Stability of plane Poiseuille Couette flows of a piezo-viscous fluid

Transcription

1 Stabilit of plane Poiseuille Couette flows of a piezo-viscous fluid Thien Duc Tran a and Serge A. Suslov b, a Computational Engineering and Science Research Centre, Universit of Southern Queensland, Toowoomba, Queensland 435, Australia b Department of Mathematics and Computing and Computational Engineering and Science Research Centre, Universit of Southern Queensland, Toowoomba, Queensland 435, Australia Abstract We examine stabilit of full developed isothermal unidirectional plane Poiseuille Couette flows of an incompressible fluid whose viscosit depends linearl on the pressure as previousl considered in [,2]. Stabilit results for a piezo-viscous fluid are compared with those for a Newtonian fluid with constant viscosit. We show that piezo-viscous effects generall lead to stabilisation of a primar flow when the applied pressure gradient is increased. We also show that the flow becomes less stable as the pressure and therefore the fluid viscosit decrease downstream. These features drasticall distinguish flows of a piezo-viscous fluid from those of its constant-viscosit counterpart. At the same time the increase in the boundar velocit results in a flow stabilisation which is similar to that observed in Newtonian fluids with constant viscosit. Ke words: pressure-dependent viscosit, piezo-viscous fluid, plane Poiseuille Couette flow, hdrodnamic stabilit Introduction Piezo-viscous fluids (PVF) form an interesting non-newtonian class. Polmer melts [3], lubricating oil in journal bearings [4,5] and magma in the Earth core are just a few examples of such fluids. In their flows, the shear stress (and effective viscosit) increases with the increasing normal stress. The properties Corresponding author address: ssuslov@usq.edu.au (Serge A. Suslov). Preprint

2 of piezo-viscous fluids were studied b a number of authors over the past two decades [6 8, e.g.]. The simplest rheological model which accounts for such a behaviour assumes that the viscosit of a fluid is proportional to the pressure. Perhaps the most comprehensive ccle of recent works studing this model both mathematicall and numericall is b Hron, Málek, Rajagopal and collaborators [,9 3]. The major goals of the analsis presented in this paper are to expose the phsical mechanism of flow instabilit in such fluids and to shed more light onto the existence of simple parallel flows of piezo-viscous fluids. Simple stead plane parallel flows of piezo-viscous shear-independent and shear-thinning fluids were first considered in []. In a later work [4] it was shown that simple parallel piezo-viscous flows can onl exist if the viscosit is a linear function of pressure. Therefore this rheological model will be used in the current work. Our previous stud [2] of such a model showed that shearthinning effects do not lead to an qualitative changes in simple flow structure in comparison with the solutions obtained for shear-independent fluids. Yet piezo-viscous effects result in completel new flow features such as choking of the pressure-driven flow and the existence of a limiting linear velocit profile for high-speed Couette flow of piezo-viscous fluid. Therefore the current work is full focused on the conceptual investigation of piezo-viscous effects on the stabilit of Poiseuille Couette flow of a fluid with shear-independent viscosit. This will enable us to compare our results directl with the well-known stabilit findings for Couette Poiseuille flows of conventional Newtonian fluids with pressure-independent viscosit (for brevit we will refer to such fluids as NF in this paper) investigated previousl [5 8]. We will report at least two new features (flow stabilisation with increasing pressure gradient and a downstream flow destabilization) which are not observed in conventional NF flows. We will also show that similarl to NF flows PVF Couette flow remains stable with respect to infinitesimal disturbances so that the transition to instabilit is detected onl for mixed Poiseuille Couette flows with a non-zero pressure gradient applied along the channel. To achieve the set goal in the most straightforward wa two major simplifications are introduced. Firstl, note that in contrast to ordinar fluids, the viscosit of PVF necessaril changes along the channel since the applied pressure gradient driving the flow leads to pressure reduction in the downstream direction. We will assume in our analsis and then confirm numericall that such a variation is sufficientl slow and occurs on a much longer length scale than the characteristic disturbance wavelength. This enables us to appl a standard normal mode expansion in which the actual location along the channel enters as a parameter. Such an approach will be discussed in detail in Section 4. Secondl, we consider onl small two-dimensional disturbances. For NF flows this is warranted b Squire s theorem which states that two-dimensional perturbations are the most dangerous (see, for example, [9]). However Squire s 2

