Oblique derivative problem with variable inclination angle: Location of the spectrum and the absence of the basis property of the system of root functions

We consider the eigenvalue problem for the Laplace equation in the disk with the oblique derivative with variable inclination angle. We show that the root functions of the problem do not form a basis in any Lebesgue space with exponent exceeding unity.     

Self-similar problem of Cauchy-Poisson waves at an inclined shore : PMM vol. 37, n6, 1973, pp. 1035–1039

We consider a two-dimensional problem concerning Cauchy-Poisson waves at an inclined shore in the case of an initial disturbance concentrated near the shore edge. We study the behavior of the solution near the shore and at large distances from it.Numerous investigations, devoted to the study of standing and progressive waves on an inclined shore, are described in [1]. A two-dimensional problem concerning nonstationary waves on a shore with an angle of inclination γ = π/2n, where n is an integer, was analyzed in [2, 3]. We consider below a case in which the angle of inclination is commensurable with λ/2, subject to the condition that the initial disturbance is concentrated in the vicinity of the shore edge, so that the problem may be considered self-similar.     

基于Hele-shaw模型的斜井偏心环空顶替流体密度差优化

针对斜井中顶替流体密度差及套管偏心引起的顶替界面突进两相流体掺混严重的问题,考虑斜井密度差引起的驱动力沿周向角和半径的变化,建立了斜井偏心环空稳定顶替界面形状模型,利用该模型分析了密度差对顶替界面形状的影响,得到了不同井斜角及套管偏心条件下顶替流体的最优密度差及变化规律:斜井中顶替界面突进的位置及程度取决于套管偏心与密度差的相对作用,存在合理的密度差使得顶替界面长度最短;最优顶替流体密度差随套管偏心度的增加而增大,随井斜角的增大而减小;固井顶替时在满足压稳防漏的条件下直井所选用的密度差越大越好,水平井所选用的密度差应相对最小。     

Quasi-steady flight to quasi-steady flight transition for abort landing in a windshear: Trajectory optimization and guidance

This paper is concerned with the optimal transition and the near-optimum guidance of an aircraft from quasi-steady flight to quasi-steady flight in a windshear. The abort landing problem is considered with reference to flight in a vertical plane. In addition to the horizontal shear, the presence of a downdraft is considered.It is assumed that a transition from descending flight to ascending flight is desired; that the initial state corresponds to quasi-steady flight with absolute path inclination of –3.0 deg; and that the final path inclination corresponds to quasi-steady steepest climb. Also, it is assumed that, as soon as the shear is detected, the power setting is increased at a constant time rate until maximum power setting is reached; afterward, the power setting is held constant. Hence, the only control is the angle of attack. Inequality constraints are imposed on both the angle of attack and its time derivative.First, trajectory optimization is considered. The optimal transition problem is formulated as a Chebyshev problem of optimal control: the performance index being minimized is the peak value of the modulus of the difference between the instantaneous altitude and a reference value, assumed constant. By suitable transformations, the Chebyshev problem is converted into a Bolza problem. Then, the Bolza problem is solved employing the dual sequential gradient-restoration algorithm (DSGRA) for optimal control problems.Two types of optimal trajectories are studied, depending on the conditions desired at the final point. Type 1 is concerned with gamma recovery (recovery of the value of the relative path inclination corresponding to quasi-steady steepest climb). Type 2 is concerned with quasi-steady flight recovery (recovery of the values of the relative path inclination, the relative velocity, and the relative angle of attack corresponding to quasi-steady steepest climb). Both the Type 1 trajectory and the Type 2 trajectory include three branches: descending flight, nearly horizontal flight, and ascending flight. Also, for both the Type 1 trajectory and the Type 2 trajectory, descending flight takes place in the shear portion of the trajectory; horizontal flight takes place partly in the shear portion and partly in the aftershear portion of the trajectory; and ascending flight takes place in the aftershear portion of the trajectory. While the Type 1 trajectory and the Type 2 trajectory are nearly the same in the shear portion, they diverge to a considerable degree in the aftershear portion of the trajectory.Next, trajectory guidance is considered. Two guidance schemes are developed so as to achieve near-optimum transition from quasi-steady descending flight to quasi-steady ascending flight: acceleration guidance (based on the relative acceleration) and gamma guidance (based on the absolute path inclination).The guidance schemes for quasi-steady flight recovery in abort landing include two parts in sequence: shear guidance and aftershear guidance. The shear guidance is based on the result that the shear portion of the trajectory depends only mildly on the boundary conditions. Therefore, any of the guidance schemes already developed for Type 1 trajectories can be employed for Type 2 trajectories (descent guidance followed by recovery guidance). The aftershear guidance is based on the result that the aftershear portion of the trajectory depends strongly on the boundary conditions; therefore, the guidance schemes developed for Type 1 trajectories cannot be employed for Type 2 trajectories. For Type 2 trajectories, the aftershear guidance includes level flight guidance followed by ascent guidance. The level flight guidance is designed to achieve almost complete velocity recovery; the ascent guidance is designed to achieve the desired final quasi-steady state.The numerical results show that the guidance schemes for quasi-steady flight recovery yield a transition from quasi-steady flight to quasi-steady flight which is close to that of the optimal trajectory, allows the aircraft to achieve the final quasi-steady state, and has good stability properties.This research was supported by NASA Langley Research Center, Grant No. NAG-1-516, by Boeing Commercial Airplane Company, and by Air Line Pilots Association.The authors are indebted to Dr. R. L. Bowles (NASA-LRC) and Dr. G. R. Hennig (BCAC) for helpful discussions.     

