Exact solution of the Navier-Stokes equations for the oscillating flow in a duct of a cross-section of right-angled isosceles triangle

The Navier-Stokes equations have been solved in order to obtain an analytical solution of the fully developed laminar flow in a duct having a cross section of a right-angled, isosceles triangle. We obtained a solution for the case of oscillating pressure gradient flow. The pulsating flow is obtained by the superposition of the steady and oscillating pressure gradient solutions.     

考虑二次梯度项影响的非线性不稳定渗流问题的精确解

考虑了二次梯度项影响的非线性径向流动问题的无限大地层和有界地层渗流模型．在井底定流量和定压生产时,对无限大地层及有界地层(包括封闭和定压地层)六种情况,利用广义Weber变换和广义Hankel变换求得了实空间的解析解,分析了非线性压力解与线性压力解的差异,发现在晚时段其差异可达8%以上．因此在试井长时要考虑二次梯度项的影响．     

Modeling unsteady flow characteristics using smoothed particle hydrodynamics

Smoothed particle hydrodynamics (SPH) method has been extensively used to simulate unsteady free surface flows. The works dedicated to simulation of unsteady internal flows have been generally performed to study the transient start up of steady flows under constant driving forces and for low Reynolds number regimes. However, most of the fluid flow phenomena are unsteady by nature and at moderate to high Reynolds numbers. In this study, first a benchmark case (transient Poiseuille flow) is simulated to evaluate the ability of SPH to simulate internal transient flows at low and moderate Reynolds numbers (Re = 0.05, 500 and 1500). For this benchmark case, the performance of the two most commonly used formulations for viscous term modeling is investigated, as well as the effect of using the XSPH variant. Some points regarding using the symmetric form for pressure gradient modeling are also briefly discussed. Then, the application of SPH is extended to oscillating flows imposed by oscillating body force (Womersley type flow) and oscillating moving boundary (Stokes’ second problem) at different frequencies and amplitudes. There is a very good agreement between SPH results and exact solution even if there is a large phase lag between the oscillating pressure difference and moving boundary and the movement of the SPH particles generated. Finally, a modified formulation for wall shear stress calculations is suggested and verified against exact solutions. In all presented cases, the spatial convergence analysis is performed.     

Analytical solutions to Stokes-type flows of inhomogeneous fluids

In this paper, we study the unsteady motion of an inhomogeneous incompressible viscous fluid, where the viscosity varies spatially according to various models. We study the Stokes-type flow for these types of fluids where in the first case the flow between two parallel plates is examined with one of the plates oscillating and in the second case when the flow is caused by a pulsatile pressure gradient. A general argument establishes the existence of oscillatory solutions to our problem. Exact solutions are obtained in terms of some special functions and comparisons are made with the cases of constant viscosity and the slow flow regimes.     

Two-layer flows of generalized immiscible second grade fluids in a rectangular channel

Unsteady one-dimensional flows of two incompressible and immiscible generalized second grade fluids in a rectangular channel are studied. A constant pressure gradient acts in the flow direction, while the channel walls have oscillating translational motions in their planes. The generalization considered in this paper consists into a mathematical model based on constitutive equations of second grade fluid with Caputo time-fractional derivative in which the history of the shear stress influences the velocity gradient. The velocity and shear stress fields in the Laplace transform domain are obtained. Numerical solutions for the real velocity and shear stress have been found by employing the Stehfest numerical algorithm for the inverse Laplace transform. The influence of the fractional parameters on the velocity and shear stress has been studied by numerical simulations and graphical illustrations. It is found that the memory effects are significant only for small values of the time t.     

平面非牛顿流体在m＞1时的径向流动

讨论平面非牛顿流体(例如高粘度高含腊量的地下石油)在m>1时的径向流动.作者首先给出了问题的数学模型,它是退缩的自由边值问题.然后得到了该问题的近似问题古典解的存在唯一性.当油井边的压力梯度是常值函数时,该问题古典解的存在唯一性也得到了.     

Non-linear peristaltic transport of a second-order fluid through a porous medium

The present paper investigates phenomena brought about into the classic peristaltic mechanism by inclusion of non-Newtonian effects through a porous space in a channel. The peristaltic motion of a second-order fluid through a porous medium was studied for the case of a planar channel with harmonically undulating extensible walls. The system of the governing nonlinear PDE is solved by using the perturbation method to second-order in dimensionless wavenumber. The analytic solution has been obtained in the form of a stream function from which the axial pressure gradient has been derived. The flow is investigated in a wave frame of reference moving with velocity of the wave. Numerical calculations are carried out for the pressure rise and frictional force. The features of the flow characteristics are analyzed by plotting graphs and discussed in detail.     

Peristaltic mechanism of a Maxwell fluid in an asymmetric channel

The peristaltic flow of a Maxwell fluid in an asymmetric channel is studied. Asymmetry in the flow is induced by taking peristaltic wave train of different amplitudes and phase. The viscoelasticity of the fluid is induced in the momentum equation. An analytic solution is obtained through a series of the wave number. The leading velocity term denotes the Newtonian result. The first and second order terms are the viscoelastic contribution to the flow. Expressions for stream function and longitudinal pressure gradient are obtained analytically. Numerical computations have been performed for the pressure rise per wavelength and discussed.     