3 theorem cannot be proven for PVF flows in which fluid viscosit changes in the streamwise direction. Yet if such a change occurs slowl, it can be shown that Squire s theorem holds locall. In order not to distract the reader from the main focus of the article and to keep the manuscript reasonabl short we leave the detailed discussion of this aspect outside the scope of this paper. We also note that full three-dimensional treatment of the problem does not change an of the conclusions regarding the phsical nature of instabilities drawn using a simplified two-dimensional approach. Therefore two-dimensional treatment is full adequate for our goal of a conceptual analsis. 2 Problem definition and governing equations Consider the flow of an incompressible fluid with pressure-dependent viscosit between two parallel horizontal plates separated b the distance 2L from each other. The top plate moves with velocit V from left to right, the bottom plate is stationar. The pressure gradient π is applied along the channel which (depending on its direction) can either enhance or partiall suppress the fluid flow caused b the motion of the upper wall. The flow is described b the equations first given in [] which, in the case of shear-independent fluid, become ρ du = π + (2µ(π) D), dt () u =, (2) where we neglect the gravit and assume that the velocit field u is twodimensional with components (u, v) in the (x, ) directions, respectivel. We choose the coordinate sstem in such a wa that the x and axes have positive right and upward directions, respectivel, and the horizontal centre-plane of a channel is located at =. In equation (), D = 2 2 u x u + v x u + v x 2 v. (3) Governing equations () (3) are complemented b the constitutive relations ρ = const., µ = aπ > (4) and the no-slip/no-penetration boundar conditions (u,v) = (, ) at = L and (u,v) = (V, ) at = L. (5) 3

4 We non-dimensionalise the equations using L, pressure π evaluated at (x,) = (, ), u = (π /ρ) /2 and t = L(ρ/π ) /2 as the scales for length, pressure, velocit and time, respectivel, to obtain u + (u )u = π + α (2πD), t (6) u =, (7) (u,v) = (, ) at = and (u,v) = (V, ) at =, (8) π = at (x,) = (, ), (9) where V = V (ρ/π ) /2, ( ) π /2 α = a () ρl 2 and all smbols now denote the corresponding non-dimensional quantities. 3 Basic flow Equations (6) (9) admit a stead parallel flow solution u = (U(), ), π = Π(x,) discussed in detail in [,2]. As will be shown in Section 5 such a flow becomes unstable when the applied pressure gradient is sufficientl strong and the wall velocit is relativel small. We refer to such flows as Poiseuilletpe or Poiseuille Couette flows. The are characterised b the existence of an extremum point Y of the velocit profile such that U for Y. In this case the solution is given b (see [,2]) e C (s Y ) ds, Y, α e U() = C (s Y ) + V + e C (s Y ) ds, Y, α +e C (s Y ) () Π(x,) = + ec (Y ) + e C Y e C (x+)/2. (2) The extremum point Y is determined using the velocit continuit condition U(Y + ) = U(Y ). The integration of () leads to U() = + 2Y α + 2 αc ln ec( Y ) + e C (+Y ) +, Y = αv C ln ec e 2 C e C + αv C 2. (3) The stabilit investigation of this famil of simple flow solutions is the major goal of the current stud. For pure Poiseuille flow (V = ) equation (3) 4

5 reduces to Y = and U() = α + 2 ln + ec αc + e C 2 ln cosh(c /2) αc cosh C /2 (4) which was originall derived in []. In the limit (C,V ) (, ) expressions (2) and (3) become U() + 2 V + C 4α (2 ) ( Π(x,) + x + V α 2 ( V 2 α 2 4 ), (5) ) C 2 + ( x αv x ) C 2 8, (6) The exact expressions for NF Poiseuille Couette flow are U() = + 2 V + C 4α (2 ), Π(x) = + C x. (7) 2 The comparison of expressions (5) (7) shows that in the limit (C,V ) (, ) the PVF and NF flows become identical and parameter C essentiall characterises the applied longitudinal pressure gradient. Without loss of generalit we assume that it is negative so that the fluid is encouraged to flow in the direction of increasing x. Also note that the quantit 4αV/ C is the ratio of the wall velocit and the maximum velocit of the Poiseuille component of the flow. This quantit will be convenientl used to characterise the shape of the basic flow velocit profile in the subsequent discussion. If αv/ C then the velocit profile becomes monotonic. We refer to such a flow as Couette Poiseuille flows. However we will show that instabilit is possible onl when αv/ C <. We call such flows Poiseuille Couette flows. For a more straightforward comparison with conventionall non-dimensionalised NF solutions we also define Renolds number Re using (): Re = ρu =u L = ρu L µ µ ( V 2 C ) = 4α We will use it instead of α in the subsequent discussion. ( V 2 C ) 4α α. (8) 4 Linearised perturbation equations Next we investigate linear stabilit of the basic states discussed in Section 3 with respect to small-amplitude velocit and pressure disturbances u = (u,v ) and π, respectivel. The linearisation about a parallel basic flow leads to the following perturbation equations 5

