This work presents new results regarding the behavior of some non-Newtonian fluids into different circumstances. After some preliminaries regarding constitutive equations, motion equations and integral transforms, new exact solutions for the ve- locity field and the shear stress corresponding to some flows with technical relevance have been established for ordinary second grade, Oldroyd-B fluids and generalized Maxwell fluids. Just as in the case of Navier-Stokes fluids, it is necessary to develop a large class of exact and approximate solutions, they serving as tests to verify nu- merical schemes that are developed to study complex unsteady flow problems. In chapter 2, by means of the Laplace transform, there are established new exact solutions corresponding to the first problem of Stokes for Oldroyd-B fluids. These solutions, in accordance with the previous results obtained using Fourier sine trans- form, can be easily specialized to give similar solutions for Maxwell fluids. The main aim of chapter 3 was to solve a very important problem, namely to determine the re- quired time to reach the steady-state for the second problem of Stokes corresponding to second grade fluids. In addition to solve this problem we also found new exact so- lutions for this problem. Our solutions, unlike the solutions obtained by Erdogan for Newtonian fluids, are written as a sum between steady state and transient solutions. Chapter 4 contains exact solutions for the unsteady flow of an Oldroyd-B fluid between two side walls perpendicular to a plate. In the absence of side walls the obtained solutions tend to the similar solutions for the flow over a flat plate (the first problem of Stokes). The influence of the pertinent parameters on the velocity of the fluid at the middle of the channel as well as on the shear stress on the bottom is underlined by graphical illustrations. In chapter 5, by means of Laplace and Hankel transforms, we obtained the solutions for unsteady flow of a generalized Maxwell fluid between two circular cylinders. These solutions, presented as a sum of the Newtonian solutions and the non-Newtonian contributions, can be easily specialized to give the similar solutions for ordinary Maxwell fluids. Finally, the influence of the pertinent parameters on the velocity of the fluid are also underlined by graphical illustrations.
اثر صہبائی (۱۹۰۱۔۱۹۶۱ء) کا اصل نام خواجہ عبد السمیع پال تھا۔ اثر ؔسیالکوٹ میں پیدا ہوئے۔ اثرؔ کے بزرگوں نے کشمیر سے ہجرت کی تھی اور سیالکوٹ میں آباد ہوئے تھے۔ آپ نے گورنمنٹ کالج لاہور سے ایم۔ اے فلسفہ اور ایل ایل بی کیا۔ ۱۹۳۱ء میں ان کی رفیقہ حیات ان سے جدا ہو گئیں تو افسردگی ‘ تاریکی اور مایوسی کے بادل ان کی زندگی پر چھا گئے۔ ۱۹۳۴ء میں آپ اس غم و اندوہ کییورش سے گھبرا کر سری نگر کشمیر چلے گئے۔ کشمیر میں ان دنوں ادبی مجلسیں اور ادبی نشستیں ہو رہی تھیں جن میں ڈاکٹر عبد الحکیم‘ نواب جعفر خان اثر لکھنوی‘ ڈاکٹر تاثیر اور پنڈت برج موہن دتاتر یہ کیفی دہلوی جیسے شعراء و ادبا شرکت کرتے تھے۔ اثر ان ادبی محفلوں کے روح رواں ہوتے تھے۔ آپ نے کشمیر ہائی کورٹ میں قائد اعظم کے ساتھ جونیئر وکیل کی حیثیت سے بھی کام کیا۔ قائد اعظم نے مقدمہ جیتنے کے بعد صہبائی کی محنت کو سراہا۔(۱) اثرؔ صہبائی کی پہلی تصنیف ’’جامِ صہبائی‘‘ ہے۔ قطعات و رباعیات پر مشتمل یہ شعری مجموعہ ۱۹۲۸ء میں دارالتالیف بیڈن روڈ لاہور سے طبع ہوا۔ ’’خمستان‘‘ اثر کا دوسرا مجموعہ کلام ہے جو غزلوں‘ نظموں‘ قطعات و رباعیات اور متفرق اشعار پر مشتمل ہے۔ اس کا پہلا ایڈیشن۱۹۳۳ء میں آزاد بک ڈپو سیالکوٹ سے شائع ہوا۔ اثر ؔکا تیسرا شعری مجموعہ ’’جامِ طہور‘‘ ۱۹۳۷ء کو تاج کمپنی لمیٹڈ لاہور نے طبع کیا۔ اس مجموعے میں رباعیات اور قطعات ہیں۔ ’’راحت کدہ‘‘ اثر ؔکا چوتھا شعری مجموعہ ہے جو ۱۹۴۲ء میں تاج کمپنی لمیٹڈ لاہور کے زیر اہتمام طبع ہو کر شائع ہوا۔’’ راحت کدہ ‘‘حضرت اثر صہبائی کے اس کلام پر مشتمل ہے جو انہوں نے اپنی جواں مرگ رفیقہ حیات راحت کی موت سے متاثر ہو کر...
Akbar’s Dream [1892] is among the last poems of Alfred, Lord Tennyson – Poet-Laureate of Britain. It was composed when the British Empire was at its apex of territorial expansion and Queen Victoria was the Empress of India. It is reflective of the position of Tennyson as PoetLaureate – whose ‘official’ task was to celebrate the achievements and other notable events of Britain. Tennyson was an ultra-conservative person who believed that Britain was doing a favour to the peoples they had conquered and subjugated. This was intended for their benefit so as to advance them in the scale of civilization.
The aim of this research work was to investigate and increase the understanding of three dimensional flow behavior under changing conditions of flow and sediments. The goal was achieved by using a 3D Computational Fluid Dynamics model FLUENT. The geometry selected for this study is that of a meandering compound channel. The parameters which were changed in this research work include the width of floodplain along with main channel width, overbank flow depth, incoming discharge, bed slope of the main channel and floodplains and roughness of the channel and floodplains. The research work was conducted by using different turbulence models. Among them are standard k - e , Re-normalization Group (RNG) theory based k - e and k - w turbulence models. These three models were applied for one case. Once the capability of k - e model was established after comparison with results of other models, then only k - e was used in the remaining part of the work. The data obtained from a physical model study in the laboratory was used for the validation of simulated results. The validation was done through a comparison of observed and calculated values of depth averaged velocities, total discharge at a section & point velocities. After validation, the CFD model was used to predict and investigate those aspects of flow which were difficult to get through experiments and for which experimentation was not conducted in the laboratory. These aspects include turbulence intensity, turbulent kinetic energy, pressure distribution on different cross sections of the channel and surface velocity magnitudes. As far as the sediments study is concerned, the Lagrangian particle tracking technique was used to predict the changes in bed levels at apex of the meander wavelength. Sediment particles with different diameters were used for this purpose. A total of eight flow cases were considered for the study of flow field. Each case varies from the other on the basis of (i) geometry or (ii) bed slope or (iii) depth of overbank flow or a combination of these. This work also proved the capability of standard k - e turbulence closure models in prediction of 3D overbank flows. Keywords: Sediment transport, CFD modeling, Flood flows, Numerical techniques, Turbulent kinetic energy.