NEW METHOD FOR IMPROVED CALCULATIONS OF UNSTEADY COMPLEX FLOWS IN LARGE ARTERIES

Using an improved computational fluid dynamics (CFD) method developed for highly unsteady three-dimensional flows, numerical simulations for oscillating flow cycles and detailed unsteady simulations of the flow and forces on the aortic vessels at the iliac bifurcation, for both healthy and diseased patients, are analyzed. Improvements in computational efficiency and acceleration in convergence are achieved by calculating both an unsteady pressure gradient which is due to fluid acceleration and a good global pressure field correction based on mass flow for the pressure Poisson equation. Applications of the enhanced method to oscillatory flow in curved pipes yield an order of magnitude increase in speed and efficiency, thus allowing the study of more complex flow problems such as flow through the mammalian abdominal aorta at the iliac arteries bifurcation. To analyze the large forces which can exist on stent graft of patients with abdominal aortic aneurysm (AAA) disease, a complete derivation of the force equations is presented. The accelerated numerical algorithm and the force equations derived are used to calculate flow and forces for two individuals whose geometry is obtained from CT data and whose respective blood pressure measurements are obtained experimentally. Although the use of endovascular stent grafts in diseased patients can alter vessel geometries, the physical characteristics of stents are still very different when compared to native blood vessels of healthy subjects. The geometry for the AAA stent graph patient studied in this investigation induced flows that resulted in large forces that are primarily caused by the blood pressure. These forces are also directly related to the flow cross-sectional area and the angle of the iliac arteries relative to the main descending aorta. Furthermore, the fluid flow is significantly disturbed in the diseased patient with large flow recirculation and stagnant regions which are not present for healthy subjects.     

低渗透非均质油藏油水两相渗流敏感系数理论模型

本文研究了低渗透油藏低速非达西油水两相渗流的反问题，建立了基于动态信息的油藏敏感系数计算模型，给出了井底压力和水油比关于渗透率和孔隙度的敏感系数的有效算法，为油藏数值模自动拟合和油藏精细描述算法提供了理论基础。     

Investigation of physiological pulsatile flow in a model arterial stenosis using large-eddy and direct numerical simulations

Physiological pulsatile flow in a 3D model of arterial stenosis is investigated by using large eddy simulation (LES) technique. The computational domain chosen is a simple channel with a biological type stenosis formed eccentrically on the top wall. The physiological pulsation is generated at the inlet using the first harmonic of the Fourier series of pressure pulse. In LES, the large scale flows are resolved fully while the unresolved subgrid scale (SGS) motions are modelled using a localized dynamic model. Due to the narrowing of artery the pulsatile flow becomes transition-to-turbulent in the downstream region of the stenosis, where a high level of turbulent fluctuations is achieved, and some detailed information about the nature of these fluctuations are revealed through the investigation of the turbulent energy spectra. Transition-to-turbulent of the pulsatile flow in the post stenosis is examined through the various numerical results such as velocity, streamlines, velocity vectors, vortices, wall pressure and shear stresses, turbulent kinetic energy, and pressure gradient. A comparison of the LES results with the coarse DNS are given for the Reynolds number of 2000 in terms of the mean pressure, wall shear stress as well as the turbulent characteristics. The results show that the shear stress at the upper wall is low just prior to the centre of the stenosis, while it is maximum in the throat of the stenosis. But, at the immediate post stenotic region, the wall shear stress takes the oscillating form which is quite harmful to the blood cells and vessels. In addition, the pressure drops at the throat of the stenosis where the re-circulated flow region is created due to the adverse pressure gradient. The maximum turbulent kinetic energy is located at the post stenosis with the presence of the inertial sub-range region of slope −5/3.     

On an adjoint initial-boundary value problem

We study an adjoint initial-boundary value problem for linear parabolic equations; which arises when modeling the unidirectional motion of two viscous fluids with a common interface under the action of a pressure gradient. Under some conditions on the pressure gradient, we obtain a priori estimates and show that the solution enters a stationary mode. For semibounded layers, we find the solution in closed form and indicate the case of a self-similar solution. We determine the volume flow rates in the layers.     

On the flow of a dusty viscous liquid through a circular pipe

Analytical solution of the flow problem of a dusty viscous liquid through a circular pipe in case of axial symmetry is obtained when pressure gradient varies harmonically with time. It is found that the effect of the fine dust is to make the velocity of sedimentation zero and when dust is sufficiently coarse, the effect of the dust is equivalent to an extra frictional force proportional to the fluid velocity.     