6 u t +U u x + v U = π [ Π x + α ( 2 ) u +Π x + 2 u 2 2 v t +U v x = π u x + v +Π + α [ Π x ( 2 v x + 2 v 2 2 ( u + v ) x + 2 Π x ] + U π + U π, ( u + v ) + 2 Π v x ) ] + U π, x u x (9) (2) =. (2) Note that unlike in NF Poiseuille Couette flows, in the present problem the basic flow pressure is a non-trivial function of both x and coordinates which is written as Π(x,) = C 2 ()e C x/2, C 2 () =, see equations (9) and (2). For small values of C the term e C x/2 remains nearl constant for a relativel large range of x in the vicinit of some reference point x so that we write approximatel Π(x,) C 2 ()E, Π x (x,) C 2 () C 2 E, Π (x,) C 2()E, (22) where E = e C x /2. For negative values of C and positive values of x which are of interest in the current stud the range of values E corresponds to spatial locations ranging from the channel entrance to its exit which ma be asmptoticall far awa from the entrance. Such an approximation significantl simplifies the subsequent analsis without sacrificing too much accurac for small C. Parametrisation (22) enables application of a standard normal mode expansion procedure where we assume the perturbation quantities in the form (u (x,,t),v (x,,t),π (x,,t)) = (u (),v (),π ())e (iβx+σt). (23) This representation remains locall valid as long as the perturbation wavenumber β C /2. (24) We will show in Section 5 that this condition is safel satisfied for all considered regimes. In transformation (23), σ = σ R +iσ I is the complex amplification rate with σ I being the frequenc of oscillations. The basic flow is linearl stable if σ R <. Upon transformation (23) the perturbation equations (9) (2) parameterised with E become 6

7 σu = { iβu + αe [ βc2 (ic β) + (C 2D + C 2 D 2 ) ]} u {U iβαe C 2} v {iβ α [U + U D]} π, (25) σv = 2 αc 2C E u + {iαβu D} π [ ( ) i {iβu αe βc 2 2 C β + 2C 2D ]} + C 2 D 2 v, (26) = iβu + v, (27) where both and D denote derivatives with respect to. 5 Numerical results We discretise the disturbance equations (25) (27) using a pseudo-spectral Chebshev collocation method of [2] and [2]. The advantage of this method is that it converges exponentiall quickl: collocation point was used to guarantee the accurac exceeding % for all reported results. Upon discretisation equations (25) (27) result in a generalised algebraic eigenvalue problem for the complex amplification rate σ (A σb)x =, (28) where A = A(C,V,E,α,β) and B are matrix operators resulting from the right- and left-hand sides of equations (25) (27), respectivel, and X is the eigenvector consisting of disturbance velocit components and pressure (u,v,π ) T. Eigenvalue problem (28) was solved using an IMSL [22] routine DGVLCG for fixed values of C, V, E and α (or, equivalentl, Re ) and over a range of wavenumbers β to determine the maximum of the amplification rate σmax R = σ R (β max ), see Figure. While the value of β max depends on the phsical parameters of the problem the qualitative behaviour of the amplification rate curves remains the same in all investigated regimes. In all cases the instabilit (i.e. σmax R > ) was detected for one mode with a single maximum σmax R (refer to the top curves in Figure (a) (d). The phsical mechanism of this instabilit will be discussed in Section 5.2. The critical value of α (or Re ) was subsequentl determined b varing it iterativel for fixed values of other governing parameters until the condition 7 was satisfied. σ R max 7

