Poiseuille Flow Research Articles

Effects of jet grouting on adjacent ground through numerical modelling

When using jet grouting for soil stabilization, effects of the installation process, such as erosion or remolding of the surrounding soil, need to be evaluated. This could be particularly important for applications in challenging ground conditions and low initial ground stability. This study investigates such effects in soft marine clays (undrained shear strength of 20-30 kPa) using numerical finite element simulations for both flat and sloping terrains. Additionally, an analytical solution using the modified Hagen-Poiseuille equation was developed to i) assess the pressure induced by the grout at any installed depth, and ii) illustrate the importance of maintaining a proper amount of return flow to avoid build-up of excessive pressures in the ground during the jet grouting process. Combining the analytical and the finite element (FE) simulations, jet grouting induced soil displacements were determined. For flat terrain, the area affected by jet grouting was approximately three to four times the column radius. By contrast, for the studied slope (inclination of 1:3), the influenced area increased to about 10 times the column radius. The influenced area could increase further for steeper slopes. The obtained displacement magnitude was also considerably larger for sloping terrain than for flat terrain. The reported findings relate to jet-grouting in marine clays (which is not the normal application of this technology) and cannot be extrapolated to other soils.

A nonlinear instability theory for a wave system inducing transition in spiral Poiseuille flow

This paper is on the transition scenario of the class of spiral Poiseuille flows that results from the onset, propagation and evolution of disturbances according to mechanisms of Tollmien-Schlichting, and Taylor, acting simultaneously. The problem is approached from the fundamental point of view of following the growth of initially infinitesimally small disturbances into their nonlinear stage when the effect of Reynolds stresses makes itself felt. To this end a set of Generalised Nonlinear Orr–Sommerfeld, Squire and Continuity Equations is set up that enables accounting for effects of growth of initially infinitesimally small disturbances into nonlinearities through a rational iteration scheme. The present proposal closely follows the method put forth for this pupose in 1971 by Stuart and Stewartson in their seminal papers on the influence of nonlinear effects during transition in the bench-mark flows of the class of spiral Poiseuille flows; which are the plane-walled channel flow and the flow in the gap between concentric circular cylinders (Taylor instability).The basic feature of the proposed method is the introduction of an Amplitude Parameter and of a slow/long- scale variable through which the effects of growing disturbances are accounted for within the framework of a rational iteration scheme. It is shown that the effect of amplified disturbances is capturable, as in the bench-mark flows, by a Ginzburg–Landau type differential equation for an Amplitude Function in terms of suitably defined slow/long-scale variables. However, the coefficients in this equation are numbers that depend upon the flow parameters of the spiral Poiseuille flow, which are a suitably defined Reynolds Number, the Swirl Number, and the geometric parameter of transverse curvature inherent in the flow geometry. The Ginzburg–Landau equation derived hints at the drastic changes in flow pattern that the spiral Poiseuille flow in transition may undergo, as its Swirl Number is taken from very small to very large values.

A model for thermocapillary driven motion of liquid drops on an inclined plane

Thermocapillary drop migration is driven by surface tension gradient. On an inclined plane, gravity acts as an additional force that can have a promoting or resisting effect depending on the thermal gradient direction. The present 2-D model extends Brochard and Chebbi models for thermocapillary migration of small drops on horizontal plates by accounting for Marangoni and Poiseuille flow components, while including the effect of gravity, acting in the case of inclines. The present treatment is based on fundamentals of conservation of mass and momentum transport, and uses the lubrication approximation and extended boundary conditions at the receding and advancing contact lines. A model for the dynamics of migration is provided, and an analytical solution is derived for drop migration dynamics and the critical angle of inclination.

