Numerical and analytical approaches to an advection-diffusion problem at small Reynolds number and large Péclet number

AbstractThe flow field created by swimming micro-organisms not only enables their locomotion but also leads to advective transport of nutrients. In this paper we address analytically and computationally the link between unsteady feeding and unsteady swimming on a model micro-organism, the spherical squirmer, actuating the fluid in a time-periodic manner. We start by performing asymptotic calculations at low Péclet number ($\mathit{Pe}$) on the advection–diffusion problem for the nutrients. We show that the mean rate of feeding as well as its fluctuations in time depend only on the swimming modes of the squirmer up to order ${\mathit{Pe}}^{3/ 2} $, even when no swimming occurs on average, while the influence of non-swimming modes comes in only at order ${\mathit{Pe}}^{2} $. We also show that generically we expect a phase delay between feeding and swimming of $1/ 8\mathrm{th} $ of a period. Numerical computations for illustrative strokes at finite $\mathit{Pe}$ confirm quantitatively our analytical results linking swimming and feeding. We finally derive, and use, an adjoint-based optimization algorithm to determine the optimal unsteady strokes maximizing feeding rate for a fixed energy budget. The overall optimal feeder is always the optimal steady swimmer. Within the set of time-periodic strokes, the optimal feeding strokes are found to be equivalent to those optimizing periodic swimming for all values of the Péclet number, and correspond to a regularization of the overall steady optimal.

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Numerical Analysis of Heat Transfer Enhancement in a Micro-Channel Due to Mechanical Stirrers

Journal of Thermal Science and Engineering Applications ◽

10.1115/1.4047170 ◽

2020 ◽

Vol 13 (1) ◽

Author(s):

M. Sreejith ◽

S. Chetan ◽

S. N. Khaderi

Keyword(s):

Heat Transfer ◽

Reynolds Number ◽

Heat Transfer Enhancement ◽

Peclet Number ◽

Periodic Case ◽

Two Dimensional ◽

Transfer Enhancement ◽

Fluidic Channel ◽

Enhancement Of Heat Transfer ◽

Péclet Number

Abstract Using two-dimensional numerical simulations of the momentum, mass, and energy conservation equations, we investigate the enhancement of heat transfer in a rectangular micro-fluidic channel. The fluid inside the channel is assumed to be stationary initially and actuated by the motion imparted by mechanical stirrers, which are attached to the bottom of the channel. Based on the direction of the oscillation of the stirrers, the boundary conditions can be classified as either no-slip (when the oscillation is perpendicular to the length of the channel) or periodic (when the oscillation is along the length of the channel). The heat transfer enhancement due to the motion of the stirrers (with respect to the stationary stirrer situation) is analyzed in terms of the Reynolds number (ranging from 0.7 to 1000) and the Peclet number (ranging from 10 to 100). We find that the heat transfer first increases and then decreases with an increase in the Reynolds number for any given Peclet number. The heat transferred is maximum at a Reynolds number of 20 for the no-slip case and at a Reynolds number of 40 for the periodic case. For a given Peclet and Reynolds number, the heat flux for the periodic case is always larger than the no-slip case. We explain the reason for these trends using time-averaged flow velocity profiles induced by the oscillation of the mechanical stirrers.

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A two-level variational multiscale meshless local Petrov–Galerkin (VMS-MLPG) method for convection-diffusion problems with large Peclet number

Computers & Fluids ◽

10.1016/j.compfluid.2017.03.023 ◽

2018 ◽

Vol 164 ◽

pp. 73-82 ◽

Cited By ~ 3

Author(s):

Zheng-Ji Chen ◽

Zeng-Yao Li ◽

Wen-Li Xie ◽

Xue-Hong Wu

Keyword(s):

Peclet Number ◽

Péclet Number ◽

Convection Diffusion ◽

Variational Multiscale ◽

Mlpg Method ◽

Large Peclet Number ◽

Diffusion Problems

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The drag of a body moving transversely in a confined stratified fluid

Journal of Fluid Mechanics ◽

10.1017/s0022112070002458 ◽

1970 ◽

Vol 43 (2) ◽

pp. 407-418 ◽

Cited By ~ 14

Author(s):

M. R. Foster ◽

P. G. Saffman

Keyword(s):

Peclet Number ◽

Slow Motion ◽

Shear Layers ◽

Stratified Fluid ◽

Rotating Fluid ◽

Péclet Number ◽

Fluid Layers ◽

Number Problem ◽

Number Flow ◽

Large Peclet Number

The slow motion of a body through a stratified fluid bounded laterally by insulating walls is studied for both large and small Peclet number. The Taylor column and its associated boundary and shear layers are very different from the analogous problem in a rotating fluid. In particular, the large Peclet number problem is non-linear and exhibits mixing of statically unstable fluid layers, and hence the drag is order one; whereas the small Peclet number flow is everywhere stable, and the drag is of the order of the Peclet number.