8 σ R.3 σ R β.5 2 β (a) Point A (b) Point B σ R.3 σ R β.5 2 β (c) Point C (d) Point D Fig.. Disturbance temporal amplification rates σ R for four leading instabilit modes as a function of wavenumber β at points A D in Figure 4 for C = 2 and E =. 5. Critical parameters at the onset of instabilit We discuss the case of small longitudinal pressure gradient C first. In this limit PVF and NF flows become identical. The critical values of the Renolds number and wavenumber obtained for both tpes of fluids for C = 4 are essentiall the same. This classical case was considered in literature in the past [5 7]. It provides an accurac check for our numerical algorithm. As seen from equation (5) in the limit V the PVF flow reduces to NF Poiseuille flow. Thus we recover the well-known critical values for such a flow: (Re,β) = ( ,.2) [23]. When V, similarl to the analsis reported in [5 7] for plane NF Poiseuille Couette flow, we find that as the wall velocit becomes larger, the value of the critical Renolds number increases reaching a local maximum, then slightl decreases before rapidl increasing again without bound, see Figure 2(a). It is found that, like NF flows, Couette-tpe PVF flows satisfing the condition U () >, are alwas stable. The value of the critical wavenumber rapidl decreases with 8

9 Re* α V/ C (a).32.3 β α V/ C (b) 4α c/ C α V/ C (c) Fig. 2. (a) The critical Renolds number Re, (b) wavenumber β and (c) scaled wavespeed 4αc/ C as functions of the scaled boundar velocit 4αV/ C for E = and C = 4 (dash-dotted lines) and C = 2 (NF solid lines, PVF dashed lines). The flow is unstable for Re > Re. the wall velocit, see Figure 2(b) which is an intuitivel clear result: a moving wall effectivel stretches the instabilit structures which will be discussed in more detail in Section 5.2. Figure 2(c) presents the ratio of the wavespeed c = σ I /β of the most amplified disturbance mode and the maximum velocit of the Poiseuille flow component C /(4α) as a function of the scaled wall velocit 4αV/ C. The behaviour of the critical wavespeed is non-monotonic. For small increasing wall velocities it also increases (the instabilit structures are slightl accelerated b the moving wall), but then the wavespeed starts decreasing relative to the wall velocit. As will be discussed in Section 5.2 this is due to the relocation of the critical laer where the condition c = U is satisfied and the major disturbance production occurs [9] towards the stationar wall. For sufficientl large wall velocities the scaled disturbance wave speed starts increasing until it reaches its asmptotic value just below.32. It remains smaller than the wall velocit so that the critical laer onl exists near the stationar wall when the ratio of 9

10 the wall velocit and the maximum Poiseuille component exceeds the value of about.245. For larger pressure gradients the qualitative behaviour of the critical parameters remains the same, but a quantitative difference between the NF and PVF critical values of Renolds number and wavespeed appears, compare the solid and dashed lines in Figure 2(a and c). PVF flows become more stable and the corresponding disturbance waves propagate slower than their NF counterparts. This is an anticipated result. Indeed the comparison of expressions (5) and (7) for PVF and NF velocities piezo-viscous effects lead to the reduction of the maximum flow velocit and, subsequentl, the velocit gradients which are responsible for instabilit. At the same time the critical values of the wavenumber remain ver close (lines in Figure 2(b) almost overlap). Overall, for both NF and PVF flows strengthening the Poiseuille component plas a destabilising role and leads to the increase of the disturbance wavenumber, while increasing the Couette component has the directl opposite effect. Our numerical results show that for both NF and PVF flows an instabilit does not occur if the ratio of the wall and Poiseuille velocities exceeds the value of 4αV/ C.74 which is in excellent agreement with the conclusion made in [5]. Despite this similarit between NF and PVF flows discussed above Figures 3 and 4 reveal two features which distinguish them drasticall. Firstl, and somewhat counter-intuitivel, increasing the pressure gradient parameter C along the channel leads to significant flow stabilisation which is demonstrated for pure Poiseuille PVF flow in Figure 3(a) (see also [24]). This stabilisation is accompanied b a slight elongation of the instabilit structures as seen from the wavenumber data presented in Figure 3(b) and b the wavespeed decrease seen in Figure 3(c). Comparison of the solid and dash-dotted lines in Figure 4(a) also shows the stabilisation of Poiseuille Couette flows for larger C. One reason for such a stabilisation is that as C increases the PVF basic flow velocit profile discussed in detail in [2] straightens near the channel walls. In other words when the pressure gradient is increased the flow profile near the walls approaches that of Couette flow which is found to be stable. The second reason for stabilisation of PVF flows becomes evident from Figure 3(c): the critical wavespeed decreases when C increases and therefore the critical disturbance production laer c = U mentioned previousl moves closer to the walls at which the disturbances are suppressed b the no-slip/no-penetration boundar conditions. It is also seen from Figure 3 that these piezo-viscous effects become noticeable for C 2. For smaller pressure gradients PVF flows behave ver similarl to ordinar fluid flows with constant viscosit. Although the fluid considered here is assumed to have a shear-independent rheolog the stabilisation of PVF Poiseuille-tpe flows in comparison with their NF counterparts is somewhat similar to that observed in shear-thickening flu-