Axial friction coefficient of turbulent spiral Poiseuille flows

Direct numerical simulations of spiral Poiseuille flows in a narrow gap geometry are performed with the aim of identifying the mechanisms governing the dynamics of the axial friction coefficient. The investigation has explored a small portion of the Reynolds number–Taylor number phase space ( $600 \leq Re \leq 5766$ and $1500 \leq Ta \leq 5000$ ), for which reference experimental results are available. The study is focused on the mechanism leading to the enhancement of the axial friction coefficient with the Taylor number when the Reynolds number is kept constant. The analysis of the spatial distribution of the Reynolds stress tensor and of the turbulent energy budget has evidenced the key role of the pressure–strain correlation in the energy transfer from the azimuthal to the axial component. The latter eventually determines the increase of the axial friction coefficient through the enhanced radial mixing of axial momentum. Data have also shown that the flow dynamics is heavily dependent on the $Ta/Re$ ratio, and different regimes develop (ranging from laminar to turbulent), each with peculiar behaviours.

Numerical simulation of an extensible capsule using regularized Stokes kernels and overset finite differences

In this paper, we present a novel numerical scheme for simulating deformable and extensible capsules suspended in a Stokesian fluid. The main feature of our scheme is a partition-of-unity (POU) based representation of the surface that enables asymptotically faster computations compared to spherical-harmonics based representations. We use a boundary integral equation formulation to represent and discretize hydrodynamic interactions. The boundary integrals are weakly singular. We use the quadrature scheme based on the regularized Stokes kernels by Tlupova and Beale 2019 (given in [34]). We also use partition-of unity based finite differences that are required for the computation of interfacial forces. Given an N-point surface discretization, our numerical scheme has fourth-order accuracy and O(N) asymptotic complexity, which is an improvement over the O(N2log⁡N) complexity of a spherical harmonics based spectral scheme that uses product-rule quadratures by Veerapaneni et al. 2011 [36]. We use GPU acceleration and demonstrate the ability of our code to simulate the complex shapes with high resolution. We study capsules that resist shear and tension and their dynamics in shear and Poiseuille flows. We demonstrate the convergence of the scheme and compare with the state of the art.

A bio-fabricated tesla valves and ultrasound waves-powered blood plasma viscometer.

Introduction: There is clinical evidence that the fresh blood viscosity is an important indicator in the development of vascular disorder and coagulation. However, existing clinical viscosity measurement techniques lack the ability to measure blood viscosity and replicate the in-vivo hemodynamics simultaneously. Methods: Here, we fabricate a novel digital device, called Tesla valves and ultrasound waves-powered blood plasma viscometer (TUBPV) which shows capacities in both viscosity measurement and coagulation monitoring. Results: Based on the Hagen-Poiseuille equation, viscosity analysis can be faithfully performed by a video microscopy. Tesla-like channel ensured unidirectional liquid motion with stable pressure driven that was triggered by the interaction of Tesla valve structure and ultrasound waves. In few seconds the TUBPV can generate an accurate viscosity profile on clinic fresh blood samples from the flow time evaluation. Besides, Tesla-inspired microchannels can be used in the real-time coagulation monitoring. Discussion: These results indicate that the TUBVP can serve as a point-of-care device in the ICU to evaluate the blood's viscosity and the anticoagulation treatment.

Numerical Study on Permeability of Reconstructed Porous Concrete Based on Lattice Boltzmann Method

The reconstruction of the porous media model is crucial for researching the mesoscopic seepage characteristics of porous concrete. Based on a self-compiled MATLAB program, a porous concrete model was modeled by controlling four parameters (distribution probability, growth probability, probability density, and porosity) with clear physical meanings using a quartet structure generation set (QSGS) along with the lattice Boltzmann method (LBM) to investigate permeability. The rationality of the numerical model was verified through Poiseuille flow theory. The results showed that the QSGS model exhibited varied pore shapes and disordered distributions, resembling real porous concrete. Seepage velocity distribution showed higher values in larger pores, with flow rates reaching up to 0.012 lattice point velocity. The permeability–porosity relationship demonstrated high linearity (the Pearson correlation coefficient is 0.92), consistent with real porous concrete behavior. The integration of QSGS-LBM represents a novel approach, and the research results can provide new ideas and new means for subsequent research on the permeability of porous concrete or similar porous medium materials.

