UNSTEADY BOUNDARY LAYER FLOW OF FRACTIONAL OLDROYD-B VISCOELASTIC FLUID PAST A MOVING PLATE IN A POROUS MEDIUM

This paper considers the unsteady boundary layer flow over a moving flat plate embedded in a porous medium with fractional Oldroyd-B viscoelastic fluid. The governing equations with mixed time-space fractional derivatives are solved numerically by using the finite difference method combined with an L1-algorithm. The effect of various physical parameters on the velocity and average skin friction are discussed and graphically illustrated in detail.Results show that the porosity € and fractional derivative α enhance the flow of Oldroyd-B viscoelastic fluid within porous medium, but fractional derivative βweakens the flow. Moreover, it is found that the average skin friction coefficient rises with the increase of fractional derivative β.     

Study on Hydrodynamics Characteristics of a Single Bubble in Viscoelastic Fluid at Low Weissenberg Numbers北大核心CSCD

The VOF method was used to numerically study the upward motion of a single bubble in viscoelastic fluid, and the Oldroyd-B model was applied to describe the fluid viscoelastic property. At low Weissenberg numbers (Wi≤1), the effects of the viscous force, the relaxation time, the surface tension and the viscosity ratio on the rising motion of the bubble were studied. The results show that, under relatively large viscous and elastic forces (such as Ga= 2, Wi≥0.5 and β = 0.2), the bubble exhibits the phenomenon of “a pointed rear end”, and this phenomenon intensifies with the increase of the elasticity and the decrease of the surface tension. Otherwise, under a relatively weak elasticity (such as Wi= 0.1), the phenomenon of “a pointed rear end” disappears, and the bubble bears a hat-like shape. For a large surface tension (such as Eo = 1), the bubble bears a longitudinally elongated ellipse-like shape without distinct tail features. The effect of the surface tension on the bubble in viscoelastic fluid is like that in viscous fluid. The bubble has 2 types of rising motions, namely, “continuous acceleration” to a stable velocity and “acceleration-deceleration-reacceleration” to a stable velocity, and the bubble rising velocity in viscoelastic fluid is higher than that in pure viscous fluid. The elastic stress around the bubble is influenced by the viscosity and the relaxation time of the fluid, and with the decrease of the fluid viscosity or/ and the increase of the relaxation time, the incidence of the elastic stress becomes wide. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Simulation Study on Dam Break Flow Based on the B-Spline Material Point Method北大核心CSCD

The dam break flow poses a common free surface flow problem in hydraulic engineering, and its ac⁃ curate simulation is of great engineering significance. The B⁃spline material point method (BSMPM), as an im⁃ proved algorithm of the material point method (MPM), has optimized accuracy and convergence in material point calculations and unique algorithmic advantages in free surface flow problems. Based on the BSMPM, a weakly compressible BSMPM (WC⁃BSMPM) was developed through introduction of an artificial equation of state. The simulation of the dam break flow problem was carried out, with the effects of the order of the B⁃ spline interpolation basis function on the simulation results analyzed. The results show that, the simulated fluid wavefront position, the wavefront velocity and the elevation variation at a given position are basically consistent with the existing experimental results. As the order of the basis function increases, the computation time will lengthen for about 1.5 times. However, the computation times of the BSMPM of different orders will uniformly increase approximately linearly with the background grid size. The validity of the WC⁃BSMPM simulation of the dam break flow problem was verified. The research provides a new idea and method for the simulation of dam break flow problems. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Numerical Simulation of Transient Non-Isothermal Viscoelastic Couette Flows Based on the SPH Method北大核心CSCD

