Yerçekimi etkili newtonyen olmayan düşen film akışı
Gravity-driven non-newtonian falling film flow
Yazar: YUSUF YEĞİNER
Danışman: PROF. DR. İBRAHİM OZKOL
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Uçak ve Uzay Mühendisliği Ana Bilim Dalı / Uçak ve Uzay Mühendisliği Bilim Dalı
Derece: Yüksek Lisans, 2014
Konu: Matematik = Mathematics ; Mühendislik Bilimleri = Engineering Sciences ; Uçak Mühendisliği = Aircraft Engineering
Dizin: Diferensiyel denklemler = Differential equations ; Newton olmayan = Non-newtonyen

Özet:
Akış temelde komşu akışkan parçacıklarının hareketi ile birbirlerinden uzağa ve yakına giderek uzama, birbiri üzerinden ve yakınından geçerek kesme, şeklinde iki türde gerçekleşir. Akış ve bu davranışı sergileyen akışkanı irdeleyen akışkanlar mekaniği, akışkan statiğini ve dinamiğini bütünlemesine inceleyerek hava hareketlerinin sebep olduğu iklimsel durumları, ulaşım araçlarının tasarımını, özel olarak tasarlanan yapıları, hidrolik özellikte makineleri, canlılara ait çoğu sayılamayacak sıvı ve gazlar ile ilgili mühendislik uygulamalarını kapsar. Akışkanlar mekaniği açısından bütün maddeler katı ve akışkan olmaları bakımından iki halde bulunmaktadırlar. Teknik açıdan bakıldığında bu iki hal arasındaki farklılık kesme gerilmesi veya teğetsel gerilme karşısında sergiledikleri tepkilerle belirlenebilmektedir. Katıya uygulanmakta olan kayma gerilmesi düşünüldüğünde bir nebze şekil değişimi ile karşılaşılır. Oysa akışkana bir gerilme uygulandığı takdirde harekete geçme ve şekil değişimi gözlenir. Newtonyen akışkanlar viskoziteyi oluşturan kesme gerilimi ile kesme hızı oranının doğrusal olduğu, Newtonyen olmayan akışkanlar ise sıfır ya da doğrusal olmadığı akışkan türüdür. Newtonyen olmayan durumda viskozite belirli bir sıcaklık ve basınç değerinde sabit olmazken akışkanın kesme hızı, akış geometrisi ve akışkan elemanının kinematik geçmişine bağlı olur. Maddenin akış özelliklerini inceleyen Reoloji, Newtonyen olmayan akışkanları sınıflandırmıştır. Newtonyen olmayan akışkanların davranışlarını ifade etmek üzere birçok model oluşturulmuştur. Bu modellerden biri olan ve birçok çalışmada kullanılan Newtonyen olmayan ve viskoziteyi hız değişiminin üssel bir ifadesi olarak ele alan Power-law modelidir. Power-law akış modelinde Newtonyen olmayan problemler içinde düşen film akışıyla alakalı çalışmalar da mevcuttur. İlk çalışmalar daha çok deneysel olarak ilerlerken, düşen film akış hidrodinamiği ile ilgili Power-law akış modelinde çalışmalar yapmıştır. Power-law modeline uyan tipte film akışlarıyla alakalı yer çekimi etkisindeki durumlar için integral metodu ve benzerlik çözümleri kullanılarak yapılan çalışmalar da mevcuttur. 
Bu çalışmada Power-law modeline uygun Newtonyen olmayan film tipi akış için süreklilik, momentum, enerji denklemleri ile sınır koşulları verilmiştir. Yeni tanımlanan benzerlik yaklaşımıyla denklemler boyutsuz hale getirilmiştir. Az grid ile kısa zamanda yeterli hassasiyette doğru sonuç verebilen yöntemler den biri olan DQ (Differential Quadrature) ile doğrusal olan ya da olmayan adi veya kısmi diferansiyel denklemlerin cebirsel denklemlere dönüştürülmesiyle daha basit şekilde çözüm arayan DT (Differential Transform) yöntemine çalışmada yer verilmiştir. Sınır şartlarına uygun olarak çözülen denklemler kaynak çalışma ile kıyaslanmış ve sonuçlar yorumlanmıştır.

Abstract:
Basically, the fluid particles move away from each other or come closer to each other. It is called elongating. Additionally they can move above the other or pass by each other so shear occurs. The study of fluids either in motion (fluid dynamics) or at rest (fluid statics) defines fluid mechanics. Both liquid and gas are capable of flowing freely so they are fluids because they can be made to move or flow. 
Lots of different fields can be thought which are releated to fluid mechanics such as blood flow, swimming, pumps, fans, turbines, airplanes, ships, rivers, windmills, pipes, engines, filters, jets etc. Almost everything on this planet is a fluid or moves near or within a fluid. 
 All matter have two states from the view of fluid mechanics as solid and fluid. The obvious difference between solid and fluid lies with their reaction to an applied tangential or shear stress. While any shear stress, no matter how small, applied to fluid a motion happens, a solid resists by a static deflection under the effect of shear stress. The fluid deformation and motion do not stop until the stress is stop. 
