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Study of Some Nonlinear Fluid Flows Between Stretching Disks

Thesis Info

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Author

Khan, Nargis

Program

PhD

Institute

The Islamia University of Bahawalpur

City

Bahawalpur

Province

Punjab

Country

Pakistan

Thesis Completing Year

2016

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/10943/1/Nargis_Khan_Maths_HSR_2016_IUB_PRR.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727393081

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This dissertation emphasis on axi-symmetric flow of Newtonian fluid, rate type fluids and nanofluids between two infinite stretching disks. The modeling of said problems is done in cylindrical coordinates. Applied magnetic field, mixed convection, viscous dissipation, joule heating and heat source/sink are taken into account in various cases. Heat transfer, chemical and material composition analysis of flow between stretching disks have been analyzed under different boundary conditions; such as slip boundary conditions and convective boundary conditions. It is also important to mention that the second order slip and second order temperature jump is also studies on both disk surfaces with homogenous and heterogeneous reactions. The Brownian motion and thermophoresis effects are investigated in the presence of radiation effect for Maxwell and Oldroyd-B nanofluids. The mathematical modeling of problem statement results in partial differential equations, which further transformed to coupled nonlinear ordinary differential equations using similarity transformations. The reliability and flexibility of homotopy analysis method has encouraged us to find the solution of system of coupled nonlinear ordinary differential equations. The convergence of derived series solutions is ensured using ℏ-curves. The numerical values of skin friction, Nusselt number and Sherwood number are discussed through tables and graphs. The effects of other important parameters like Archimedes number, Eckert number, Prandtl number, Biot number, Schmidt number and Brownian motion parameters on velocity, pressure, temperature and concentration profiles are discussed and analyzed graphically
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ساتواں باب: فرقے

قدیم فرقے

باب ہفتم کے اہم نکات

  1. یہودی فرقوں کا تعارف و ابتدا۔
  2. یہودی فرقوں کے عقائد۔
  3. یہودی فرقوں کی کتب مقدسہ۔
  4. یہودی فرقوں کا تقابل۔
  5. یہودی فرقوں کا نظریہ اسرائیل۔
  6. یہودی اداروں کا تعارف۔
  7.  عصر حاضر میں یہودیت کا ارتقا۔
یہودیت میں تاریخ کو بنیادی حیثیت حاصل ہے تاہم جس بے کسی کی زندگی یہودیوں کا مقدر رہی ہے اس سے ان کی تاریخ کا ہر گوشہ متاثر ہوا ہے۔ یہودی فرقوں کی تاریخ کو بھی اس ضمن میں استثنا حاصل نہیں ہے۔ نیز فرقوں کی تقسیم کے بیان کرنے میں بھی یہودی مؤرخین منفرد مزاج کے حامل ہیں، مثلاً پرانے وقتوں میں بارہ یہودی قبائلشمالی اور جنوبی ریاستوں میں بٹ گئے تھے۔ شمالی ریاست میں بتوں کی عبادت کو رواج دیا جانے لگا تھا۔ اول سلاطین میں اس حوالے سے آیا ہے:

۔۔۔ یُربعام نے سو نے کے دو بچھڑے بنوائے۔ بادشاہ یربعام نے لوگوں سے کہا، 'تمہیں یروشلم کو عبادت کے لئے نہیں جانا چاہیے اے اسرائیلیو! یہی سب دیوتا ہیں جو تمہیں مصر سے باہر لائے۔ بادشاہ یُر بعام نے ایک سونے کا بچھڑا بیت ایل میں رکھا۔ اس نے دوسرا سونے کا بچھڑا شہر دان میں رکھا لیکن یہ گناہ عظیم تھا۔ بنی اسرائیلیوں نے بیت ایل اور دان کے شہروں میں بچھڑوں کی پرستش کر نے کے لئے سفر کیے لیکن یہ بہت بڑا گناہ تھا۔[1]

 شمالی ریاست نے نہ صرف ایک خدا پر یقین کے عقیدے کو بدل ڈالا اور دو بچھڑوں کو معبود بنا لیا۔ ان واقعات کو ایک نئے فرقے کی شروعات کے طور پر دیکھا جا سکتا ہے اس کے برعکس یہودی محققین نہ صرف ان عوامل کو بلکہ موسیؑ، داؤدؑ...

مسئلہ حجاب: فرانسیسی مسلمان خواتین اور اسلامی تعلیمات

Human history is replete with preposterous and unjustifiable incidents of unearned sufferings against the women. Sometimes they were maltreated and molested harshly and sometimes they were abused, persecuted bestially. Contrary to these incidents occasionally they were considered superior and super angelic but on the contrary Islam has bestowed a dignified status to them regarding their rights and responsibilities. In this regard a comprehensive manifestation has been introduced by the Islam and until this manifestation was being followed by the Muslims no single complain was lodged by any woman against the violation of her basic in the Islamic societies till the climax of Islamic regime. But today some European countries are holding discussions to impose illegal sanctions against the veil of women and girls. The parliament of France has approved a discriminatory law against veil of the Muslim women or girls. It is amazing that Christian nun is at her liberty to cover her head with scarf or not but if Muslim women consider themselves safe in veil they are contemptuously scorned with derision and disdained. In this article views of France and Islamic teachings have been brought under discussion.

