Sonsuz uzunlukta silindirik tüp içerisinde sıkışmaz ikinci derece akışkanın daimi akımı
The steady flow of second order fluid in an infinitely long cylindirical tube
Yazar: SAADET SEHER ÖZER
Danışman: PROF. DR. ERDOĞAN S. ŞUHUBİ
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Mekanik Ana Bilim Dalı
Derece: Yüksek Lisans, 1998
Konu: Mühendislik Bilimleri = Engineering Sciences
Dizin: Akış modelleri = Flow models ; Akışkanlar = Fluids

Özet:
Bu çalışmada bir Newtonyen olmayan akışkan modeli olan, ikinci derece Rivlin-Ericksen akışkanının sonsuz uzunlukta bir silindirik tüp içerisinde sabit karakteristik basınç gradyanı altında hareketi incelenmiştir. Giriş bölümünde genel olarak non Newtonyen akışkan modellerinden kısaca bahsedildikten sonra, Rivlin-Ericksen akışkan modeli elde edilmiş ve özellikle termodinamik uyum ve stabilite tartışmalarının tarihsel gelişimi üzerinde durulmuş tur. İkinci Bölümde termodinamik uyum şartlarının malzemenin bünye denk lemlerine getirdiği kısıtlar elde edilmiştir. Akışkana ait hareket denklemleri 3. Bölümde elde edildikten sonra 4. Bölümde silindirik tüp içinde akım için düzenlen miştir. Bu aşamada akışkanın ikinci derece boyutsuz malzeme sabitinin yeterince küçük olduğu, yani ya malzeme sabitinin çok küçük, yada karakteristik çapın çok büyük olduğu durumlar gözönüne alınarak pertürbasyon açılımı yapılmıştır. Çözüm lerde analitik fonksiyonlar kullanılmıştır. Problem özelinde analitik fonksiyonların basit bağımlı bölgelerde ve çok bağımlı bölgelerde yapılan belirlenmiştir. Bölüm 4.4'de basit bağımlı dairesel bölgede önce homojen sınır koşulu altında, yani tüpün hareketsiz olduğu ve tüp yüzeyinde hızların sıfır olduğu halde ardından da akışkanın tüpün yüzeylerinde emildiği durum için çözümler aranmıştır. Homojen sınır koşullan ele alındığı takdirde problemin pertürbasyon açılımının yeni bir çözüm vermediği ve sadece klasik hız alanını ürettiği gözlenmiştir. Bölüm 4.5 de ise konform dönüşüm aracılığıyla dairesel bölgenin keyfî basit bağımlı bölgelere dönüşümü ele alınmıştır.. 5. Bölümde Poiseuille akımının stabilitesi incelenmiş, akım fonksiyonunun ^ye bağlı olmayan çözümleri araştırılmıştır.

Abstract:
Some materials, in Nature, show some fluid characteristics that can not be defined by a linear viscous flow model. Such fluids are generally called non- Newtonian fluids. The non-Newtonian fluid model is defined by the relation between the stress and velocity gradient tensor at a point not being linear in the constitutive equation, which defines the character of the fluid. A model, which has not a viscoelastic component in non-Newtonian fluids, is Rivlin-Ericksen fluids which is defined by t = -pI + a,I + a2A, + a3A2 +a4Af +a5Aj + ar6(A,A2 + A2A,) + «7(Ai2A2 +A2A2i) + as(A1A22+A22Ai) The fluid which is obtained by second order truncation of the expansion (1) represented by t = -pi + juA^+ or, A2 + a2 A2 (2) is called second order fluid. Here t represents the Cauchy stress tensor at a point, an, 0:2 and fx are some materials constants, which may depend on temperature. The tensors Aı > A2... Ar are defined as follows A,=2d = (L + LT) A2 =A, +LAj +A,LT Ar+1 = Ar + LAr + ArLTand is called Rivlin-Ericksen tensors. There are many studies inwhich the thermodynamic analysis and the stability of Rivlin-Ericksen fluids are treated. These can be seen in the list of references. One of them, Dunn and Fosdick [23] have shown that the following conditions must be satisfied a, +«2=0, //^0, a,i>0 (3) for the second order fluid to be consistent with thermodynamics. For this case the constitutive equation (2) for second order fluid can be written as t = -pl + juAl + a(A2-A2) (4) The equation of motion for a second order Rivlin-Ericksen fluid can be obtained in vectorial notation as follows v, + wxv + grad- - f = -gradjp + //V2v (5) + a V2v, +V2wxv + grad(v-V2v + i|A1|2) Here p is the pressure, (.) denotes the material derivate, f shows the external forces, w is the spin vector which is defined as: w = Vxv To obtain non-dimensionalized equations of steady motion let us define