A mathematical model for Bingham-like fluids with visco-elastic core

Transcription

1 Z. angew. Math. Phys. 55 (24) /4/ DOI 1.17/s c 24 Birkhäuser Verlag, Basel Zeitschrift für angewandte Mathematik und Physik ZAMP A mathematical model for Bingham-like fluids with visco-elastic core Lorenzo Fusi and Angiolo Farina Abstract. In this paper we present a new mathematical model for Bingham-like materials in which the core behaves as a visco-elastic Maxwell fluid. We deduce the model in a general 3D framework, using a thermodynamical approach based on the theory of natural configurations. We apply the model to the case of a plane Poiseuille flow driven by a time-dependent pressure gradient. The mathematical formulation of the latter case turns out to be a free boundary problem in which a parabolic equation and a dissipative wave equation are coupled together. Keywords. Bingham fluid, visco elastic fluid, constitutive equations, dissipation equation, free boundary problems. 1. Introduction According to the classical model Bingham fluids are non-newtonian fluids that behave like rigid solids if the applied stress is below a certain threshold (usually called yield stress and denoted by τ o ) and as viscous fluids if the stress exceeds that threshold. For the mathematical modelling of Bingham fluids we refer the reader to Bird, Stuart and Lightfoot [1], Rubinstein [2], Duvaut and Lions [3] and to numerous papers on the subject (see, for instance, [4], [5], [6], [7], [8]). Drilling muds, waxy crude oils, coal water slurries, toothpaste and ceramics are typical examples of materials that are usually modeled as Bingham fluids (see [9], [1], [11], [12]). We remark however that the Bingham model, even though in some cases provides a good schematization of the behavior of certain materials, fails in its basic assumption since a perfect rigid body can never exist. In other words, we never have a rigid fluid transition. For this reason in a previous paper [13] we tried to modify the usual model considering the material in the solid region no longer rigid, but deformable. In particular, we modeled the solid as a nonlinear neo-hookean elastic material. In this work we further extend the model developed in [13] treating the material in the non fluid region as a visco-elastic fluid. As in [13] we assume that the body is divided into two domains by a sharp interface whose evolution is not a priori known. We suppose that in one region the material behaves like a viscous incom-

2 Vol. 55 (24) Bingham-like fluids with visco-elastic core 827 pressible fluid, whereas in the other like a visco-elastic upper convected Maxwell fluid (see [14]). The transition between the viscous and the visco elastic behavior is governed by a parameter a that we call transition parameter. If a exceeds a certain threshold a o (related to the yield stress τ o ) the continuum is modeled as a viscous fluid while if a<a o a visco elastic constitutive equation applies. In Section 2 we deduce the general 3D model operating in the context of the theory of multiple natural configurations (see Rajagopal et al. [15], [16], [17], [18], [19]). Then, in Section 3, we consider the simple case of a channel flow driven by a pressure gradient (Poiseuille flow in planar geometry). The flow in the viscous domain is governed by a parabolic equation, while in the visco-elastic domain by a hyperbolic equation with a dissipative term (telegraph equation). We do not consider the mathematical analysis of the problem concerning existence, uniqueness etc., but we limit ourselves to an analysis of some simplified cases. 2. The general model 2.1. Preliminary results Let us assume that the material a time t = occupies the domain K R and that the motion is given by a continuously differentiable mapping x = χ( p, t), where p denotes the Lagrangian coordinates while x the Eulerian coordinates. In particular, x K t, where K t is the spatial domain occupied by the system at time t. We introduce the deformation gradient 1 F :=Gradχ( p, t), the velocity gradient 2 and its symmetric part We introduce also D := 1 2 L := v, ( L + L T ). B := FF T, C := F T F, 1 Grad denotes the gradient operator with respect to coordinates p, while denotes the gradient operator with respect to coordinates x. 2 Recall that it can be shown L = ḞF 1. The superposed dot means differentiation along a particle path, i.e. the material derivative f = f + v f. t

3 828 L. Fusi and A. Farina ZAMP Figure 1. Schematic diagram of the reference, actual and natural configurations. called left and right Cauchy-Green stretch tensors, respectively. Assuming incompressibility, we have v =trd =. (2.1) We suppose that there exists a surface Γ(t) K t, evolving in time, that separates the viscous region from the visco-elastic region. In other words, denoting by 1 and 2 the viscous phase and the visco elastic phase, respectively, we assume that K t = K 1t K 2t, and K 1t K 2t =, where K it, i =1, 2, are the current configurations of the viscous and visco elastic phases, respectively. Looking at the picture in Fig. 1, we introduce, for each phase, the relative natural configuration K ip, i =1, 2 (see [15], [19]). Roughly speaking the above means that we consider a decomposition of the deformation tensor F in the product F 1 = F 1p G 1, (2.2)

