A STABILITY STUDY OF NON-NEWTONIAN FLUID FLOWS

Transcription

1 U.P.B. Sci. Bull., Series A, Vol. 71, Iss. 4, 009 ISSN A STABILITY STUDY OF NON-NEWTONIAN FLUID FLOWS Corina CIPU 1, Carmen PRICINĂ, Victor ŢIGOIU 3 Se stuiază problema e curgere a unui flui Olroy B generalizat între ouă plăci paralele. Ne interesează acă soluţiile e tip von Karman sunt amisibile pentru acest flui. Precizăm un caru al problemei in punct e veere al stabilităţii. Prezentăm restricţii asupra parametrilor constitutivi, ca rezultat al unei anumite inegalităţi. Punem în evienţă caracterul stabil al unei curgeri e bază pentru fluiul Olroy B cu parametri e material constanţi. The problem of the flow of a generalize Olroy B flui between two parallel plates is stuie. We are intereste if a von Karman type solutions are amissible for this flui. We precise a frame of the problem from the point of stability view. We iscuss some restrictions by certain inequality upon constitutive parameters. We etermine the stability character of non trivial base flows for Olroy B flui with constant material mouli. Keywors: Olroy-B flui, von Karman s type solutions, stability character. 1. Introuction By a stability stuy for an incompressible secon grae flui, from Clausius- Duhem inequality are obtaine restrictions for the constitutive mouli of Cauchy stress tensor: μ 0, α1 + α = 0, (1.1) an α 1 0 if the free energy is to be minimum in equilibrium(see J.E. Dunn, R.L Fosick [1]). In the paper of R. L. Fosick an B. Straughan (see []), for instance, was investigate the instability in a flui of thir grae. Employing the Clausius- Duhem inequality an emaning that the free energy be a minimum in equilibrium, Fosick an Rajagopal [3] have shown that the corresponing constitutive equation for an incompressible flui of thir grae is: μ 0, β 0, 4μβ α1 + α 4μβ, α1 0. (1.) 1 Department of Mathematics, Faculty of Applie Sciences, University POLITEHNICA of Bucharest, Romania Faculty of Internal an International Commercial an Financial-Banking Relations, Romanian American University 3 Faculty of Mathematics an Informatics, University of Bucharest, Romania

2 16 Corina Cipu, Carmen Pricină, Victor Ţigoiu In [], the authors assume that the inequalities ( 1.) 1 are strict inequalities. They show that the conition α 1 < 0, which is compatible with the Clausius- Duhem inequality but not with the free energy, being a minimum in equilibrium an thus they leas to behavior which may not be physically acceptable.. The flow problem The paper eals with the problem concerning the flow of a generalize Olroy B flui between two parallel plates. The Cauchy stress tensor is: DT 1 DA T = -pi + T, E T A A 1 E + E = μ 1 + α + α1, (.1) Dt 1 Dt where the convective erivative is expresse by (see Fetecau [4], [5]): DA T = A + AL + L A. (.) Dt In the equation (.1), T E is the extra stress tensor (effective stress tensor), - p I enotes the ineterminate spherical stress, L is the velocity T graient, A 1 = L + L is the first Rivlin Ericksen tensor, is the relaxation time, μ is the ynamic viscosity, α 1 an α are the constant constitutive coefficients. We have the constitutive restrictions (see Tigoiu [6], [7]): μ 0, α1 + α = 0. (.3) The flui flows between two parallel plates. The upper plate is suppose to be porous an the flui passes through with constant vertical velocity, meaning: v(x, y) y = = v0 j, (.4) an the lower plate moves with the velocity: v(x, y) y= 0 = cxi, (.5) where is istance between the two plates, i an j are the unit vector in the horizontal an respective vertical irections an c is a given constant. We remark that the origin is preserve at rest (see Fig. 1). Fig. 1. Flow omain

3 A stability stuy of non-newtonian flui flows 17 We shall suppose, like in [8], that the amissible velocity fiel is of von Karman s type: ' u = cxf ( η), v = cf ( η), y η. (.6) We stuy if a generalize Olroy -B flui accept a von Karman s solutions for the flow problem escribe above. This flow fiel satisfies the constraint of incompressibility. Since the velocity fiel is inepenent of z, the stress fiel will also be inepenent of z. Therefore from the constitutive equation (1.1) we obtain the following system: c x cμ f ' α1 (c ff '' + f '' ) = cxf ' cf + cf ' T E 1 1 x c x cx cμ f '' + α1 (3 f ' f '' f f ''') = cxf ' cf + f '' T + E TE cxf ' cf + cf ' TE = 0 c x T ' 1( '' '' ) ' E T cμ f + α c f f + f = cxf cf E + (.7) cx f '' TE cf ' TE 3 3 T 1 ' E T E cx cxf cf + f '' TE cf ' TE = T 1 33 ' E T cxf cf E = 0. The equations of the motion are: ρ a = ρb + ivt. The acceleration is given by: a =c x (f ' ff '') i + c ff ' j. If we consier b = 0, then the flow equations are: 1 11 T T p c x f ff E E ρ ( ' '') = +, (.8) T T p ρc ff = E + E T ', E T + E = 0. We suppose that the effective stress is of the form: n ij x ij TE = TE n ( η). (.9) n n= 0

