حالات و زمانہ کی رعایت سے متعلق جمہور فقہاء کرام کی تصریحات (عصر حاضر کے تناظر میں ایک مطالعہ)

Human life is the subjective of change with the changing circumstances. If we look at the condition of our late grandfather or great – grandfather and look at the contemporary situation; it will not be difficult to understand this change. The rules about some important issues had been changed with the passage of time. In the initial descendants of Adam the marriage of brothers and sisters was allowed, after the flood of Noah, due to the scarcity of animals, the meat of all kinds of animals was allowed, two sisters were married to Hazrat Yaqub (A.S) at the same time, but the rules had been reversed afterwards. Even in the time of the Holy Prophet and the rightly guided caliphs, such changes took place and they were associated with the changing conditions. Initially it was banned to visit the graves and the sacrificial meet was not allowed to keep for more than three days and it was mandatory upon the women to attend the congregational prayer but afterwards the permissions were granted contrarily. Fortunately, we have a body of jurisprudence in the light of which not only contemporary issues can be solved but also guidance can be provided for the future. Our leading Jurists, who are followed, have done a remarkable job in this regard. The following article reviews the teachings of the three jurists, namely Imam Shafi’i, Imam Malik, and Imam Ahmad ibn Hanbal and teachings of their glean jurists. It will make clear that under changing circumstances these jurists seemed to agree on change. Therefore, by using the efforts of these jurists, we can try to resolve the problems of our contemporary changes and requirements throughout the Muslim World.

Flow and Heat Transfer over Stretching and Shrinking Surfaces

Flow and Heat Transfer Over Stretching and Shrinking Surfaces Finding the numerical solutions of ordinary and partial differential equations of nonlinear nature has become somewhat possible, during the last few decades, due to the evolution of efficient computing. However, the governing equations for fluid flow are difficult to address in terms of finding analytical solutions. This difficulty lies in the highly nonlinear nature of these equations. Thus, it has always been a challenging task for mathematicians and engineers to find possible exact/approximate analytical and numerical solutions of these equations. The analytical results have great advantage in the sense that; it helps to make comparison with exact numerical solution ensuring the reliability of the two results and also helps to explain the underlying physics of fluid. The analytical solutions are further useful to develop an insight for the development of new analytical techniques and for the modeling of new exciting fluid flow problems in both Newtonian and non-Newtonian fluids. There has been a continuously increasing interest of the researchers to investigate the boundary layer fluid flow problems over a stretching/shrinking surface. It is now known that surface shear stress and heat transfer rate for both viscous and non-Newtonian fluid are different. These stretching and shrinking velocities can be of various types such as liner, power law and exponential. Thus our main objective in this thesis is to analyze some boundary layer flow problems due to stretching/shrinking sheet with different types of velocities analytically. Both transient and steady forced and mixed convection flows are considered. The present thesis is mainly structured in two parts. Chapters 3 to 5 consist of transient and steady mixed convection boundary layer flow of Newtonian and some classes of non-Newtonian fluids with linear stretching and shrinking cases. Chapters 6 to 8 present the investigation of exponential stretching case. The chapters of the thesis are arranged in the following fashion. Chapter 1 dealt with the previous literature related to boundary layer stretched flows of viscous and non- Newtonian fluids. Chapter 2 includes the basic equations of fluid flow and heat transfer. Definitions of dimensionless physical parameters are also presented here. Chapter 3 explores unsteady mixed convection flow of a viscous fluid saturating porous medium adjacent ixto a heated/cooled semi-infinite stretching vertical sheet. Analysis is presented in the presence of a heat source. The unsteadiness in the flow is caused by continuous stretching of the sheet and continuous increase in the surface temperature. Both analytical and numerical solutions of the problem are given. The effects of emerging parameters on field quantities are examined and discussed. The magnetohydrodynamic boundary layer flow of Casson fluid over a shrinking sheet with heat transfer is investigated in Chapter 4. Interesting solution behavior is observed with multiple solution branches for a certain range of magnetic field parameter. Laminar two- dimensional unsteady flow and heat transfer of an upper convected, an incompressible Maxwell fluid saturates the porous medium past a continuous stretching sheet is studied in Chapter 5. The velocity and temperature distributions are assumed to vary according to a power-law form. The governing boundary layer equations are reduced to local non-similarity equations. The resulting equations are solved analytically using perturbation method. Steady state solutions of the governing equations are obtained using the implicit finite difference method and by local non- similarity method. A good agreement of the results computed by different methods has been observed. Chapter 6 addresses the steady mixed convection boundary layer flow near a two-dimensional stagnation- point of a viscous fluid towards a vertical stretching sheet. Both cases of assisting and opposing flows are considered. The governing nonlinear boundary layer equations are transformed into ordinary differential equations by similarity transformation. Implicit finite difference scheme is implemented for numerical simulation. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. Chapter 7 examines the boundary layer flow of a viscous fluid. The flow is due to exponentially stretching of a surface. Unsteady mixed convection flow in the region of stagnation point is considered. The resulting system of nonlinear partial differential equations is reduced to local non-similar boundary layer equations using a new similarity transformation. A comparison of the perturbation solutions for different time scales is made with the solution obtained for all time through the implicit finite difference scheme (Keller-box method). Attention is focused to investigate the effect of emerging parameters on the flow quantities. Chapter 8 aims to investigate the boundary layer flow of nanofluids. The flow here is induced by an exponentially stretching surface with constant temperature. The mathematical formulation xof this problem involves the effects of Brownian motion and thermophoresis. Numerical solution is presented by two independent methods namely nonlinear shooting method, and finite difference method. The effects of embedded parameters on the flow fields are investigated. Chapter 9 summaries the research material presented in this thesis.