4 Vol. 55 (24) Bingham-like fluids with visco-elastic core 829 F 2 = F 2p G 2, (2.3) where F i is simply the restriction of F to the i-th phase. Both decompositions have an interesting physical interpretation. Focusing, for instance, on the viscous phase, the natural configuration K 1p can be thought as the configuration to which the body would tend if it were instantaneously unloaded. However, to be precise, in general we should not speak of configurations because decompositions (2.2) and (2.3) do not automatically imply that there exist global mappings χ ip and χ ig, i =1, 2, such that 3 χ i = χ ip χ ig and Gradχ ip = F ip,gradχ ig = G i. So, whenever we refer to natural configurations we just intend decompositions (2.2) and (2.3). We recall also that tensors F ip are defined up to rigid rotations (see [19]) so that we can write F ip = V ip = U ip, i =1, 2, (2.4) where V ip, U ip are positive definite symmetric tensors such that Vip 2 = FT ip F ip and U 2 ip = F ipf T ip (see the decomposition Theorem in [2]). Following standard kinematical results on natural configurations theory we introduce and L ig :=Ġ i G 1 i D ig := 1 2 (L ig + L T ig), i =1, 2, It can be shown (see [19]) that B ip :=F ip F T ip, C ip =: F T ipf ip, i =1, 2. Ḃ ip = LB ip + B ip L T 2F ip D ig F T ip, i =1, 2, (2.5) and that 4 I Ḃ ip =2B ip (D D ig), i =1, 2. (2.6) Following [14], we introduce the frame indifferent upper-convected time derivative Bip= Ḃ ip LB ip B ip L T, i =1, 2, (2.7) that recalling (2.4) and (2.5) gives Bip= 2V ip D ig V ip, i =1, 2. (2.8) 2.2. Constitutive assumptions Following [21] we write the dissipation equation 3 χ i is the restriction of the mapping χ to the i-th phase. 4 Recall that A B =tr ( A T B ). ξ = T D ψ q T, (2.9)

5 83 L. Fusi and A. Farina ZAMP where ξ is the rate of dissipation, ψ is the stored energy function, T D is the stress power, i.e. the rate at which internal work is done, q is the heat flux and T is the absolute temperature. Assuming isothermal conditions, (2.9) reduces to ξ = T D ψ. (2.1) A proper choice of ξ and ψ will allow us to derive the constitutive equations for the two phases and to model the transition at the interface. Since we are dealing with a Bingham like material, there exists a yield stress τ o to be overcome in order to have purely viscous behavior. We therefore introduce a parameter a (function of some quantities characterizing the system), that we call transition parameter, and a threshold value a o, which in turn will be a function of τ o. If a a o we model the material as a viscous incompressible fluid, while if a<a o as a visco-elastic fluid (more in detail, we model the visco-elastic phase on the basis of the upper convected Maxwell model). The above, operating in the context of the natural configurations theory, means that the transition from the viscous behavior to the visco elastic behavior is essentially due to a different evolution of the body s natural configurations. In particular, we stipulate ( ) 2 η a := D ig D ig, i =1, 2, (2.11) µ and set a o := τ o 2 2µ 2, (2.12) where η and µ are two positive constants 5 representing the viscosity and the elastic modulus, respectively. We assume that the dissipation ξ and the stored energy ψ are 6 ξ = H(a o a)[2ηd 2G B 2pD 2G]+H(a a o )[2ηD 1G D 1G], (2.13) [ µ ] ψ = H(a o a) 2 (I B 2p 3) + H(a a o )ψ o, (2.14) where ψ o > is a constant and H(x) is the Heaviside function 1 if x, H (x) := if x<. To model the evolution of the system we need to determine the law that governs the evolution of the natural configurations, namely we need to determine tensors D 1G and D 2G. The dissipation equation (2.1), considered separately in each phase, does not suffice itself to isolate D 1G or D 2G. To select such tensors we use the so-called maximization of the rate of dissipation criterion (see [17] [19]). 5 [η] =Pa s and [µ] =Pa. 6 We underline that constitutive relations (2.13) and (2.14) are assumptions. We are assuming that the material dissipates and stores energy according to (2.13) and (2.14).