Analytical solution of the Poiseuille flow of a De Kee viscoplastic fluid

We provide an explicit analytical solution of the planar Poiseuille flow of a viscoplastic fluid governed by the constitutive equation proposed by De Kee and Turcotte (1980). Formulae for the velocity and the flow rate are derived, making use of the Lambert W function. It is shown that a solution does not always exist because the flow curve is bounded from above and hence the rheological model can accommodate stresses only up to a certain limit. In fact, the flow curve reaches a peak at a critical shear rate, beyond which it exhibits a negative slope, giving rise to unstable solutions.

Hybrid Fiber Influence on the Crack Permeability of Cracked Concrete Exposed to Freeze-Thaw Cycles.

This paper describes hybrid fiber's influence on the crack permeability of cracked concrete exposed to freeze-thaw cycles. A permeability setup and a laser-scanning setup have been designed to measure the crack permeability and the fractured surface roughness of cracked hybrid fiber-reinforced concrete, containing polypropylene fiber and steel fiber, under a splitting tensile load. The results show that, when the effective crack width of the specimens is less than 25 μm, the rough crack surface significantly reduces the concrete's crack permeability. As the crack width increases, the effect of the concrete crack surface on crack permeability gradually decreases, and the crack permeability of the concrete is closer to the Poiseuille flow model. The permeability parameter α derived from the Poiseuille flow model is effective for assessing the crack permeability of concrete. Compared to the modified factor ξ of crack permeability, the permeability parameter α can effectively evaluate and quantify the development trend of crack permeability within a certain range of crack widths. The permeability parameter α of SF20PP2.3, subjected to the same freeze-thaw cycles, decreases by 16.3-94.8% compared to PP4.6 and SF40, and SF20PP2.3 demonstrates a positive synergistic effect on the crack impermeability of cracked concrete. The crack impermeability of SF40PP2.3, subjected to the same freeze-thaw cycles, lies between that of PP6.9 and SF60. The roughness of crack surface (X) and the crack permeability (Y) are highly correlated and follow an exponential curve (Y = 1.0415 × 107·e-6.025·X) in concrete. This demonstrates that hybrid fibers enhance crack impermeability by increasing the crack surface roughness. Furthermore, the combination of polypropylene fiber and steel fiber effectively promotes the formation of micro-cracks and facilitates the propagation of multiple cracks in the concrete matrix. This combination increases the head loss of water flow through the concrete and decreases the crack permeability.

Investigation of the Newtonian fluids flow in circular horizontal tubes at low inlet pressures

The issues of increasing hydrodynamic efficiency, improving the operational and technical characteristics of various apparatus and devices used in the fi eld of heat and mass transfer, as well as ensuring the required regime and flowow conditions of liquids of different viscosities do not lose their relevance. To solve these issues, in addition to studying the nature of the flowow of various liquids in round horizontal tubes (capillaries), it is necessary to determine the conditions when the flowow of liquid in capillaries and round tubes of small diameter is laminar and can be described by the Poiseuille equation. Experimental data on determining water flow ow through horizontal round tubes of various diameters are presented. Determining the patterns and features of such flow ows is mostly aimed at increasing hydrodynamic efficiency, improving the operational and technical characteristics of various apparatus and devices, as well as ensuring the required flow ow regime and conditions. Experimental data on determining the flow ow rate of water through horizontal tubes of circular cross-section with different diameters are presented in the article. The dependence of the volumetric flow ow rate on the pressure drop has been determined; it has been shown that the main parameters that determine the nature of the flow ow of liquids in horizontal tubes are the tube radius and the dynamic viscosity of the liquid. The flow ow of distilled water in tubes with diameters of 0.95, 1.6 and 2.0 mm at an overpressure of 0.266 kPa to 4 kPa is considered. It is shown that the dependence of the volumetric flow ow rate on the excess pressure remains linear at 0.95 mm in the entire considered pressure range. An increase in the tube radius increases the likelihood of velocity flow uctuations and the appearance of a radial velocity component, i.e. the occurrence of elements of a turbulent structure. In this regard, with tube diameters of 1.6 and 2.0 mm, a deviation of the water flow ow regime from the laminar character was established at pressures of more than 1.3 kPa and 1 kPa, respectively. The dependence of the volumetric flow ow rate on pressure for a 40 % aqueous solution of CaCl2, transformer, transmission and engine oils with dynamic viscosity coefficients from 0.002 Pa·s to 0.182 Pa·s remains linear up to pipe diameters of 5–6 mm. The experimental results are shown in the form of a nomogram of the dependence of the coefficient K* on the tube radius at different values of the viscosity coefficient. The results of studies of liquids of various viscosities are presented in the form of a nomogram of the dependence of the ratio of the radius of the tube to the fourth power to the viscosity of the liquid on the radius of the tube. The analysis of which makes it possible to predict the nature of the flow ow of the investigated liquid at the given values of the tube radius and the dynamic viscosity coefficient. Heat exchange devices for the design of which the presented results can be used include a variety of radiators that include tubes through which coolant circulates.

