We study the linear stability of a plane Poiseuille flow of an incompressible fluid whose viscosity depends linearly on the pressure. It is shown that the local critical Reynolds number is a sensitive function of the applied pressure gradient and that it decreases along the channel. While in the limit of small pressure gradients conventional results for a pressure-independent Newtonian fluid are recovered, a significant stabilisation of the flow and an elongation of the critical disturbance wavelength are observed when

the longitudinal pressure gradient is increased. These features drastically distinguish the stability characteristics of a piezo-viscous flow from its pressure-independent Newtonian counterpart

Suslov, Sergey A.

Tran, Thien Duc

English

University of Southern Queensland ePrints

XXII ICTAM, 25–29 August 2008, Adelaide, AustraliaSTABILITY OF PLANE POISEUILLE FLOW OF A FLUID WITH PRESSURE-DEPENDENTVISCOSITYThien Duc Tran∗ and Sergey A. Suslov∗∗∗Institute of Applied Mechanics, Vietnamese Academy of Science and Technology, Vietnam andComputational Engineering and Science Research Centre, University of Southern Queensland, Toowoomba,Queensland 4350, Australia∗∗Department of Mathematics and Computing and Computational Engineering and Science Research Centre,University of Southern Queensland, Toowoomba, Queensland 4350, AustraliaSummary We study the linear stability of a plane Poiseuille flow of an incompressible fluid whose viscosity depends linearly onthe pressure. It is shown that the local critical Reynolds number is a sensitive function of the applied pressure gradient and that itdecreases along the channel. While in the limit of small pressure gradients conventional results for a pressure-independent Newtonianfluid are recovered, a significant stabilisation of the flow and an elongation of the critical disturbance wavelength are observed when thelongitudinal pressure gradient is increased. These features drastically distinguish the stability characteristics of a piezo-viscous flowfrom its pressure-independent Newtonian counterpart.PROBLEM DEFINITION AND GOVERNING EQUATIONSWe consider a flow of an incompressible fluid with a pressure-dependent viscosity. Such fluids are found, for example,in geological, planetary and lubrication applications. It was shown in [1, 2] that steady unidirectional plane flows canexist only when the fluid viscosity is a linear function of the pressure. Therefore here we consider the flow of such afluid between two parallel horizontal plates separated by distance 2L. The x and y axes have the left-to-right and upwardpositive directions, respectively. The horizontal centre-plane of a channel is located at y = 0. The negative pressuregradient ∇pi is applied along the channel. With the gravity neglected the governing equations are [1]ρdudt= −∇pi +∇ · (2µ(pi)D) , ∇ · u = 0 (1)which are complemented with the constitutive equations for density ρ and viscosity µρ = const. , µ = api > 0 (2)and the no-slip/no-penetration boundary conditions(u, v) = (0, 0) at y = ±L . (3)Here the velocity field u = (u(x), v(x)), the pressure pi = pi(x), the coordinate vector x = (x, y), and D = 12 (∇u +∇uT ). Following [3], in order to capture the piezo-viscous nature of the flow in the most explicit way we non-dimensionalisethese equations using the pressure pi∗ evaluated at (x, y) = (0, 0) at time t = 0, the characteristic speed u∗ = (pi∗/ρ)1/2and time t∗ = L(ρ/pi∗)1/2 as the scales for length, pressure, velocity and time, respectively, to obtain∂u∂t + (u · ∇)u = −∇pi + α∇ · (2piD) , ∇ · u = 0 , (4)(u, v) = (0, 0) at y = ±1 , pi = 1 at (x, y, t) = (0, 0, 0) , (5)where parameter α = a(pi∗ρL2) 12= api∗ρ(pi∗/ρ)1/2L= µ∗ρu∗L plays the role of effective inverse Reynolds number. Note thatall unstarred symbols now denote the corresponding non-dimensional quantities.BASIC FLOWThe governing equations (4), (5) admit a steady unidirectional solution for the velocity and a two-dimensional solutionfor the pressure which were first reported in [1] and are given byU(y) =2αC0lncosh C0y2cosh C02, Π(x) =1 + e−C0y2eC0(x+y)/2 . (6)Parameter C0 < 0 plays a dual role. Firstly, it characterises the strength of the applied pressure gradient. Secondly, itmeasures the piezo-viscous effects which distinguish the considered fluid from its conventional Newtonian counterpart.This is best seen in the limit of C0 → 0 when the series expansion of (6) leads toU(y) ≈y2 − 14αC0 −y4 − 196αC30 , Π(x) ≈ 1 +x2C0 + (x2 + y2)C208. (7)−10−1 −10−2 −10−3 −10−42000300040005000600070008000900010000 C0 Re*(a)−10−1 −10−2 −10−3 −10−40.950.960.970.980.9911.011.021.031.041.05 C0 β(b)Figure 1. The critical values of Reynoldsnumber (a) and wavenumber (b) as functionsof the pressure parameter C0 for two lo-cations along the channel corresponding toE = 1 (solid lines) and E = 1/2 (dashedlines). The critical wavenumber curves forthese spatial locations overlap in plot (b).The maximum flow speed achieved at y = 0 in this limit isUmax ≈|C0|4α(1−C2024). (8)In the above expressions, the terms