Effect of the Dimensions of Mines and Angles of Inclination of Beds on the Stress Distribution in Mountain Masses

The effect of the angle of inclination of a bed and the geometric dimensions of a mine with unloading crevices on the stressed state of a massif is studied numerically. A mine with a vaulted cross section and different angles of inclination of the sides to the base plane is considered.     

Decomposition technique and optimal trajectories for the aeroassisted flight experiment

The aeroassisted flight experiment (AFE) refers to a spacecraft to be launched and then recovered by the space shuttle in 1994. It simulates a transfer from a geosynchronous Earth orbit (GEO) to a low Earth orbit (LEO). Specifically, the AFE spacecraft is released from the space shuttle and is accelerated by means of a solid rocket motor toward Earth, so as to achieve atmospheric entry conditions close to those of a spacecraft returning from GEO. Following the atmospheric pass, the AFE spacecraft ascends to the specified LEO via an intermediate parking Earth orbit (PEO). The final maneuver includes the rendezvous with and the capture by the space shuttle. The entry and exit orbital planes of the AFE spacecraft are identical with the orbital plane of the space shuttle. In this paper, with reference to the AFE spacecraft, an actual GEO-to-LEO transfer is considered and optimal trajectories are determined by minimizing the total characteristic velocity. The optimization is performed with respect to the time history of the controls (angle of attack and angle of bank), the entry path inclination and the flight time being free. Two transfer maneuvers are considered: (DA) direct ascent to LEO; (IA) indirect ascent to LEO via PEO. While the motion of the AFE spacecraft in a 3D-space is described by a system of six ODEs, substantial simplifications are possible if one exploits these facts: (i) the instantaneous orbital plane is nearly identical with the initial orbital plane; (ii) the bank angle is small; and (iii) the Earth's angular velocity is relatively small. Under these assumptions, the complete system can be decoupled into two subsystems, one describing the longitudinal motion and one describing the lateral motion. The angle of attack history, the entry path inclination, and the flight time are determined via the longitudinal motion subsystem; in this subsystem, the total characteristic velocity is minimized subject to the specified LEO requirement. The angle of bank history is determined via the lateral motion subsystem; in this subsystem, the difference between the instantaneous bank angle and a constant bank angle is minimized in the least square sense subject to the specified orbital inclination requirement. It is shown that both the angle of attack and the angle of bank are constant. This result has considerable importance in the design of nominal trajectories to be used in the guidance of AFE and AOT vehicles.     

Quasi-steady flight to quasi-steady flight transition in a windshear: Trajectory optimization and guidance

