Linear stability of plane Poiseuille flow of a Bingham fluid in a channel with the presence of wall slip

Highlights

• Stability analysis for plane Poiseuille flow of a Bingham fluid with wall slip.

• Eigenvalue problem solved by Chebyshev collocation method at Gauss-Lobatto points.

• Streamwise slip shows stabilizing effects on the flow.

• Spanwise slip causes flow to be linearly unstable for large spanwise slip numbers.

Abstract

In the present paper, the effects of a channel symmetric wall slip are investigated on the stability of plane Poiseuille flow of a Bingham fluid, based on the modal and non-modal linear stability analysis approaches. Both streamwise and spanwise slip conditions are considered and their results are compared with those of the no-slip condition, in terms of the flow stability. The results are obtained for different dimensionless groups that govern the stability picture, for example the Bingham number, the Reynolds number and the streamwise/spanwise slip numbers. The linearized perturbation equations are obtained and the corresponding eigenvalue problem is solved based on the Chebyshev collocation method at the Gauss-Lobatto grid points, using the QZ algorithm. In the modal analysis, the focus is on the changes made by the channel wall slip on the eigenvalue spectra. The flow is shown to be linearly stable for no-slip and streamwise slip conditions. However, for the spanwise slip condition, an unstable eigenvalue is found for large values of the spanwise slip number. The non-modal stability analysis is based on the transient energy growth of the perturbations, which is calculated using the eigenfunctions of the eigenvalue problem representing the amplitude of the perturbations. Increasing the streamwise slip decreases the maximum energy growth, eventually leading to a flow for which the transient energy only decays in time. Thus, two distinct flows showing the energy growth and energy decay regimes can be identified, for a range of Bingham, Reynolds and streamwise slip numbers.

Introduction

Yield stress materials belong to a branch of non-Newtonian fluids for which a threshold value for the applied stress, called the yield stress, is considered. Below the yield stress value, the material behaves as a solid and above this threshold it deforms as a viscous fluid. Therefore, yield stress fluids are also known as viscoplastic fluids. The most inclusive and widely used model that addresses many of the complexities of a yield stress non-Newtonian fluid, such as yield stress, shear thinning, shear thickening, and viscous effects, is the Herschel–Bulkley model. However, several other well-known models such as the Bingham model and the Casson model also describe the yield stress fluid rheology. Due to a variety of reasons, viscoplastic fluid flows may experience wall slip over the flow geometry walls, i.e. a phenomenon frequently observed in real applications, affecting the dynamics of these flows. In the present study, we investigate the effects of wall slip on the dynamics of a simple flow (i.e. plane Poiseuille flow) of a Bingham fluid, focusing on the flow stability picture.

There are many fluids with proven yield stress behavior in a number of industrial applications, for example, molten polymers in polymer extrusion and co-extrusion processes [1,2], cement slurry in oil and gas well cementing processes [3,4], and waxy crude oil transported through pipelines [5]. In addition, hair gel and many other cosmetic products such as moisturizing creams exhibit viscoplastic behavior [6,7]. Besides, many food products, for example chocolate cream, butter and jam are yield stress fluids [7]. Interestingly, some biofluids such as mucus and blood also show yield stress rheology [7,8].

New controlled experiments have suggested that the no-slip boundary condition assumption may not be always valid for fluid flows in contact with solid walls [9]. In fact, slippery flows have been observed in numerous situations, including microfluidics [10], [11], porous media [12], [13], [14], biological processes [14], [15], electro-osmotic flows [16], [17], extrusion through dies [1], lubrication [18], [19], and sedimentation [20]. In the aforementioned applications, the fluid rheology may vary from simple Newtonian fluids to more complex non-Newtonian fluids.

There are specific applications where viscoplastic fluid rheology and wall slip effects coexist. One example is the cementing process in oil and gas well constructions, using foamed cement. A foamed cement slurry, as a mixture of cement slurry, foaming agents and gas, is known to be a viscoplastic fluid and it is believed to slip during the cementing process (within the casing or over the formation walls) [21]. Another example is the flow of waxy crude oil through underwater pipelines, showing complex rheology including viscoplastic behavior, and this flow is prone to slip over the pipe walls. In some microfluidic applications, the rheology of fluids flowing through microfluidic devices exhibit yield stress characteristics, for example in giant electrorheological fluids, which can also exhibit wall slip effects [22], [23]. It is believed that wall slip characteristics may play an important role on the fluid flow dynamics in microfluidic devices [24].

Wall slip has a significant effect on the flow dynamics and, in particular, on the velocity profiles of viscoplastic fluids. Panaseti et al. [25] have analytically investigated the flow of Herschel–Bulkley fluids in a channel with slip on one of the walls. They have presented velocity profile solutions for different flow regimes caused by the flow profile asymmetry, due to the asymmetric wall slip effects. In a followup study [26], they have considered a larger slip at the upper wall, compared to the lower wall, and developed analytical solutions to calculate the velocity profiles. Ferras et al. [2] have developed analytical solutions for symmetric wall slip in Newtonian and inelastic non-Newtonian fluids, using four slip models including the linear Navier, non-linear Navier, empirical asymptotic, and Hatzikiriakos slip models. They have considered a generalized Newtonian constitutive equation, with the viscosity modeled via the power-law, Bingham, Herschel–Bulkley, Sisko and Robertson-Stiff models. Kalyon and Malik [27] have analytically analyzed axial laminar flows of viscoplastic fluids in a concentric annulus with wall slip, providing solutions applicable to Bingham plastic, power-law, and Newtonian fluids. Taghavi [21] has theoretically studied buoyant displacement flows of two generalized Newtonian fluids, flowing through a two-dimensional channel with wall slip boundary conditions. The author has considered three slip cases, including slip only at the top wall, slip only at the bottom wall, and slip at both walls, showing the significance of wall slip conditions on the velocity profiles and the overall displacement flow behavior.