Based on the smoothed particle hydrodynamics (SPH) method, the transient non-isothermal viscoelastic flows were numerically simulated. First, the viscoelastic Couette flow based on the Oldroyd-B model under isothermal condition was simulated. Then, the simulation was extended to the non-isothermal case, in which the Reynolds exponential model was adopted to evaluate the dependence of the viscosity and the relaxation time on the temperature. The accuracy and effectiveness of the SPH method for simulating transient non-isothermal viscoelastic flows were verified through comparison with the finite volume method and evaluation of numerical convergence. The different flow characteristics of the non-isothermal flow compared with those of the isothermal flow were discussed. The effects of the temperature dependence coefficient and the Péclet number on the flow physics were analyzed. The numerical results demonstrate that, the SPH method can accurately and effectively simulate transient non-isothermal viscoelastic flow problems. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Shape optimization in two-dimensional viscous compressible fluids

We present a method for solving the optimal shape problems for profiles surrounded by viscous compressible fluids in two space dimensions. The class of admissible profiles is quite general including the minimal volume condition and a constraint on the thickness of the boundary. The fluid flow is modelled by the Navier-Stokes system for a general viscous barotropic fluid with the pressure satisfying p（o） = aQlog^d（o） for large Q. Here d 〉 1 and a 〉 0.     

On the Rayleigh-Taylor Instability for Two Uniform Viscous Incompressible Flows

The authors study the Rayleigh-Taylor instability for two incompressible immis- cible fluids with or without surface tension, evolving with a free interface in the presence of a uniform gravitational field in Eulerian coordinates. To deal with the free surface, instead of using the transformation to Lagrangian coordinates, the perturbed equations in Eule- rian coordinates are transformed to an integral form and the two-fluid flow is formulated as a single-fluid flow in a fixed domain, thus offering an alternative approach to deal with the jump conditions at the free interface. First, the linearized problem around the steady state which describes a denser immiscible fluid lying above a light one with a free interface separating the two fluids, both fluids being in （unstable） equilibrium is analyzed. By a general method of studying a family of modes, the smooth （when restricted to each fluid domain） solutions to the linearized problem that grow exponentially fast in time in Sobolev spaces are constructed, thus leading to a global instability result for the linearized problem. Then, by using these pathological solutions, the global instability for the corresponding nonlinear problem in an appropriate sense is demonstrated.     

A VARIATIONAL METHOD FOR RESEARCH ON AN UNSTEADY TUBE FLOW OF UPPER CONVECTED MAXWELL FLUID

The unsteady flow of non-Newtonian fluid is an important problem in the industries of chemistry, polymer, food and power. For Newtonian fluid flow there are some solutions of special cases given in books but no general solution has been obtained. For non-Newtonian fluid there are solutions only for the power law fluid. In the present paper using a variational approach of Kantoroviach an unsteady flow     

Numerical exploration of new features on three-dimensional transition of a cylinder wake

The three-dimensional transition of the wake flow behind a circular cylinder is studied in detail by direct numerical simulations using 3D incompressible N-S equations for Reynolds number ranging from 200 to 300. New features and vortex dynamics of the 3D transition of the wake are found and investigated. At Re = 200, the flow pattern is characterized by mode A instability. However, the spanwise characteristic length of the cylinder determines the transition features. Particularly for the specific spanwise charac-     

EIGENVALUE THEORY OF STRETCHING FLUID SHEETS

Using the constitutive equation for slightly disturbed extensional flow, an eigenvalue theory of stretching fluid sheets has been developed. The present stability analysis is given from the Lagrangean point of view. The dependent variables are written as functions of material particles and time. An equation for the amplitude of the disturbance stream function φ(η, t) is derived. A general eigenvalue theory is discussed to explain the instability of Newtonian and non-Newtonian stretching fluid sheets. For the non-Newtonian fluid case the theory is specialized to a Maxwell model. The present theory predicts the influence of wave number on the stability of stretching fluids.     

Mixed Method for Compressible Miscible Displacement with Dispersion in Porous Media

Compressible miscible displacement of one fluid by another in porous media is modelled by a nonlinear parabolic system. A finite element procedure is introduced to approximate the concentration of one fluid and the pressure of the mixture. The concentration is treated by a Galerkin method while the pressure is treated by a parabolic mixed finite element method. The effect of dispersion, which is neglected in [1], is considered. Optimal order estimates in L2 are derived for the errors in the approximate solutions.     

