The paper presents a semi-implicit time-stepping Taylor-Galerkin pressure correction primitive variable finite element algorithm to simulate fluid flow for two dimensional planar combined mixing and separating geometry. Two cases; one with reversed channel flows interacting through a gap in the common separating walls filled with Newtonian fluids in both arms of the channels and other with unidirectional flows were modeled in order to examine the performance of the scheme. Steady solutions were obtained using unsteady finite element scheme. The influence of increasing inertia on variation in flow directions and varying flow rate configurations in both channel arms are studied in detail. The scheme is found to be fast, robust and stable for varying Reynolds number.

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Various flow bifurcations are investigated for two dimensional combined mixing and separating geometry. These consist of two reversed channel flows interacting through a gap in the common separating wall filled with porous media of Newtonian fluids and other with unidirectional fluid flows. The Steady solutions are obtained through an unsteady finite element approach that employs a Taylor-Galerkin/pressure-correction scheme. The influence of increasing inertia on flow rates are all studied. Close agreement is attained with numerical data in the porous channels for Newtonian fluids.

Non-linear Eq.s occur as a sub-problem in a wide variety of engineering and scientific domains. To deal with the complexity of Non-linear Eq.s, it is often required to use numerical procedures, which are the most suitable method to employ in certain circumstances. Many classic iterative approaches have been regularly employed for various situations; nevertheless, the convergence rate of those methods is low. In many cases, an iterative approach with a faster convergence rate is needed. This is something that classical methods like the Newton-Raphson Method (NRM) cannot provide. As part of this investigation, a modification to the NRM has been suggested to speed up convergence rates and reduce computational time. Ultimately, this research aims to improve the NRM, resulting in a Modified Iterative Method (MIM). The proposed method was thoroughly examined. According to the research, the convergence rate is higher than that of NRM. The proposed method delivers more accurate results while reducing computational time and requiring fewer iterations than earlier methods. The numerical findings confirm that the promised performance is correct. The results include the number of iterations, residuals, and computing time. This innovative technique, which is appropriate to any Non-linear equation, produces more accurate approximations with less iteration than conventional methods, and it is appropriate to any Non-linear equation.

A meteorological drought study is performed using monthly time scale data from three separate locations in southern Sindh, Pakistan. Rainfall and temperature have been used to identify the drought. These data were transformed into drought indices known as the standardized precipitation evapotranspiration index (SPEI) and standardized precipitation index (SPI), which were derived using (the Hargreaves equation). In this study, two indices are compared for three separate meteorological stations Chhor, Mithi, and Badin where most socioeconomic livelihoods depend heavily on water. The SPEI is produced through a simple water balance combining precipitation and temperature, in distinction to the SPI, it just considers precipitation. In conclusion, our study showed that both indices were capable of detecting droughts that fluctuated in time across the reference period of 2004–2021. SPI and SPEI's direction of change was similar, however the impact on the drought condition varied. SPEI discovered more droughts with longer durations and greater with 13 moderate droughts at SPEI-3 for Chhor and Badin Station while Mithi indicated 8 moderate droughts during 2004-2021 and SPI-3 indicated 4 moderates for Chhor, Mithi and Badin indicated 6 moderate drought. Conversely, SPEI discovered more moderate-level droughts than SPI, however they were of shorter length and less frequent occurrence than the severe to moderate droughts. The findings imply that drought characteristics are significantly influenced by temperature variability.