Analyzing the flow stability with slip boundary conditions dates back at least to 1974 when Gersting [28] investigated the stability of Newtonian flows through two-dimensional slippery channels with a constant pressure gradient. The author has applied the finite difference scheme to solve the Sheppard’s equation, finding that the wall slip condition stabilizes the flow as the value of the slip parameter increases. Lauga and Cossu [29] have studied both symmetric and asymmetric wall slip effects on the stability of pressure-driven channel flows of Newtonian fluids, confirming the results of Gersting [28] and stating that wall slip increases the critical Reynolds number significantly. Min and Kim [30] have investigated the effects of hydrophobic surfaces (represented by slip boundary conditions) on the flow stability of wall-bounded Newtonian shear flows. Their stability analysis has indicated that a streamwise slip increases the critical Reynolds number but the spanwise slip accelerates the transition to turbulence. Several studies have been also performed on the stability analysis of two-layer Newtonian fluids with streamwise slip boundary conditions, assuming a hydrophobic wall nature [31], [32], [33], [34], [35], [36], finding that wall slip in the streamwise direction generally has a stabilizing effect on these flows.

In the context of the stability of non-Newtonian fluid flows, most studies have investigated the flow stability with no-slip boundary conditions, mainly using the linear stability analysis. Pavlov et al. [37] have studied for the first time the linear stability of Poiseuille flow of a viscoplastic fluid, revealing that the flow is linearly stable under infinitesimal perturbations. Frigaard et al. [38] have derived the Orr-Sommerfeld equation and boundary conditions for Poiseuille flow of Bingham fluids with no-slip conditions. Nouar and Frigaard [39] have performed the non-linear stability analysis of a Bingham flow, with the use of the Reynolds-Orr energy equation, finding that the critical Reynolds number (R_cr) increases with the Bingham number (B), following R_cr ~ B^1/2 as B → ∞. Adding the Bingham term into the stability equation, Frigaard and Nouar [40] have conducted a three-dimensional stability analysis of plane Poiseuille flow of Bingham fluids, finding that R_cr ~ B^3/4 as B → ∞. In another study [41], they have used viscosity regularization models to solve the stability problem for viscoplastic fluid flows and they have reported that the usage of regularization models leads to spurious eigenvalues that must be eliminated from the stability analysis. Peng and Zhu [42] have considered the linear stability of Bingham fluids in a spiral Couette flow, with concentric cylinders of independent rotation speeds, where the inner cylinder is able to slide axially. They have found that the islands of instability, a characteristic of spiral Couette flow of Newtonian fluids, do not emerge due to the yield stress.

While most studies consider the modal analysis to evaluate the flow stability of viscoplastic fluids, a number of works also develop the non-modal stability analysis to capture the temporal growth of the kinetic energy of the perturbations. For instance, Nouar et al. [43] have analyzed the modal and non-modal stability of plane Poiseuille flow of Bingham fluids, finding that, while the flow is linearly stable in the modal analysis, the non-modal analysis indicates the presence of oblique wave numbers associated to the optimum energy growth. Liu et al. [44] have performed the non-modal stability analysis of plane Poiseuille flow of Herschel–Bulkley fluids, focusing on the effects of shear-thinning rheology on the flow stability. They have found that, as the power-law index decreases (representing an increase in shear-thinning behavior), the flow becomes more unstable. Métivier et al. [45] have analyzed the linear stability of the Rayleigh–Benard Poiseuille flow of Bingham fluids when the Bingham number approaches zero. By comparing their results with those of Newtonian fluids, they have noticed discontinuous behavior between the critical stability conditions, associated to the intact plug zone in the Bingham fluid flow. In a followup study [46], they have shown that increasing the Bingham number delays the onset of convection in the Rayleigh–Benard Poiseuille flow. For low Reynolds and high Bingham numbers, their results have shown that the critical Rayleigh and wave numbers follow B² and B^1/4, respectively.

There is only a limited number of studies that have investigated the wall slip effects on the stability of non-Newtonian inelastic fluid flows. As an example, Bouteraa et al. [47] have presented the linear and weakly nonlinear stability analysis of Rayleigh–Benard convection for shear-thinning fluids, using the Carreau model. Considering symmetric slip conditions at the flow geometry walls, they have shown that with an increase in slip the critical Rayleigh number decreases and the critical wave number increases. Métivier and Magnin [48] have studied the effects of wall slip on the stability of Rayleigh–Benard channel flows of viscoplastic fluids, by considering Poiseuille flow of Bingham fluids with symmetric wall slip conditions. Their linear stability analysis has revealed that the critical Rayleigh number decreases with an increase in slip at the walls. In summary, these studies suggest that wall slip has a destabilizing effect on Rayleigh–Benard flows and it promotes the onset of convection.

Based on the brief overview of the literature above, it is clear that there is a dearth of understanding of the effect of wall slip on the stability of viscoplastic fluids. Therefore, our work aims to contribute to this field, via developing the modal and non-modal linear stability analysis for a simple flow (i.e. plane Poiseuille flow) of a Bingham fluid. Both streamwise and spanwise wall slip effects are considered for the first time, in order to evaluate the wall slip effects on the flow stability picture.

The present paper is organized as follows. First, the governing equations of the flow stability problem are brought in Section 2, presenting the analytical solution for the base flow as well as the equations for the modal and non-modal stability analysis. The results are presented in Section 3, for the modal and non-modal stability analysis with the presence of streamwise and spanwise wall slips. Finally, Section 4 briefly summarizes the main findings.