A nemato-hydrodynamic theory of shear thinning in solutions of lyotropic nematic-liquid-crystal polymers (LCP's) consisting of semiflexible polymer molecules is developed. It is based on an effective two-media theory assuming a percolative flow of solvent through globularly shaped polymer arrays, which are supposed to form a medium of "porous" spheres with permeability k0. A single porous sphere may be viewed as a randomly coiled and closed disclination loop in an otherwise aligned LCP. The key step of the theory consists of mapping the ensemble of "porous" spheres onto the "domains" displayed by the foamlike texture of a LCP. This map is parameterized by the shear rate gamma and replaces many "porous" spheres by a renormalized and massless object in the form of a domain of linear dimension R and uniform order parameter line S. Borrowing ideas from the theory of critical phenomena, where sigma (=R/[k0(R)]1/2 approximately R1-alpha/2) plays the role of a scale parameter, and a characteristic exponent alpha > 2 is assumed, a fundamental equation for the apparent viscosity eta(gamma) in the limit of small shear rates gamma is derived, based on a relation between the average of R as a function of gamma, obtained as a consequence of Ericksen stress. Various cases of interest are discussed leading for regime I (preceding the Newtonian regime II) to the scaling laws eta approximately gamma-1/2, and eta approximately gamma-2/3, and extrapolate for regime II to eta approximately const. The prediction of a -1/2 slope in regime I of the curve 1n-eta versus 1n-gamma compares favorably with the behavior displayed by many LCP's [K. F. Wissbrun, in Lecture Notes in Mathematics, edited by J. L. Ericksen (Springer, New York, 1984), Vol. 1063, p. 1], as well as the prediction eta approximately const for regime II, although the latter is experimentally only a narrow plateau. Crossover laws between power-law behavior of the curve eta versus gamma for gamma greater-than-or-equal-to gamma-c and eta congruent-to const for 0 less-than-or-equal-to gamma less-than-or-similar-to gamma-c, displaying a Newtonian plateau at the origin, are also derived, and may apply to regimes II and III of a LCP, where alpha = 2 and alpha < 2, respectively, apply.