4 18 Corina Cipu, Carmen Pricină, Victor Ţigoiu Then we will be able to suppose that the pressure has the same type like ij function T E oes: p ( x, η) = p0 ( η) + ( x / ) p1( η) + ( x / ) p ( η). (.10) If we use the relations (.7) 3,(.7) 5, (.8) 3 we can etermine the 13 expressions for the components TE n, TE n 3 an the equation for the function f (η). We remark that employing the relations: 3 T 1 0, 13 T 0, we arrive at the following equation for f: cf '' f + 6cf ' 3 f ' = 0. (.11) Thus the problem is to solve the equation (.11) uner conitions obtaine from the escribe mechanical problem, which are: f (0) = 0, v0 f (1) =, c f '(0) = 1, f '(1) = 0. (.1) The problem is if the equation obtaine for f (η ) has a solution if we consier any two point problem of type (.1). Using (.1) 1 (.1) 3 we foun: c = 1/. (.13) Thus the equation (.11) becomes: f '' f + 6 f ' 6 f ' = 0. (.14) For the stuy of the above problem, we first evelop the function f in power series, in orer to etermine the coefficient of secon orer in η : 3 n f ( η ) = a0 + a1η + aη + a3η +... a n η +... (.15) From (.1) 1 (.1) 3 we calculate that: a 0 = 0, a1 = 1, a = 0.5 = v0 / c. (.16) For the secon approximation (of f) in η we make a change of the function introucing a new function h ( 1 η) : f ( η) = η + a η h(1 η) η, (.17) getting for h (t), t 1 η, a ifferential Cauchy problem: 5 3 (1 t ) ( ( t) h 3 ) hh'' = ( 1 h + 4(1 t) h' )(1 t) h( (1 t) h 3 + t) ( t (1 t) h + (1 t) h' ) 6 ( t (1 t) h + (1 t ) h' ) (.18) h ( 0) = 1, h (0) = 0. By a numerical calculus we etermine the function h only for η [0, 0.6], fit the ata of function h, an obtain a seventh egree polynom (see Fig.). The value η = 0. 6 express the first point for which h ( η) = 0. For η [ 0.6, 1] we shall use the fitte polynom, h being a continuous function.

5 A stability stuy of non-newtonian flui flows 19 Fig. Compute function h an fitte polynom Fig. 3 Function f: numerical compute an using interpolate polynom The approximation of h over the interval [0, 0.6] is: h ( t) 840t 100t + 540t 48t 4t + 4.4t 0.4t + 1, (.19) an the f approximation by polynom over the same interval is: f η + 0.5η η [840(1 η) 100(1 η) + 540(1 η) (1 η ) 4(1 η) + 4.4(1 η) 0.4(1 η) + 1]. (.0) 3. Stability of the solution by numerical analysis For the flui stuie, now we consier a small perturbation of the base flow. The perturbe flow is given by the following expressions: u ~ = cx( f ( η ) + εg'( η )) = u + u, v ~ = c( f ( η ) + εg( η )) = v + v. (3.1) Since u ~, v ~ are given by the same equations of motion like u an v oes, the small perturbation u, v, must satisfies:

6 130 Corina Cipu, Carmen Pricină, Victor Ţigoiu g''( f + ε fg) + 6g'( f 1+ εg ) + g f ( f + εg) = 0, g(0) = ε, g'(0) = 0,(3.) neglecting O ( ε g ) in respect with O ( f ). 6. Conclusions Fig. 4 The small perturbation g. Our stuy conclue that existence of von Karman type solution for the Olroy B flui implies a certain constants for obtaining imensionless values of the velocity: c = 1/, an v0 = c. Also, we observe that small perturbations of the base flow are numerically stable ( O( g) O( ε ) as were impose initially in η = 0, see Fig. 4). B I B L I O G R A P H Y [1]. J. Ernest, R.L.Fosick, thermoynamics, stability, an bouneness of Fluis of complexity an fluis of secon Grae,Archs Rational Mech.Anal.56, 191-5,(1974). []. R. L. Fosick, B. Straughan, Catastrophic instabilities an relate results in a flui of thir grae, Int.J.Non-Linear Mechanics,16, , (1981). [3]. R. L. Fosick,K. R. Rajagopal, Thermoynamics an stability of fluis of thir grae,proc.roy.soc. A339, 351, (1980). [4]. C. Fetecau, Analytical solutions for non-newtonian flui flows in pipe-like omains, Int.J.Non-LinearMech. 39, 5-31, (004). [5]. C. Fetecau, Corina Fetecau, Decay of a potential vortex in a Maxwell flui, Int.J.Non- LinearMech. 38, , (003). [6]. V. Tigoiu, Prepr.ser.Math.69.INCREST,Bucharest,(1984). [7]. V. Tigoiu, Stuii si Cercetari Mat. 39(4), , (1987). [8]. V. Tigoiu, The flow of a viscoelasic flui between two parallel plates with heat transfer,int.j.engng Sci., 9, 1, , (1991).