Time-dependent Poiseuille Flow Research Articles

A transient model integrating the nanoconfinement effect and pore structure characteristics of oil transport through nanopores

Understanding the integrated transport behavior of oil in shale nanopores is critical to efficient shale oil development. In this paper, based on the time-dependent Poiseuille flow momentum equation, we present a novel transient model to describe oil transport in unsteady and steady states. The model incorporates the effect of the critical shift density, apparent viscosity, slip length, and alkane property, as well as pore tortuosity and surface roughness. We evaluated our model through a comparison with other models, experiments, and molecular dynamics simulations. The results show that the development rates of the volume flows of C6–C12 alkane confined in inorganic nanopores and C12 alkane confined in organic nanopores were faster than that of the corresponding bulk alkane. In addition, the critical drift density positively promoted the volume flow development rate in the unsteady state and negatively inhibited the mass flow rate in the steady state. This effect was clearest in pores with a smaller radius and lower-energy wall and in alkane with shorter chain lengths. Furthermore, both the nanoconfinement effect and pore structure determined whether the volume flow enhancement rate was greater than or less than 1. The rate increased or decreased with time and was controlled mainly by the nanoconfinement effect. Moreover, as the wall energy increased, the flow inhibition effect increased; as the carbon number of alkane increased, the flow promotion effect increased. The results indicate that the proposed model can accurately describe oil transport in shale nanopores.

Assessment of polymer feedback coupling approaches in simulation of viscoelastic fluids using the lattice Boltzmann method

The lattice Boltzmann method (LBM) has emerged as a popular approach to simulate complex non-Newtonian fluid mechanics problems, such as those involving viscoelastic fluids. In these cases, the LBM solver often requires modelling additional non-linear and non-ideal hydrodynamic contributions to the flow resulting from the addition of polymer molecules. These additional contributions, commonly referred to as the polymer feedback, are either incorporated directly in the Boltzmann stress (pressure method) or alternatively in the momentum field as a forcing contribution (forcing method). The choice of coupling approach has in previous works been strongly speculated to impact the numerical accuracy and stability of LBM solvers for viscoelastic fluids, however no formal comparison has been conducted. In this work, we compare the two popular polymer feedback coupling approaches, as well as the various schemes commonly used in the forcing method, namely, the explicit forcing (EF) scheme (He et al., 1998), the Guo et al. scheme (Guo et al., 2002), and the exact-difference-method (EDM) (Kupershtokh et al., 2009). First, through a theoretical comparison in recovering the macroscopic equations, we show that both the EF and Guo et al. forcing schemes retain the exact hydrodynamic representation up to second-order, whereas the EDM scheme includes additional lattice error terms. Similarly, the pressure method retains additional unphysical terms, which are known to violate Galilean invariance. Through numerical tests, involving the steady and time-dependent Poiseuille flow cases, as well as the four-roll mill problem under both steady and transient regimes, it is shown that the forcing method has enhanced numerical accuracy and stability at varying elastic effects. Alternatively, owing to the violation of Galilean invariance, the pressure method proved to be unstable even for moderate elastic effects. The obtained results are useful in determining the optimal polymer feedback coupling approach when simulating viscoelastic fluids using LBM, as well as further testing the generality of previous LBM findings on non-ideal coupling approaches for viscoelastic fluids.

A Comparative Review of Smoothed Particle Hydrodynamics, Dissipative Particle Dynamics and Smoothed Dissipative Particle Dynamics

Smoothed particle hydrodynamics (SPH), dissipative particle dynamics (DPD) and smoothed dissipative particle dynamics (SDPD) are three typical and related particle-based methods. They have been increasingly attractive for solving fluid flow problems, especially for the biofluid flow, because of their advantages of ease and flexibility in modeling complex structure fluids. This work aims to review what the exact similarities and differences are among them, by studying four simple fluid flows: (i) self-diffusion of quiescent flow, (ii) time-dependent Coutte flow, (iii) time-dependent Poiseuille flow, and (iv) lid-driven cavity flow. The simulations show that SPH, DPD and SDPD can give the similar results. SPH generates quite smooth results and has zero system temperature due to the absence of thermal fluctuations, suitable for macroscale problems. However, DPD and SDPD have fluctuating results around the reference results and nonzero system temperature with considerable thermal fluctuations, suitable for mesoscale problems. SDPD is more convenient than DPD to some extent, because it is not required to pre-define the force coefficients. SDPD can adopt more diverse equation of state (EOS) than DPD, because its EOS is user-defined unlike the EOS of DPD, inbuilt in the formulations.

Incompressible Navier–Stokes flows with time-dependent prescribed fluxes in domains with cylindrical outlets to infinity

We consider instationary Navier–Stokes equations with prescribed time-dependent fluxes in a cylindrical domain \(\Omega \subset {\mathbb R}^n,n\ge 3,\) with several exits to infinity. First, we prove existence and uniqueness of time-dependent Poiseuille flow in \(L^q\)-spaces on infinite straight cylinders over time interval \((0,T)\), \(0 2, q\in (\frac{n-1}{2},\infty )\), with an exponential weight along the axial directions of \(\Omega \).