linear in C0 correspond to the flow of aNewtonian fluid with pressure-independent viscosity while the higher order inC0 terms describe piezo-viscous effects. For a more straightforward compari-son of our results with the conventionally non-dimensionalised solutions for aNewtonian fluid we introduce Reynolds number based on the maximum speedRe∗ =ρUmaxu∗Lµ∗=ρ|C0|u∗L4αµ∗=|C0|4α2. (9)STABILITY RESULTSEquations (4) and (5) are linearised about the basic flow solution assuming thedisturbed flow in the formu(x, t) = U(y) + u′(x, t) , pi(x, t) = Π(x) + pi′(x, t) , (10)where U, u′, Π, pi′ are the basic and disturbance velocity and the basic anddisturbance pressure, respectively. The disturbance quantities then are writtenin a normal mode form (u′(x, t), pi′(x, t)) = (u′(y), pi′(y))eσt+iβx. Upondiscretisation using Chebyshev pseudo-spectral method with 100 collocationpoints, the resulting algebraic generalized eigenvalue problem is solved for thecomplex amplification rate σ over a range of wavenumbers β and Reynoldsnumbers Re∗. Note that since the basic flow pressure and thus the fluid vis-cosity depend on x the above normal mode expansion is local in its nature. Itis only valid if the characteristic length over which the pressure changes sig-nificantly is much longer than the disturbance wavelength, i.e. if |C0| ≪ β.This condition is safely satisfied in the current analysis, see Figure 1(b). Thelocality of the solution is parametrised by E = eC0x/2.As expected from equations (7) and (8), when C0 → 0, the basic velocityprofile reduces to that of a conventional Poiseuille flow and we recover thecritical values of Reynolds and wavenumber (Re∗c , βc) ≈ (5772, 1.02) for a Newtonian fluid. However increasing thepressure gradient parameter |C0| gives rise to a significant stabilisation of the flow, see Figure 1(a). This behaviour iscompletely opposite to that observed in flows of fluids with pressure-independent viscosity. Physically, the larger valuesof |C0| correspond to a larger pressure difference between the channel ends. In order to increase the pressure gradientthe pressure upstream has to be raised. For tested rheological model this leads to the increase of the fluid’s viscosity and,subsequently, to the decrease of the maximum speed of the flow (see the expression for Umax above). Both these effectsresult in the observed stabilisation. It is found that the flow remains stable near x = 0 regardless of the strength of theapplied pressure gradient for the values of |C0| ≥ 0.35. This can be traced back to the channel-choking effect discoveredin [3], an essentially piezo-viscous effect when increasing the pressure gradient leads to the proportional increase in thefluid viscosity so that the flow maximum speed remains constant.At the same time, the flow is destabilised downstream where the local pressure and the fluid viscosity decrease, seethe dashed line in Figure 1(a). The critical Reynolds number drops below the classical value of 5772 even in the limitof C0 → 0. This signifies the essential differences between piezo-viscous and Newtonian fluids. The present resultssuggest that in a sufficiently long channel the instability will develop near the channel exit regardless of the entranceflow conditions. This instability will destroy a unidirectional flow before the fluid reaches the channel end. Such a findingprovides a possible resolution of the concern expressed in [4]. There the authors noted that due to the exponential decreaseof the pressure along the channel the fluid becomes essentially inviscid, which is physically unlikely. Therefore accordingto [4] the steady plane unidirectional solutions for piezo-viscous flows might have limited physical relevance. The currentstudy shows that such flows can exist at least in a relatively short channel. They never become fully inviscid because theinstability inevitably sets once the viscosity reaches a sufficiently low level and then the developing pressure disturbancesguarantee (via the constitutive law (2)) that the viscosity remains non-zero.References[1] J. Hron, J. Málek and K. R. Rajagopal: Simple flows of fluids with pressure-dependent viscosities. Proc. R. Soc. Lond. A 457:1603–1622, 2001.[2] M. Renardy: Parallel shear flows of fluids with a pressure-dependent viscosity. J. Non-Newtonian Fluid Mech. 114:229–236, 2003.[3] S. A. Suslov and T. D. Tran: Revisiting plane Couette-Poiseuille flow of a piezo-viscous fluid. Submitted in J. Non-Newtonian Fluid Mech., 2007.[4] R. R. Huilgol and Z. You: On the importance of the pressure dependence of viscosity in steady non-isothermal shearing flows of compressibleand incompressible fluids and in the isothermal fountain flow, J. Non-Newtonian Fluid Mech., 136:106–117, 2006.

On the importance of the pressure dependence of viscosity in steady non-isothermal shearing ﬂows of compressible and incompressible ﬂuids and in the isothermal fountain ﬂow,

Stability of plane Poiseuille flow of a fluid with pressure-dependent viscosity

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