Rivlin-Ericksen akışkanının dalgalı bir kanal içindeki hareketi
The flow of a Rivlin-Ericksen fluid in a waving channel
- Tez No: 39173
- Danışmanlar: DOÇ.DR. GÜLGÜN YALÇINKAYA
- Tez Türü: Yüksek Lisans
- Konular: Makine Mühendisliği, Mechanical Engineering
- Anahtar Kelimeler: Akışkanlar, Kanal modelleri, Rivlin-Ericksen akışkanı, Fluids, Channel models, Rivlin-Ericksen fluid
- Yıl: 1993
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
Bu calxsmada, sınırları dalga şeklinde olan bir kanalda Rivlin-Ericksen akışkanına ait akış, düzlemsel ve daimi hal için incelenmiştir. Kanal sınırlarını ve ren fonksiyon en genel halde ele alınmış, daha sonra sinüzoidal cidar için çözüm yapılmıştır. Kanal yarıçapının kanal uzunluğuna oranı olan 6 pertürbasyon parametresi olarak seçilmiş, çözüm için gerekli değişken dönüşümleri ve boyutsuz landırma işlem leri yapılarak akım fonksiyonu, 5' nın küçük değerleri için uygunluk denkleminin seriye açılmasıyla hesaplanmış tır. Seri çözümdeki yakınsama kontrol edilerek yeterli düzeyde olduğu gösterilmiştir. Newtonien ve non-Newtonien haller ve değişik Reynolds sayıları için akım çizgileri çizilerek mukaye se edilmiştir. Viskoelastik katsayının çok küçük olduğu durumda akışkanın Newtonien akışkana benzer davranış gös terdiği, geri akımın oluşturduğu vorteksin az miktarda büyüdüğü görülmüştür. Viskoelastik katsayının daha bü yük değerinde geri akım bariz bir şekilde artmıştır. Düşük hızlarda ise vorteksin ortadan kaybolduğu ve akış profilinin düzgünleştiği belirlenmiştir. Akışkana ait gerilme tansörünün bileşenleri hare ket denkleminden hesaplanmış, kayma gerilmesi değeri için belirli bir noktada çözüm yapılarak viskoelastik katsayı nın etkisi gösterilmiştir. Basınç dağılımını veren denk lemler elde edilmiştir.
Özet (Çeviri)
The Flow of a Rivlin-Ericksen Fluid in a Waving Channel Analysis of forced convection and mass transfer in Laminar flow inside a conduit has been the subject of extensive study. Hydrodynamic solutions on steady laminar flow with moderate Reynolds numbers in conduits with irregular surfaces are obtained. In these surfaces the spread of roughness has been large, compared with the mean radius of the conduit. These results are of considerable interest, especially for blood flow in arteries with stenoses and for membrane Oxygenators using parallel plates with wavy surfaces. In this thesis, the fluid model is taken to be Rivlin-Ericksen which is a non-Newtonian fluid. To give an idea, some non-Newtonian effects are summarized below. One of the most important non-Newtonian effects is that viscosity changes due to the change in the velocity field and normal stress differences affects the motion of the fluid. A series of experiments are made to see these effects. Some of these are tube flow where shear thinning or shear thickening occurs; rod climbing where the non-Newtonian Liquid shows the Weissenberg effect ; extrudate swell where a non-Newtonian liquid shows diameter increase upon emergence from the capillary tube? occurrence of secondary flows in the disk and cylinder system; the tubeless siphon; the uebler effect and flow through a sudden contraction. In this study the steady flow of a Rivlin-Ericksen fluid in a channel with waving surfaces is investigated. The fluid is incompressible, the flow is two dimensional and body forces are negligible. It differs from Sturges' work in the way that Sturges' study is unsteady and is investigated only for sinusoidal boundary. In our study, the boundary function can vary so that the problem can be solved for different forms of the boundary, and the problem is taken into account for steady motion. The geometry of the flow is shown below. &- Ay* 11**!In our study the boundary of the channel is given by, n+ (x*) = d(l-eh(x*)), n* (x*) s-da + ehCx*) ) Here h (x ) is chosen as a sinusoidal function. The wavelength is shown to be 2H£. <S is defined as the ratio of channel radius to its length» &<<l and the average channel diameter is 2d. The velocity field of the flow is defined as, # * * * u = u (x,y ) v a v (x,y ) The constitutive equation of a Rivlin-Ericksen fluid is given as, T = -pI+o^Ai+aı A2+a2A!2, where T is the stress tensor, p is pressure, I is the tensor of unity, Aj and A2 are the first and the second order Rivlin-Ericksen tensors respectively. ^0 is the coefficient of apparent viscosity, aj is the