Unsteady flow of Oldroyd-B fluids in an uniform rectilinear pipe using 1D models

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1 Proceedings of the 2006 IAME/WEA International Conference on Continuum Mechanics, Chalkida, Greece, May -3, 2006 pp6-66 Unsteady flow of Oldroyd-B fluids in an uniform rectilinear pipe using D models FERNANDO CARAPAU Uniersidade de Éora Dept. Matemática and CIMA/UE R. Romão Ramalho, 59, , Éora PORTUGAL flc@ueora.pt ADÉLIA EQUEIRA Instituto uperior Técnico Dept. Matemática and CEMAT/IT A. Roisco Pais, , Lisboa PORTUGAL adelia.sequeira@math.ist.utl.pt Abstract: Using the one-dimensional approach based on the director theory which reduces the exact threedimensional equations to a system depending only on time and on a single spatial ariable, we analyze the axisymmetric unsteady flow of an incompressible iscoelastic fluid of Oldroyd type in an uniform rectilinear pipe with circular cross-section. From this system we obtain the relationship between aerage pressure gradient and olume flow rate oer a finite section of the pipe and the corresponding equation for the wall shear stress. Attention is focused on the steady flow case with rigid and impermeable walls. Key Words: Cosserat theory, Oldroyd-B fluid, steady solution, axisymmetric motion, olume flow rate, aerage pressure gradient. Introduction In this paper we introduce a D model for iscoelastic non-newtonian Oldrody-B flows in an axisymmetric pipe with circular cross-section, based on the director approach Cosserat theory with nine directors deeloped by Caulk and Naghdi 5]. The theoretical basis of this approach see Cosserat 6], 7], based on the work of Duhen 8] is to consider an additional structure of deformable ectors called directors assigned to each point on a space cure the Cosserat cure. With this approach and integrating the axial component of linear momentum for the flow field oer the pipe cross-section, the 3D system of equations is replaced by a system of partial differential equations which, apart from the dependence on time, depends only on a single spatial ariable. Using this one-dimensional Cosserat theory we can predict some of the main properties of the three-dimensional problem. For additional background information, we refer that the Cosserat theory has been used in studies of rods, plates and shells, see e.g. Ericksen and Truesdell 9], Truesdell and Toupin 8], Green et al. 4], 3] and Naghdi 6]. Later, this theory has been deeloped by Caulk and Naghdi 5], Green and Naghdi 5], and Green et al. 2] in studies of unsteady and steady flows, related to fluid dynamics. Recently, the nine-director approach has been applied to blood flow in the arterial system by Robertson and equeira 7] and also by Carapau and equeira 2], 3], 4], considering Newtonian and non-newtonian flows, respectiely. In this paper we are interested in studying the initial boundary alue problem of an incompressible homogeneous Oldroyd-B fluid model in a straight circular rigid and impermeable pipe with constant radius where the fluid elocity field, gien by the director theory, can be approximated by the following finite series : with = k x α...x αn W α...α N, N= = i z, te i, W α...α N = W i α...α N z, te i. 2 Here, represents the elocity along the axis of symmetry z at time t, x α...x αn are the polynomial weighting functions with order k the number k identifies the order of hierarchical theory and is related to the number of directors, the ectors W α...α N are the director elocities which are completely symmetric with respect to their indices and e i are the associated unit basis ectors. From this elocity field approach, we obtain Latin indices subscript take the alues, 2, 3, Greek indices subscript, 2. ummation conention is employed oer a repeated index.