Hamiltonian Properties of Directed Toeplitz Graphs

To determine whether or not a given graph has a hamiltonian cycle is much harder than deciding whether it is Eulerian, and no algorithmically useful characterization of hamiltonian graphs is known, although several necessary conditions and many suf- ficient conditions (see [6]) have been discovered. In fact, it is known that determining whether there are hamiltonian paths or cycles in arbitrary graphs is N P-complete. The interested reader is referred in particular to the surveys of Berge ([5], Chapter 10), Bondy and Murty ([10], Chapters 4 and 9), J. C. Bermond [6], Flandrin, Faudree and Ryj ́ a c ˇ stek [21] and R. Gould [27]. Hamiltonicity in special classes of graphs is a major area of graph theory and a lot of graph theorists have studied it. One special class of graphs whose hamiltonicity has been studied is that of Toeplitz graphs, introduced by van Dal et al. [13] in 1996. This study was continued by C. Heuberger [32] in 2002. The Toeplitz graphs investigated in [13] and [32] were all undirected. We intend to extend here this study to the directed case. A Toeplitz matrix, named after Otto Toeplitz, is a square matrix (n × n) which has constant values along all diagonals parallel to the main diagonal. Thus, Toeplitz matrices are defined by 2n − 1 numbers. Toeplitz matrices have uses in different areas in pure and applied mathematics, and also in computer science. For example, they are closely connected with Fourier series, they often appear when differential or inte- gral equations are discretized, they arise in physical data-processing applications, in viiviii the theories of orthogonal polynomials, stationary processes, and moment problems; see Heinig and Rost [31]. For other references on Toeplitz matrices see [26], [28] and A special case of a Toeplitz matrix is a circulant matrix, where each row is ro- tated one element to the right relative to the preceding row. Circulant matrices and their properties have been studied in [14] and [28]. In numerical analysis circulant matrices are important because they are diagonalized by a discrete Fourier trans- form, and hence linear equations that contain them may be quickly solved using a fast Fourier transform. These matrices are also very useful in digital image processing. A directed or undirected graph whose adjacency matrix is circulant is called cir- culant. Circulant graphs and their properties such as connectivity, hamiltonicity, bipartiteness, planarity and colourability have been studied by several authors (see [8], [11], [15], [25], [35], [38], [41] and [24]). In particular, the conjecture of Boesch and Tindell [8], that all undirected connected circulant graphs are hamiltonian, was proved by Burkard and Sandholzer [11]. A directed or undirected Toeplitz graph is defined by a Toeplitz adjacency matrix. The properties of Toeplitz graphs; such as bipartiteness, planarity and colourability, have been studied in [18], [19], [20]. Hamiltonian properties of undirected Toeplitz graphs have been studied in [13] and [32]. For arbitrary digraphs the hamiltonian path and cycle problems are also very dif- ficult and both are N P-complete (see, e.g. the book [22] by Garey and Johnson). It is worthwhile mentioning that the hamiltonian cycle and path problems are N P- complete even for some special classes of digraphs. Garey, Johnson and Tarjan shows [23] that the problem remains N P-complete even for planar 3-regular digraphs. Some powerful necessary conditions, due to Gutin and Yeo [10], are considered for a digraphix to be hamiltonian. For information on hamiltonian and traceable digraphs, see e.g. the survey [2] and [3] by Bang-Jensen and Gutin, [9] by Bondy, [29] by Gutin and [39] by Volkmann. In this thesis, we investigate the hamiltonicity of directed Toeplitz graphs. The main purpose of this thesis is to offer sufficient conditions for the existence of hamil- tonian paths and cycles in directed Toeplitz graphs, which we will discuss in Chapters 3 and 4. The main diagonal of an (n × n) Toeplitz adjacency matrix will be labeled 0 and it contains only zeros. The n − 1 distinct diagonals above the main diago- nal will be labeled 1, 2, . . . , n − 1 and those under the main diagonal will also be labeled 1, 2, . . . , n − 1. Let s 1 , s 2 , . . . , s k be the upper diagonals containing ones and t 1 , t 2 , . . . , t l be the lower diagonals containing ones, such that 0 < s 1 < s 2 < · · · < s k < n and 0 < t 1 < t 2 < · · · < t l < n. Then, the corresponding di- rected Toeplitz graph will be denoted by T n s 1 , s 2 , . . . , s k ; t 1 , t 2 , . . . , t l . That is, T n s 1 , s 2 , . . . , s k ; t 1 , t 2 , . . . , t l is the graph with vertex set 1, 2, . . . , n, in which the edge (i, j), 1 ≤ i < j ≤ n, occurs if and only if j − i = s p or i − j = t q for some p and q (1 ≤ p ≤ k, 1 ≤ q ≤ l). In Chapter 1 we describe some basic ideas, terminology and results about graphs and digraphs. Further we discuss adjacency matrices, Toeplitz matrices, which we will encounter in the following chapters. In Chapter 2 we discuss hamiltonian graphs and add a brief historical note. We then discuss undirected Toeplitz graph, and finally mention some known results on hamiltonicity of undirected Toeplitz graphs found by van Dal et al. [13] and C. Heuberger [32].x Since all graphs in the main part of the thesis (Chapters 3 and 4) will be directed, we shall omit mentioning it in these chapters. We shall consider here just graphs without loops, because loops play no role in hamiltonicity investigations. Thus, un- less otherwise mentioned, in Chapters 3 and 4, by a graph we always mean a finite simple digraph. In Chapter 3, for k = l = 1 we obtain a characterization of cycles among directed Toeplitz graphs, and another result similar to Theorem 10 in [13]. Directed Toeplitz graphs with s 1 = 1 (or t 1 = 1) are obviously traceable. If we ask moreover that s 2 = 2, we see that the hamiltonicity of T n 1, 2; t 1 depends upon the parity of t 1 and n. Further in the same Chapter, we require s 3 = 3 and succeed to prove the hamiltonicity of T n 1, 2, 3; t 1 for all t 1 and n. In Chapter 4 we present a few results on Toeplitz graphs with s 1 = t 1 = 1 and s 2 = 3. They will often depend upon the parity of n. Chapter 5 contains some concluding remarks.