رسم عثمانی میں غیر موجود اور صحیح سند سے ثابت قراءات کا حکم

Some of the variant readings of the Holy Quran having a sound chain of narration are not included in the Uthmanic Maṣāḥif (Codices). Hence, following three probabilities can be deduced about these readings; First: Those were abrogated in ʿArḍah Akhīrah (the last revision). Second: Those might be among those explanatory notes of the Holy Text by Prophet Muhammad (SWA) that were erroneously written by a few companions within the actual text of Quran considering them a part of the Quran. Third: Those may belong to such Aḥruf (readings) that were authentically transmitted from the Messenger of Allah (SWA) but, they were not mentioned in the orthography of the Uthmanic Maṣāḥif by the compilers due to any possible reason. To us, if we come across any authentically narrated recitation of the senior Qurrāʾ companions that seems contrary to the orthography of the Uthmanic Maṣāḥif and there is no proof of their being from the second category, then, it is better to consider them from the third category instead of the first one.

Numerical Solutions of Unsteady Models of Non-Newtonian Fluids

The aim of this thesis is to examine the unsteady effects in the two or three dimensional boundary layer flow of non-Newtonian fluids along with the presence of different physical factors. Heat and mass transfer is involved with first order chemical reaction. In some cases, soret and dufour effects are also investigated. The flow models are explained in terms of continuity, momentum, energy, concentration equations which are converted into dimensionless form by using similarity transformation. Solutions are then obtained by an analytic-numeric technique Homotopy analysis method (HAM) and then comparison is shown by using Runge-Kutta-Fehlberg algorithm with shooting method. Convergence of solution and graphs of error analysis are also included to check the authenticity. Study of motion of the fluid’s behavior with heat and diffusion processes are depicted vividly through graphs and tables against different pertinent parameters. It is observed that magnetic force always slows down the velocity rate and presence of Casson and Maxwell parameter enhances the thermal and solute boundary layer as compared to the steady model. In Chapter 2, an endeavor is applied to observe the approximate solutions of unsteady two-dimensional boundary layer flow with heat and mass transfer behavior of Casson fluid past a stretching sheet in presence of wall mass transfer and 1st order chemical reaction. After adopting the similarity transformation; Homotopy analysis method (HAM) is applied to solve these ordinary differential equations. Numerical work is done carefully with a wellknown software MATHEMATICA for the examination of non-dimensional velocity, temperature, and concentration profiles, and then results are presented graphically. It is observed that involvement of time dependent factor reduces the thermal but enhances the solute boundary layer as compared to the steady model. v The Chapter 3 is useful to investigate the analytic-numeric solutions of an unsteady 3-D MHD Casson fluid flow which is embedded in a porous medium past a stretching sheet under the heat transfer influence. The similarity transformation is adopted to convert the partial differential equations of current model into its dimensionless form. Then, it is solved by employing an algorithm of a powerful technique Homotopy analysis method (HAM). After that, solutions are compared by invoking Runge-Kutta based shooting method. The theme of Chapter 4 is to study the first-order chemically reacting Maxwell fluid past a stretching sheet embedded in a porous medium with the discussion of wall mass transfer and Soret, Dufour effects. HAM is invoked on the dimensionless form of flow model for the series solution; where numerical simulation is carried out by a powerful software MATHEMATICA. Furthermore, the impact of disparate physical parameters on the flow, temperature and concentration profiles are shown vividly. Errors graphs are also there to show the authenticity of results. At the end, in Chapter 5, a precise note or conclusion related to whole of my thesis is provided.