Branching of the Falkner-Skan solutions for λ < 0

The Falkner-Skan equation f'" + ff" + λ(1 - f'^2) = 0, f(0) = f'(0) = 0, is discussed for λ < 0. Two types of problems, one with f'(∞) = 1 and another with f'(∞) = -1, are considered. For λ = 0- a close relation between these two types is found. For λ < -1 both types of problem allow multiple solutions which may be distinguished by an integer N denoting the number of zeros of f' - 1. The numerical results indicate that the solution branches with f'(∞) = 1 and those with f'(∞) = -1 tend towards a common limit curve as N increases indefinitely. Finally a periodic solution, existing for λ < -1, is presented.

Exact Calculation of Laminar Boundary Layer in Longitudinal Flow over a Flat Plate with Homogeneous Suction

/2 ), less than -2.0. ° This is verified in both Pigs. 1 and 2. However, for plane stagnation-point flow, Pigs. 1 and 2 show a considerable discrepancy between actual and asymptotic shear for values of the suction parameter well below -2.0. And, of course, as the suction parameter approaches zero (small condensation rates), the asymptotic expression gives way to the dry-wall value of the shear for the particular geometry involved. This suggests that a simple interpolation formula between the two extremes could be used for the entire range. A formula of the form The numerical results o

Morphometric Relationship, Phylogenetic Correlation, and Character Evolution in the Species-Rich Genus Aphis (Hemiptera: Aphididae)

The species-rich genus Aphis consists of more than 500 species, many of them host-specific on a wide range of plants, yet very similar in general appearance due to convergence toward particular morphological types. Most species have been historically clustered into four main phenotypic groups (gossypii, craccivora, fabae, and spiraecola groups). To confirm the morphological hypotheses between these groups and to examine the characteristics that determine them, multivariate morphometric analyses were performed using 28 characters measured/counted from 40 species. To infer whether the morphological relationships are correlated with the genetic relationships, we compared the morphometric dataset with a phylogeny reconstructed from the combined dataset of three mtDNA and one nuclear DNA regions.Based on a comparison of morphological and molecular datasets, we confirmed morphological reduction or regression in the gossypii group unlike in related groups. Most morphological characteristics of the gossypii group were less variable than for the other groups. Due to these, the gossypii group could be morphologically well separated from the craccivora, fabae, and spiraecola groups. In addition, the correlation of the rates of evolution between morphological and DNA datasets was highly significant in their diversification.The morphological separation between the gossypii group and the other species-groups are congruent with their phylogenetic relationships. Analysis of trait evolution revealed that the morphological traits found to be significant based on the morphometric analyses were confidently correlated with the phylogeny. The dominant patterns of trait evolution resulting in increased rates of short branches and temporally later evolution are likely suitable for the modality of Aphis speciation because they have adapted species-specifically, rapidly, and more recently on many different host plants