11 8.2 Re * 6 β C C (a) (b).28 4α c/ C (c) Fig. 3. (a) The critical Renolds number Re, (b) wavenumber β and (c) scaled wavespeed 4αc/ C for plane Poiseuille flow (V = ) as functions of the pressure gradient parameter C for E = (solid lines) and E =.5 (dashed lines). The flow is unstable for Re > Re. Solid and dashed lines overlap in plot (c). ids. As follows from equation (2) the pressure distribution across the channel is such that it increases towards the walls (see also Figures 7 9(a) in Section 5.2). According to (4) this leads to the viscosit increase in the near-wall high-shear regions where instabilit is produced. The associated increase in viscous dissipation suppresses arising disturbances similarl to what happens in shear-thickening fluids even though such an effect is due to a larger viscosit and not due to a stronger shear. The second feature which distinguishes NF and PVF flows is that the latter destabilise along the channel as the downstream coordinate x increases (or, equivalentl, parameter E decreases), compare the solid and dashed lines in Figures 3(a) and 4(a) for C = 4 and the dash-dotted and dotted lines lines for C = 2 in Figure 4(a). The reason for such a behaviour is traced back to the fact that the fluid viscosit is a linear function of pressure. Since according to equation (2) the pressure decreases along the channel, so does the fluid viscosit. The respective reduction in viscous dissipation encourages C

12 5.8 A B Re * 4 B C D β.6.4 C D A α V/ C (a) α V/ C (b).32 4α c/ C αV/ C (c) Fig. 4. (a) The critical Renolds number Re, (b) wavenumber β and (c) scaled wavespeed 4αc/ C as functions of the scaled boundar velocit 4αV/ C for C = 4 (solid and dashed lines), C = 2 (dash-dotted and dotted lines) and E = (solid and dash-dotted lines) and E =.5 (dashed and dotted lines). The flow is unstable for Re > Re. The dotted and dash-dotted lines overlap in plots (b) and (c). the development of instabilit at the downstream locations at lower values of Renolds number. Figure 5 confirms this. It shows that the decrease of the critical Renolds number associated with the reduction of viscosit along the channel is essentiall linear in E (i.e. exponential in x) for pure Poiseuille flow (V = ) for both C = 4 and C = 2, see the solid lines in Figure 5(a) and (b). In the limit of an infinitel long channel (i.e. for E ) PVF Poiseuille flow is inherentl unstable because the critical Re. This means that no matter how small the actual flow Renolds number is, PVF 2

13 Re* x 4.5 E (a) Re* E (b) β E (c) β E (d).5 c..5.5 E (e) c E (f) Fig. 5. (a) and (b) Critical Renolds number Re, (c) and (d) wavenumber β and (e) and (f) wavespeed c as functions of E defined in equation (22) for C = 4 ((a), (c), (e)) and C = 2 ((b), (d), (f)) and V = (solid lines), V =. (dashed lines), V =.5 (dash-dotted lines), V =. (dotted lines). Poiseuille flow will become unstable and will eventuall break down far enough from the entrance to the channel. This is a remarkable distinction from NF Poiseuille flow which remains stable regardless of the length of the channel provided the flow is sufficientl slow at the channel entrance. However in drawing such a conclusion care should be taken since if the pressure along 3