6 Vol. 55 (24) Bingham-like fluids with visco-elastic core The viscous phase In the viscous phase a a o, H(a o a) =andh(a a o ) = 1, so that From (2.1) we get ξ = 2ηD 1G D 1G, (2.15) ψ = ψ o. (2.16) T D =2ηD 1G D 1G, that, assuming incompressibility (i.e. tr(d) = ), gives { T = P I+2ηD, (2.17) D = D 1G, where P is the Lagrange multiplier arising from incompressibility. The second of (2.17) tells us that in the viscous phase the natural configuration coincides with the actual one, i.e. at time t the natural configuration is simply the one occupied by the material. Notice that in this simple situation there is no need to apply the maximization criterion. Recalling (2.11) and (2.12) we have a = η2 µ 2 D D τ o 2 2µ 2, (2.18) that is D D τ o 2 2η 2. (2.19) Thus in the viscous phase (i.e. in K 1 t ) the above inequality has to be satisfied The visco-elastic phase In the visco-elastic phase a<a o, H(a o a) =1andH(a a o )=. Wehave ξ = ξ (D 2G, B 2p )=2ηD 2G B 2p D 2G, (2.2) ψ = ψ (B 2p )= µ 2 (I B 2p 3). (2.21) Equation (2.1) becomes ξ (D 2G, B 2p )=T D µ 2 I Ḃ 2p. By virtue of (2.6) (T µb 2p ) D =ξ (D 2G, B 2p ) µb 2p D 2G, from which, since the right-hand side does not depend on D, we immediately get T = P I+µB 2p, (2.22)

7 832 L. Fusi and A. Farina ZAMP and ξ (D 2G, B 2p )=(T+PI) D 2G, (2.23) where P is the Lagrange multiplier due to incompressibility. Following [17] [19], the maximization of the rate of dissipation criterion says that the tensor D 2G that regulates the evolution of the natural configurations is the one that maximizes the dissipation ξ given by (2.2) under constraint (2.23) and constraint I D 2G =. (2.24) The latter has to be fulfilled since the material is incompressible in every configuration and K 2 p is one of the possible configurations of the body. By standard techniques of constrained maximization, the problem of finding D 2G reduces to solving the following implicit equation ( ) ξ ξ + λ 1 T + λ 2 I =, (2.25) D 2G D 2G where λ 1 and λ 2 have to be determined imposing (2.23) and (2.24) and where B 2p is kept fixed. Recalling (2.2) and (2.22), (2.25) rewrites 2η (1 + λ 1 )(D 2G B 2p + B 2p D 2G )=λ 1 T λ 2 I. Since D 2G and B 2p commute (see appendix A of [19]), we have T = 4η (1 + λ 1) B 2p D 2G + λ 2 I. (2.26) λ 1 λ 1 Inserting (2.26) in (2.23) we find that λ 1 = 2, so that T =2ηB 2p D 2G λ 2 I. (2.27) 2 Substituting (2.22) into (2.27) yields 2 η µ D 2G= λb 1 2p + I, (2.28) where λ := 1 ( P λ ) 2. µ 2 Applying the trace operator to both sides of (2.28) and recalling (2.24) we obtain 3 λ = tr B 1, (2.29) 2p that is λ 2 =2P 6µ tr B 1. 2p Eventually, by virtue of (2.4) and (2.8), from (2.28) we have B 2p + B2p =λ I, (2.3)

8 Vol. 55 (24) Bingham-like fluids with visco-elastic core 833 where 7 = η µ. (2.31) So, in the visco-elastic phase the stress tensor is given by (2.22) with B 2 p evolving according to (2.3). At this stage the analogy with the upper convected Maxwell model is not clear, since in the latter the deviatoric part of the stress S evolves according to a constitutive equation of the type S + α S =2βD, (2.32) where α, β > are two given constants (see [14]). By the way, following section 5.3 of [19], we see that the constitutive equation (2.32) can be put in a form analogous to (2.3), so that the only difference between the two constitutive relations is the coefficient λ, which in (2.3) is not constant but depends on B 2p (see (2.29)). Such a difference will be eliminated (see next section) by linearizing the elastic response from the natural configuration. We eventually recall that Definition (2.11) gives [ a = 9 B 1 2p ] B 1 2p ( ) 4 tr B 1 2 1, (2.33) 3 2p which, since we are in the visco elastic phase where a<a o, implies [ 9 B 1 2p ] B 1 2p ( ) 2 tr B ( ) 2 τo <. (2.34) 3 µ 2p 2.5. Linearized elastic response Let us set 8 (B 2p ) ij = δ ij + B ij, i, j =1, 2, 3, namely 1+B 11 B 12 B 13 B 2p = B 12 1+B 22 B 23, (2.35) B 13 B 23 1+B 33 and assume B ij = O (ε) i, j =1, 2, 3, with ε 1. (2.36) From (2.24) ) I L 2G =tr (Ġ2 G 1 2 =, and since ) det (G 2 )= det (G 2 )tr (Ġ2 G 1 2 =, 7 [] =s. 8 δ ij denotes the Kronecker delta.

