علوم حدیث پر برصغیر کی اردو کتب کا تعارفی و تجزیاتی مطالعہ

The Hadith were account usually brief of the words and actions of the beloved Prophet, [May Allah Bless him and grant him peace]. As Such, they were subjected to intense security by generations of Muslim Scholars. The Principles to authenticate and document this literature along with it peculiar terminology called Usool-e-Hadith. This unique Science is a historic achievement of early Muslim scholars, having and history of centuries contributing to its evolution. In the opinion of the Late 'Allama Rashid Rida of Egypt, "The Indian Muslims are playing the leading role in the diffusion and dissemination of Hadith learning in the world to-day. As a matter of fact, according to him, but for the painstaking labour of the Indian Muslims towards the cultivation of the science of al-Hadith, it would have well nigh died down." A number of Scholars in the Indo-Pak sub-continent have produced an extensive work on the subject in Urdu language as well, during last century. My Research work focuses on analytical study of the same books on Usool-e-Hadith.

An Artificial Compressibility Formulation for Phase-Field Model and its Application to Two-Phase Flows

An Artificial Compressibility Formulation for Phase-field Model and its Application to Two-phase Flows With the advent of high-speed computers, computational methods have become a very useful tool for solving problems in science and engineering along with analytical and ex perimental approaches. The starting point of computational methods is the mathematical model, the form, and origin of which depends on the particular field of study. Many im portant physical processes in nature are governed by partial differential equations (PDE’s). For this reason, it is important to understand the physical behavior of the PDE’s. Also, the knowledge of mathematical character, properties, and solution of the equations are re quired. A proper mathematical model and a good numerical method can provide realistic answerstocomplexphysicalphenomenaforwhichanalyticalsolutionmaynotbeavailable in a finite time. Two-phase flow occurs in nature and many areas of physical and biological sciences like oil recovery processes (water and oil), blood flow (plasma and blood cell), mud-flow (wa ter and suspended particles), atmosphere and ocean system (air and water), cloud and fog (water and air). In dealing with the two-phase flow, an important consideration is how to modelthemovinginterface/surface. Nevertheless,thePDEsdescribingthetwo-phaseflow are highly nonlinear and stiff so it is difficult to solve them analytically and a numerical simulation is an alternate option. The numerical solution obtained may only approximate that of original problem or at least within some required tolerance of the true solution. However,accuratesimulationofmovinginterfacepresentsaproblemofconsiderablediffi cultyandisthereforeverychallengingfordevelopingnumericalmethodsusinglarge-scale computation. Also, the boundary conditions need a particular treatment near the moving interface during numerical simulations. Shocks in the compressible flows, vortex sheets in inviscid flows, and boundaries between immiscible fluids are some of the very known examples. Mathematicalmodelsadoptedinbothanalyticalandnumericalstudiesforavarietyoftwo phase flow with moving interface are classified into two types, i.e., sharp interface models and diffuse interface models. Sharp interface models like level set method assume that the interface has zero thickness. However, in the phase transition, the existence of a transition region introduces the idea of the diffuse interface that allows the interface to have finite thickness. Onetypeofdiffuse-interfacemodelsofparticularinterestisaphase-fieldmodel by an introduction of a phase-field variable that represents the interface. In this approach, the phase-field variable is a continuous function in space and time. Phase-field models are numerically attractive for not tracking the interface explicitly but can be obtained as a part of the solution processes. InordertosolveunsteadyincompressibleNavier-Stokesequations,severalnumericalmeth ods are developed, including the artificial compressibility method. In this research work, a numerical algorithm based on artificial compressibility formulation of the phase-field modelisusedforsimulatingtwo-phaseflowsproblems. Thecoupledhydrodynamicalsys tem consists of the incompressible Navier-Stokes equations and volume preserving Allen Cahntypephase-fieldequationarerecastintoconservativeformswithsourceterms,which are suited for implementing high-order and high-resolution discretization schemes. The Boussinesq approximation is used for buoyancy effects in the flow with moderately differ ent densities. The performance of the numerical method is demonstrated by its application to some benchmark two-phase flow problems.