discussed in Section 5.2, where a similar solution for f 6) is developed. The ordinary differential equation that describes the scaled velocity / does not directly involve pressure. The objective of this exercise is to recover the pressure field from the similar solution. 

Fig. 6.13 Comparison of streamlines from rotating-disk solutions at two rotation rates. Both cases are for air flow at atmospheric pressure and T = 300 K. The induced inlet velocity is greater for the higher rotation rate. In both cases the streamlines axe separated by 27tA4< = 1.0 x 10-6 kg/s. The solutions are illustrated for a 2 cm interval above the stagnation plane and a 3 cm radius rotation plane. The similarity solutions themselves apply for the semi-infinite half plane above the surface.

This is clearly a Beltrami equation, but what is more amazing is that the field result (88) describes a solution to the free-space Maxwell equations that, in contrast to standard PWS, the electric (E0) and magnetic (Bo) vectors are parallel [e.g., Eo x Bo = 0, where Eo x Bo = i(E0 A Bo)], the signal (group) velocity of the wave is subluminal (v < c), the field invariants are non-null, and as (91) clearly shows, this wave is not transverse but possesses longitudinal components. Moreover, Rodrigues and Vaz found similar solutions to the free-space Maxwell equations that describe a superluminal (v > c) situation [71]. 

The basic similarity solution for this ignition problem is derived from the slow flow (6,7J approximation, characterized by (1) flow velocities which are small compared to the speed of sound, and (2) an essentially constant pressure field. The energy and velocity equations may then be written as... 

The similarity solution uses as a starting point the assumption that the boundary layer profiles are similar at all values of x, i.e., that the basic form of the velocity profiles at different values of jc, as shown in Fig. 3.3, are all the same [1],[2],[3],[4],[5]. 

Similarity solutions for a few cases of flow over a flat plate where the plate temperature varies with x in a prescribed manna can also be obtained. In an such cases the solution for the velocity profile is, of course, not affected by the boundary condition... 

In the preceding sections, the solution for boundary layer flow over a flat plate wav obtained by reducing the governing set of partial differential equations to a pair of ordinary differential equations. This was possible because the velocity and temperature profiles were similar in the sense that at all values of x, (u u ) and (Tw - T)f(Tw - T > were functions of a single variable, 17, alone. Now, for flow over a flat plate, the freestream velocity, u, is independent of x. The present section is concerned with a discussion of whether there are any flow situations in which the freestream velocity, u 1, varies with Jr and for which similarity solutions can still be found [1],[10]. 

Thus, the velocity distribution and the function, e, for similar solutions must be such that d e2u )ldx is a constant equal to (2a—/3)C/. Now one possibility is that (2a p) be equal to zero. However, the variation of jq with x that provides this situation seems to have little practical significance and it will not be considered here. Therefore, (2a - P) will be assumed to be nonzero. 

In terms of m, the freestream velocity distributions for which similarity solutions can be found, these being given in Eq. (3.95), can be written as ... 

Having established the form of the freestream velocity distribution that gives similarity solutions, consideration can be given to the similarity variable function e(x). This is found by using Eq. (3.90). Noting that a has already been set equal to one, this equation gives ... 

Before discussing any solutions to this equation, it is useful to consider what body shapes will give a velocity distribution of the type for which similarity solutions can be found, i.e., of the type given in Eq. (3.99). A velocity distribution of this type will exist with flow over wedge-shaped bodies having an included angle, 0, which is equal to rr0 as shown in Fig. 3.10. 

Having established that similarity solutions for the velocity profile can be found for certain flows involving a varying ffeestream velocity, attention must now be turned to the solutions of the energy equation corresponding to these velocity solutions. The temperature is expressed in terms of the same nondimensional variable that was used in obtaining the flat plate solution, i.e., in terms of 8 = (Tw - T)f(Tw -Tt) and it is assumed that 0 is also a function of ij alone. Attention is restricted to flow over isothermal surfaces, i.e., with Tw a constant, and T, of course, is also constant. 

It should be realized that there is no real purpose in comparing this velocity profile with that given by the exact similarity solution since the integral equation method does not seek to accurately predict the details of the velocity and temperature profiles. The method seeks rather, by satisfying conservation of mean momentum and energy, to predict with reasonable accuracy the overall features of the flow. 

Air flows at a velocity of 9 m/s over a wide flat plate that has a length of 6 cm in the flow direction. The air ahead of the plate has a temperature of 10°C while the surface of the plate is kept at 70°C. Using the similarity solution results given in this chapter, plot the variation of local heat transfer rate in W/m2 along the plate and the velocity and temperature profiles in the boundary layer on the plate at a distance of 4 cm from the leading edge of the plate. Also calculate the mean heat transfer rate from the plate. 

Solution. The similarity solution given above indicates that the temperature and velocity distributions are given by ... 

Using the similarity solution results, derive an expression for the maximum velocity in the natural convective boundary layer on a vertical flat plate. At what position in the boundary layer does this maximum velocity occur ... 

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