The concept of super line graph was introduced in the year 1995 by Bagga, Beineke and Varma. Given a graph with at least r edges, the super line graph of index r, Lr(G), has as its vertices the sets of r-edges of G, with two adjacent if there is an edge in one set adjacent to an edge in the other set. The line completion number lc(G) of a graph G is the least positive integer r for which Lr(G) is a complete graph. In this paper, we  nd the line completion number of Grid graph Pn  Pm for various cases of n and m.

This article addresses the abilities and limitations of the Lattice Boltzmann (LB) method in solving advection-dominated mass transport problems. Several schemes of the LB method, including D2Q4, D2Q5, and D2Q9, were assessed in the simulation of two-dimensional advection-dispersion equations. The concept of Single Relaxation Time (SRT) and Multiple Relaxation Time (MRT) in addition to linear and quadratic Equilibrium Distribution Functions (EDF) were taken into account. The results of LB models were compared to the well-known Finite Difference (FD) solutions, including Explicit Finite Difference (EFD) and Crank-Nicolson (CN) methods. All LB models are more accurate than the aforementioned FD schemes. The results also indicate the high potency of D2Q5 SRT and D2Q9 SRT in describing advection-controlled mass transfer problems. The numerical artificial oscillations are observed when the Grid Peclet number (GPN) is greater than 10, 25, 20, 25, and 10 regarding D2Q4 SRT, D2Q5 SRT, D2Q5 MRT, D2Q9 SRT and D2Q9 MRT, respectively, while the corresponding GPN values obtained for the EFD and CN methods were 2 and 5, respectively. Finally, a coupled system of groundwater and solute transport equations were solved satisfactorily using several LB models. Considering computational time, all LB models are much faster than CN method.

In this research, a low speed wind tunnel test has been carried out to investigate the effects of adding Grid winglet, its size and number of fingers on the flow pattern over the upper surface of a wing model and its aerodynamic coefficients. The investigation has been performed for different angles of attack ranging from -2 to 21 degrees and at a free stream velocity of 24.7 m/sec, corresponding to the Reynolds number of 101000. The model has a rectangular shape with an aspect ratio of 4.75. Wing section airfoil is NACA 23016. Flow visualization is performed using tuft over the upper surface of the wing and lift & drag coefficients are measured. The obtained results show that at low angles of attack (lower than 6 degree), adding the Grid winglet has no effect on the lift & drag coefficients. However, at higher angles of attack, its effect increases. Maximum amount of lift and minimum amount of drag is for the case that the Grid winglet has 4 fingers sorted from long to short.

A mathematical model was proposed to appropriately describe the fate of multicomponent substrates in porous media, especially soil. The model utilized appropriate biodegradation kinetic expressions that better describe the consumption or degradation rate of the substrate. The Equation, with the second and third type boundary conditions in nondimensionalized form, was solved using the Finite Volume method and simulated in the Matlab environment. An experiment, using a 5 cm (inside diameter) x 60 cm (height) glass column packed with severally autoclaved soil spiked with 2 % substrate (a mixture of hexadecane, heneicosane, 1-methylnaphthalene, 2-methylnaphthalene, and 1, 3-dimethylnapthalene) and a consortium of organisms (Providential rettgeri, Streptococcus salivarius, Trichoderma harzianum, Aspergillus flavipes, and Candida famata) was set up to validate the model. The result showed that the model describes the fate of each component within the multicomponent substrate. It also indicates that both Peclet and Thiele numbers affect the biodegradation of the substrate. It was observed that a small Peclet number should be allowed for effective biodegradation of the substrate. The model was validated with data obtained for an experiment where a mixture of hydrocarbons was degraded with a mixed culture of microorganisms. The results of the experiment were well described by the model indicating that the model can be used to predict the compositions of components of a mixture during biodegradation.

The present paper aims at investigating the applicability of hybrid cells in the series model for pollution transport inside the layered porous media. For this purpose, four layers of rock material have fallen inside the experimental flume, with eight sensors installed longitudinally inside the media to obtain experimental BTCs. In order to measure time parameters of the model, named, two different methods of LSCF and MM have been examined. The model's sensitivity as well as its temporal equations with different parameters have been assessed. Finally, results show that at the fixed time step, the model is more sensitive to parameter (advection zone time parameter) rather than residential time parameters, to which the moment relations are more sensitive. A detailed computation of the related transport parameters has been operated and the Peclet number, crossing velocity, dispersion coefficient, time to the max, and maximum concentration have been calculated. Eventually, the model's applicability for large-scale porous media has been proven with only one unit of the cells.