characteristic velocity £/and characteristic length L and write the equation of motion (5) in non-dimensionalized form as follows: Vp = - V2v + s (V2w x v) - (w x v) Re (6) Here XIII2 I a I2 İv A t p = p+ U - £ (v. V2 v+i- U- ) y 2 4 will be called the characteristic pressure. Furthermore we have also defined Re = - -, s = >" pi2 ' By taking into consideration that the curl of the gradient is zero, the equation of motion can be written as: Vx[ - V2v+£-(V2 wxv)-wxv]=0 (7) Re This is the equation of motion of steady flow for a second order incompres sible fluid. Flow in a Cylindrical Tube In the case of the steady flow of a second order fluid in a cylindrical tube, if we assume that u=u(x,y), it is possible to define the velocity field in terms of a stream function yas follows dw dw., s, i u = -1-, v = - - u=m i+v j+w k 8y dx where the unit vectors i, j and k are along the cartesian axes jc, y, Ç. Ç axis concides with the axis of the cylinder whereas x,y axes are chosen within the cross section. Then the equations of motion (7) become: [Wxwy -vywx\ = -^(v24 +*(mv2*0, -r,(V2w) J, (9) -ivV+4(^H - w,(v2h»" +^(v», -r,(vV) J = ¥x(y2¥)y-¥y^¥)x xuHere O*^ shows the partial differentiation with respect to coordinates. Considering the equations (8) and (9) together, it is seen that the following equation can be written as: ¥,*>, -v^y ~W+4(VH -^(V2h0J= C (1 1) Where the constant C is actually characteristic pressure gradient assumed to be constant. We can see that by using equation (6). As a special case in which the non-dimensional second order constant e is sufficiently small namely either the material constant a is small or the characteristic diameter L is very large will be taken into account. We can then employ perturbation expansions for the stream function and longitudinal velocity component as follows: w = w0 +ewt + s2w2i - ır = sıy1+£2v2+ - C = P=P0+ePt+e2P2+- When the equations (1 1) (10) and (6) are considered together the following equations can be obtained in accordance to various orders of the e : V2w0=-PeP0 _ (12) V2w, = Re{w0xVly -w0yylx -Pt) V2w2 =Relp0jr2y+wlxyly -w0yiy2x -w,yy/lx -P2) V>2 =HwoyV2Wu +VxyV2Vu -p"VVi, ~w0xV2wly) (13) (14) Here equations (12), (13)ı, (14)i are Poisson's equations and the others are biharmonic equations. In this article, the solutions in terms of analytic functions of these equations will be examined. The equation (12) can be solved as by using analitycal functions w0 = - - - zz + $(z)+$(z) xniwhere z = x+iy and z is its conjugate, here ^(z) is analytical function. In a similar fashion, the first order of stream function can be expressed as y/x = zQ(z)+ zQ(z)+ Mz", p(:) = J]i/ n=0 n=0 here the coefficients a" and b" are constants which will be determined by using the boundary conditions. Analytical Functions in a Multiply Connected Domain Let us consider a multiply connected domain which is bounded with the curve Co externally and with the curves C\,Ci C" internally (Fig.4.3). In this case the velocities should be single- valued even if the functions Q(z) and cp(z) are not single-valued. The conditions for complex velocity expressed by co=u+iv to be the single- valued imply that: [zQ'(z) + Q(z) + 0(z)(lS) Here the functions Qo(z) and .<* -**)". *.«= ZM*-**)' The problem now is to combine the separate Laurent expansions written around the contours and find a solution for the whole domain. This seems rather difficult. In order to give an example to our problem we consider flow in an impermeable pipe. It is straightforward to see that all the terms of perturbation expansion for stream function are identically zero. At this stage the only difference between the solution of this problem and the solution for ideal viscous fluid problem is that a constant pressure gradient inducing the longitudinal velocity distribution is replaced now by a constant characteristic pressure gradient. For second grade and third grade Rivlin-Ericksen fluids solutions are possible for which the particle velocities are those of the longitudinal flow. However, if the fourth grade constitutive