In order to interpret the physical feature of Bessho form translating-pulsating source Green function, the phase function is extracted from the integral representation and stationary-phase analysis is carried out in t...In order to interpret the physical feature of Bessho form translating-pulsating source Green function, the phase function is extracted from the integral representation and stationary-phase analysis is carried out in this paper. The complex characteristics of the integral variable and segmentation of the integral intervals are discussed in m complex plane. In θ space, the interval [-π/2+φ,-π/2+φ-iε] is dominant in the near-field flow, and there is a one-to-one correspondence between the real intervals in m space and the unsteady wave patterns in far field. If 4τ>1(τ is the Brard number), there are three kinds of propagation wave patterns such as ring-fan wave pattern, fan wave pattern and inner V wave pattern, and if 0<4τ<1, a ring wave pattern, an outer V and inner V wave pattern are presented in far field. The ring-fan or ring wave pattern corresponds to the interval [-π+α,-π/2+φ] for integral terms about k2, and the fan or outer V wave pattern and inner V wave pattern correspond to [-π+α,-π/2) and(-π/2,-π/2+φ] respectively for terms about k1. Numerical result shows that it is beneficial to decompose the unsteady wave patterns under the condition of τ≠0 by converting the integral variable θ to m. In addition, the constant-phase curve equations are derived when the source is performing only pulsating or translating.展开更多
Consider the following Schr？dinger-Poisson-Slater system, （P） where ω 〉 0, λ 〉 0 and β 〉 0 are real numbers, p ∈ （1, 2）. For β=0, it is known that problem （P） has no nontrivial solution if λ 〉 0 suit...Consider the following Schr？dinger-Poisson-Slater system, （P） where ω 〉 0, λ 〉 0 and β 〉 0 are real numbers, p ∈ （1, 2）. For β=0, it is known that problem （P） has no nontrivial solution if λ 〉 0 suitably large. When β 〉 0, -β/｜x｜ is an important potential in physics, which is called external Coulomb potential. In this paper, we find that （P） with β 〉 0 has totally different properties from that of β = 0. For β 〉 0, we prove that （P） has a ground state and multiple solutions if λ 〉 cp,ω, where cp,ω 〉 0 is a constant which can be expressed explicitly via ω and p.展开更多
Many marine plankton species are motile and perform daily vertical migrations,traveling across water columns over distances of tens of meters.It is intriguing that these tiny and slow swimmers can travel in a certain ...Many marine plankton species are motile and perform daily vertical migrations,traveling across water columns over distances of tens of meters.It is intriguing that these tiny and slow swimmers can travel in a certain direction within a turbulent environment.One way to do that is by exploiting gravitaxis,which is a form of taxis characterised by the directional movement of an organism in response to gravity.Many plankton species are able to generate a gravitational torque(e.g.,due to a nonuniform mass distribution)that reorients them upwards.However,the swimming direction is disturbed by the shearing motions and the velocity fluctuations that characterise oceanic turbulence:these can generate a viscous torque that may destabilize the swimmer.The directed locomotion resulting from the combination of gravitational and viscous torques in a flow is termed gyrotaxis,which is known to lead to a non-uniform spatial accumulation of swimmers in patches or layers.These phenomena depend strongly on the non-linear dynamics that originate from the fluid motions,and the study of gyrotactic swimmers in complex flows is attracting growing attention.Numerical simulations of the Navier-Stokes equations coupled with suitable models of gyrotactic swimmers have proven their capability to provide valuable insight into the dynamical and statistical properties of self-propelled organisms.In this paper,we review recent studies and key findings on gyrotactic swimmers in turbulent flows.First,we introduce the most recent results concerning the orientation and vertical migration of gyrotactic swimmers in isotropic turbulence.Second,we discuss the findings on the accumulation of the swimmers.Last,we review recent progresses concerning the behaviour of gyrotactic swimmers in free-surface turbulence.展开更多
基金financial support from the National Natural Science Foundation of China under Grant No. 50879090the Key Program of Hydrodynamics of China under Grant No.9140A14030712JB11044
文摘In order to interpret the physical feature of Bessho form translating-pulsating source Green function, the phase function is extracted from the integral representation and stationary-phase analysis is carried out in this paper. The complex characteristics of the integral variable and segmentation of the integral intervals are discussed in m complex plane. In θ space, the interval [-π/2+φ,-π/2+φ-iε] is dominant in the near-field flow, and there is a one-to-one correspondence between the real intervals in m space and the unsteady wave patterns in far field. If 4τ>1(τ is the Brard number), there are three kinds of propagation wave patterns such as ring-fan wave pattern, fan wave pattern and inner V wave pattern, and if 0<4τ<1, a ring wave pattern, an outer V and inner V wave pattern are presented in far field. The ring-fan or ring wave pattern corresponds to the interval [-π+α,-π/2+φ] for integral terms about k2, and the fan or outer V wave pattern and inner V wave pattern correspond to [-π+α,-π/2) and(-π/2,-π/2+φ] respectively for terms about k1. Numerical result shows that it is beneficial to decompose the unsteady wave patterns under the condition of τ≠0 by converting the integral variable θ to m. In addition, the constant-phase curve equations are derived when the source is performing only pulsating or translating.
基金supported by National Natural Science Foundation of China(Grant Nos.11071245,11171339 and 11201486)supported by the Fundamental Research Funds for the Central Universities
文摘Consider the following Schr？dinger-Poisson-Slater system, （P） where ω 〉 0, λ 〉 0 and β 〉 0 are real numbers, p ∈ （1, 2）. For β=0, it is known that problem （P） has no nontrivial solution if λ 〉 0 suitably large. When β 〉 0, -β/｜x｜ is an important potential in physics, which is called external Coulomb potential. In this paper, we find that （P） with β 〉 0 has totally different properties from that of β = 0. For β 〉 0, we prove that （P） has a ground state and multiple solutions if λ 〉 cp,ω, where cp,ω 〉 0 is a constant which can be expressed explicitly via ω and p.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.11911530141 and 91752205).
文摘Many marine plankton species are motile and perform daily vertical migrations,traveling across water columns over distances of tens of meters.It is intriguing that these tiny and slow swimmers can travel in a certain direction within a turbulent environment.One way to do that is by exploiting gravitaxis,which is a form of taxis characterised by the directional movement of an organism in response to gravity.Many plankton species are able to generate a gravitational torque(e.g.,due to a nonuniform mass distribution)that reorients them upwards.However,the swimming direction is disturbed by the shearing motions and the velocity fluctuations that characterise oceanic turbulence:these can generate a viscous torque that may destabilize the swimmer.The directed locomotion resulting from the combination of gravitational and viscous torques in a flow is termed gyrotaxis,which is known to lead to a non-uniform spatial accumulation of swimmers in patches or layers.These phenomena depend strongly on the non-linear dynamics that originate from the fluid motions,and the study of gyrotactic swimmers in complex flows is attracting growing attention.Numerical simulations of the Navier-Stokes equations coupled with suitable models of gyrotactic swimmers have proven their capability to provide valuable insight into the dynamical and statistical properties of self-propelled organisms.In this paper,we review recent studies and key findings on gyrotactic swimmers in turbulent flows.First,we introduce the most recent results concerning the orientation and vertical migration of gyrotactic swimmers in isotropic turbulence.Second,we discuss the findings on the accumulation of the swimmers.Last,we review recent progresses concerning the behaviour of gyrotactic swimmers in free-surface turbulence.