coefficient of elastic viscosity and a2 is the coefficient of cross viscosity. The basic equation of the problem is the compatibility equation of a Rivlin-Ericksen fluid and is obtained by putting the constitutive equation into the equation of motion. The compatibility equation is as follows, D(v2Ay*) _r D(vH*,y*) _ _“”* D(x*,y*) D(x*,y*) The velocity components is terms of the stream function ¥ are, u% ^ 8y*3x* The boundary conditions for the problem are defined as the following: 1) Velocity is zero at the boundaries. So, 2) The value of the flow rate is constant in the channel. So that, 4 f u. dy = constant In the solution of our problem non-dimensional values are used so that after the necessary non- dimensionalizations are made, the compatibility equation comes out to be, 6Re D(V2^) _Yi6 Re D(7^,f) =v,, D(x,y) D(x,y) d 5i Here 6 = - and yi = - yPerturbation method is used in the solution of our problem and compatibility equation is expanded in series in terms of 6. The transformation of the independent variables are also made in the way, X = x n(x) The compatibility equation is perturbated up to the second order and solutions are made by integrating equations using boundary conditions. The zeroth order solution of the stream function is, f = - i (Y3-3Y) o 2 The first order solution is, u, It? ' t _2 Y?+ _2 *5~ 2L y3+_2 y\ *l = 4Keri I“ 70 10 70 14 J The value of the second order solution is, ?9 -(4n.2-T1T1.,){ _3 Y5- _2 *3+_2 Y - Ref{.2(98Yll ^2 -l*n nn M 20 10 20 ' 431200in ^ -1155y9+4488y7-8778y5+8222Y3-2875Y) +nn”(-35Y1 2 + 385Y<3-1518Y7 + 3234Y5-3279Y3 + 1213) } T?<=2 -.«2=(4r)' 2-nn") (5Y9-36Y7 + 126Y5-164Y3 + 69Y) } 1120n2 For small values of 6, the stream function is, ¥=.¥ + 6Yi+62,}S o -1 z The streamlines are plotted for Newtonian (yi=0) and Non-Newtonian (Yi*0) cases for different Reynolds numbers. In the Newtonian case a reverse flow is formedat f = 1.0 01 due to the sudden decrease in velocity in the climbing part of the boundary. This flow shows itself in form of a vortex. In the Non-Newtonian flow, the vortex area gets larger pointing out that the effect of the reverse flow is much more important. When yi gets the value -0.001 at Re = 25, the streamlines up to the value ¥ = 1 are nearly the same, and at the value ¥= 1.001 the vortex enlarges slightly. However, at the value y= -0.01, the reverse flow becomes more effective and causes the vortex to enlarge significantly. At lower values of the Reynolds number (Re = 10, Re = 1) vortex disappears and the flow adapts itself to channel geometry more easily. It is observed that the streamlines move upwards as the Reynolds number decreases. The components of the stress tensor are calculated in terms of the stream function. Solution is made for the shear stress at the point, X = 0 Y = 1 and it's found out to be, TxY I X = 0 = Ri +0/086+9,Q5Y1 Y m 1 where the effect of the viscoelastic coefficient is clearly seen. The pressure equations are also obtained from the equation of motion.
Benzer Tezler
- Üçüncü mertebeden Rivlin-Ericksen akışkanının biri sabit diğeri hareketli olan kesişen iki düzlem arasındaki yavaş akımı
The Creeping flow of the third-order Rivlin-Ericksen fluid between intersecting planes, one of which is stationary and the other moving
SERDAR BARIŞ
- Farklı eksenlerde aynı açısal hız ile dönen gözenekli ve paralel iki düzlem levha arasındaki rivlin-ericksen akışkanının hareketi
The Flow of a rivlin-ericksen fluid between two parallel, porous planes rotating with the same angular velocity at different
NECMİ AYDIN ÜNVERDİ
- Sonsuz uzunlukta silindirik tüp içerisinde sıkışmaz ikinci derece akışkanın daimi akımı
The steady flow of second order fluid in an infinitely long cylindirical tube
SAADET SEHER ÖZER
Yüksek Lisans
Türkçe
1998
Mühendislik Bilimleriİstanbul Teknik ÜniversitesiMekanik Ana Bilim Dalı
PROF. DR. ERDOĞAN S. ŞUHUBİ
- İkinci derece akışkanlar için sınır tabakası denklemlerinin benzerlik çözümleri
Similarity solutions of boundary layer equations for second order fluids
MEHMET PAKDEMİRLİ
- Kauçuk türü malzemelerin malzeme sabitlerinin eş-iki eksenli çekme deneyi ile belirlenmesi
Determination of material constant for rubber like material using equi-biaxial test several
VAHAP VAHAPOĞLU
Yüksek Lisans
Türkçe
1998
Makine MühendisliğiKaradeniz Teknik ÜniversitesiMakine Mühendisliği Ana Bilim Dalı
PROF. DR. SAMİ KARADENİZ