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Convection-Diffusion Problems occur very frequently in applied sciences and engineering. In this paper, the cru x of research articles published by numerous researchers during 2007-2011 in referred journals has been presented and this leads to conclusions and recommendations about what methods to use on Convection-Diffusion Problems. It is found that engineers and scientists are using finite element method, finite volu me method, finite volu me element method etc. in flu id mechanics. Here we discuss real life problems of fluid engineering solved by various numerical methods .which is very useful for finding solution of those type of governing equation, whose analytical solution are not easily found.

The fuzzy variable fractional differential equations (FVFDEs) play a very important role in mathematical modeling of COVID-19. The scientists are studying and developing several aspects of these COVID-19 models. The existence and uniqueness of the solution, stability analysis are the most common and important study aspects. There is no study in the literature to establish the existence, uniqueness, and UH stability for fuzzy variable fractional (FVF) order COVID-19 models. Due to high demand of this study, we investigate results for the existence, uniqueness, and UH stability for the considered COVID-19 model based on FVFDEs using a fixed point theory approach with the singular operator. Additionally, discuss the maximal/minimal solution for the FVFDE of the COVID-19 model.

This article introduces the computational analytical approach to solve the m-dimensional space-time variable Caputo fractional order advection–dispersion equation with the Dirichlet boundary using the two-step Adomian decomposition method and obtain the exact solution in just one iteration. Moreover, with the help of fixed point theory, we study the existence and uniqueness conditions for the positive solution and prove some new results. Also, obtain the Ulam–Hyers stabilities for the proposed problem. Two generalized examples are considered to show the method’s applicability and compared with other existing numerical methods. The present method performs exceptionally well in terms of efficiency and simplicity. Further, we solved both examples using the two most well-known numerical methods and compared them with the TSADM solution.

One of the traditional methods used to improve the efficiency of a gas turbine is to increases the inlet temperature; thereby increasing the power output and in turn, the efficiency. The internal cooling passages of blades are roughened by artificial roughness to improve the cooling performance. The present study investigates the convective heat transfer and friction factor (pressure drop) characteristics of a rib-roughened square duct. The test section of the duct is roughened on its top and bottom wall with V and ᴧshaped square ribs. In the study, the Reynolds number (Re) varied from 10,000 to 40,000, the relative roughness height (e/Dh) is 0.060, relative roughness pitch (p/e) is 10 and rib attack angle (α) is taken as 45 0 and 60 0. The comparison of heat transfer and pressure drop for V and ᴧshaped ribs is presented in the form of Nusselt Number and friction factor. The results show that the Nusselt number enhancement decreases when the Reynolds number increases. The friction factor ratio is found to increases as Reynolds number increases. The thermal performance decreases when the Reynolds number increases. It seems that the ribs disturb the main flow resulting in the recirculation and secondary flow near the ribbed wall. The heat transfer coefficient and friction factor for all the cases are higher than that of the smooth duct. The V-shaped ribs have higher Nusselt number and friction factor than the ᴧ-shaped ribs. It is also concluded that the V-shaped ribs perform better than the ᴧshaped ribs. Ribs with α = 60 0 produces higher heat transfer and friction factor than α = 45 0. The results obtained in the study will help the turbine blade designer to design the effective cooling methods to cool the blades.