Mass Transfer Mechanism Within Commercial PTFE Membranes In Spacer-Filled Direct Contact Membrane Distillation

The mass transfer mechanism within commercial PTFE membrane with various nominal pore sizes in spacer-filled direct contact membrane distillation (DCMD) was determined. Mass fluxes and membrane permeability (MP) values for each commercial PTFE membrane were experimentally measured. The MP values increased insignificantly with 2.3% and 4% when the inlet temperature at feed side rose from 400C to 500C in different membrane pore sizes. All investigated mass transfer models except from Dusty Gas model were good enough to simulate the mass transfer inside smaller membrane pore sizes (0.22 µm, and 0.45 µm). Compared to combined diffusion model, the predicted mass fluxes using the overall mass transfer model including the contribution of Poiseuille flow obtained better agreement with experimental results for larger membrane pore size (1 µm). The mean absolute percentage error (MAPE) values for combined diffusion model were up to 16.5% compared with the Ding et al. model or Schofield et al. model (under 3%). Regarding the root mean square error (RMSE), the combined diffusion model obtains larger values than the mass transfer model considering the Poiseuille flow contribution. Consequently, the contribution of Poiseuille flow in mass transfer mechanism within smaller PTFE membrane pores (0.22 µm, and 0.45 µm) could be ignored, however, the contribution of Poiseuille flow to the overall mass transfer should be included in case of larger membrane pores (1 µm), or the case of applied transmembrane hydrostatic pressure.

Modal stability analysis of the density-stratified plane Couette–Poiseuille flow

Shear and density stratification strongly affect the flow mechanism of the different atmospheric and ocean flows. In this paper, we investigate the stability characteristics of plane Couette–Poiseuille (CP) flow with stable density stratification in the vertical direction. A modal stability analysis is carried out to examine the exponentially growing instability of stratified plane CP flow under different controlling parameters. The domain of the flow is periodic in streamwise and vertical directions. The stability problem is solved numerically using the spectral collocation method. The present analysis is carried out for Reynolds number Re=104 with different speeds of moving wall and different strengths of stratification. The results show that the mass diffusivity impact on the flow instability mechanism is almost negligible beyond the Schmidt number Sc≥20. The three-dimensional mode is generally the most unstable mode for density-stratified CP flow. However, the most unstable mode in unstratified CP flow is always two-dimensional. In contrast to unstratified CP flow, the density-stratified CP flow is unstable even moving wall velocity exceeds 70% of the center velocity of the plane Poiseuille flow component. It is also observed that the moving wall velocity and density stratification simultaneously affect the stability of the flow, which shows the importance of the shear and stratification in the flow.

