Acta I~{eehaniea39, 127--138 (1981) ACTA MECHANICA @ by Springer-Verlag t981 On Statistical Theories of Turbulent Flow of Micropolar Fluids By G. Ahmadi, Shiraz, Iran (Received December 4, 1978; revised July 20, 1979) Summary - Zusammenfassung On Statistical Theories of Turbulent Flow of Micropolar Fluids. The theory of turbulen~ flow of micropolar fiuids is considered. I-Iopf's characteristic functional is introduced arid the equation governing its time developmen~ is derived and the equivalence of this fuuctionM equation to the hierarchy of the moment equations is discussed. The equation for dynamics of the probability density functions is also obtained. Lewis-Kraichnan space-Lime functional formulation is presented and its solution for the final period of decay turbulence is obtained and discussed. |~:ber statistisehe Theorien turbulenter Striimungen mikropolarer Fluide. Es wird die Theorie turbulenter Str6mungen mikropolarer Fluide betrachtet. /)as charakteristische Funktional yon Hopf wird eingeffihrt und die Gleichung hergeleitet, welchc seine Zeit- entwicklung beschreibt und die Gleichwertigkeit dieser Fur~ktionalgleichung mit der Hierarchie der Momentengleichungen wird diskutiert. Es wird auch die Gleichung ffir die Dynamik tier Wahrscheinlichkeitsdiehtefunktion erhalten. Die Levis-Kraiehnan Raum-Zeit- Funktionalformulierung wird eingefiihrt und ihre L6sung ffir den endlichen Abschnitt der Turbulenzabnahme erhalten und untersucht. 1. Introduction The flow of fluids with microstructure.s have attracted considerable attention in the past decade. Excellent reviews are provided by Ariman, Turk and Sylvester [1], [2]. No attempt shall be made here to survey the related literature. However, some of the recent work directly related to the present problem should be men- tioned. The theory of mieropolar fluids has been developed by Eringen [3], [4]. A wealth of papers concerning the application of this theory to complicated fluids flow- problem has since appeared in the pertinent literature. For instance, Lee and Eringen [5]--[7] developed a micropolar theory of liquid crystals and Kline and Allen [8]--[10], Ariman [11] and Turk, Sylvester and Ariman [12] studied blood flows based on mieropolar model. The application of micropolar theory in low concentration suspension flows has been pointed out by Ahmadi [13], [14] among others [1], [2]. The theory of turbulent shear flows of mieropolar fluid was considered by Liu [15], Nikolaeviskii [16], and Ahmadi [17]. In the present study the statistical theory of turbulent flows of micropolar fluids via a functional approach is considered. The functional formulation of 0001-5970/81/0039/0127/802.40