This paper is concerned with the near-optimum guidance of an aircraft from quasi-steady flight to quasi-steady flight in a windshear. The take-off problem is considered with reference to flight in a vertical plane. In addition to the horizontal shear, the presence of a downdraft is considered. It is assumed that the power setting is held at the maximum value and that the aircraft is controlled through the angle of attack. Inequality constraints are imposed on both the angle of attack and its time derivative.First, trajectory optimization is considered. The optimal transition problem is formulated as a Chebyshev problem of optimal control: the performance index being minimized is the peak value of the modulus of the difference between the absolute path inclination and a reference value, assumed constant. Two types of optimal trajectories are studied: type 1 is concerned with gamma recovery (recovery of the initial value of the relative path inclination); and type 2 is concerned with quasisteady flight recovery (recovery of the initial values of the relative velocity, the relative path inclination, and the relative angle of attack). The numerical results show that the type 1 trajectory and the type 2 trajectory are nearly the same in the shear portion, while they diverge to a considerable degree in the aftershear portion of the optimal trajectory.Next, trajectory guidance is considered. A guidance scheme is developed so as to achieve near-optimum quasi-steady flight recovery in a windshear. The guidance scheme for quasi-steady flight recovery includes three parts in sequence. The first part refers to the shear portion of the trajectory and is based on the result that this portion of the trajectory depends only mildly on the boundary conditions; therefore, any of the guidance schemes already developed for type 1 trajectories can be employed (for instance, variable gamma guidance). The second part (constant gamma guidance) refers to the initial aftershear portion of the trajectory and is designed to achieve almost velocity recovery. The third part (constant rate of climb guidance) refers to the final aftershear portion of the trajectory and is designed to achieve almost complete restoration of the initial quasi-steady state.While the shear guidance and the initial aftershear guidance employ constant gain coefficients, the final aftershear guidance employs a variable gain coefficient. This is done in order to obtain accuracy and prompt response, while avoiding oscillations and overshoots. The numerical results show that the guidance scheme for quasi-steady flight recovery yields a transition from quasi-steady flight to quasi-steady flight which is close to that of the optimal trajectory, ensures the restoration of the initial quasi-steady state, and has good stability properties.This paper is based on Refs. 1 and 2.This research was supported by NASA-Langley Research Center, Grant No. NAG-1-516, and by Boeing Commercial Aircraft Company. The authors are indebted to Dr. R. L. Bowles, NASA-Langley Research Center, for helpful discussions.     

Investigation of secondary flow on an infinite yawed cylinder in a compressible stream

In this paper a study of the secondary flow on an infinite yawed cylinder in a compressible stream is made at stagnation for unit Prandtl number and for a given Mach number. The secondary flow profiles are calculated for different edge inclination angles. The chordwise and spanwise velocity profiles given by Crabtree¹ are used here in the calculation. It is found that the secondary flow profiles are the same for complementary edge inclination angles and they are maximum for the edge inclination angle of 45°. A comparison with the incompressible flow profiles² is made. It is found that the magnitude of the secondary flow profiles in a compressible stream is less than that in an incompressible stream for the same edge inclination angle.     

The optimality of the velocity-gradient method in the problem of controlling the escape from a potential well

The problem of controlling the escape of a particle from a potential well for a nonlinear system with friction is considered. The velocity-gradient method [Polushin IG, Fradkov AL, Hill, D. Passivity and passivation in non-linear systems. Avtomatika i Telemekhanika 2000;3:3–37] is proved to be optimal in the sense that if it does not guarantee escape from the well, then this is also impossible with any other control law. Nonlinear Duffing and Helmholtz oscillators with one degree of freedom and negative stiffness are considered. For each of them a curve is constructed separating the parameter plane of the problem into two parts: one where escape is feasible and one where it is not. An estimate is obtained for the inclination angle of the tangent to that curve near the origin.     

Modeling the join curve between two co-axial carbon nanotubes

Making use of an applied mathematical model, we employ a calculus of variations technique to join two co-axial nanotubes. Due to the axial symmetry of the tubes, the three-dimensional problem can be reduced to a problem in two dimensions. The curvature squared for the join region is minimized for a prescribed join length and given tube radii. In this model, a certain non-dimensional parameter B arises, which approximately has the same numerical value when compared with the standard method for the joining between any two carbon nanotubes of different radii. This value occurs in consequence of adopting an angle of inclination of 9.594°, which occurs in the conventional method for joining two carbon nanotubes of different radii and which is necessary to accommodate a single pentagon. The simple calculus of variations model described here provides a general framework to connect nanotubes or other nanostructures.     

