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Viscous heating effects on the linear stability of Poiseuille flow of an inelastic fluid Ahmet Pinarbasi ∗ , Muharrem Imal Cukurova University, Mechanical Engineering Department, 01330 Adana, Turkey Received 20 August 2004; received in revised form 2 February 2005; accepted 12 February 2005

Abstract In this paper, the effect of viscous heating on the stability of a non-Newtonian fluid flowing between two parallel plates under the effect of a constant pressure gradient is investigated. The viscosity of the fluid depends on both temperature and shear rate. Exponential dependence of viscosity on temperature is modeled through Arrhenius law. Non-Newtonian behavior of the fluid is modeled according to the Carreau rheological model. Motion and energy balance equations that govern the base flow and the stability of the flow are coupled and the solution to the problem is found iteratively using a pseudospectral method based on the Chebyshev polynomials. In the presence of viscous heating, the effect of activation energy parameter, Prandtl and Brinkman numbers, material time and power-law constants on the stability of the flow is presented in terms of neutral stability curves. © 2005 Elsevier B.V. All rights reserved. Keywords: Viscous heating; Poiseuille flow; Linear stability

1. Introduction The interaction between viscous heating and fluid flow is of great importance in variety of applications that involves the flow of viscous fluids with temperature and shear ratedependent properties. These application areas include polymer processing, tribology and lubrication, food processing, on-line measurements, instrumentation and viscometry [1]. Since polymeric solutions and melts typically have zero shear rate viscosities that are three to eight orders of magnitude greater than the viscosity of water, viscous heat generation is often important in industrial polymer processing applications [2]. Viscous flows with variable viscosity effects in the presence of viscous heating have been the subject of a number of studies. Papathanasiou [1] investigated circular Couette flow of a Newtonian fluid with temperature-dependent viscosity and thermal conductivity in the presence of viscous heating. He found that the thermal conductivity has a small effect on the perturbed velocity profile, which is governed ∗

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0377-0257/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2005.02.004

by the coefficients of the viscosity model. Davis and Kriegsmann [3] studied steady parallel flows of Newtonian liquids that have temperature-dependent viscosities and substantial viscous heating generation. They presented shear stress versus shear rate characteristics and found that activation energy parameter affects the results considerably. In a recent study, Pinarbasi and Imal [4] considered non-isothermal channel flow of a non-Newtonian fluid with viscous heating. They found that while pressure gradient–flow rate graph is monotonic for certain values of flow controlling parameters, there is a large jump in the graph under certain values of these parameters. Stability of non-Newtonian fluids with viscous heating has attracted the attention of many researchers in recent years. Becker and McKinley [5] investigated the stability of two-dimensional viscoelastic creeping Couette and Poiseuille flows with viscous heating. They found that viscous heating has a stabilizing/destabilizing tendency for Couette/Poiseuille flow at long to moderate disturbance wavelengths, and a stabilizing effect at short wavelengths, but no instabilities were found in the inertialess flow limit. Al-Mubaiyedh et al. [6] considered energetics effects on the stability of viscoelastic Dean flow under creeping flow

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condition. They found that when temperatures at walls are kept at constant values, the critical Deborah number increases with increasing Brinkman number indicating that viscous heating stabilizes the flow in this case. Olagunju et al. [7] studied the effect of viscous heating on linear stability of viscoelastic cone-and-plate flow in axisymmetric case. They showed that viscous heating has a stabilizing effect on both long wave and short wave disturbances. In a recent study, Thomas et al. [8] investigated the roles of the base flow temperature and inertia on the stability boundary of the Taylor–Couette flow of viscoelastic polymer solutions. They showed that increasing the fluid temperature results in multi-valued stability boundary in the We–Na number space. In this study, we investigate the influence of viscous heating on the linear stability of a non-Newtonian fluid for Reynolds number values up to 30,000. The viscosity of the fluid depends on both temperature and shear rate. Exponential dependence of viscosity on temperature is modeled through Arrhenius law. Non-Newtonian behavior of the fluid is modeled according to the Carreau rheological model. Motion and energy balance equations that govern the base flow and the stability of the flow are coupled and the solution to the problem is found iteratively using a pseudospectral method based on the Chebyshev polynomials. In the presence of viscous heating, the effect of activation energy parameter, Prandtl and Brinkman numbers, material time and power-law constants on the stability of the flow is presented in terms of neutral stability curves. The goal of the present study is to compare the quantitative and qualitative, if any, changes in stability diagrams of an inelastic fluid including viscous heating effects with those of a Newtonian fluid without viscous heating.