A percolation model of unsaturated hydraulic conductivity using three-parameter Weibull distribution

Accurate estimation of unsaturated hydraulic conductivity, K(Sw), is crucial for modeling water and solute transport in variably-saturated soils. In this study, a new model for the estimation of K(Sw) based on percolation theory and critical path analysis is presented, in which the pore-throat size distribution is described by the three-parameter Weibull probability density function. To evaluate the performance of the proposed K(Sw) model, two datasets including more than three hundred samples are analyzed. The first dataset includes 240 pore-network simulations, while the second dataset consists of 101 soil samples from the UNSODA database. For the pore-network simulations, for which both pore-throat size distributions and capillary pressure curves are available, we estimate K(Sw) from the capillary pressure curve with RMSLE = 0.54 slightly better than the estimations obtained from the pore-throat size distribution with RMSLE = 0.6. For the soil samples, since only the capillary pressure curve is measured, we estimate K(Sw) from the capillary pressure curve and evaluate three previously proposed approaches for determining the exponent, α, in Poiseuille's law. We find that RMSLE = 0.98 for α = 3, RMSLE = 0.87 for α = 2(4 − D) − (3 − D)/(2D − 3), and RMSLE = 1.06 for α = 4 − 0.74D, in which D is the fractal dimension characterizing the pore space. Comparing the results with previous studies in which the power-law probability density function was used to characterize capillary pressure curve and pore sizes shows that the proposed three-parameter Weibull distribution, in conjunction with the CPA analysis, may more accurately estimate K(Sw) for a wide variety of soils ranging from very fine to very coarse textures.

Extension of finite particle method simulating thermal-viscoelastic flow and fluid–rigid body interactional process in weakly compressible smoothed particle hydrodynamics scheme

In computational non-Newtonian fluid dynamics, heat transfer has obvious effects on motions of viscoelastic fluids, mechanical mechanism of elasticity, and flow regimes. This study suggests an extended numerical scheme of smoothed particle hydrodynamics and finite particle method within density smoothing (SPH_DSFPM), which involves the discretization of smoothed particle hydrodynamics (SPH) and finite particle method (FPM) within density smoothing (DS) in the weakly compressible flow scheme. A corrected particle shifting technique is incorporated to eliminate tensile instability and inhomogeneity near solid boundaries. A corrected dynamic solid coupled boundary is introduced to deal with casting molding within high-pressure operations, which has a good compatibility between virtual particle method and repulsive force model. Numerical results show that the present scheme has the nearly lower relative error (0.5%) than conventional SPH (2.6%) in the case of evolutionary thermal-viscoelastic Poiseuille flow and heat effects have active influences on velocity, pressure variations for viscoelastic fluid flow around periodic circular cylinders. Three different printing modes of moving printers significantly generate into differentiated forming regimes through high-pressure extrusion. Adaptive particle distributions possess robust flow evolutions, by which the shocked jets can be tracked well and the sinking velocities of wedge entering into solutions can be numerically probed well considering different cuspidal biting angles. In the case of macroscopic fluid–rigid body interactions, the statistical degree of deviation on probed forces with experiments is relatively 4.35% and that is 12.5% for SPH. The proposed numerical scheme has a good performance on improved accuracy, convergence, and stability for simulating transient thermal-viscoelastic flows.

Стационарное протекание несжимаемой вязкой жидкости сквозь двухмерный разветвленный канал

The results of a numerical experiment for the problem of an incompressible viscous fluid stationary flow through a branched planar (two-dimensional) channel are presented. The region in which the flow occurs simulates either blood vessels or a river delta. The finite element method and a modification of the penalty method, as well as the splitting method, are used for calculations. The implementation of the calculation algorithm is using with the help the package FreeFem++. The main goal, in addition, of course, to study the properties and structure of the stationary flow, is to demonstrate the effectiveness of the proposed modification of the penalty method. It is assumed that the region has one input boundary section through which the liquid flows into the region, and several (five) boundary sections through which the liquid flows out of the region. The remaining sections of the region boundary are considered impermeable to liquid. The boundary condition corresponding to the Poiseuille flow in a plane rectilinear channel is set at the input section. Three types of boundary conditions are considered on the output boundary secctions. 1. The boundary conditions correspond to the conditions of conservation of motion of luid particles i. e. the material derivative of the velocity is zero. 2. The boundary conditions correspond to the setting of the flow velocity. 3. The boundary conditions correspond to the setting of the same pressure. The stationary solution is constructed by the relaxation method. In fact, a non-stationary problem is solved over a sufficiently large time interval. As an initial condition for a non-stationary problem, a flow is chosen that is a Poiseuille flow in some region near the input boundary. The dependence of the convergence rate of the relaxation method on the initial data is investigated. It is found that the Poiseuille flow, given at the input section of the boundary of the region, induces similar Poiseuille flows at the output boundaries sections of the region in all these cases of boundary conditions. For a region with five sections of the output boundaries for some configuration of the region, the presence of stationary vortex flows (‘maelstorm’) is found.