14 Table Critical values of the Renolds number in the limit of ver long channel (E = 4 ) for C = 4 and C = 2. V C = 4 C = 2 4α C V 4α C c Re 4α β C V 4α C c Re β the channel becomes too small the rheological model (4) is unlikel to remain valid [25]. As seen from Figure 5(a and b) parallel PVF Poiseille Couette flow (V ) becomes more stable with increasing V at an location along the channel for the selected parameters. The critical value of the Renolds number still decreases along the channel, but this decrease is not linear an more. More importantl, for V the limiting value of Re as E is not zero, see the dotted line for V =. in Figure 5(a) as well as data computed for E = 4 which is presented in Table. It increases with V. This means that for PVF Poiseuille Couette flows there ma exist a range of Renolds numbers for which parallel basic flow remains stable even in ver long channels. However this range is significantl narrower than that for conventional fluids with pressure-independent viscosit and it is further reduced as the applied pressure gradient increases, see Figure 5(b) and Table. This is consistent with our computational observations that PVF Couette-tpe flows with large wall velocit V remain stable. Note that in the limit E the wavenumber corresponding to the most dangerous disturbances tends to, see Figure 5(c) and (d), meaning that disturbance structures elongate down the channel to such a degree that the become effectivel non-periodic and modif the mean flow. The data in Table shows that despite the significant reduction of the critical wavenumber β the condition (24) is safel satisfied even for ver small values of E confirming the validit of parametrisation (22). Figure 5(c f) also shows that the elongation of the PVF flow disturbance structures in the downstream direction is accompanied b the decrease of their critical wavespeed for an fixed wall velocit. This is et another feature distinguishing PVF flow from its NF counterpart for which the disturbance wavespeed remains constant along the channel. The wavespeed reduction is more pronounced for larger values of the pressure gradient parameter C. 4

15 5.2 Instabilit mechanism To identif the phsical mechanism of instabilit we consider the disturbance energ balance. We multipl the momentum equation (25) and (26) b the complex conjugate velocit components ū and v, respectivel, add them together, integrate b parts across the laer and use the continuit condition (27) and boundar conditions (u,v ) = (ū, v ) = (, ) to obtain where σ R Σ k = Σ α + Σ α2 + Σ uv, (29) Σ k E k d = ( u 2 + v 2) d, Σ α E α d = α Π [ ( u 2 + v 2 ) + β ( 2 u 2 + v 2)] d [ Π +α R (v v + v v ) Π ] x R(u v ) d, Σ α2 E α2 d = α [U R (π ū + iβπ v ) + U R(π ū )] d, Σ uv E uv d = U R(v ū )d and R( ) denotes the real part of the term in parentheses. The above terms have a straightforward phsical meaning. Namel: Σ k represents the disturbance kinetic energ. Σ k is positivel defined. Therefore the sign of σ R and therefore whether the flow is stable or unstable is determined b the sign of the sum of the three terms in the right-hand side of equation (29). Subsequentl, the positive (negative) terms in the right-hand side pla destabilising (stabilising) role. Σ α represents the viscous dissipation. The first integral in its definition is similar to that in pressure-independent Newtonian fluids and is negativel defined. The second integral takes into account the variation of a basic flow viscosit with pressure. The numerical contribution of the second integral is found to be small (it is proportional to C and C 2, respectivel) so that Σ α remains negative signifing the stabilising role of the basic flow viscosit. Since within a linearised analsis framework the eigenfunctions (u,v,π ) are defined up to a multiplicative constant we scale them in such a wa that Σ α = for all computed regimes. Σ α2 also represents the contribution to viscous dissipation however it is caused b the perturbation of viscosit due to the pressure disturbances. This term has no analogue in pressure-independent fluids and therefore it is instructive to consider it separatel. 5

16 Table 2 Critical parameters and disturbance energ integrals for selected points in Figure 4 for C = 2 and E =. V 4αV/ C α 4 Re β 4αc/ C E α2 E uv A B C D Σ uv represents the energ exchange between the basic and disturbance velocit fields. Numerical results for selected marginal stabilit regimes for C = 2 and E = are presented in Table 2. The show that flow destabilization is alwas caused b the interaction between the disturbance and basic flow velocit fields. Namel, the velocit perturbations draw the energ from the basic flow so that E uv >. At the same time term E α2 remains negative. This means that piezo-viscous effects enhance viscous dissipation. In turn this leads to the increase of the critical Renolds number for PVF observed in Figures 2(a) and 4(a). Note that the relative role of piezo-viscous dissipation is quite significant even for small values of C : it reaches 2% of the corresponding NF value for C = 2. The results for different values of C and E are found to be qualitativel similar and thus the are not discussed here. In order to gain insight into the spatial structure of instabilit, in Figure 6 we present the distribution of disturbance energ integrands across the channel. The results show that for small wall velocities the disturbance energ production occurs in the narrow regions near the channel walls, see the dotted line in Figure 6(a). These correspond to a pair of classical Tollmien Schlichting waves similar to those observed in pure Poiseuille NF flow, see, for example, [9]. The occur in the critical laers where the condition c = U is satisfied. This is clearl seen from Figure 6(a and b): the horizontal solid lines show the locations of two critical laers near the walls which coincide exactl with the maximum of the disturbance energ production integrand E uv. When the top wall velocit increases so that it exceeds the value of the critical wavespeed, see entries for points C and D in Table 2, onl one Tollmien Schlichting wave near the stationar wall survives while the disturbance energ production near the moving wall becomes negative. This means that the disturbance energ is fed back to the basic flow near the moving wall. As expected, the viscous dissipation integrand of Σ α corresponding to the basic flow viscosit remains negative throughout the complete flow domain (see the solid lines in Figure 6). The strongest dissipation effects are observed ver close to the channel walls which suppress velocit perturbations. This behaviour is tpical for NF Poiseuille-tpe flows. However the distribution of the Σ α2 integrand (see the 6