Optimal penetration landing trajectories in the presence of windshear

This paper is concerned with optimal flight trajectories in the presence of windshear. The penetration landing problem is considered with reference to flight in a vertical plane, governed by either one control (the angle of attack, if the power setting is predetermined) or two controls (the angle of attack and the power setting). Inequality constraints are imposed on the angle of attack, the power setting, and their time derivatives.The performance index being minimized measures the deviation of the flight trajectory from a nominal trajectory. In turn, the nominal trajectory includes two parts: the approach part, in which the slope is constant; and the flare part, in which the slope is a linear function of the horizontal distance. In the optimization process, the time is free; the absolute path inclination at touchdown is specified; the touchdown velocity is subject to upper and lower bounds; and the touchdown distance is subject to upper and lower bounds.Three power setting schemes are investigated: (S1) maximum power setting; (S2) constant power setting; and (S3) control power setting. In Scheme (S1), it is assumed that, immediately after the windshear onset, the power setting is increased at a constant time rate until maximum power setting is reached; afterward, the power setting is held constant; in this scheme, the only control is the angle of attack. In Scheme (S2), it is assumed that the power setting is held at a constant value, equal to the prewindshear value; in this scheme, the only control is the angle of attack. In Scheme (S3), the power setting is regarded as a control, just as the angle of attack.Under the above conditions, the optimal control problem is solved by means of the primal sequential gradient-restoration algorithm (PSGRA). Numerical results are obtained for several combinations of windshear intensities and initial altitudes. The main conclusions are given below with reference to strong-to-severe windshears.In Scheme (S1), the touchdown requirements can be satisfied for relatively low initial altitudes, while they cannot be satisfied for relatively high initial altitudes; the major inconvenient is excess of velocity at touchdown. In Scheme (S2), the touchdown requirements cannot be satisfied, regardless of the initial altitude; the major inconvenient is defect of horizontal distance at touchdown.In Scheme (S3), the touchdown requirements can be satisfied, and the optimal trajectories exhibit the following characteristics: (i) the angle of attack has an initial decrease, which is followed by a gradual, sustained increase; the largest value of the angle of attack is attained near the end of the shear; in the aftershear region, the angle of attack decreases gradually; (ii) initially, the power setting increases rapidly until maximum power setting is reached; then, maximum power setting is maintained in the shear region; in the aftershear region, the power setting decreases gradually; (iii) the relative velocity decreases in the shear region and increases in the aftershear region; the point of minimum velocity occurs at the end of the shear; and (iv) depending on the windshear intensity and the initial altitude, the deviations of the flight trajectory from the nominal trajectory can be considerable in the shear region; however, these deviations become small in the aftershear region, and the optimal flight trajectory recovers the nominal trajectory.A comparison is shown between the optimal trajectories of Scheme (S3) and the trajectories arising from alternative guidance schemes, such as fixed controls (fixed angle of attack, coupled with fixed power setting) and autoland (angle of attack controlled via path inclination signals, coupled with power setting controlled via velocity signals). The superiority of the optimal trajectories of Scheme (S3) is shown in terms of the ability to meet the path inclination, velocity, and distance requirements at touchdown. Therefore, it is felt that guidance schemes based on the properties of the optimal trajectories of Scheme (S3) should prove to be superior to alternative guidance schemes, such as the fixed control guidance scheme and the autoland guidance scheme.Portions of this paper were presented at the AIAA 26th Aerospace Sciences Meeting, Reno, Nevada, January 11–14, 1988 (Paper No. AIAA-88-0580).This research was supported by NASA-Langley Research Center, Grant No. NAG-1-516, by Boeing Commercial Airplane Company (BCAC), and by Air Line Pilots Association (ALPA).The authors are indebted to Dr. R. L. Bowles, NASA-Langley Research Center, and to Dr. G. R. Hennig, Boeing Commercial Airplane Company, for helpful discussions.     

Two-phase modeling of batch sedimentation

A two-phase model for the simulation of sedimentation processes is presented. The model solves the continuity and momentum equations for the pure-clear liquid and the sludge phases, and it is verified against a well-known benchmark problem, for which analytical solutions exist. Numerical simulations of a typical 1-D batch sedimentation process for mono-dispersed particles are carried out and results are found to be in satisfactory agreement with experimental data and model predictions of other researchers. A further expansion of the model to two-dimensions leads to predictions of the dynamic behavior of settling tanks and the effect of the inclination angle on the sedimentation process.     