Viscous-flow theory was unexploited, since Newtonian viscous terms had been added by Navier (1785–1836) and Stokes (1819–1903) to the equations of motion successfully. For arbitrary flows the resulting Navier-Stokes equations were too difficult to analyze. In 1904, a very important paper was written by a German engineer, Ludwig Prandtl (1875–1953). He stated that fluid flows which have small viscosity can be divided into a thin viscous or boundary layer, near solid surfaces and interfaces, closed to the nearly inviscid outer layer, so the Euler and Bernoulli equations could be used. In modern flow analysis, boundary-layer theory, supplying better understanding about many flow phenomena, was a very important tool. Many different texts were written about boundary-layer theory. 
Rheology is the study of deformation and flow of fluids in response to stress. As mentioned before fluids include both gases and liquids. If the stress versus strain rate curve is linear and passes through the origin, then the fluid is Newtonian fluid. A non-Newtonian fluid is a fluid, not obeying Newton's law of viscosity, whose flow properties differ in any way from those of Newtonian fluids. Therefore, shear stress cannot be defined as a linear function of the shear rate, or shear stress versus shear rate does not pass through the origin. 
The apparent viscosity, defined as shear stress divided by shear rate, depends on flow geometry shear rate, kinematics of fluid element etc. It is not constant at a given temperature and pressure.
There has been an immense amount of study performed on Non-Newtonian power-law fluids. While the first studies were carried out experimentally, numerical calculation techniques for non-Newtonian fluid flow have been also studied. Hydrodynamics of falling film flow of power law fluids was reviewed. Similarity analysis and integral method approach have been applied in the theoretical studies of the hydrodynamics of gravity-driven power-law films. The accelerating film flow was divided into three regions as the boundary layer region, the fully viscous region, and the developed flow region by applying similarity transformations.
In this research, the objective is to gain a better understanding of the behavior of a gravity-driven non-Newtonian Power-law film flow on an inclined plane.  For this project, first, power law non-Newtonian fluids characteristics are introduced and then the equations of motion are driven. Moreover, by applying the new similarity approach, the equations are nondimensionalized. Then, the boundary conditions have been changed to solve the new equations. 
In order to solve these partial differential equations previous researchers have used numerical method, a standard shooting technique based on classical 4th-order Runge–Kutta integration, in combination with a Newton iteration procedure. Also they have written the partial equations as a system of first order differential equations and solved again adopting the shooting method means of fifth-order Runge–Kutta integration which utilize variable grid spacing . After that a Newton iteration procedure has been applied to satisfy the boundary conditions. In this research, to solve the equations of the problem two different methods have been investigated.
Differential quadrature method was developed to provide solutions to differential equations of any systems. Then, for boundary or initial value conditions for engineering problems DQM became an alternative approach to the standard methods like finite elements or finite difference. The main point in DQM is, the partial derivative of a function on a discrete point respect to a variable can be approximated as a weighted linear sum of unknown function values at specific region which contains all discrete points of this variable. The most important part is to define the weighted coefficients. 
The method varies according to the function which is chosen for calculation of weight coefficients. The function is approximated by a (n-1)th degree polynomial in Polynomial Differential Quadrature (PDQ), a Fourier series expansion in the Fourier Expansion Base Differential Quadrature (FDQ), employing harmonic functions named the Harmonic Differential Quadrature (HDQ). 
DQM performance is dependent on not only the sampling grid points but also boundary conditions. Without restricting the choice of grid meshes to find simple algebraic expressions for the weighting coefficients, the Generalized Differential Quadrature Method (GDQM) has also been developed. Moreover, in the previous researches, it can be seen that proper grid points and weighting coefficients effect the results. In the problems having linear equations and homogenous boundary conditions, for solution, equal grids are adequate. 
Grid points through the Chebyshev-Gauss-Lobatto method is appropriate for vibration problems. For the problem in this research a nonuniform mesh like in Chebyshev-Gauss-Lobatto method have been thought. But the boundary of variables are on one side so the density of grid points had to be high near only the boundary. Therefore, it has been necessary to define new grid types which dense near only the boundary conditions. For this purpose two different grid types have been formulated. 
The differential transform method (DTM) is an efficient technique, having a considerable accuracy and easiness, for solving differential equations. The concept of DTM was first introduced in 1980s. It applied to electrical circuit analysis solving not only linear but also non-linear initial value problems. At a point in terms of known and unknown boundary conditions, the differential transform method gives exact values of the nth derivative of an analytical function. DTM constructs an analytical solution in a polynomial form for differential equations. 
Differential transform is a simulation method, depending on on Taylor Series expansion, contrary to higher order Taylor series, this method does not need the symbolic calculation of derivatives. For large orders The Taylor series method is computationally taken long time. Obtaining analytical Taylor series solutions of differential equations by iteration, desired results can be found with great accuracy.
The nondimensionalized equations for the problem in tihs work are solved by using Generalized Differential Quadrature Method and results are compared with the reference work. Also Differential transform method results are obtained for the problem and compared with GDQM results.