Time-dependent Poiseuille flow of a viscous compressible fluid confined between two planar walls with dynamic partial slip boundary condition

Time-dependent Poiseuille flow of a viscous compressible fluid confined between two planar walls is studied for a partial slip boundary condition with frequency-dependent slip length. After an initial uniform impulse parallel to the walls, the flow pattern quickly becomes nearly parabolic. For a narrow gap, a dynamic slip length can lead to damped oscillations of total fluid momentum.

Solvability in weighted spaces of the three-dimensional Navier-Stokes problem in domains with cylindrical outlets to infinity

The nonstationary Navier-Stokes problem is studied in a three-dimensional domain with cylindrical outlets to infinity in weighted Sobolev function spaces. The unique solvability of this problem is proved under natural compatibility conditions either for a small time interval or for small data. Moreover, it is shown that the solution having prescribed fluxes over cross-sections of outlets to infinity tends in each outlet to the corresponding time-dependent Poiseuille flow.

Global Solvability in W 2 2,1 -Weighted Spaces of the Two-Dimensional Navier–Stokes Problem in Domains with Strip-Like Outlets to Infinity

The time-dependent Navier–Stokes system is studied in a two-dimensional domain with strip-like outlets to infinity in weighted Sobolev function spaces. It is proved that under natural compatibility conditions there exists a unique solution with prescribed fluxes over cross-sections of outlets to infinity which tends in each outlet to the corresponding time-dependent Poiseuille flow. The obtained results are proved for arbitrary large norms of the data (in particular, for arbitrary fluxes) and globally in time.

Investigation of Mechanisms of Excitation and Growth of Unstable Perturbations in Pulsating Flows

General laws of the processes of generation and amplification of secondary perturbations in oscillating viscous fluid flows are studied theoretically. The stability and receptivity are analyzed with reference to perturbations generated by fluctuations of the flow rate of Poiseuille flow induced by small two-dimensional roughnesses of the channel walls. It is shown that the presence of roughness leads to excitation in the flow of perturbations at all multiples of the main flow oscillation frequency. Using the Fourier transform along the streamwise coordinate, the problem of calculating the frequency harmonics is reduced to a system of equations of the Orr-Sommerfeld type interrelated via the oscillatory component of the main flow. On the basis of an investigation of the analytic properties of the Fourier-image it is shown that upstream and downstream of the roughness the perturbation can be represented in the form of a superposition of modes of the time-dependent Poiseuille flow. The modes are classified and their spectrum is calculated. The structure of the mean-square fluctuations generated by free perturbations is investigated. Examples of calculating the evolution of forced perturbations are given for cases in which the scattering of the oscillations of the main flow on the roughness leads to the generation of one or two modes growing downstream.

SPH simulations of time-dependent Poiseuille flow at low Reynolds numbers

A working Smoothed Particle Hydrodynamics (SPH) formalism for solving the equations of motion of a viscous fluid is presented. The method is based on a standard symmetrized SPH expression for the viscous forces, which involves only first-order derivatives of the kernel through a direct evaluation of the viscous stress tensor. Therefore, the interpolation can be performed using low-order kernels of compact support without compromising the accuracy and stability of the results. In principle, the scheme is suitable for treating compressible fluids with arbitrary shear and bulk viscosities. Here, we demonstrate that when it is combined with the pressure-gradient correction proposed by Morris et al., the method is also suitable for solving the Navier–Stokes equations for incompressible flows without any further assumptions. Simulations using the method show close agreement with the analytic series solutions for plane Poiseuille and Hagen–Poiseuille flows at very low Reynolds numbers. At least for these specific tests, the results obtained are essentially independent of employing either a cubic or a quintic spline kernel.

Classroom Note:Time-Dependent Poiseuille Flow

The importance of including eigenfunctions in any time-dependent fluid motion is illustrated through a simple example.

Existence and asymptotic stability of time-dependent poiseuille flows of isothermal bipolar fluids

The existence of both weak and classical solutions, for an initial-boundary value problem associated with the time-dependent Poiseuille flow of a non-linear isothermal bipolar fluid in a channel, is established along with companion results on uniqueness and asymptotic stability. The existence proof is obtained by combining an iteration scheme for the quasilinear parabolic equation governing the velocity of the dipolar fluid with a priori estimates for a class of associated linear prarbolic initial-boundary value problems.

Dilute solutions of macromolecules in a time-dependent Poiseuille flow

We have extended our previous results on the rheological properties of dilute solutions of elastic and rigid dumbbells to the general time-dependent Poiseuille flow. We present here the results for a few specific time-dependent flows that are typical of rheological experiments: Startup, cessation, and oscillatory flows. In all the cases considered for the rigid dumbbells, the inhomogeneity of the Poiseuille flow decreases the value of the viscosity at all times at the center of the channel.