wavelet collocation method for the numerical solution of boundary layer fluid flow problems

Haar wavelets System of coupled second-order ordinary differential equations Boundary layer flow Boundary-value problems (BVPs) Numerical method Collocation method Similarity transformations a b s t r a c t Based on Haar wavelets an efficient numerical method is proposed for the numerical solution of system of coupled Ordinary Differential Equations (ODEs) related to the natural convection boundary layer fluid flow problems with high Prandtl number (Pr). The numerical study of these flow models is necessary as the existing literature is more focused on the flow problems with small values of Pr. In this work, the problem of natural convection which consists of coupled nonlinear ODEs is solved simultaneously. The ODEs are obtained from the Navier Stokes equations through the similarity transformations. The effects of variation of Pr on heat transfer are investigated. Performance of the Haar Wavelets Collocation Method (HWCM) is compared with the finite difference method (FDM), RungeeKutta Method (RKM), homotopy analysis method (HAM) and exact solution for the last problem. More accurate solutions are obtained by wavelets decomposition in the form of a multi-resolution analysis of the function which represents solution of the given problems. Through this analysis the solution is found on the coarse grid points and then refined towards higher accuracy by increasing the level of the Haar wavelets. Neumann's boundary conditions which are problematic for most of the numerical methods are automatically coped with. A distinctive feature of the proposed method is its simple applicability for a variety of boundary conditions. Efficiency analysis of HWCM versus RKM is performed using Timing command in Mathematica software. A brief convergence analysis of the proposed method is given. Numerical tests are performed to test the applicability, efficiency and accuracy of the method

Starch-grafted polyacrylic acid copolymer with acrylamide: an advanced adsorbent for Victoria green B dye removal and environmental remediation

A new adsorbent, starch grafted polyacrylic acid copolymer with polyacrylamide (SG@AA-co-AM), was synthesized using free radical polymerization techniques. Proton nuclear magnetic resonance (1H NMR), Fourier-transform infrared spectrophotometry (FTIR), X-ray diffraction analysis (XRD), thermogravimetric analysis (TGA), scanning electron microscopy (SEM), particle size distribution (PSD), and fluorescent electron microscopy (FEM) were employed to elucidate the structure, crystalline nature, thermal stability, and surface morphology of SG@AA-co-AM. SG@AA-co-AM was utilized as an adsorbent for the removal of Victoria green B (VGB) dye from wastewater. SG@AA-co-AM exhibited a removal percentage (% R) of 97.6% towards VGB under optimized conditions: a contact time of 30 min, temperature of 25 °C, adsorbent dose of 20 mg, pH of 8, concentration of dye solution of 20 ppm, and volume of dye solution of 30 mL. The point of zero charge (PZC) for SG@AA-co-AM was determined to be 5.2. Nonlinear pseudo-second-order (PSO) and Langmuir adsorption isotherm models best fitted to the experimental data, with regression coefficients (R2) of 0.95 and 0.99, respectively. The results confirmed the chemi-sorption and monolayer adsorption of VGB onto SG@AA-co-AM. Thermodynamic studies revealed that the adsorption of VGB onto SG@AA-co-AM is endothermic and spontaneous. Furthermore, the regeneration experiment showed a decline of 3.9% after five cycles, confirming the economical and reusable nature of SG@AA-co-AM.<br/

Synergizing date palm seeds-derived oxidized activated carbon: Sustainable innovation for enhanced water retention, efficient wastewater treatment, and synthetic dye removal

Herein, this study explores the efficacy of oxidized activated carbon from date palm seeds (OACDS) as a multifaceted solution for sustainable wastewater treatment and water retention improvement. Through oxidation synthesis, OACDS demonstrates exceptional capability in adsorbing contaminants from water sources, offering a promising eco-friendly option for environmental remediation. Batch adsorption experiments highlight the rapid and efficient removal of Rh-6 G dyes, achieving an impressive 88.06% adsorption rate under optimized conditions of pH 6 and a 70-minute contact time. Moreover, OACDS exhibits remarkable adsorption capacities, reaching 91.3% at a 60 mg adsorbent dose and 94.23% at 55°C. Kinetic and adsorption analyses align with PSO and Langmuir models, indicating chemisorption and mono-layered adsorption of Rh-6 G on OACDS. Thermodynamic evaluation suggests the process's spontaneity and endothermic nature. The regeneration experiment revealed a notable 10.9% decrease in the adsorption capacity of OACDS towards Rh-6 G after five cycles. These findings substantiate the reusability and cost-effectiveness of OACDS, underscoring its potential economic advantages. Beyond wastewater treatment, OACDS showcases notable water retention properties, promising for agricultural applications. Its integration into clay and sandy soil enhances water retention capacities, with the clay soil-OACDS mixture displaying a peak of 16.8 mL. The material's rough and porous surface positively impacts water retention in crops, benefiting agricultural yields. Comprehensive characterization analyses using SEM, FTIR, and XRD support OACDS's effectiveness in adsorption, highlighting its amorphous structure and suitability for environmental applications. This study positions OACDS as a comprehensive solution addressing wastewater treatment and water conservation challenges, encouraging further exploration of agricultural byproducts for sustainable environmental solutions.<br/