On the Flow of a Cement Suspension: The Effects of Nano-Silica and Fly Ash Particles.

Additives such as nano-silica and fly ash are widely used in cement and concrete materials to improve the rheology of fresh cement and concrete and the performance of hardened materials and increase the sustainability of the cement and concrete industry by reducing the usage of Portland cement. Therefore, it is important to study the effect of these additives on the rheological behavior of fresh cement. In this paper, we study the pulsating Poiseuille flow of fresh cement in a horizontal pipe by considering two different additives and when they are combined (nano-silica, fly ash, combined nano-silica, and fly ash). To model the fresh cement suspension, we used a modified form of the power-law model to demonstrate the dependency of the cement viscosity on the shear rate and volume fraction of cement and the additive particles. The convection-diffusion equation was used to solve for the volume fraction. After solving the equations in the dimensionless forms, we conducted a parametric study to analyze the effects of nano-silica, fly ash, and combined nano-silica and fly ash additives on the velocity and volume fraction profiles of the cement suspension. According to the parametric study presented here, larger nano-silica content results in lower centerline velocity of the cement suspension and larger non-uniformity of the volume fraction. Compared to nano-silica, fly ash exhibits an opposite effect on the velocity. Larger fly ash content results in higher centerline velocity, while the effect of the fly ash on the volume fraction is not obvious. For cement suspension containing combined nano-silica and fly ash additives, nano-silica plays a dominant role in the flow behavior of the suspension. The findings of the study can help the design and operation of the pulsating flow of fresh cement mortars and concrete in the 3D printing industry.

Modelling Turbulent Flow of Superfluid $$^{4}$$He Past a Rough Solid Wall in the $$T =$$ 0 Limit

AbstractWe present a numerical study, using the vortex filament model, of vortex tangles in a flow of pure superfluid $$^4$$ 4 He in the $$T = 0$$ T = 0 limit through a channel of width $$D = 1$$ D = 1  mm for various applied velocities V. The flat channel walls are assumed to be microscopically rough such that vortices terminating at the walls are permanently pinned; vortices are liberated from their pinned ends exclusively through self-reconnection with their images. Sustained tangles were observed, for a period of 80 s, above the critical velocity $$V_c \sim 0.20$$ V c ∼ 0.20  cm s$$^{-1} = 20 \frac{\kappa }{D}$$ - 1 = 20 κ D . The coarse-grained velocity profile was akin to a classical parabolic profile of the laminar Poiseuille flow, albeit with a nonzero slip velocity $$\sim$$ ∼ 0.20 cm s$$^{-1}$$ - 1 at the walls. The friction force was found to be proportional to the applied velocity. The effective kinematic viscosity was $$\nu ' \sim 0.1\kappa$$ ν ′ ∼ 0.1 κ , and effective Reynolds numbers within $$\mathrm {Re'} < 200$$ Re ′ < 200 . The fraction of the polarised vortex length varied between zero in the middle of the channel and $$\sim$$ ∼ 60% within the shear flow regions $$\sim D/4$$ ∼ D / 4 from the walls. Therefore, we studied a state of statically polarised ultraquantum (Vinen) turbulence fuelled at short length scales by vortex reconnections, including those with vortex images due to the relative motion between the vortex tangle and the pinning rough surface.

