The Generalized Integral Transform Technique (G.I.T.T.) is extended to handle the incompressible NavierStokes equations for two-dimensional steady laminar¯ow in cylindrical geometries. Hybrid numerical-analytical solutions with controlled accuracy are obtained, as a result of an appropriate choice of the associated eigenfunction expansion basis, extracted from the diffusion operator of the stream function-only formulation for this class of problems. The approach is illustrated for developing laminar¯ow within an annular channel and numerical results are obtained to demonstrate the excellent convergence characteristics of this hybrid method. Critical comparisons against the boundary layer formulation are provided, and a set of benchmark results is produced, for different values of Reynolds number and aspect ratio.

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SUMMARYA hybrid numerical-analytical solution based on the generalized integral transform technique is proposed to handle the two-dimensional Navier-Stokes equations in cylindrical coordinates, expressed in terms of the streamfunction-only formulation. The proposed methodology is illustrated in solving steady-state incompressible laminar flow of Newtonian fluids in the developing region of a circular tube. The flow modeling also considers two limiting inlet conditions, namely, uniform velocity profile representing a parallel flow, and zero vorticity that characterizes irrotational inlet flow. The integral transform analysis for such a full cylindrical region brings up singularities at the channel centerline, and, as previously described in a work dealing with the boundary-layer formulation, a way to alleviate this difficulty is to adopt a recently introduced fourth-order eigenvalue problem as the basis for the eigenfunction expansion. A thorough convergence analysis of the proposed expansion is then undertaken, for different values of Reynolds number, and a set of reference results for the velocity distributions and friction factors are then presented in tabular and graphical forms.

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Selection indexes can be applied to simultaneous analysis of traits, increasing the efficiency of breeding in choosing the genotypes that meet the needs of both the market and the consumer. While it is a technic applied to several cultures, there are few studies about it in lettuce lines. For this reason, the aim of this study was to comparate assess the efficiency of different selection indexes for selecting biofortifiedleaf lettuce lines with good agronomic traits. The experiment was carried out at the Estação Experimental de Hortaliças of Federal University of Uberlândia (UFU), campus Monte Carmelo, in 2018. The experiment was designed in full randomized blocks of 3 repetitions with 25 leaf lettuce genotypes (22 lines from cultivars Pira 72 versus Uberlândia 10000 and 3 commercial cultivars – cv. Grand Rapids, UFU-Biofort and Uberlândia 10000). The lines are part of the UFU biofortified lettuce breeding program and were assessed for the total green mass (g), the stem diameter (cm), commercial leaves count, plant diameter (cm), foliar temperature (°C), SPAD index, anthocyanin content (mg 100g-1 sample), and bolting (days after sowing). To estimate selection gains, 10 genotypes were selected using both direct and indirect selection, as well as the traditional index, the sum of ranks index, Willians base index, and the multiplicative index. Mulamba and Mock sum of ranks index and Subandi multiplicative index provided the biggest total gains for the traits assessed in biofortified leaf lettuce. Both indexes were similar in selecting genotypes.

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The so-called generalized integral transform technique (GITT) is employed in the hybrid numerical± analytical solution of two-dimensional fully-developed laminar¯ow of non-Newtonian power-law¯uids inside rectangular ducts. The characteristic of the automatic and straightforward global error control procedure inherent to this approach, permits the determination of fully converged benchmark results to assess the performance of purely numerical techniques. Therefore, numerical results for the product Fanning friction factor-generalized Reynolds number are computed for different values of powerlaw index and aspect ratio, which are compared with previously reported results in the literature, providing critical comparisons among them as well as illustrating the powerfulness of the integral transform approach. The resulting velocity pro®les computed by using this methodology are also compared with those calculated by approximated methods for power-law¯uids, within the range of governing parameters studied.