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.     

Elastic Analysis of Anisotropic Rotating Sandwich Circular Ring With a Functionally Graded Transition Region北大核心CSCD

The elastic analysis of anisotropic rotating sandwich ring with a functionally graded transition region was carried out. Like the shell sandwich structure in nature, the ring is composed of 3 well⁃bonded regions, of which the inner and outer regions are made of homogeneous anisotropic materials, and the intermediate transi⁃ tion region is made of a material with arbitrary⁃gradient properties along the radial direction. Based on the boundary conditions and the continuity conditions at the interface, the 2nd Fredholm integral equation for the radial stress was obtained with the integral equation method, then the stress and displacement fields of the sandwich ring structure were obtained through numerical solution. The distributions of the stress and displace⁃ ment fields in the sandwich ring structure were given. Different gradient changes encountered in engineering practice can be solved only through substitution of the corresponding function model. The effectiveness and ac⁃ curacy of the integral equation method were verified through comparison of the numerical solutions with the ex⁃ act ones for a special power function gradient variation form. The more general Voigt function model was adopt⁃ ed for the intermediate transition region, and the influences of the anisotropy degree, the gradient parameter, and the thickness on the stress and displacement fields were analyzed. The proposed Fredholm integral equation method provides a powerful tool for the optimal design of anisotropic functionally graded materials and sand⁃ wich ring structures. The numerical results make a theoretical guidance for the safety design of anisotropic functionally graded sandwich ring structures. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

FORMATION OF SINGULARITY FOR COMPRESSIBLE VISCOELASTICITY Dedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthday

The formation of singularity and breakdown of classical solutions to the threedimensional compressible viscoelasticity and inviscid elasticity are considered.For the compressible inviscid elastic fluids,the finite-time formation of singularity in classical solutions is proved for certain initial data.For the compressible viscoelastic fluids,a criterion in term of the temporal integral of the velocity gradient is obtained for the breakdown of smooth solutions.     

Identification of Pipeline Inner Wall Geometry Based on the POD⁃RBF Method北大核心CSCD

Based on the proper orthogonal decomposition⁃radial basis function (POD⁃RBF), a geometric identification method for pipeline inner wall was proposed to solve the internal corrosion detection problem of natural gas and oil pipelines. In view of the static magnetic field, the simplified finite element model for the pipelines was established, and the variable⁃geometry sample library was constructed, to realize the response prediction of arbitrary geometry by the POD⁃RBF. The proposed method achieves reduced⁃order analysis and avoids repeated solution of the stiffness matrix due to the geometrical change during the identification process. Hence, it can significantly improve the computation efficiency. Finally, the grey wolf optimization (GWO) algorithm was used to optimize the objective function and avoid the calculation of the sensitivity in the process of geometry change. The numerical examples show that, the proposed method has high efficiency and accuracy in the geometric identification of the pipeline inner wall, with good stability even under introduced noises. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

NUMERICAL ANALYSIS OF A FEM FOR A TRANSIENT VISCOELASTIC FLOW

We present the numerical analysis of a coupled method for the numerical simulation of transient viscoelastic flow obeying a differential constitutive equation with a Newtonian viscosity. The scheme used is based on Euler implicit method in time and maintains at each time step a couple of the velocity u and the viscoelastic part of the stress σ. Approximation in space is made by finite element method. The approximate stress, velocity and pressure are, respectively, P1-continuous, p2-continuous, and P1-continuous. Upwinding needed for convection of σ is made by a ““““Streamline Upwind Petrov Galerkin““““ method (SUPG).     

APPROXIMATION OF THE CONCENTRATION BY A VISCOSITY SPLITTING METHOD FOR TWO-PHASE MISCIBLE DISPLACEMENT PROBLEM

The miscible displacement of one incompressible fluid by another in a porous medium is considered in this paper. The concentration is split in a first-order hyberbolic equation and a homogeneous parabolic equation within each lime step. The pressure and Us velocity field is computed by a mixed finite element method. Optimal order estimates are derived for the no diffusion case and the diffusion case.     

