scholarly journals Transient dispersion process of active particles

2021 ◽  
Vol 927 ◽  
Author(s):  
Weiquan Jiang ◽  
Guoqian Chen
Keyword(s):  
Integral Transform ◽  
Temporal Evolution ◽  
Dispersion Model ◽  
Practical Interest ◽  
Expansion Method ◽  
Local Distribution ◽  
Transform Method ◽  
Dispersion Process ◽  
Active Particles ◽  
Transient Dispersion

Active particles often swim in confined environments. The transport mechanisms, especially the global one as reflected by the Taylor dispersion model, are of great practical interest to various applications. For the active dispersion process in confined flows, previous analytical studies focused on the long-time asymptotic values of dispersion characteristics. Only several numerical studies preliminarily investigated the temporal evolution. Extending recent studies of Jiang & Chen (J. Fluid Mech., vol. 877, 2019, pp. 1–34; vol. 899, 2020, A18), this work makes a semi-analytical attempt to investigate the transient process. The temporal evolution of the local distribution in the confined-section–orientation space, drift, dispersivity and skewness, is explored based on moments of distributions. We introduce the biorthogonal expansion method for solutions because the classic integral transform method for passive transport problems is not applicable due to the self-propulsion effect. Two types of boundary condition, the reflective condition and the Robin condition for wall accumulation, are imposed respectively. A detailed study on spherical and ellipsoidal swimmers dispersing in a plane Poiseuille flow demonstrates the influences of the swimming, shear flow, initial condition, wall accumulation and particle shape on the transient dispersion process. The swimming-induced diffusion makes the local distribution reach its equilibrium state faster than that of passive particles. Although the wall accumulation significantly affects the evolution of the local distribution and the drift, the time scale to reach the Taylor regime is not obviously changed. The shear-induced alignment of ellipsoidal particles can enlarge the dispersivity but impacts slightly on the drift and the skewness.

Reflection of an Acoustic Step Wave From an Elastic Cylinder

1958 ◽  
Vol 25 (1) ◽  
pp. 103-108
Author(s):  
Richard Skalak ◽  
M. B. Friedman
Keyword(s):  
Integral Transform ◽  
Formal Solution ◽  
Elastic Cylinder ◽  
Transform Method ◽  
Pressure Fields ◽  
Plane Step ◽  
Single Integral ◽  
Short Time ◽  
Acoustic Fluid

Abstract An elastic cylinder, circular in section and infinite in length, is considered in an infinite acoustic fluid. The object of this paper is the determination of the reflected and diffracted pressure fields at large distances resulting from a plane step wave of pressure impinging on the cylinder and moving in a direction normal to the axis of the cylinder. A formal solution is obtained for the general case of an elastic cylinder. Numerical results are computed for rigid, fixed cylinders, and for rigid, floating cylinders. Two different methods are used to achieve results in the different ranges of time which are of interest. A short time approximation is developed by the use of a double integral-transform method. A mode approach and a single integral transform are used for later times. The results show that the reflected pulse decays quickly, within a time on the order of the transit time of the original wave across the cylinder.

Acceleration of the Convergence of the Integral Transform Method

1976 ◽  
Vol 61 (3) ◽  
pp. 431-434 ◽  
Author(s):  
V. C. Boffi ◽  
V. G. Molinari ◽  
G. Spiga
Keyword(s):  
Integral Transform ◽  
Transform Method ◽  
Integral Transform Method

scholarly journals Extended Stokes' Problems for Relatively Moving Porous Half-Planes

2009 ◽  
Vol 2009 ◽  
pp. 1-10 ◽  
Author(s):  
Chi-Min Liu
Keyword(s):  
Exact Solutions ◽  
Integral Transform ◽  
Initial Conditions ◽  
Integral Transforms ◽  
Momentum Equation ◽  
Transform Method ◽  
Stokes Problems ◽  
Flow Phenomena ◽  
Mass Influx ◽  
Mathematical Techniques

A shear flow motivated by relatively moving half-planes is theoretically studied in this paper. Either the mass influx or the mass efflux is allowed on the boundary. This flow is called the extended Stokes' problems. Traditionally, exact solutions to the Stokes' problems can be readily obtained by directly applying the integral transforms to the momentum equation and the associated boundary and initial conditions. However, it fails to solve the extended Stokes' problems by using the integral-transform method only. The reason for this difficulty is that the inverse transform cannot be reduced to a simpler form. To this end, several crucial mathematical techniques have to be involved together with the integral transforms to acquire the exact solutions. Moreover, new dimensionless parameters are defined to describe the flow phenomena more clearly. On the basis of the exact solutions derived in this paper, it is found that the mass influx on the boundary hastens the development of the flow, and the mass efflux retards the energy transferred from the plate to the far-field fluid.

scholarly journals Unsteady Helical Flows of a Size-Dependent Couple-Stress Fluid

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Qammar Rubbab ◽  
Itrat Abbas Mirza ◽  
Imran Siddique ◽  
Saadia Irshad
Keyword(s):  
Stress Tensor ◽  
Couple Stress ◽  
Integral Transform ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Fluid Velocity ◽  
Axial Flow ◽  
Couple Stress Fluid ◽  
Stress Vector ◽  
Transform Method

The helical flows of couple-stress fluids in a straight circular cylinder are studied in the framework of the newly developed, fully determinate linear couple-stress theory. The fluid flow is generated by the helical motion of the cylinder with time-dependent velocity. Also, the couple-stress vector is given on the cylindrical surface and the nonslip condition is considered. Using the integral transform method, analytical solutions to the axial velocity, azimuthal velocity, nonsymmetric force-stress tensor, and couple-stress vector are obtained. The obtained solutions incorporate the characteristic material length scale, which is essential to understand the fluid behavior at microscales. If characteristic length of the couple-stress fluid is zero, the results to the classical fluid are recovered. The influence of the scale parameter on the fluid velocity, axial flow rate, force-stress tensor, and couple-stress vector is analyzed by numerical calculus and graphical illustrations. It is found that the small values of the scale parameter have a significant influence on the flow parameters.

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