equation is valid, for the fluid, a steady flow in transverse planes is superposed on the longitudinal flow [31]. As another example, the problem of suction or injection of the fluid along the boundary of the tube is studied. If a layer of a permeable solid material is used as the boundary of a region of fluid flow, fluid may be sucked or injected through the boundary. In this case the normal component of the velocity will be the function of a suction function at the boundary and the tangential component of the velocity will be zero [37]. Therefore, the boundary conditions for the case of suction will be as follows: u"=u.n=j(s), «t=0 (19) If we expand the suction function, which is the function of the arc length s to a perturbation series in e the boundary conditions (19) can be written as xvdVt ds w'l«=0 ' = 0,1,2,... = /,(*) ¥t\m=Ft{3) i = 1,2,.. (20) a» V, iz dz ds Wn 8R dz ~ds i = 1,2,... Here Ft (s) defined as, F( (s) = J / (s) ds Here the problem will be examined in a circular cylindrical tube. Then the parameter s will be equal to 6. Since F(0) should a 2% periodic function it is can be expanded in Fourier Series. By employing the boundary conditions (20) the component of velocity and stream function are determined as follows w0(z,z) = (l-zz), w0(x,y) = (l-x2-y2) 0v=c> + E «=1 ((n + 2)-nzz) ^(x,y) = C0 + f\Uc"(x + iyy+C_n(x-iyy\(n + 2)-n(x2+y2)) n=l »i(*,.)0 = Z B=l ^(-2y(Cn(x+iyr +C_.(x- w)f /(C"(x+/>r1 -C B(x-/>r'X(« + 2)-W(x2 +y2)] v,(*,^) = -£ B=l L |(-2x(C.(x + ^)"+C^(x-W)f (c.cx+^r1 +c "(x-/>riX(«+2)-«(x2 +/)] wi(*,*) = Z /wife n=\ 8(l+«X2+») +(l-zz) [zf^(l+w>2-2(2+»)2)+ (w2 +7n + S)\Cnzn -C_nzn) Wi(*,>0 = E iriRe - [(w2 + In + 8) + (x2 + y2 )[«(! + «)(x2 + y2 ) - tf8(l+»)(2+«)' 2(2+»)2](Cn(x + />r-Cn(x-/»")+(l-x2-/) XVIHere C" are the constants of F(0) function in the Fourier expansion. For a special case, if F{0) function only has first two degree terms the functions above will be simplified. The graphics of these functions are given in appendices. Con formal Transformation After having obtained the solution corresponding to the circular tube, the solution for an arbitrary simply connected domain can be obtained through the well- known conformal transformation Fig (4.9). The expression z=h(d) establishes a correspondence between the points of complex planes a and z. Consequently the analytical functions under this transformation will be written as follows: Q(z) = Q[h(a)] = Qt (o-), { dy ydxJ V\ Here N2 is defined as rf=£Re2. If we consider circular cross section as an example, the stability equation becomes f V4+2N2 B02 ^i=0 ( a2 d2 Id Id2 -+ +- + ik- drz rdr r* d0l 30 r & d2 Id Id2 -+ +- -ik- drl rdr rl d0l 60) 2 a/i2 Vi=0 Here IN2 = k2. The stream function for linear stability analysis in Fourier modes can be written as: \f/x = T(r)e" Here m shows the wave number. Because of the operator obtained in this case is being linear, the factors are being commutative, the spectrums are being discrete and simple eigenvalues being entirely different, the general solution can be expressed as an arbitrary sum of separate solutions of factors: xviu¥(/?)= ¥,(/.)+%(#?) Where the function ^,(r)and ^(r) satisfy f J2 d1 1 rf -+ + dr r dr km m 2\ *,(/.) = o, ^rf2 1 d (, m + km + - - JJ dr r dr r\\ J) %(r) = 0 So, the solution can be expressed in terms of Bessel's functions as follows : V(r) = AJS^r) + CYm(jtor) + BIS^r) + DKm{4km~r) (22) ¥ (r) has to be finite. Since r-»0. Ym(Jkmr)-*co, Km(Jkmr)^>ao then we have to take C=D=0. The boundary conditions are applied for a circle with radius equal to 1 Ur=-V0> 1h=-¥r> »rU=K*L,=0 the equation (22) can be written as: AJm(Jtoi)+BIm(^) = 0, AJ'm(Jhn)+BI'm(^hü) = 0 In this case, in order to have a non-zero solution the determinant of coefficient matrix must vanish. Here the smallest root was reached for m=3J A=17.01001 182792537309334832959 The other values are given in Table 1. It is possible to say that the instability begins after this value. The graphic of the stability are given in Figure 5. 1 and the other instability graphics can be seen on Appendices.