By L.P. Yarin
This quantity offers functions of the Pi-Theorem to fluid mechanics and warmth and mass move. The Pi-theorem yields a actual motivation in the back of many move strategies and hence it constitutes a worthy instrument for the clever making plans of experiments in fluids. After a quick creation to the underlying differential equations and their remedies, the writer provides many novel techniques the way to use the Pi-theorem to appreciate fluid mechanical matters. The publication is a smart price to the fluid mechanics group, because it cuts throughout many subdisciplines of experimental fluid mechanics.
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Additional resources for The Pi-Theorem: Applications to Fluid Mechanics and Heat and Mass Transfer
The proof of this theorem, as well as numerous instructive examples of its applications for the analysis of various scientific and technical problems are contained in the monographs by Bridgman (1922), Sedov (1993), Spurk (1992) and Barenblatt (1987). Referring the readers to these works, we restrict our consideration by applications of the Pi-theorem to problems of hydrodynamics and the heat and mass transfer only. The study of thermohydrodynamical processes in continuous media consists in establishing the relations between some characteristic quantities corresponding to a particular phenomenon and different parameters accounting for the physical properties of the matter, its motion and interaction with the surrounding medium.
34) rÃ vÃ ðiÞ @v , A1 ¼ . 36) j¼kþ1 jÃ QðiÞ where jÃ ¼ AjÃ =AkÃ are the dimensionless groups. To illustrate the general approach described above, we render dimensionless the Navier–Stokes equations, the energy and species balance equations, as well as the continuity equation. 40) where r; v T; P and cx are the density, velocity vector, the temperature, pressure and the concentration of the species x. In particular, let us use the Cartesian coordinate system where vector v has components u; v and w in projections to the x; y and z axes.
7. 1. Transform the van der Waals equation (Kestin 1966; Jones and Hawkis 1986) to the dimensionless form. Show that such form is universal for any van der Waals gas if one uses the critical values of the pressure, volume and temperature as the characteristic scales. 1) where A1 ¼ P; A2 ¼ a=V 2 ; A3 ¼ V; A4 ¼ Àb; A5 ¼ RT with P; V and T being pressure, molar volume and temperature, respectively. 3) Problems 31 . where A1 ¼ A1 =A1Ã ¼ P=P; A2 ¼ A2 A2Ã; ¼ ðV =VÃ ÞÀ2 ; A3 ¼ A3 =A3Ã ¼ ðV=VÃ Þ; A4 ¼ A4 =A4Ã ¼ 1 A5 ¼ A5 =A5Ã ¼ T=TÃ are the dimensionless variables, and a ¼ À Á A2Ã =A1Ã ¼ a= VÃ2 PÃ ; b ¼ A3Ã =A1Ã ¼ VÃ =PÃ ; g ¼ A4Ã =A1Ã ¼ Àb=PÃ ; and e ¼ A5Ã =A1Ã ¼ RTÃ =PÃ are the dimensionless constants.
The Pi-Theorem: Applications to Fluid Mechanics and Heat and Mass Transfer by L.P. Yarin