The best known transformation is the "Prandtl stretching" defined, for small e, byx-+x: y-+sy: u-+u: v-+sv: s 2 = l/R ^ 0 (5) The transformation preserves the form of the equations and the coefficient of u yy in Eq. (1) becomes unity. This "stretching" has been thoroughly discussed elsewhere/ BirkhofT 2 suggests the following one parameter group:Under this transformation the form of Eqs. (1) and (2) is preserved with all terms having a multiplying factor of I//? 2 . Thus for ^ ^ 0 or oo the group defined by Eq. (6) leaves the boundarylayer equations invariant. Finally we discuss the general four-parameter group 2 described by x-»oo;: y-+/3y-u-^yu\ v-+dv (7) Birkhoff discusses the application to flow past an infinite wedge for which U = cx m . This analysis reduces the boundary-layer equations to the usual Falkner-Skan ordinary differential equation. We are interested in generalized flow around a cylinder of the form U = m sin xWe wish to know for what subgroups of (7) the boundary-layer equations are invariant with respect to variations in m. We also wish to preserve the form of 17. Transforming the equations by 2-0 1-8 1-6 1-4 /m" 1-0 OS 0-6 0-4 0-2 D TERRILL ~ SOLUTION (REF. 3) O SCHONAUER -SOLUTION (REF. 4)means of (7) and equating the coefficients of different terms in each equation it is found that a=l: )5=l/m 1/2 : y = m: d = m 1/2 (9) where m^Q, oo. Thus the required transformation is a oneparameter subgroup of (7) uniquely defined byx->x: y-+y/m l/2 : u^nnu: v-+m 1/2 v (10) It is worth noting that u y -»m(m) ll2 u y . These results have the following application. It is often useful to be able to compare the boundary-layer solutions of different workers who have used different values of m for the external flow. The transformation (10) gives the scaling that must be used on the coordinates and velocities to compare solutions that have different m values. Typical values are m = 1 for Terrill's solution (Ref. 3) and m = 2 for Schonauer's solution (Ref. 4). The velocity profiles, at x = 1.0, obtained from Refs. 3 and 4 are compared by plotting u/U vs y/m 1/2 on Fig. 1. It is seen that the two solutions fall on the same curve, confirming the prediction of Eq. (10).
High‐strength cast iron is used for manufacturing the supporting elements of minerals and cement clinker grinding machines. The dross layer remains inside large dimension castings after manufacturing. It has worse mechanical properties and resistance to fatigue crack formation and propagation. In this paper, the results of an experimental investigation of cyclic bending strength of semi‐natural specimens from cast iron with a dross layer of 10–11 mm thickness are presented. The mechanical properties and structures of the base and dross layers differed considerably. Before testing the methods of defectoscopy were applied to detect technology defects. The stress ratio was r = ‐0,62 and the stress alternating range varied from 70 to 280 MPa. To 1*108cycles, crack growth was insignificant. Further crack propagation was controlled. The 1st specimen was broken after 3*108 and the 2nd one after 2*108 cycles. The dependencies of crack propagation on cycle number and stress intensity factor range were estimated. The dross layer stops crack propagation, when crack front passes from dross to base metal.
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