17 (a) Point A (b) Point B (c) Point C (d) Point D Fig. 6. Disturbance energ integrands for points A D in Figure 4. The solid, dashed and dotted lines correspond to the integrands of E α, E α2 and E uv, respectivel. The horizontal solid lines indicate the location of the critical laer. 7

18 dashed lines in Figure 6) which represents piezo-viscous effects shows that it is onl negative near the stationar wall, but becomes positive near the moving wall when its velocit is increased. Thus in contrast to their NF counterparts, in PVF flows piezo-viscous effects tend to make a flow near the moving wall less stable and a flow near the stationar wall more stable even though the overall effect of viscosit remains stabilising. The Σ α2 data in Table 2 suggests that the overall piezo-viscous effects tend to become less stabilising as the wall velocit increases. This is indeed seen in Figure 4(a): as V becomes larger the curves for C = 2 approach those for C = 4 i.e. the NF limit. In concluding this paper, in Figures 7 we present the basic flow velocit and pressure profiles and the most dangerous disturbance fields for selected points marked in Figure 4. For small wall velocities and C = 2 the basic flow velocit profile is close to parabolic and the basic flow pressure (and so the viscosit) is almost constant across the channel, see Figure 7(a). The instabilit pattern takes the form of two periodic sets of instabilit rolls located near the upper and lower walls shifted b a half of a wavelength with respect to each other, see Figure 7(b). As the upper wall velocit progressivel increases two effects are observed as demonstrated in Figures 8 (b). Firstl, the flow stabilises near the moving wall so that the top row of instabilit rolls seen in Figure 7(b) disappears in Figures 8. Secondl, the sstem of instabilit rolls near the bottom stationar wall survives, but the distance between the neighbouring rolls increases. Figures 8 (b) also reveal a noteworth correlation between the disturbance velocit and pressure fields for Poiseuille Couette flows: the disturbance pressure achieves its maxima inside the vortices rotating counter-clockwise and minima inside the clockwise rotating vortices. Therefore the pressure is higher if disturbance vorticit has the sign opposite to that of the basic flow. The piezo-viscous fluid inside such vortices is slightl more viscous than inside the vortices rotating clockwise. 6 Conclusions The stabilit of plane Poiseuille Couette flows of a fluid whose viscosit increases linearl with pressure has been investigated. It was shown that in the limit of small longitudinal pressure gradients PVF flow solutions reduce to those for common NF fluids with constant viscosit. However for larger values of the pressure gradient PVF flows generall become more stable than their NF counterparts. Another significant distinction of PVF flows is that the tend to destabilise in the downstream direction even if the are stable in the vicinit of the channel entrance. The phsical mechanism of instabilit is identified as Tollmien Schlichting waves which appear in the critical laers near both channel walls when the upper wall velocit is small. When the wall velocit increases onl one disturbance wave near the stationar 8

19 U ( Π) 6 (a) x (b) Fig. 7. (a) Basic flow velocit and pressure profiles and (b) disturbance velocit and pressure fields for point A in Figure 4. Lighter shading corresponds to higher pressure. plate survives. It was also shown that the instabilit waves draw their energ from the basic velocit field and that the piezo-viscous effects pla an overall stabilising role. Onl stabilit of simple flows of shear-independent fluids has been considered in the current work. Analsis of shear-thinning or shear-thickening piezo-viscous fluids would proceed in a similar wa and is planned to be presented elsewhere. However our preliminar results do not indicate that shear-thinning/thickening effects can lead to results qualitativel different from those presented here for shear-independent fluids. References [] J. Hron, J. Málek, K. R. Rajagopal, Simple flows of fluids with pressuredependent viscosities, Proc. R. Soc. Lond. A 457 (2) [2] S. A. Suslov, T. D. Tran, Revisiting plane Couette-Poiseuille flows of a piezoviscous fluid, J. Non-Newtonian Fluid Mech. 54 (28) [3] F. Koran, J. M. Deal, A high pressure sliding plate rheometr for polmer melts, J. Rheol. 43 (999)