2 Proceedings of the 2006 IAME/WEA International Conference on Continuum Mechanics, Chalkida, Greece, May -3, 2006 pp6-66 the unsteady relationship between aerage pressure gradient and olume flow rate, and the correspondent equation for the wall shear stress. The goal of this paper is to deelop a ninedirector theory k = 3 in for the steady flow of an Oldroyd-B fluid in a straight pipe with constant radius, to compare the aerage pressure gradient for different alues of both Reynolds and Weissenberg numbers. 2 Model Problem Let us consider a homogeneous fluid inside a circular straight and impermeable pipe, the domain Ω R 3, with boundary Ω composed by the proximal cross-section Γ, the distal cross-section Γ 2 and the lateral wall Γ w, see Fig.. with the initial condition x, 0 = 0 x in Ω, 4 and the boundary condition x, t = 0 on Γ w 0,T, 5 where = i e i is the elocity field and ρ is the constant fluid density. Equation 3 represents the balance of linear momentum and 3 2 is the incompressibility condition. In equation 3 3, p is the pressure and σ ij are the components of the symmetric extra stress tensor gien by σ ij = µ n i,j j,i σeij, where σ eij are the components of its iscoelastic part. Here t denotes the stress ector on the surface whose outward unit normal is ϑ = ϑ i e i, and t i are the components of t. In equation 3 4 the symbol σ eij represents the objectie Oldroyd deriatie of the tensor σ eij gien by see e.g. 9] σ eij = σ eij t k σ eij x k σ eik k,j j,k i,k k,i σekj a i,k k,i σekj Figure : Fluid domain Ω with the components of the surface traction ector τ,τ 2 and p e.γ w is the lateral wall of the pipe with equation φz,t, and Γ,Γ 2 are the upstream part and downstream districts of the pipe, respectiely. Let x i i =, 2, 3 be the rectangular Cartesian coordinates and for conenience set x 3 = z. Consider the axisymmetric motion of an incompressible fluid without body forces, inside a surface of reolution, about the z axis and let φz, t denote the instantaneous radius of that surface at z and time t. The components of the threedimensional equations goerning an Oldroyd type fluid motion are gien in Ω =Ω 0,Tby 2 ρ t,i i = t i,i, i,i =0, in Ω, t i = p e i σ ij e j, t = ϑ i t i, σ eij λ σeij = µ e i,j j,i, 3 2 We use the notation i,j = i / x j and,i i = i / x i adopted in Naghdi et al. 5], ]. σ eik k,j j,k ], 6 where a, ] is a real gien parameter. The initial elocity field 0 is assumed to be known. Finally, µ n =µλ 2 /λ is the Newtonian iscosity and µ e = µ λ 2 /λ = µλ is the elastic iscosity, with µ = µ n µ e denoting the iscosity coefficient, and the constants λ and λ 2 with 0 λ 2 <λ being the relaxation and retardation times, respectiely. Models with λ 2 =0are called of Maxwell type and those with λ 2 > 0 of Jeffreys type. Oldroyd-B fluids correspond to Jeffreys type fluids with a = in equation 6 and Oldroyd-A fluids correspond to a =, see e.g. 0]. The lateral surface Γ w of the axisymmetric domain is defined by φ 2 = x α x α, 7 and the components of the outward unit normal to this surface are x α ϑ α = φ φ 2 /2, ϑ 3 = z φ 2 /2, 8 z where a subscript ariable denotes partial differentiation. ince equation 7 defines a material φ z