Peridynamic modeling of seepage in multiscale fractured rigid porous media

AbstractThe numerical simulation of seepage in fractured porous media holds significant relevance for subsurface energy development. The presence of fractures at various scales profoundly influences the hydraulic properties of porous media during seepage. A peridynamic (PD)‐based frame is proposed for the seepage problem analysis of multiscale fractured porous media, in which micro‐fractures are implicitly represented like the porous media, macro‐fractures are explicitly represented. For fluid flow modeling, the peridynamic bonds are divided into three types: pore seepage bond, fracture seepage bond, and fluid exchange bond. The micro‐permeabilities of different bonds are derived by Darcy's law, Poiseuille's law, and fluid exchange model, respectively. The proposed peridynamic model can naturally simulate the seepage problem of porous rock with multiscale fractures in a unified framework. Then, four cases of pore seepage and fracture seepage are analyzed using the proposed model, fractured porous media seepage with micro‐fractures and macro‐fractures are respectively considered. The predicted results are compared with the available theoretical solutions, the close agreement shows that the wide validity and applicability of the proposed model in multiscale seepage problems.

Impact of wall-slip on the soluto-marangoni instability in a two-fluid system in a channel–Creeping flow scenario

The role of wall-slip on the soluto-Marangoni instability that emerges in a two-fluid creeping two-dimensional Poiseuille flow is assessed within the framework of Orr-Sommerfeld analysis and species-transport system. Different wall-slip scenarios, namely S1 (slip at the upper wall), S2 (slip at the lower wall), S1S2 (symmetric slip at the upper and lower walls) are considered. The system instability is characterized by identifying unstable and stable zones in the m-n (viscosity ratio - depth ratio) plane for each of the slip conditions at the wall using asymptotic analysis for long-wave disturbances. Numerical simulations based on Chebyshev spectral collocation method reveals the emergence of two modes (M1 and M2) and corresponding stability features are displayed for typical values of Marangoni numbers, n, m and slip lengths. The occurrence of short-wave modes and the features of exchanging dominance between the short-wave modes in the presence of wall-slip for different slip condition are elucidated and the physical mechanism responsible for triggering the instability in the system are revealed through energy budget analysis. The channel system hosting less viscous fluid adjacent to the upper wall demonstrates interesting characteristics such as (among other things): stabilizing/destabilizing the emerging M1/M2 modes in system with no-slip by imposing S1/S2/S1S2 wall conditions and increasing the slip length; exchange of dominance of unstable modes beyond a threshold slip length for S1 slip type. The study thus presents possible means of stabilizing/destabilizing a creeping channel flow system by designing the walls of the channel with appropriate slip type relevant to the application.

Derivation of extended-OBurnett and super-OBurnett equations and their analytical solution to plane Poiseuille flow at non-zero Knudsen number

In this work, we analyse wall-bounded flows in the continuum to transition regime with the help of higher-order transport equations (super-set of the Navier–Stokes equation). Towards this, we incorporate second-order in Knudsen number accurate terms in the single-particle distribution function, and with this complete representation, we first derive the second-order accurate extended-OBurnett (EOBurnett) and third-order accurate super-OBurnett (SOBurnett) equations. We then demonstrate that these newly derived equations exhibit unconditional linear stability. We finally validate the equations by solving for plane Poiseuille flow and derive closed-form analytical solutions for the pressure and velocity fields. The pressure and velocity results thus obtained have been compared with direct simulation Monte Carlo (DSMC) data in the transition regime. The results from both the EOBurnett and SOBurnett equations are found to yield better agreement with DSMC data than that obtained from the Navier–Stokes equations. This improved agreement is attributed to the presence of additional terms in the proposed equations, which effectively capture the effect of the Knudsen layer near the wall. The obtained higher-order transport equations and the closed-form solution presented in this work are novel. The ability of the equations to describe the flow in the transition regime should form the basis for conducting further realistic analytical studies of wall-bounded flows in the future.