20 U 5 ( Π) 6 (a) x (b) Fig. 8. Same as Figure 7 but for point B. [4] T. W. Bates, J. A. Williamson, J. A. Spearot, C. K. Murph, A correlation between engine oil rheolog and oil film thickness in engine journal bearing, Societ of Automotive Engineers (986) [5] D. R. Gwnllw, A. R. Davies, T. N. Phillips, On the effects of a piezoviscous lubricant on the dnamics of a journal bearing, J. Rheol. 4 (996) [6] M. Renard, Some remark on the Navier-Stokes equations with a pressuredependent viscosit, Comm. Partial Differential Equations (986) [7] F. Gazzola, A note on the evolution of Navier-Stokes equations with a pressuredependent viscosit, Z. Angew. Math. Phs. 48 (997) [8] G. Ha, M. E. Macka, K. M. Awati, Y. Park, Pressure and temperature effects in slit rheometr, J. Rheol. 43 (999) [9] J. Málek, J. Necas, K. R. Rajagopal, Global analsis of the flows of fluids with pressure-dependent viscosities, Arch. Rational Mech. Anal. 65 (22) [] J. Málek, J. Necas, K. R. Rajagopal, Global existence of solutions for flows of fluids with pressure and shear dependent viscosities, Appl. Math. Lett. 5 (22) [] J. Hron, J. Málek, J. Necas, K. R. Rajagopal, Numerical simulations and global existence of solutions of two-dimensional flows of fluids with pressure and shear dependent viscosities, Math. Comput. Simulations 6 (23)

21 U 5 5 ( Π) 6 (a) x (b) Fig. 9. Same as Figure 7 but for point C. [2] M. Franta, J. Málek, K. R. Rajagopal, On stead flows of fluids with pressureand shear-dependent viscosities, Proc. R. Soc. A 46 (25) [3] K. R. Rajagopal, G. Saccomandi, Unstead exact solution for flows of fluids with pressure-dependent viscosities, Math. Proc. R. Irish Academ 6A (26) 5 3. [4] M. Renard, Parallel shear flows of fluids with a pressure-dependent viscosit, J. Non-Newtonian Fluid Mech. 4 (23) [5] M. C. Potter, Stabilit of plane Couette-Poiseuille flow, J. Fluid Mech. 24 (966) [6] W. C. Renolds, M. C. Potter, Finite-amplitude instabilit of parallel shear flows, J. Fluid Mech. 27 (967) [7] F. D. Hains, Stabilit of plane Couette-Poiseuille flow, The Phs. Fluids (967) [8] A. M. Sagalakov, The spectrum of small perturbations in plane Couette- Poiseuille flow, J. Appl. Mech. Tech. Phs. 2 (97) [9] P. G. Drazin, W. H. Reid, Hdrodnamic Stabilit, Cambridge Universit Press, 98. 2

22 U 5 5 ( Π) 6 (a) x (b) Fig.. Same as Figure 7 but for point D. [2] H.-C. Ku, D. Hatziavramidis, Chebshev expansion methods for the solution of the extended Graetz problem, J. Comput. Phs. 56 (984) [2] D. Hatziavramidis, H.-C. Ku, An integral Chebshev expansion method for boundar-value problems of O.D.E. tpe, Comp. & Maths with Appls (6) (985) [22] IMSL Inc., IMSL mathematical librar, Tech. rep., Houston, TX (995). [23] S. A. Orszag, Accurate solution of the Orr-Sommerfeld stabilit equation, J. Fluid Mech. 5 (97) [24] T. D. Tran, S. A. Suslov, Stabilit of plane Poiseuille flow of a fluid with pressure dependent viscosit, in: Proceedings of ICTAM8, International Congress on Theoretical and Applied Mechanics, Adelaide, Australia, ISBN , 28. [25] R. R. Huilgol, Z. You, On the importance of the pressure dependence of viscosit in stead non-isothermal shearing flows of compressible and incompressible fluids and in the isothermal fountain flow, J. Non-Newtonian Fluid Mech. 36 (26)