9 834 L. Fusi and A. Farina ZAMP decomposition (2.3) and (2.1) yield tr (Ḟ2p F 1 2p ) =, or (see [2]) det (B 2p )=1. At this point, exploiting (2.35), we have which implies We thus conclude 3 1=1+ B ii + O ( ε 2), i=1 3 B ii = O ( ε 2). i=1 tr B 2p =3+O ( ε 2). In an analogous manner, setting ( ) B 1 2p = δ ij ij + B ij, i, j =1, 2, 3, we have B ij = O (ε), and 3 B ii = O ( ε 2), i=1 Neglecting O ( ε 2) terms in (2.29) yields tr B 1 2p =3+O ( ε 2). (2.37) λ =1, and λ 2 =2(P µ). (2.38) We have therefore recovered the usual upper-convected Maxwell model. Remark 2.1. Evaluating B 1 2p B 1 2p under condition (2.36) we get B 1 2p B 1 2p =3+O ( ε 2), which inserted in (2.33), with the aid of (2.37), entails Condition (2.34) is thus verified. a =+O ( ε 2).

10 Vol. 55 (24) Bingham-like fluids with visco-elastic core 835 Figure 2. Geometry of the channel upper part. 3. A one-dimensional Poiseuille flow In this section we consider a 1D Poiseuille flow driven by a time dependent pressure gradient. We suppose that the fluid is confined between two parallel planes whose distance is 2L. Due to symmetry reasons we study only the upper part of the channel (see Fig. 2) Kinematics and motion equations Considering laminar flow, the viscous and visco elastic region are separated by the horizontal time-dependent interface y = s (t). In particular, the viscous domain is s(t) <y<l, while the visco-elastic one is <y<s(t). We will treat the latter assuming (2.36), i.e. assuming that the elastic response from the natural configuration K 2 p can be linearized. In both phases the velocity field is directed along the x axis and is a function of coordinate y and time t, thatis v = v(y, t) e x, s(t) <y<l, (3.1) v = u(y, t) e x, <y<s(t). (3.2) We recall that the surface s(t) is a priori unknown.

11 836 L. Fusi and A. Farina ZAMP Let us consider the viscous phase first. We have 9 L = v y, D = 1 v y v y, D 2 = 1 vy 2 vy From (2.18) Inequality (2.19) thus implies where D 1 denotes the domain a = 1 2 ( ) 2 ηvy. µ. η v y τ o (y, t) D 1, (3.3) D 1 = {s (t) <y<l, <t}. (3.4) At the interface v y (s(t),t) = τ o η. (3.5) Writing the equation of motion (in the absence of body forces) component-wise, we get Conditions (3.7) and (3.8) yield p = p(x, t). From (3.6) ρv t ηv yy = P x, (3.6) = P y, (3.7) = P z. (3.8) P = f(t)x + g(t), (3.9) for some f and g. In particular, f can be determined once we know the pressure drop between two fixed points, say x 1 and x 2. We assume f (t). The dynamics of the fluid in the viscous phase is governed by the following parabolic equation ρv t ηv yy = f(t), (y, t) D 1. (3.1) Let us now consider the visco-elastic phase. Since the velocity field depends only upon y and is directed along the x axis, we assume B 2p = B 2p (y, t). We get Ḃ 2p = B 2p t +( v ) B 2p = B 2p, t since ( v ) B 2p =. Equation (2.3) reduces to B 2p + ( B2p t LB 2p B 2p L T ) = λi, 9 The y and t suffices stand for partial differentiation, i.e. f y = f y and ft = f t.