3 Proceedings of the 2006 IAME/WEA International Conference on Continuum Mechanics, Chalkida, Greece, May -3, 2006 pp6-66 surface, the elocity field must satisfy the condition φφ t φφ z 3 x α α = 0 9 at the boundary 7. Let us consider z, t as a generic axial section of the domain at time t defined by the spatial ariable z and bounded by the circle defined in 7 and let Az, t be the area of this section z, t. The olume flow rate Q is defined by Qz, t = 3x,x 2,z,tda, 0 z,t and the aerage pressure p is defined by pz, t = p x,x 2,z,tda. Az, t z,t In what follows, this general framework will be applied to the specific case of the nine-director theory in a rigid pipe, i.e. φ = φz. Using condition, with k = 3, it follows from Caulk and Naghdi 5] that the approximation for the threedimensional elocity field is gien by x 2 = x x2 2 2φ z Q ] φ 2 πφ 3 e x 2 x 2 x2 2 φ 2 2φ z Q ] πφ 3 e 2 2Q πφ 2 x 2 x2 2 φ 2 ] e 3 2 where the olume flow rate Qt is Qt = π 2 φ2 z 3 z, t. 3 We remark that the initial condition 4 is satisfied when Q0 = ct. Also, from Caulk and Naghdi 5] the stress ector on the lateral surface Γ w is gien by t w = φ φ 2 z /2 τ x φ z p e x τ 2 x 2 φ 2 z /2] e φ φ 2 τ x 2 φ z p e x 2 z /2 τ 2 x φ 2 z /2] e 2 φ 2 z /2 τ p e φ z ]e 3, 4 where τ represents the wall shear stress in the axial direction of the flow. Instead of satisfying the momentum equation 3 pointwise in the fluid, we impose the following integral conditions z,t ] t i,i ρ t,ii da =0, 5 ] t i,i ρ z,t t,ii x α...x αn da =0, 6 where N =, 2, 3. Using the diergence theorem and integration by parts, equations 5 6 for nine directors, can be reduced to the four ector equations: m α...α N z n z f = a, 7 l α...α N = k α...α N b α...α N, 8 where n, k α...α N, m α...α N are resultant forces defined by n = t 3 da, k α = t α da, 9 k αβγ = k αβ = t α x β t β x α da, 20 t α x β x γ t β x α x γ t γ x α x β da, 2 m α...α N = The quantities a and b α...α N defined by b α...α N = a = t 3 x α...x αn da. 22 are inertia terms ρ t,i i da, 23 ρ t,ii x α...x αn da, 24 and f, l α...α N, which arise due to surface traction on the lateral boundary, are defined by l α...α N = f = φ 2 z /2tw ds, 25 φ 2 z /2tw x α...x αn ds. 26 The equation relating the aerage pressure gradient with the olume flow rate will be obtained using these quantities 9 26.

4 Proceedings of the 2006 IAME/WEA International Conference on Continuum Mechanics, Chalkida, Greece, May -3, 2006 pp Results and Discussion Replacing the solutions of equations 9 26 into equations 78 the relationship between aerage pressure and olume flow rate in a rigid axisymmetric straight pipe with constant radius 3 φ, is gien by p z z, t = 8µ n πφ 4 Qt 4ρ 3πφ 2 2 φ 2 ψ 33 z 4 φ 4 ϖ 33 z Qt φ 2 ψ z 4 φ 4 ϖ z, 27 where the functions ψ ij and ϖ ij are defined by 4 ψ ij = σ eij da, ϖ ij δα β = σ eij x α x β da, the iscoelastic part of the stress tensor due to compatibity conditions takes the particular form σ e = σ e σ e σ e33, 28 and, the corresponding wall shear stress τ is gien by τ = 4µ n πφ 3 Qt ρ 6πφ Qt 2πφ ψ 33 z 2 πφ 3 ϖ 33 z 2πφ ψ z 2 πφ 3 ϖ z. 29 Now, let us consider the following dimensionless ariables 5 ˆQ = ˆx = x φ, ˆφ =, ˆt = ω 0 t, 2ρ πφµ Q, ˆ p = φ2 ρ µ 2 p, ˆσ e = φ2 ρ µ 2 σ e, where ω 0 is a characteristic frequency for unsteay flow. ubstituting these dimensionless ariables 3 Equation 3 4 introduces some difficulties in handling the general case φ = φz. 4 δα β is the two-dimensional Kronecker symbol. 