2. Governing equations 2.1. Base ﬂow We consider two-dimensional, non-isothermal, steady, hydrodynamically and thermally fully developed flow of an incompressible, non-Newtonian fluid between two infinite parallel plates. The channel is assumed to be infinite in depth direction. The distance between the upper and lower walls of the channel is l and the coordinate axis is located on the lower wall. Both channels are kept at the same constant temperature, and the effect of viscous heating is included in the analysis. The flow is driven by a constant pressure gradient acting along the channel axis. Fluid viscosity depends on the local temperature and shear rate. The inelastic fluid is modeled by a two-parameter Carreau equation and temperature dependence of the fluid is modeled through Arrhenius law. Dimensionless momentum balance and energy balance equations take the following form: 2 (n−1)/2 dP d −(βθ/θ+1) du du 1 + λ2 = e (1) dx dy dy dy

2 (n−1)/2 2 du du d2 θ + (Br) e−(βθ/θ+1) 1 + λ2 =0 dy dy dy2 (2) subject to the following boundary conditions: u(0) = u(1) = 0 θ(0) = θ(1) = 0

(3)

where µ = e−(βθ/θ + 1) is the viscosity and Reynolds number is defined as Re = ρ¯ U¯ 0¯l/µ ¯ 0. The above non-linear flow governing equations are made dimensionless using u, v =

u, v , U0

T¯ − T0 θ= , T0

x, y =

x, y,

l ¯ p¯ l p= µ ¯ 0U0

,

µ=

µ ¯ , µ ¯0 (4)

where bars denote dimensional quantities. In the above equations, x shows the flow direction, y the coordinate normal to the planes, u the velocity parallel to the planes, P the pressure, β the dimensionless activation energy parameter, θ the dimensionless temperature, λ the material time constant, n the dimensionless power-law index, U¯ 0 the average velocity, µ ¯ 0 the viscosity at the reference temperature T¯ 0 and Br is the Brinkman number defined as 2

Br =

µ0 U 0 kT0

where k¯ is the thermal conductivity. Base flow governing coupled boundary value problem given above was solved numerically by an iterative approach using pseudospectral method based on the Chebyshev polynomials. The reader is referred to Ref. [4] for details of the solution procedure. 2.2. Linear stability analysis Performing a linear stability analysis is a straightforward and well know procedure and can be found in many references (see Refs. [9,10], for example). Therefore, only the final stability governing equations are given below. Momentum equation takes the following form: iαRe (Ub − c)(φ − α2 φ) − φUb

= ηξφ + 2ηξ + 2ξη φ + −4α2 η+2α2 ηξ+2η ξ + ηξ + ξη − iαReUb φ + −4α2 η + 2α2 ηξ + 2α2 η ξ φ + 2α2 η ξ + α2 η ξ + α2 ηξ + α4 ηξ − iαReUb −iαReUb φ (5)

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and the energy equation becomes θ − (α2 + iαPeUb )θ + (iαPeθb + BrηUb α2 (1 + ξ)) φ +BrηUb α2 (1 + ξ)φ + ciαPeθ = 0

(6)

The associated boundary conditions are: φ = φ = 0

and θ = 0

at y = 0

and

y=1

(7)

In the above equations, Ub and θ b denote steady base flow velocity and temperature, respectively, α the wave number of disturbances, c the complex wave speed of disturbances, Pe the Peclet (RePr) number, φ and θ are velocity and temperature disturbances, respectively, and 2 (n−1)/2

η = µ[1 + λ2 (Ub ) ]

Fig. 2. Neutral stability curve for various values of Brinkman number (Br). Other flow parameters: β = 2, λ = 2, n = 0.75, Pr = 7 (S: stable, U: unstable).

while ξ =1+

(n − 1)λ2 (Ub )2 1 + λ2 (Ub )2

Primes in Eqs. (5)–(7) represent differentiation with respect to y. The stability problem given above results in a generalized complex matrix eigenvalue problem of the form Ax = cBx and solved with an IMSL subroutine software.