Numerical Simulation of Hydraulic Fractures Intersecting Natural Fractures in Shale With Plastic Deformation北大核心CSCD

The plastic deformation and numerous natural joints of shale pose a great challenge for the predic⁃ tion of the hydraulic fracture geometry extension. Based on the finite element method, a fully coupled numeri⁃ cal model for elastoplastic hydraulic fractures was established with natural fractures and bedding planes consid⁃ ered. The numerical model was validated with the KGD analytical solution and Blanton’s curve. The numerical results show that, compared with the numerical model solution of linear elasticity, the hydraulic fractures are prone to enter the natural weak interface due to the rock plastic deformation. The rock plastic deformation area mainly lies in the reservoir layer during the fracture propagation. In the case of rock ductile damage, the hy⁃ draulic fracture is more likely to penetrate the bedding plane. Hydraulic fractures can directly penetrate natural fractures and bedding planes at high injection rates due to large driving forces. The study provides new insights in terms of hydraulic fracture extension in elastoplastic formations. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

DIFFUSIVE—DISPERSIVE TRAVELING WAVES AND KINETIC RELATIONS IV. COMPRESSIBLE EULER EQUATIONS

The authors consider the Euler equations for a compressible fluid in one space dimension when the equation of state of the fluid does not fulfill standard convexity assumptions and viscosity and capillarity effects are taken into account. A typical example of nonconvex constitutive equation for fluids is Van der Waals' equation. The first order terms of these partial differential equations form a nonlinear system of mixed (hyperbolic-elliptic) type. For a class of nonconvex equations of state, an existence theorem of traveling waves solutions with arbitrary large amplitude is established here. The authors distinguish between classical (compressive) and nonclassical (undercompressive) traveling waves. The latter do not fulfill Lax shock inequalities, and are characterized by the so-called kinetic relation, whose properties are investigated in this paper.     

MODELING,SIMULATION,AND OPTIMIZATION OF SURFACE ACOUSTIC WAVE DRIVEN MICROFLUIDIC BIOCHIPS

We will be concerned with the mathematical modeling, numerical simulation, and shape optimization of micro fluidic biochips that are used for various biomedical applications. A particular feature is that the fluid flow in the fluidic network on top of the biochips is in- duced by surface acoustic waves generated by interdigital transducers. We are thus faced with a multiphysics problem that will be modeled by coupling the equations of piezoelectricity with the compressible Navier-Stokes equations. Moreover, the fluid flow exhibits a multiscale character that will be taken care of by a homogenization approach. We will discuss and analyze the mathematical models and deal with their numerical solution by space-time discretizations featuring appropriate finite element approximations with respect to hierarchies of simplicial triangulations of the underlying computational domains. Simulation results will be given for the propagation of the surface acoustic waves on top of the piezoelectric substrate and for the induced fluid flow in the microchannels of the fluidic network. The performance of the operational behavior of the biochips can be significantly improved by shape optimization. In particular, for such purposes we present a multilevel interior point method relying on a predictor-corrector strategy with an adaptive choice of the continuation steplength along the barrier path. As a specific example, we will consider the shape optimization of pressure driven capillary barriers between microchannels and reservoirs.     

Factors Influencing the Disturbed Flow Patterns Downstream of Curved Atherosclerotic Arteries

Pulsatile blood flows in curved atherosclerotic arteries are studied by com- puter simulations.Computations are carried out with various values of physiological parameters to examine the effects of flow parameters on the disturbed flow patterns downstream of a curved artery with a stenosis at the inner wall.The numerical re- suits indicate a strong dependence of flow pattern on the blood viscosity and inlet flow rate,while the influence of the inlet flow profile to the flow pattem in downstream is negligible.