The flow of a supersonic ideal gas with “weak” and “strong” shocks over a wedge

The problem of the flow of a uniform supersonic ideal (inviscid and non-heat-conducting) gas over a wedge is considered. If the turning angle of the flow, which is equal to the angle of inclination of the generatrix of the wedge, is less than the maximum value, the problem has two solutions. In the solution with an oblique low-intensity (“weak”) shock, the uniform flow between the shock and the wedge is almost always supersonic. One exception is a small vicinity of the maximum turning angle. For an ideal gas this vicinity does not exceed a fraction of a degree at all Mach numbers. Behind a high-intensity (“strong”) shock, the flow of an ideal gas is always subsonic. “Weak” shocks are observed in all experiments with finite wedges. Some researchers attribute this preference to the “downstream” boundary conditions (“on the right at infinity” for a flow incident on the wedge from the left), and others attribute it to the instability (“Lyapunov” instability) of a flow with a strong shock when it flows over the wedge and to the stability of flow with a weak shock. The results presented below from calculations of the flows that occur for finite wedges within the two-dimensional unsteady Euler equations, when the parameters behind the strong shock are specified on the right-hand boundary, i.e., on the arc of a circle between the wedge and the shock, demonstrate the correctness of the conclusion of the first group of researchers and the incorrectness of the conclusion of the other group. In these calculations, after both small and fairly large perturbations, the flows investigated (which are, in fact, Lyapunov unstable!) return to the solution with a strong shock. In addition, the problem of steady flow over a wedge was regarded as the limit of the two-dimensional non-steady problems at infinite time. Simplification of one of them leads to the problem of the submerged over-expanded supersonic steady outflow. In the ideal gas model this problem is equivalent to flow over a wedge with both weak and strong shocks. All the solutions considered are stable.     

Vibration behavior of simply supported inclined single-walled carbon nanotubes conveying viscous fluids flow using nonlocal Rayleigh beam model

A mathematical model is proposed to investigate the dynamic response of an inclined single-walled carbon nanotube (SWCNT) subjected to a viscous fluid flow. The tangential interaction of the inside fluid flow with the equivalent continuum structure (ECS) of the SWCNT is taken into account via a slip boundary condition. The dimensionless equations of motion describing longitudinal and lateral vibrations of the fluid-conveying ECS are obtained in the context of nonlocal elasticity theory of Eringen. The unknown displacement fields are expressed in terms of admissible mode shapes associated with the ECS under simply supported conditions with immovable ends. Using Galerkin method, the discrete form of the equations of motion is derived. The time history plots of the normalized longitudinal and transverse displacements as well as the nonlocal axial force and bending moment of the midspan point of the SWCNT are provided for different levels of the fluid flow speed, small-scale parameter, and inclination angle of the SWCNT. The effects of small-scale parameter, inclination angle, speed and density of the fluid flow on the maximum dynamic amplitude factors of longitudinal and transverse displacements as well as those of nonlocal axial force and bending moment of the SWCNT are then studied in some detail.     

倾斜层中的对流斑图及其临界条件

通过二维流体力学基本方程的数值模拟，探讨了Prandtl(普朗特)数Pr=6.99时，倾斜矩形腔体中的对流斑图和斑图转换的临界条件.根据倾角θ和相对Rayleigh(瑞利)数Ra_r的变化，倾斜矩形腔体中的对流斑图可以分为:单滚动圈对流斑图、充满腔体的多滚动圈对流斑图和过渡阶段的多滚动圈对流斑图.当θ一定时，随着Ra_r的减小，系统由充满腔体的多滚动圈对流斑图过渡到单滚动圈对流斑图.这时，对流振幅A和Nusselt(努塞尔)数Nu随着Ra_r的增加而增加.当Ra_r=9时，随着θ的增加，系统由充满腔体的多滚动圈对流斑图过渡到单滚动圈对流斑图，这时对流振幅A随着θ的增加而减小，Nusselt数Nu随着θ的增加而增加.在θ_c-Ra_r平面上对多滚动圈到单滚动圈对流斑图过渡的模拟结果表明， 在Ra_r=2时, 腔体中没有发现多滚动圈对流斑图.在Ra_r为2.5左右时，腔体中出现多滚动圈到单滚动圈对流斑图的过渡.当多滚动圈到单滚动圈对流斑图过渡的临界倾角θ_c<10°时，θ_{c随着Rar的减小而增加.当θc>10°时，θc随着Rar的增加而增加，在Rar≤5时，θc随着Rar的增加而迅速增加；当Rar>5时，θc随着Rar的增加而缓慢增加.θc与Rarθ的关系与Rar类似}     