5 In cases where a steady flow rate is specified, the nondimensional flow rate ˆQ is identical to the classical Reynolds number used for flow in pipes, see Robertson and equeira 7]. into equations 27 and 3 4, we obtain, respectiely ˆ pẑ = 4 λ ˆQˆt 2 3 W2 0 ˆQˆt2 ˆψ33 ẑ 4 ˆϖ 33 ẑ ˆψ ẑ 4 ˆϖ ẑ, 30 and ˆσ eij W e ˆσeij =2λ ˆD ij 3 where W 0 = φ 2 ρω 0 /µ is the Womersley number, which reflect the unsteady flow phenomena, W e = λ ω 0 is the Weissenberg number, related with the flow iscoelasticity and D ij = µ φ 2 ρ ˆD ij, where D ij = 2 i,j j,i is the rate of deformation tensor. ubstituting the gien dimensionless ariables into equation 29, we obtain ˆτ = 2 λ ˆQˆtW0 2 ˆQˆt 2 2π ˆψ 33 ẑ 2 π ˆϖ 33ẑ 2π ˆψ ẑ 2 π ˆϖ ẑ, 32 where ˆτ = φ2 ρ µ 2 τ. Integrating condition 30 oer the interal ẑ, ẑ], with ẑ fixed, we obtain the following relationship between aerage pressure gradient and olume flow rate ˆ ppẑ, t = ˆ pẑ,t ˆ pẑ,t 33 = 4 λ Â ẑ ˆQˆt 2 3 W2 0Â2ẑ ˆQˆt 2 ˆψ33 ẑ,t ˆψ 33 ẑ,t ˆB 4 ẑ 4 ˆϖ 33 ẑ, t ˆϖ 33 ẑ,t ˆB5 ẑ ˆψ ẑ,t ˆψ ẑ,t ˆB6 ẑ 4 ˆϖ ẑ, t ˆϖ ẑ,t ˆB 7 ẑ, where Â ẑ =Â 2 ẑ = ˆB 5 ẑ = ˆB 6 ẑ =ẑ ẑ and ˆB 4 ẑ = ˆB 7 ẑ =ẑ ẑ. Now, considering equations 32 and 33 in the steady case and fixing a = in 6 we deal with the Oldrody-B fluid model. From, 28 and 3 dimensionless forms, we obtain ˆσ e = ˆσ e22 =0, 34

5 Proceedings of the 2006 IAME/WEA International Conference on Continuum Mechanics, Chalkida, Greece, May -3, 2006 pp6-66 Figure 2: Nondimensional aerage pressure gradient 33 in the steady case of an Oldroyd-B fluid for different alues of the Reynolds number ˆQ s =0.00, 0.5,, 5, 0, 20 and Weissenberg number W e =0.0, 0.,, 50, 00. Figure 3: Nondimensional wall shear stress 32 of an Oldroyd-B fluid in the steady case for different alues of the Reynolds number ˆQ s =5, 0, 20 and Weissenberg number W e =0.000, 0.00, 0.0, 0.. and ẑ ˆσ e33 = exp W e ˆQ s ˆx 2 ˆx Using 34, 35 and the approximation ẑ exp W e ˆQ s ζ 2 exp ẑ W e ˆQ s ζ 2 W e ˆQs exp ẑ, W e ˆQs we get ˆψ =ˆϖ =0, ˆψ 33 = π exp ẑ πẑ exp ẑ, W e ˆQ s 2 W e ˆQ s W e ˆQ s and ẑ W e ˆQs πẑ exp 6 W e ˆQ s ˆϖ 33 = 4 π exp ẑ. W e ˆQ s Again due to compatibility conditions, these results are only alid when λ 0, i.e. λ λ 2. hown in Fig.2 is the normalized nine-director aerage pressure gradient steady solution 33 for an Oldroyd-B fluid for different alues of Reynolds and Weissenberg numbers in 0, ẑ]. We conclude that the behaior of the steady solution with fixed Reynolds number does not change when we increase the Weissenberg number. Howeer, with fixed Weissenberg number we can obsere a slight change of the steady solution behaior, with increasing Reynolds number. Also, we compare the corresponding wall shear stress 32 for different alues of the Reynolds and Weissenberg numbers, see Fig.3, and conclude that it undergoes a small perturbation for ẑ close to zero and W e Howeer, for higher alues of the Weissenberg number the wall shear stress becomes constant. 4 Conclusion Contrarily to Newtonian, generalized Newtonian and second order fluids see e.g. 5], 7], 2], 3], 4], respectiely where the D director approach has been applied without restrictions to rectilinear flows, in the case of Oldroyd-B fluids, the D theory is only possible when the relaxation and retardation times are close to each other, i.e. λ λ 2. This is due to compatibility conditions imposed to system 3, as described aboe. One of the possible extensions of this work is the application of the D nine-director approach to other iscoelastic models, including generalized Oldroyd type fluids with shear-dependent iscosity and blood flow models in both rigid and flexible walled straight and cured essels as well as in essels with branches or bifurcations. This is the object of ongoing research. More detailed discussion of some of these issues can be found in ]. Acknowledgements: This work has been partially supported by the research cen-