12 Vol. 55 (24) Bingham-like fluids with visco-elastic core 837 where is given by (2.31) and where λ = 1, since we are working under assumption (2.36). By calculations and recalling (2.35), we get B 2p + 1 t B 2p u y 2B 12 1+B 22 B 23 1+B 22 = (3.11) B 23 1 Since tensor B 2p is symmetric, system (3.11) can be written as B t B 11 2u y B 12 =, (3.12) B t B 12 u y (B 22 +1)=, (3.13) B t B 13 u y B 23 =, (3.14) B t B 22 =, (3.15) B t B 23 =, (3.16) B t B 33 =, (3.17) As initial condition for B 2p, we consider B 2p (y, ) = I, that is Integration of system (3.12)-(3.17) gives t B 11 (y, t) =2 B ij (y, ) =, i, j =1, 2, 3. B 12 (y, t) = ( θ t u y (y, θ) B 12 (y, θ)exp t ) dθ, (3.18) ( ) θ t u y (y, θ)exp dθ, (3.19) B 13 =, (3.2) B 22 =, (3.21) B 23 =, (3.22) B 33 =. (3.23) Let us now write the equation of motion for the visco-elastic phase. Bearing in mind that (T) ij = Pδ ij + µb ij, i,j =1, 2, 3,

13 838 L. Fusi and A. Farina ZAMP where B ij are given by (3.18) (3.23) and where ( P + µ) has been replaced with P,weget t ρu t = P x + µ ( θ t u yy (y, θ)exp ) dθ, (3.24) = P y, (3.25) = P z. (3.26) For the same arguments previously used we can show that P = P (x, t) = F (t) x + G (t). (3.27) So, in the visco-elastic phase the velocity field u satisfies the following equation 1 c 2 u t = F (t) t ( ) θ t µ + u yy (y, θ)exp dθ, (y, t) D 2, (3.28) where D 2 = { <y<s(t), <t}, (3.29) and where µ c = ρ. (3.3) After multiplying both sides by exp (t/), equation (3.28) can be differentiated with respect to time yielding 1 c 2 ( u tt + 1 u t ) u yy = 1 [ µ F (t)+ 1 ] F (t), (3.31) which is a dissipative wave or telegraph equation (see [22]). The mathematical model (3.31) has been deduced operating under assumption (2.36). We now have to point out which conditions ensure the validity of (2.36). Scaling the variables y, t and u with L, and U (U is the characteristic velocity), respectively, from (3.18) and (3.19) we have that B 12 = O (ε) andb 11 = O ( ε 2) if ε = U 1, (3.32) L and if u y remains bounded. Remark 3.1. We notice that letting η and keeping the elastic modulus µ finite (which is equivalent to letting ), equation (3.31) becomes 1 c 2 u F tt u yy = (t) µ. (3.33) If ϕ(y, t) denotes the displacement in the visco-elastic region, then ϕ t = u and (3.33) becomes 1 c 2 ϕ tt ϕ yy = 1 (F (t)+c), µ

14 Vol. 55 (24) Bingham-like fluids with visco-elastic core 839 for some constant C. So in this limiting case the region a<a o behaves as a neo-hookean material and we recover the model extensively treated in [13] Boundary conditions From classical Rankine-Hugoniot conditions (see [23]) we know that any quantity ζ satisfying a balance equation of the type (ρζ) + (ρζ v) = π + ω, t obeys to the following jump relation across a discontinuity surface [ρζ ( v w) π ] + n =[ρζ ( v w) π ] n, (3.34) where w is the velocity of the surface, n is the normal to the surface and [ ] +,[ ] stand for the limits evaluated in the regions that contain n and n, respectively. In our specific case the discontinuity surface is y = s(t) and n = e y, w = ṡ (t) e y, so that we identify the viscous and the visco-elastic phase with + and respectively. In the sequel we write (3.34) for the cases in which ζ represents density, momentum along the x axis, momentum along the y axis and energy. Density. In this case so that equation (3.34) rewrites yielding ζ =1, [ v ] + = v e x, [ v ] = u e x, π =, ρ + (v e x ṡ e y ) e y = ρ (u e x ṡ e y ) e y, ρ + = ρ, i.e. the continuity of density across the interface. Momentum along the x direction. Now we have and equation (3.34) becomes [ζ] + = v, [ζ] = u, [ π ] + = T + e x, [ π ] = T e x. [ρv (v e x ṡ(t) e y )+P + e x ηv y e y ] e y = [ρu (u e x ṡ(t) e y )+(P µb 11 ) e x µb 12 e y ] e y, or equivalently ρvṡ(t) ηv y = ρuṡ(t) µb 12.