Existence and uniqueness of rectilinear slit maps

We consider a generalization of the parallel slit uniformization in which the angle of inclination of each image slit is assigned independently. Koebe proved that for domains of finite connectivity there is, up to a normalization, a unique rectilinear slit map achieving any given angle assignment. Koebe's theorem is partially extended to domains of infinite connectivity. A uniqueness result is shown for domains of countable connectivity and arbitrary angle assignments, and an existence result is proved for arbitrary domains under the assumption that the angle assignment is continuous and has finite range. In order to prove the existence result a new extremal length tool, called the crossing-module, is introduced. The crossing-module allows greater freedom in the family of admissible arcs than the classical module. Several results known for the module are extended to the crossing-module. A generalization of Jenkins' module condition for the parallel slit problem is given for the rectilinear slit problem in terms of the crossing-module and it is shown that rectilinear slit maps satisfying this crossing-module condition exist.

     

Influence of Multiple Micro Cracks on the Damage Behavior of a Macro-Crack Tip北大核心CSCD

The solution of an infinite plane containing a macro crack and a cluster of micro cracks under uniaxial tensile load was presented based on Muskhelishvili’s complex function method and the stepwise recursive method. The stress field and stress intensity factor K were obtained. Combined with the damage mechanics, damage parameter D of the macro-crack tip and the micro-crack tip under uniaxial tension was redefined, and the influence of different damage zone forms on the damage of the crack tip was analyzed. The results show that, both the chain-distribution and the reverse-chain-distribution micro cracks have an amplifying effect on the macro crack growth, and the damage parameter increases with the decrease of the inclination angle of the micro crack and the reduction of the distance between the macro crack and the micro cracks. For a relatively small inclination angle of the micro crack, the damage parameters of the macro crack and the micro crack heightens, and the damage parameter of the macro crack increases with the micro-crack length. For evenly distributed micro cracks in the continuous damage zone, the micro cracks have an amplifying effect on the macro-crack growth, and the damage parameter of the macro crack increases with the micro-crack number. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

An interval uncertain optimization method for vehicle suspensions using Chebyshev metamodels

This paper proposes a new design optimization framework for suspension systems considering the kinematic characteristics, such as the camber angle, caster angle, kingpin inclination angle, and toe angle in the presence of uncertainties. The coordinates of rear inner hardpoints of upper control arm and lower control arm of double wishbone suspension are considered as the design variables, as well as the uncertain parameters. In this way, the actual values of the design variables will vary surrounding their nominal values. The variations result in uncertainties that are described as interval variables with lower and upper bounds. The kinematic model of the suspension is developed in software ADAMS. A high-order response surface model using the zeros of Chebyshev polynomials as sampling points is established, termed as Chebyshev metamodel, to approximate the kinematic model. The Chebyshev meta-model is expected to provide higher approximation accuracy. Interval uncertain optimization problems usually involve a nested computationally expensive double-loop optimization process, in which the inner loop optimization is to calculate the bounds of the interval design functions, while the outer loop is to search the optimum for the deterministic optimization problem. To reduce the computational cost, the interval arithmetic is introduced in the inner loop to improve computational efficiency without compromising numerical accuracy. The numerical results show the effectiveness of the proposed design method.     

Angles in Complex Vector Spaces

The article reviews some of the (fairly scattered) information available in the mathematical literature on the subject of angles in complex vector spaces. The following angles and their relations are considered: Euclidean, complex, and Hermitian angles, (Kasner's) pseudo-angle, the Kähler angle (synonyms for the latter used in the literature are angle of inclination, characteristic deviation, holomorphic deviation, holomorphy angle, Wirtinger angle, slant angle).     

Numerical Simulation of Red Blood Cell Suspensions Behind a Moving Interface in a Capillary

Computational modeling and simulation are presented on the motion of red blood cells behind a moving interface in a capillary. The methodology is based on an immersed boundary method and the skeleton structure of the red blood cell (RBC) membrane is modeled as a spring network. As by the nature of the problem, the computational domain is moving with either a designated RBC or an interface in an infinitely long two-dimensional channel with an undisturbed flow field in front of the computational domain. The tanking-treading and the inclination angle of a cell in a simple shear flow are briefly discussed for the validation purpose. We then present and discuss the results of the motion of red blood cells behind a moving interface in a capillary, which show that the RBCs with higher velocity than the interface speed form a concentrated slug behind the moving interface.