3. Results and discussions Recent theoretical and experimental studies show that thermal effects induced by viscous heating can significantly alter the stability characteristics of polymeric solutions [5–8]. In this section, we present the neutral stability maps of a nonisothermal inelastic fluid with the inclusion of viscous heating in energy equation. Fig. 1 shows the effect of activation energy parameter β on the neutral stability curve. Constant parameters in Fig. 1 are n = 0.75, λ = 2, Br = 5 and Pr = 7 while β values are changed as

Fig. 1. Neutral stability curve for various values of activation energy parameter β. Other flow parameters: n = 0.75, λ = 2, Br = 5, Pr = 7 (S: stable, U: unstable).

1, 5 and 7. Activation energy parameter signifies the sensitivity of viscosity variations in temperature. Fig. 1 shows that as activation energy parameter increases, critical Reynolds number (defined as the value of Re where transition to instability occurs) decreases considerably and unstable region in the stability map increases. It can then be concluded that viscosity variations due to temperature has a strong destabilizing effect on a non-Newtonian fluid. Fig. 2 presents the effect of Brinkman number (Br) on the neutral stability curve. Constant flow parameters in Fig. 2 are given as n = 0.75, λ = 2, β = 2 and Pr = 7. Brinkman number was changed as 5, 10 and 25. As can be seen in Fig. 2, Brinkman number that is a measure of the magnitude of viscous heating, has a similar effect to the activation energy parameter given in Fig. 1. As Brinkman number increases, the critical Reynolds number decreases and the unstable region in the stability map increases. This result shows that viscous heating has a destabilizing effect on the linear stability of plane Poiseuille flow. This is in contrast to the results of viscoelastic cone-and plate flow [7] that was performed under creeping flow conditions. Becker and McKinley [5] report that viscous heating has a stabilizing/destabilizing tendency for Couette/Poiseuille flow at long to moderate disturbance wavelengths, and stabilizing effect at short wavelengths under creeping flow conditions. We show here that without creeping flow assumption (for large values of Reynolds numbers), viscous heating has a stabilizing effect for small wavenumbers (between 0.5 and 1.0, approximately) and destabilizing effect for large wavenumbers (between 1.5 and 2.0, approximately) but the overall effect of viscous heating on Poiseuille flow of an inelastic fluid is destabilization of the flow. Fig. 3 shows the effect of Prandtl number (Pr) on the neutral stability curve. Constant flow parameters in Fig. 3 are given as n = 0.75, λ = 2, β = 2 and Br = 5 while Pr was changed as 7 and 50. Prandtl (or Peclet) number can be interpreted as the dimensionless number that signifies axial conduction of thermal energy. Fig. 3 shows that axial conduction of thermal energy has an overall stabilizing effect on the neutral stability

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Fig. 3. Neutral stability curve for various values of Prandtl number (Pr). Other flow parameters: β = 2, λ = 2, Br = 5, n = 0.75 (S: stable, U: unstable).

curve. As Pr increases, critical Reynolds number decreases and unstable area decreases as well in the neutral stability map. Becker and McKinley [5] report that for Poiseuille and Couette flow of viscoelastic fluid under creeping flow conditions, increasing Peclet number has a destabilizing effect on the flow. Our study for high Reynolds number shows that increasing Pr (or Pe) number has a stabilizing effect on the flow. Fig. 4 presents the effect of power-law index n on the neutral stability map. Constant flow parameters in Fig. 4 were selected as λ = 2, β = 2, Br = 5 and Pr = 7 while n takes the values of 1, 0.75 and 0.5. Note that n = 1.0 corresponds to Newtonian fluid case. It can be seen in Fig. 4 that as the shear thinning of the fluid increases, the flow becomes more un-

Fig. 5. Neutral stability curve for various values of material time constant λ. Other flow parameters: β = 2, n = 0.75, Br = 5, Pr = 7 (S: stable, U: unstable).

stable. The same effect was observed by earlier studies [9] in which the effect of viscous heating was neglected. However, when the results of this study are compared with those published earlier, we can conclude that with the inclusion of viscous heating, critical Reynolds number decreases even further compared to the case where there is no viscous heating effects. Fig. 5 shows the effect of material time constant λ on the neutral stability map. Constant flow parameters in Fig. 5 were selected as n = 0.75, β = 2, Br = 5 and Pr = 7 while λ takes the values of 2, 5 and 10. Similar to the effect of power-law index n, material time constant has a destabilizing effect on the flow. However, compared to the power-law index, destabilizing effect of material time constant is more significant. Pinarbasi and Liakopoulos [9] studied isothermal stability of a Carreau fluid and observed that material time constant has a destabilizing effect on the flow. However, Fig. 5 shows that destabilizing effect of material time constant is more dominant when viscous heating effects are included in the analysis.

