Paralel iki sabit plak arasında ve rijid silindirik tüpte viskozitenin  basınca bağlı olduğu akım
The flow of fluid with a pressure depedent viscosity between two parallel plates and in a rigid tube
Yazar: GÜRKAN KARAGÖZ
Danışman: DOÇ. DR. ALİ ERCENGİZ
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Matematik Mühendisliği Ana Bilim Dalı
Derece: Yüksek Lisans, 2014
Konu: Matematik = Mathematics ; Mühendislik Bilimleri = Engineering Sciences

Özet:
Bu tez çalışmasında paralel iki sabit plak arasında ve rijid dairesel silindirik tüpte viskozitenin basınca bağlı olduğu tek yönlü akım problemi incelenmiştir. Viskozitenin basınca bağlılığı  ,   ve    şeklinde üç farklı durum olarak alınmış, her bir durum için hız profileri ve basınç dağılımları elde edilmiş ve grafikleri çizilmiştir.
Birinci bölümde konunun tarihsel gelişimi üzerinde durulmuş, bu alanda yapılan çalışmalardan ve elde edilen sonuçlardan söz edilmiştir. 
İkinci bölümünde Sıkışmaz –Newtonian akışkan için genel bünye denklemleri elde edilmiştir.Bu denklemler momentumun korunumu denkleminde kullanılarak hareket denklemleri dik kartezyen ve silindirik koordinatlarda açık olarak yazılmıştır.
Üçüncü bölümde kartezyen koordinatlarda paralel ve sabit iki plak arasında tek yönlü akım için viskozitenin basınca bağlılığının  ,   ve    şeklinde olduğu üç viskozite basınç ilişkisi gözönüne alınarak hareket denklemleri elde edilmiş ve çözülerek basınç dağılımları ve  hız fonksiyonları belirlenmiştir. Her bir durum için bu fonksiyonların değişimleri grafik olarak gösterilmiş ve tartışılmıştır.
Dördüncü bölümde silindirik koordinatlarda, rijid tüp içinde tek yönlü akım için viskozitenin basınca bağlılığının   ,  ve    şeklinde olduğu üç viskozite basınç ilişkisi gözönüne alınarak hareket denklemleri elde edilmiş ve çözülerek basınç, hız fonksiyonları belirlenmiştir. Her bir durum için bu fonksiyonların değişimleri grafik olarak gösterilmiş ve tartışılmıştır.
Beşinci bölümde ise tez çalışmasında elde edilen sonuç ve grafiklerin genel  değerlendirmesi ve yorumu yapılmıştır.

Abstract:
In this thesis, the flow of an incompressible Newtonian fluid whose viscosity can depend on the pressure is studied between two parallel plates and in a rigid tube.
In general, the viscosity ca be a function of the pressure p, however in many practiced situations the flow conditions are such that it is essentially a constant leading to the classical incompressible Navier-Stokes model. 
In a simple shear flow, the constituve equation is given by
 Where  is the stress tensor,  is the pressure,  is the viscosity and  is the deformation rate tensor.It follows from this equation that the shear stress is proportional to the shear rate, the constant of proportionally being the viscosity. Put another  way, the viscosity is the ratio of the shear stress (resistance) to the shear rate (velocity gradient). The resistance to the motion is due to the friction between the adjacent layers of the fluids. 
As the normal stress increases, the resistance to the sliding motion increases. Extending this idea of the resistance due to the relative motion of adjacent layers prompts us to suppose that the viscosity in a fluid should increase with normal stress or in a three dimension setting, the mean normal stress. That the viscosity in a fluid could depend on the presure was considered by Stokes[1].
Barus[4] suggested the following exponential relationship between viscosity  and  .
 Where  has unit  . Barus' formula can be used to get a rough estimate of the variations of the viscosity with respect to pressure for some common organic liquids. For Naphthenic mineral oil  has been determined experimentally to be 23.4  at  . Based on the Barus' formula for Naphthenic mineral oil we then have
 
On the other hand, using the Dawson-Higginson[5] empirical formula, the density varies with respect to pressure as


 Where  is in  . Using the Dawson-Higginson formula we have 
 

Many other experiments have also indicated that change in density due to changes in pressure at high pressure is indeed negligible.
In the flows that are usually encounters in pipes and channels, due to pressure gradient, the variations in pressure are not sufficiently great to cause significant changes in viscosity. However, if the fluid is forced through very narrour regions from a domain of much greater thickness, such as the geometry relevant to elastohydrodynamics, the increase in the pressure is so dramatic that it can influence the viscosity significantly.
The importance of including the effect of the viscosity on the pressure cannot be overemphasized. For instance, in theory of elasthohydrodynamic lubrication(i.e., hydrodynamic lubrication applied to solid surfaces capable of eleastic deformation), the existence of continuous films under severe loading cannot be explained unless the dependence of the fluid viscosity on the pressure and elastic deformation of the solid are taken into account. Also, pressures which are large enough for pressure dependence of viscosity to become a serious concerns are sometimes reached in polymer melt processing.
In this thesis, we consider the following form for the pressure viscosity function   . 
                                                     a)   

b)   

c)   

Where   and   are constant. Experimental studies show that   increases with increasing pressure, and thus we assume that    is positive.
For each cases, we have studied the flow between parallel plates (Plane Poiseuille Flow) and in a rigid tube (Axisymmetric  Poiseuille Flow).
In the flow between two parallel plates, for cases (a), (b), (c), we obtain velocity and pressure as a function of x,y and  . The constant    that appear in the solutions is related pressure gradient along x-direction. We also observe in two cases that as the parameter   approaches that corresponding to the profile for constant viscosity. For larger values of  , we obtain profiles that are "V" shaped and we see that we need a very large pressure difference to maintain such a flow. In case (c), we show that the flow is a trivial (no motion). But, defining a parameter  , we  expend pressure and velocity to the asymptotic series, and we could obtain only first three terms of the series.

In the flow in a rigid tube, for cases (a) and (b), we obtain similar result to the flow between plates. In the case (c), we use asymtotic series, became the series are convergent we are able to obtain the analytical solutions for velocity and pressure.
In the determining analytical solution, the relations between constants  that appear in terms of the series play important role, each different relation gives a different analytical solutions for velocity and pressure.