15 84 L. Fusi and A. Farina ZAMP Assuming a no-slip condition at the interface u (s (t),t)=v(s(t),t), (3.35) and recalling (3.19) we get t ηv y (s (t),t)=µ ( θ t u y (s (t),θ)exp ) dθ, (3.36) that, taking (3.5) into account, yields t ( ) τ o = µ θ t u y (s (t),θ)exp dθ. (3.37) Momentum along the y direction. In this case ζ =, [ π ] + = T + e y, [ π ] = T e y, that is ηv y µb 12 [ π ] + = P +, [ π ] = P. We get [ ηvy e x P + e y ] ey = [ µb 12 e x P e y ] ey, that expresses the continuity of the pressure, namely P + = P. (3.38) From the above, recalling (3.9) and (3.27), we obtain f(t) =F (t). Energy. According to (2.16) and (2.21) we have [ζ] + = 1 2 v 2 + ψ o, [ζ] = 1 2 u 2 + µ 2 (I B 2p 3), and [ π ] + = T + v = P + v ηv y v, [ π ] = T u = Substituting into equation (3.34) we get ( ) 1 ρ ṡ 2 v 2 + ψ o ηv y v = ρ ṡ P v + µvb 11 µvb 12 [ 1 2 u 2 + µ 2 (I B 2p 3). ] µ vb 12, which, together with (3.18), (3.19), (3.35), (3.36) and (3.38) fixes the internal energy of the fluid ψ o = τ 2 o µ.

16 Vol. 55 (24) Bingham-like fluids with visco-elastic core 841 We conclude the section specifying the boundary conditions on y = L and on y =. Ony = L we assume a no slip condition v (L, t) =, (3.39) while on y =, for symmetry reasons, we impose u y (,t)=. (3.4) 3.3. The mathematical model We are now able to state the mathematical problem for the one-dimensional case. We do not perform a mathematical study concerning existence, uniqueness, continuous dependence on data etc., but limit ourselves to write the problem in a non dimensional form, studying some limiting cases (see next section). We give two formulations of the problem. In the first formulation, referred to as problem (P int ), the equation for u and the boundary condition on s(t) are given in an integral form. We thus have ρv t ηv yy = f(t), (y, t) D 1, v(l, t) =, <t, v(s(t),t)=u(s(t),t), <t, v(y, ) = v o (y), s o <y<l, v y (s(t),t) = τ o η, <t, (P int ) 1 c 2 u t = f(t) t ( ) θ t µ + u yy (y, θ)exp dθ, (y, t) D 2, t ( ) θ t u y (s (t),θ)exp dθ = τ o µ, <t, u y (,t)=, <t, u(y, ) = u o (y), <y<s o, where s() = s o,<s o <L,and where v o (y) andu o (y) are the initial conditions for v and u, respectively. We remark that if we assume 1 v o (L) =, v o (s o )= τ o η the boundary conditions (3.5) and (3.36) become and v o (y) τ o η y [s o,l], v y (s(t),t)= τ o η, (3.41) 1 The choice v o (so) = τo/η is reasonable from a physical point of view since velocity is expected to decrease as y approaches the wall y = L, where a no slip condition is imposed.

17 842 L. Fusi and A. Farina ZAMP and t ( ) θ t u y (s (t),θ)exp dθ = τ o µ, (3.42) respectively. If we assume enough regularity for the velocity u, problem (P int )canbereformulated differentiating with respect to time equation (3.28) and boundary condition (3.42). The above allows to formulate the problem in a differential form, namely (P diff ) ρv t ηv yy = f(t), (y, t) D 1, v (L, t) =, <t, v (s (t),t)=u(s(t),t), <t, v(y, ) = v o (y), s o <y<l, v y (s(t),t)= τ o ( η, <t, 1 (P diff ) c 2 u tt + 1 ) u t u yy = 1 [ f(t)+ 1 ] µ f(t), (y, t) D 2, u y (s(t),t)+ 1 c 2 u t(s(t),t)ṡ(t) = 1 [ f (t)ṡ (t) τ ] o, <t, µ u y (,t)=, <t, u(y, ) = u o (y), <y<s o, u t (y, ) = f() ρ, <y<s o. Notice that the last condition comes from setting t = in (3.28). Remark 3.2. To prove that condition (3.3) is fulfilled in D 1, we consider the problem for z(y, t) :=v y (y, t), namely ρz t ηz yy =, (y, t) D 1, z y (L, t) = f (t) η, <t, z (s (t),t)= τ o η, <t, z(y, ) = v o(y), s o <y<l. By the maximum principle, if (y,t ) D 1, t > is a point such that z(y,t ) > τ o η, i.e. v y (y, t) > τ o, we deduce that z achieves a maximum at the point ( η L, t ), < t t. The parabolic version of Hopf s Lemma requires z y (L, t ) >