6 Proceedings of the 2006 IAME/WEA International Conference on Continuum Mechanics, Chalkida, Greece, May -3, 2006 pp6-66 ters CEMAT/IT and CIMA/UE, through FCT s funding program and by projects POCTI/MAT/4898/200, HPRN-CT of the European Union. References: ] F. Carapau, Deelopment of D Fluid Models Using the Cosserat Theory. Numerical imulations and Applications to Haemodynamics, PhD Thesis, IT, ] F. Carapau, and A. equeira, Axisymmetric flow of a generalized Newtonian fluid in a straight pipe using a director theory approach, Proceedings of the 8th WEA International Conference on Applied Mathematics, 2005, pp ] F. Carapau, and A. equeira, D Models for blood flow in small essels using the Cosserat theory, WEA Transactions on Mathematics, Issue, Vol.5, 2006, pp ] F. Carapau, and A. equeira, Axisymmetric motion of a second order iscous fluid in a circular straight tube under pressure gradients arying exponentially with time, Proceedings of the ixth Wessex International Conference on Adances in Fluid Mechanics, May 2006, in press. 5] D.A. Caulk, and P.M. Naghdi, Axisymmetric motion of a iscous fluid inside a slender surface of reolution, Journal of Applied Mechanics, 54, 987, pp ] E. Cosserat, and F. Cosserat, ur la théorie des corps minces, Compt. Rend., 46, 908, pp ] E. Cosserat, and F. Cosserat, Théorie des corps déformables, Chwolson s Traité de Physique, 2nd ed. Paris, 909, pp ] P. Duhem, Le potentiel thermodynamique et la pression hydrostatique, Ann. École Norm, 0, 893, pp ] J.L. Ericksen, and C. Truesdell, Exact theory of stress and strain in rods and shells, Arch. Rational Mech. Anal.,, 958, pp ] C. Guillopé, and J.C. aut, Mathematical problems arising in differential models for iscoelastic fluids, in: Mathematical Topics in Fluid Mechanics, Pitman Research Notes in Mathematics series, edited by J.F. Rodrigues an A. equeira, Vol.274, pp.64, 992. ] A.E. Green, and P.M. Naghdi, A direct theory of iscous fluid flow in pipes: I Basic general deelopments, Phil. Trans. R. oc. Lond. A, 342, 993, pp ] A.E. Green, and P.M. Naghdi, A direct theory of iscous fluid flow in pipes: II Flow of incompressible iscous fluid in cured pipes, Phil. Trans. R. oc. Lond. A, 342, 993, pp ] A.E. Green, N. Laws, and P.M. Naghdi, Rods, plates and shells, Proc. Camb. Phil. oc., 64, 968, pp ] A.E. Green, P.M. Naghdi, and M.L. Wenner, On the theory of rods II. Deelopments by direct approach, Proc. R. oc. Lond. A, 337, 974, pp ] A.E. Green, and P.M. Naghdi, A direct theory of iscous fluid flow in channels, Arch. Ration. Mech. Analysis, 86, 984, pp ] P.M. Naghdi, The Theory of hells and Plates, Flügg s Handbuch der Physik, Vol. VIa/2, Berlin, Heidelberg, New York: pringer-verlag, pp , ] A.M. Robertson, and A. equeira, A Director Theory Approach for Modeling Blood Flow in the Arterial ystem: An Alternatie to Classical D Models, Mathematical Models & Methods in Applied ciences, 5, nr.6, 2005, pp ] C. Truesdell, and R. Toupin, The Classical Field Theories of Mechanics, Handbuch der Physik, Vol. III, pp , ] J.G. Oldroyd, On the Formulation of Rheological Equations of tate, Proc. Roy. oc. London, A200, 950, pp

Axisymmetric Flow of a Generalized Newtonian Fluid in a Straight Pipe Using a Director Theory Approach

Proceedings of the 8th WEA International Conference on APPLIED MATHEMATIC, Tenerife, pain, December 6-8, 2005 pp303-308) Axisymmetric Flow of a Generalized Newtonian Fluid in a traight Pipe Using a Director