层流管流入口压力突然升高引起的水力瞬变的理论分析

研究了层流状态下管道入口压力突然升高引起的水力瞬变过程,建立了瞬态压力分布的偏微分方程和初边值条件,用分离变量法求得了压力的理论解．根据压力和流量间的约束关系,得到了关于流量的偏微分方程和初边值条件,用分离变量求得了瞬变过程流量分布理论解．最后,用特征线法(MOC)对该问题进行了数值求解,理论解和数值解吻合很好．     

Non-linear peristaltic flow of a non-Newtonian fluid under effect of a magnetic field in a planar channel

This paper is devoted to the study of peristaltic flow of a fourth grade fluid in a channel under the considerations of long wavelength and low-Reynolds number. The flow is examined in a wave frame of reference moving with velocity of the wave. The analytic solution has been obtained in the form of a stream function from which the axial velocity and axial pressure gradient have been derived. The results for the pressure rise and frictional force per wavelength have also been computed numerically. The computational results indicate that the pressure rise and frictional force per wavelength are increased in case of non-Newtonian fluid when compared with Newtonian fluid. Several graphs of physical interest are displayed and discussed.     

Steady flow of incompressible fluid between two co-rotating disks

This article provides an analytical solution of the Navier–Stokes equations for the case of the steady flow of an incompressible fluid between two uniformly co-rotating disks. The solution is derived from the asymptotical evolution of unknown components of velocity and pressure in a radial direction – in contrast to the Briter–Pohlhausen analytical solution, which is supported by simplified Navier–Stokes equations. The obtained infinite system of ordinary differential equations forms recurrent relations from which unknown functions can be calculated successively. The first and second approximations of solution are solved analytically and the third and fourth approximations of solutions are solved numerically. The numerical example demonstrates agreements with results obtained by other authors using different methods.     

Exact analytical solution of a generalized multiple moving boundary model of one-dimensional non-Darcy flow in heterogeneous multilayered low-permeability porous media with a threshold pressure gradient

A nonlinear generalized multiple moving boundary model of one-dimensional non-Darcy flow in heterogeneous multilayered low-permeability porous media with a threshold pressure gradient is constructed, in which the total rate of fluid injection into the porous media remains constant. The number of layers in the model can be arbitrary, and thus the generalized model will be very suitable for describing the one-dimensional non-Darcy flow characteristics in low-permeability reservoirs with strong heterogeneity. Through the similarity transformation method, the exact analytical solution of the multiple moving boundary model is obtained, and the formula for the subrate of fluid injection into every layer is provided. Moreover, it is strictly proved that the exact analytical solution can reduce to the solution of Darcy flow as the threshold pressure gradient in different layers simultaneously tends to zero. Through the exact analytical solution, the effects of the layer threshold pressure gradient, the layer permeability ratio, and the layer elastic storage ratio on the moving boundaries, the spatial pressure distributions, the transient pressure, and the layer subrate in low-permeability porous media are discussed. Through comparison of the exact analytical solutions, it is also demonstrated that incorporation of the multiple moving boundary conditions is very necessary in the modeling of non-Darcy flow in heterogeneous multilayered porous media with a threshold pressure gradient, especially when the threshold pressure gradient is large. In particular, an explicit formula is presented for estimating the relative error of the transient pressure introduced by ignoring the moving boundaries in the modeling. All in all, solid theoretical foundations are provided for non-Darcy flow problems in stratified reservoirs with a threshold pressure gradient. They can be very useful for strictly verifying numerical simulation results, and for giving some guidance for project design and optimization of layer production or injection during the development of heterogeneous low-permeability reservoirs and heavy oil reservoirs so as to enhance oil recovery.     

Dusty effects on pulsating viscous flow of two immiscible fluids between two parallel plates

The flow of two immiscible incompressible dusty viscous fluids between two parallel plates generated by a pulsating pressure gradient is investigated. Velocity fields for the fluid-particle system along with the expressions for the skin friction drag at the plates are obtained and studied graphically. It is found that there is an immediate response to pressure fluctuations in the first stream at low frequency range 0<σ≤4 being maximum at σ=4. On the contrary, the second stream is more responsive to fluctuations at relatively higher frequencies. The maximum response in this case is shifted to σ=16.     

基于摄动法的压敏储层非线性渗流问题研究

研究地层压力变化对渗流特征的影响,对低渗透储层、碳酸盐岩储层,或其他致密性储层的油气开采和储层改造都具有重要意义.在考虑渗流压力梯度平方项存在的前提下,运用摄动法求解的相关理论,将渗透率随压力的变化融入到渗流问题的求解过程中,有效地求解了该类压敏型储层的非线性渗流问题.结果表明,在实际应用中,尤其是在储层压敏性较弱的情况下,可考虑直接用0阶摄动解即可满足较好的计算精度;储层的压敏性越强,越适宜通过用摄动解的修正,来达到精确求解的目的.     

Calculation of pressure in the linear problem of vibrator in a supersonic boundary layer : PMM vol. 43, no. 6, 1979, pp. 1014–1028

The supersonic flow over a body consisting of a triangular oscillating plate the vibrator— mounted between two flat plates is investigated. The body is assumed to be thermally insulated, and the vibrator dimensions and the oscillation frequencies to be such that the flow can be defined by equations of a boundary layer with self-induced pressure [1 — 5]. The oscillation amplitude is assumed small so that these equations can be linearized. The Fourier transform of the longitudinal coordinate is used for solution derivation. The inverse Fourier transform is obtained by numerical methods. It is shown that the perturbations of flow parameters induced by the vibrator are damped upstream and downstream in accordance with an exponential law.