18 Vol. 55 (24) Bingham-like fluids with visco-elastic core 843 which contradicts f (t) η domain D 1.. Thus, condition (3.3) is fulfilled everywhere in the 3.4. Two particular cases Recalling the scaling introduced in Section 3.1, let us consider v = Uṽ, u = Uũ, t = T t, y = Lỹ, s = L s, f = f o f. and 11 T v = ρl2 η, (3.43) T e = L c, that can be interpreted as the characteristic times of the viscous and elastic regions, respectively. Problem (P diff ) becomes ( T v T v fo L 2 ) t v yy = f, ηu v(1,t)=, v(s(t),t)=u(s(t),t), v(y, ) = v 1 (y) v y (s(t),t)= τ ol ηu, ( ) 2 Te u tt + T ( v T T u fo L 2 )( df t u yy = µut dt + T ) f (3.44), ( ) 2 ( Te fo L 2 ) u y (s(t),t)+ ṡu t = ṡf τ ol T µut ηu, u y (,t)=, u(y, ) = u ( 1 (y), ) fo T u t (y, ) = f (), ρu where we have omitted the tildes and where v 1 (ξ) = v o(lξ) U, u 1(ξ) = u o(lξ) U. We now select T = T v and consider 11 Notice that Tv T 2 e =1. U = f ol 2 η, (3.45)

19 844 L. Fusi and A. Farina ZAMP as characteristic velocity. Bearing in mind that condition (3.32) has to hold, we have f o L µ 1. Assuming T e 1, T v problem (3.44) reduces to 12 v t v yy = f (t), v(y, ) = v 1 (y), v(1,t)=, v y (s(t),t)= τ o f o L, u t u yy = df T v dt + f (t), u(y, ) = u 1 (y), u y (,t)=, u(s(t),t)=v(s(t),t), ṡf(t) =u y (s(t),t)+ τ o T v f o L, s () = s o, (3.46) which is a free boundary involving two parabolic equations (we refer the reader to [24], [25], [26], and [27] for classical literature). We remark that (3.46) presents a peculiar characteristic: The datum on the free boundary s(t) is not prescribed. Indeed, such a problem cannot be included in any of the two-phase free boundary problems classified in [28]. The qualitative analysis of such a problem will be presented in a forthcoming paper. Assuming (3.45), if we select T = T e and suppose that T e 1, T v the equation for v in (3.44) reduces to whose solution is v(y, t) = f (t) 2 v yy = f (t), ( 1 y 2 ) ( + f (t) s (t) τ ) o (y 1). f o L 12 Notice that if f is constant the equations for the velocity fields v and u are the same.

20 Vol. 55 (24) Bingham-like fluids with visco-elastic core 845 Thus problem (3.44) becomes u tt u yy = df T e dt + f (t), u(s(t),t)= f (t) ( s 2 1 ) τ o (s 1), 2 f o L u y (,t)=, u(y, ) = u 1 (y), u t [ (y, ) = f(), ṡ u t (s(t),t) ] f (t) = u y (s(t),t) τ o T e f o L, s () = s o, (3.47) which is a free boundary problem involving a single hyperbolic equation. Existence and uniqueness in the large of problem (3.47) can be proved following [13]. As a matter of fact, problem (3.47) is a simplified version of the free boundary problem we studied in [13], where the hyperbolic equation was coupled with a parabolic one. Here we deal with only an hyperbolic equation coupled with an explicit evolution equation for s(t). The only difference from [13] is that now the hyperbolic equation has a source term. Obvious changes in the assumptions on the data and in the representation formula lead to classical solvability of problem (3.47). 4. Conclusions In this paper we have developed a new model for Bingham-like materials whose core undergoes deformations. In particular, the core has been treated as a viscoelastic upper convected Maxwell fluid. The model has been deduced in a general 3D framework operating within the context of the theory of multiple natural configurations. Key point of the model is that the transition from the viscous to the visco-elastic behavior is governed by a parameter depending on the natural configurations evolution. We have selected a threshold value (tied to the yield stress of the material) such that the viscous or visco-elastic behavior depends on whether the parameter is larger, or not, than such a threshold. To derive the constitutive equations of the medium we have used a thermodynamical approach, selecting, as constitutive assumptions, the way in which energy is stored and dissipated. We have then considered the special case of a one-dimensional Poiseuille flow driven by a time-dependent pressure gradient. The resulting mathematical problem has turned out to be a free boundary problem - with non-standard boundary conditions - in which a dissipative wave equation (or telegraph equation) is coupled with a parabolic equation. Such a problem has been put in a dimensionless form and some special simplified situations have been considered.

