Von Mises transformation

There are computational advantages to be realized by transforming the cross-stream coordinate to the stream function, namely the Von Mises Transformation. In a subsequent section this transformation is done for the general boundary-layer setting, and it includes 

At this juncture it is prudent to review briefly the form of an independent variable transformation. Consider a transformation from an (x, y) coordinate system to a (rj, j ) system. Some dependent variable, say T(x, y), may be represented as 

Return now to the Von Mises transformation. With the z coordinate being unchanged, note that the necessary transformation metrics are 

Notice several important aspects about the transformed equations. Of course, xjr is now an independent variable, with the relationship between r and x/r coming from the stream-function definition 

This relationship is captured in differential-equation form as Eq. 7.60. Since the momentum and energy equations (Eqs. 7.59 and 7.62) explicitly involve r2, the radial coordinate has become a dependent variable, not an independent variable. A consequence of the Von Mises transformation is that the radial velocity v is removed as a dependent variable and the radial convective terms are eliminated, which is a bit of a simplification. However, the fact that the group of dependent variables pur2 appear within the diffusion terms is a bit of a complication. The factor pur2 plays the role of an apparent variable diffusion coefficient.  

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The Von Mises transformation offers certain advantages in solving the channel-flow boundary-layer problems. However, it is certainly possible to solve the equations in their primative form. For a cylindrical channel, using constant fluid properties, consider the differences between the primative equations (Eqs. 7.44—7.47) and the Von Mises form (Eqs. 7.59-7.62). 

Spalding [89] abandoned the Couette flow hypothesis and solved Eq. 6.179 through the use of the von Mises transformation obtaining... 

The Bi-Sn and Sn-Cu solders were best represented by single, but different, mechanisms. It is noted that the Sn-Cu creep data reflects a shear test configuration. Assuming a Von Mises transformation, the conversion between shear stress (x) and tensile stress (a), as well as that between the minimum shear strain rate dyldt, and minimum tensile strain rate dzfdt, , can be calculated by Eq 29 and 30, respectively. 

Further, by passing from 8, y to the von Mises variables 8, ip, we obtain The transformation... 

However, there is one very important difference between the martensitic transformation in inherently ductile materials such as iron and brittle materials such as zirconia. As for the latter the von Mises criteria for plastic deformation are not satisfied the deformation process is accommodated by a change in shape as manifest in a serrated surface akin to that shown in Figure 4.5, panel c on a microscopic... 

Transforming disability from an existential disaster into an economic bonanza has, of course, far-reaching financial, personal, political, and social consequences. Long before the advent of Medicare, Medicaid, and the AWDA, Ludwig von Mises warned As a social institution it [social insurance] has thus made the neurosis of the insured a dangerous public disease. Should the institution be extended and developed the disease will spread.. . . We cannot weaken or destroy the will to health without producing illness. 7... 

As a first step Eqs. (2.18) and (2.20b) are transformed into the von Mises system of coordinates. The transformation is defined for the present system by... 

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