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Passive scalar statistics in high-Péclet-number grid turbulence

Journal of Fluid Mechanics ◽

10.1017/s0022112097008161 ◽

1998 ◽

Vol 358 ◽

pp. 135-175 ◽

Cited By ~ 196

Author(s):

L. MYDLARSKI ◽

Z. WARHAFT

Keyword(s):

Reynolds Number ◽

Velocity Field ◽

Peclet Number ◽

Passive Scalar ◽

Structure Functions ◽

Order Structure ◽

Grid Turbulence ◽

Temperature Derivative ◽

Péclet Number ◽

Transverse Temperature

The statistics of a turbulent passive scalar (temperature) and their Reynolds number dependence are studied in decaying grid turbulence for the Taylor-microscale Reynolds number, Rλ, varying from 30 to 731 (21[les ]Peλ[les ]512). A principal objective is, using a single (and simple) flow, to bridge the gap between the existing passive grid-generated low-Péclet-number laboratory experiments and those done at high Péclet number in the atmosphere and oceans. The turbulence is generated by means of an active grid and the passive temperature fluctuations are generated by a mean transverse temperature gradient, formed at the entrance to the wind tunnel plenum chamber by an array of differentially heated elements. A well-defined inertial–convective scaling range for the scalar with a slope, nθ, close to the Obukhov–Corrsin value of 5/3, is observed for all Reynolds numbers. This is in sharp contrast with the velocity field, in which a 5/3 slope is only approached at high Rλ. The Obukhov–Corrsin constant, Cθ, is estimated to be 0.45–0.55. Unlike the velocity spectrum, a bump occurs in the spectrum of the scalar at the dissipation scales, with increasing prominence as the Reynolds number is increased. A scaling range for the heat flux cospectrum was also observed, but with a slope around 2, less than the 7/3 expected from scaling theory. Transverse structure functions of temperature exist at the third and fifth orders, and, as for even-order structure functions, the width of their inertial subranges dilates with Reynolds number in a systematic way. As previously shown for shear flows, the existence of these odd-order structure functions is a violation of local isotropy for the scalar differences, as is the existence of non-zero values of the transverse temperature derivative skewness (of order unity) and hyperskewness (of order 100). The ratio of the temperature derivative standard deviation along and normal to the gradient is 1.2±0.1, and is independent of Reynolds number. The refined similarity hypothesis for the passive scalar was found to hold for all Rλ, which was not the case for the velocity field. The intermittency exponent for the scalar, μθ, was found to be 0.25±0.05 with a possible weak Rλ dependence, unlike the velocity field, where μ was a strong function of Reynolds number. New, higher-Reynolds-number results for the velocity field, which smoothly follow the trends of Mydlarski & Warhaft (1996), are also presented.

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Flow Rate Prediction for a Semi-permeable Membrane at Low Reynolds Number in a Circular Pipe

Transport in Porous Media ◽

10.1007/s11242-021-01716-w ◽

2021 ◽

Author(s):

Suguru Miyauchi ◽

Shuji Yamada ◽

Shintaro Takeuchi ◽

Asahi Tazaki ◽

Takeo Kajishima

Keyword(s):

Reynolds Number ◽

Osmotic Pressure ◽

Membrane Permeability ◽

Flow Rate ◽

Peclet Number ◽

Concentration Polarization ◽

Circular Pipe ◽

Permeate Flux ◽

Permeable Membrane ◽

Péclet Number

AbstractA concise and accurate prediction method is required for membrane permeability in chemical engineering and biological fields. As a preliminary study on this topic, we propose the concentration polarization model (CPM) of the permeate flux and flow rate under dominant effects of viscosity and solute diffusion. In this model, concentration polarization is incorporated for the solution flow through a semi-permeable membrane (i.e., permeable for solvent but not for solute) in a circular pipe. The effect of the concentration polarization on the flow field in a circular pipe under a viscous-dominant condition (i.e., at a low Reynolds number) is discussed by comparing the CPM with the numerical simulation results and infinitesimal Péclet number model (IPM) for the membrane permeability, strength of the osmotic pressure, and Péclet number. The CPM and IPM are confirmed to be a reasonable extension of the model for a pure fluid, which was proposed previously. The application range of the IPM is narrow because the advection of the solute concentration is not considered, whereas the CPM demonstrates superior applicability in a wide range of parameters, including the permeability coefficient, strength of the osmotic pressure, and Péclet number. This suggests the necessity for considering concentration polarization. Although the mathematical expression of the CPM is more complex than that of the IPM, the CPM exhibits a potential to accurately predict the permeability parameters for a condition in which a large permeate flux and osmotic pressure occur.

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