21 846 L. Fusi and A. Farina ZAMP Acknowledgment We are indebted with Prof. A. Fasano (Università degli Studi di Firenze) for several fruitful discussions about the content of this work. This research was funded by the project P.R.I.N. Problemi a Frontiera Libera from the Italian University and Research Ministry (MIUR). References [1] R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, Wiley, New York 196. [2] L. Rubinstein, The Stefan Problem, American Mathematical Society Translation, 27, American Mathematical Society, Providence RI [3] G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219, Springer Verlag, Berlin [4] E. Comparini, A one-dimensional Bingham flow, J. Math. Anal. Appl. 169 (1992), [5] E. Comparini and E. De Angelis, Flow of a Bingham fluid in a concentric cylinder viscometer, Adv. Math. Sci. Appl. 6 (1996), [6] A. Fasano, M. Primicerio and F. Rosso, On quasi steady axisymmetric laminar flows of Bingham type with stress induced degradation, Computing 49 (1992), [7] J.U. Kim, On the Cauchy problem associated with the motion of a Bingham fluid in the Plane, Trans. Amer. Math. Soc. 298 (1986), [8] J.U. Kim, On the initial boundary value problem for a Bingham fluid in a three dimensional domain, Trans. Amer. Math. Soc. 34 (1987), [9] L.T. Wardhaugh and D.V. Boger, Measurement of the unique flow properties of waxy crude oils, Chem. Eng. Res. Des. 65 (1987), [1] L.T. Wardhaugh and D.V. Boger, Flow characteristics of waxy crude oils: Application to pipeline design, AIChE 37 (1991), [11] R.N. Tuttle, High-pour point and asphaltic crude oils and condensates, Journal of Petroleum Technology 94 (1983), [12] A. Terenzi, E. Cariniani, E. Donati and D. Ercolani, Problems of nonlinear fluid dynamics in industrial plants, in Complex Flows in Industrial Processes, Ed. A. Fasano, Birkhäuser, Boston 2, pp [13] L. Fusi and A. Farina, An extension of the Bingham model to the case of an elastic core, Advances Math. Sci. Appl., 13 (23), [14] J.G. Oldroyd, On the formulation of rheological equations of state, Proc. Roy. Soc. London A2 (195) [15] K.R. Rajagopal, Multiple configurations in continuum mechanics, Reports of the Institute for Computational and Applied Mechanics, University of Pittsburgh, 6 (1995). [16] K.R. Rajagopal and A.S. Wineman, A constitutive equation for non linear elastic materials which undergo deformation induced microstructural changes, Int. J. Plasticity 8 (1992), [17] K.R. Rajagopal and A.R. Srinivasa, Mechanic of inelastic behavior of materials. Part I: Theoretical underpinnings, Int. J. Plasticity 14 (1998), [18] K.R. Rajagopal and A.R. Srinivasa, Mechanic of inelastic behavior of materials. Part II: Inelastic response, Int. J. Plasticity 14 (1998), [19] K.R. Rajagopal and A.R. Srinivasa, A thermodynamic frame work for rate type fluid models, J. Non Newtonian Fluid M. 88 (2), [2] M.E. Gurtin, An Introduction to Continuum Mechanics, Mathematics in Science and Engineering, 158, Academic Press, Boston 1981.

22 Vol. 55 (24) Bingham-like fluids with visco-elastic core 847 [21] C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Flgge s Handbuch der Physik, III/3, Springer Verlag, Berlin [22] I. Rubinstein and L. Rubinstein, Partial Differential Equations in Classical Mathematical Physics, Cambridge University Press, Cambridge [23] J. Coirer, Mcanique des Milieux Continus, Dunod, Paris [24] A. Fasano and M. Primicerio, General free boundary problems for heat equation, I, J. Math. Anal. Appl. 57 (1977), [25] A. Fasano and M. Primicerio, General free boundary problems for heat equation, II, J. Math. Anal. Appl. 58 (1977), [26] A. Fasano and M. Primicerio, General free boundary problems for heat equation, III, J. Math. Anal. Appl. 59 (1977), [27] A. Fasano and M. Primicerio, Free boundary problems for nonlinear parabolic equations with nonlinear free boundary conditions, J. Math. Anal. Appl. 72 (1979), [28] A. Fasano and M. Primicerio, Classical solutions of general two-phase parabolic free boundary problems in one dimension, in Free Boundary Problems: Theory and Applications, Eds. A. Fasano and M. Primicerio, Research Notes in Mathematics, 79/II, Pitman, Boston 1983, pp Lorenzo Fusi and Angiolo Farina Dipartimento di Matematica U. Dini Università degli Studi di Firenze Viale Morgagni 67/A, 5134 Firenze Italy fusi@math.unifi.it, farina@math.unifi.it (Received: March 31, 23)