4. Conclusion

Fig. 4. Neutral stability curve for various values of power-law index n. Other flow parameters: β = 2, λ = 2, Br = 5, Pr = 7 (S: stable, U: unstable).

This study investigates the linear stability of twodimensional plane Poiseuille flow between two parallel plates under the effect of a constant pressure gradient. The viscosity of the fluid depends on both temperature and shear rate. Exponential dependence of viscosity on temperature is modeled through Arrhenius law. Non-Newtonian behavior of the fluid is modeled according to the two-parameter Carreau rheological model. Motion and energy balance equations that govern the base flow and stability are coupled due to dependence of viscosity on temperature, and the solution to the problem is found iteratively using a pseudospectral method based on the Chebyshev polynomials. Linear stability analysis leads to a generalized complex eigenvalue problem, which is solved by an IMSL subroutine.

A. Pinarbasi, M. Imal / J. Non-Newtonian Fluid Mech. 127 (2005) 67–71

In the presence of viscous heating, the effect of activation energy parameter, Prandtl and Brinkman numbers, material time and power-law constants on the stability of the flow is presented in terms of neutral stability curves. It was found that activation energy parameter and Brinkman number has an overall destabilizing effect on the flow. On the other hand, the Prandtl number has a stabilizing effect. Carreau rheological model parameters, power-law index and material time constant, both have an overall destabilizing effect on the flow. It was seen that compared to isothermal stability analysis, viscous heating included non-isothermal flow magnifies the destabilizing effect of these two parameters. Overall, it can be concluded that viscous heating effects are strong enough not to be neglected in the stability analysis of the flow of polymeric solutions and melts. Observed destabilization with the inclusion of viscous heating is simply due to the fact that viscous heating decreases the viscosity and increases the effective Reynolds number. In addition, although there are major quantitative differences, the stability diagrams presented in this study show no qualitative difference from those for a Newtonian fluid without viscous heating, suggesting that the stability observed is shear instability and no new instability mechanisms operate here.

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References [1] T.D. Papathanasiou, Circular Couette flow of temperature-dependent materials: asymptotic solutions in the presence of viscous heating, Chem. Eng. Sci. 52 (1997) 2003. [2] J.R.A. Pearson, Mechanics of Polymer Processing, Elsevier, London, 1985. [3] S.H. Davis, G.A. Kriegsmann, Multiple solutions and hysteresis in steady parallel viscous flows, Phys. Fluids 26 (1983) 1177. [4] A. Pinarbasi, M. Imal, Non-isothermal channel flow of a nonNewtonian fluid with viscous heating, Int. Comm. Heat Mass Transfer 29 (2002) 1099. [5] L.E. Becker, G.H. McKinley, The stability of viscoelastic creeping plane shear flows with viscous heating, J. Non-Newt. Fluid Mech. 92 (2000) 109. [6] U.A. Al-Mubaiyedh, R. Sureshkumar, B. Khomami, Energetics effects on the stability of viscoelastic Dean flow, J. Non-Newt. Fluid Mech. 95 (2000) 277. [7] D.O. Olagunju, L.P. Cook, G.H. McKinley, Effect of viscous heating on the linear stability of viscoelastic cone-and-plate flow: axisymmetric case, J. Non-Newt. Fluid Mech. 102 (2002) 321. [8] D.G. Thomas, R. Sureshkumar, B. Khomami, Effect of inertia on thermoelastic flow instability, J. Non-Newt. Fluid Mech. 120 (2004) 93. [9] A. Pinarbasi, A. Liakopoulos, Stability of two-layer Poiseuille flow of Carreau–Yasuda and Bingham-like fluids, J. Non-Newt. Fluid Mech. 57 (1995) 227. [10] A. Pinarbasi, Formulation and computational issues for stability of two-layer inelastic fluids, Comp. Fluids 29 (2000) 935.