Locomotion inside a surfactant-laden drop at low surface Péclet numbers

2018 ◽  
Vol 851 ◽  
pp. 187-230 ◽  
Author(s):  
Vaseem A. Shaik ◽  
Vishwa Vasani ◽  
Arezoo M. Ardekani
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Stable Configuration ◽  
Far Field ◽  
Small Reynolds Number ◽  
Drop Surface ◽  
Field Representation ◽  
Small Surface ◽  
Bipolar Coordinates ◽  
Drag Forces

We investigate the dynamics of a swimming microorganism inside a surfactant-laden drop for axisymmetric configurations under the assumptions of small Reynolds number and small surface Péclet number $(Pe_{s})$. Expanding the variables in $Pe_{s}$, we solve the Stokes equations for the concentric configuration using Lamb’s general solution, while the dynamic equation for the stream function is solved in the bipolar coordinates for the eccentric configurations. For a two-mode squirmer inside a drop, the surfactant redistribution can either increase or decrease the magnitude of swimmer and drop velocities, depending on the value of the eccentricity. This was explained by analysing the influence of surfactant redistribution on the thrust and drag forces acting on the swimmer and the drop. The far-field representation of a surfactant-covered drop enclosing a pusher swimmer at its centre is a puller; the strength of this far field is reduced due to the surfactant redistribution. The advection of surfactant on the drop surface leads to a time-averaged propulsion of the drop and the time-reversible swimmer that it engulfs, thereby causing them to escape from the constraints of the scallop theorem. We quantified the range of parameters for which an eccentrically stable configuration can be achieved for a two-mode squirmer inside a clean drop. The surfactant redistribution shifts this eccentrically stable position towards the top surface of the drop, although this shift is small.

Steady approach of unsteady low-Reynolds-number flow past two rotating circular cylinders

2013 ◽  
Vol 736 ◽  
pp. 414-443 ◽  
Author(s):  
Y. Ueda ◽  
T. Kida ◽  
M. Iguchi
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Stokes Equations ◽  
Low Reynolds Number ◽  
Vortex Method ◽  
The Self ◽  
Circular Cylinders ◽  
Far Field ◽  
Flow Past ◽  
Low Reynolds

AbstractThe long-time viscous flow about two identical rotating circular cylinders in a side-by-side arrangement is investigated using an adaptive numerical scheme based on the vortex method. The Stokes solution of the steady flow about the two-cylinder cluster produces a uniform stream in the far field, which is the so-called Jeffery’s paradox. The present work first addresses the validation of the vortex method for a low-Reynolds-number computation. The unsteady flow past an abruptly started purely rotating circular cylinder is therefore computed and compared with an exact solution to the Navier–Stokes equations. The steady state is then found to be obtained for $t\gg 1$ with ${\mathit{Re}}_{\omega } {r}^{2} \ll t$, where the characteristic length and velocity are respectively normalized with the radius ${a}_{1} $ of the circular cylinder and the circumferential velocity ${\Omega }_{1} {a}_{1} $. Then, the influence of the Reynolds number ${\mathit{Re}}_{\omega } = { a}_{1}^{2} {\Omega }_{1} / \nu $ about the two-cylinder cluster is investigated in the range $0. 125\leqslant {\mathit{Re}}_{\omega } \leqslant 40$. The convection influence forms a pair of circulations (called self-induced closed streamlines) ahead of the cylinders to alter the symmetry of the streamline whereas the low-Reynolds-number computation (${\mathit{Re}}_{\omega } = 0. 125$) reaches the steady regime in a proper inner domain. The self-induced closed streamline is formed at far field due to the boundary condition being zero at infinity. When the two-cylinder cluster is immersed in a uniform flow, which is equivalent to Jeffery’s solution, the streamline behaves like excellent Jeffery’s flow at ${\mathit{Re}}_{\omega } = 1. 25$ (although the drag force is almost zero). On the other hand, the influence of the gap spacing between the cylinders is also investigated and it is shown that there are two kinds of flow regimes including Jeffery’s flow. At a proper distance from the cylinders, the self-induced far-field velocity, which is almost equivalent to Jeffery’s solution, is successfully observed in a two-cylinder arrangement.

Transient inertial hydrodynamic interaction between two identical spheres settling at small Reynolds number

2008 ◽  
Vol 605 ◽  
pp. 263-279 ◽  
Author(s):  
B. U. FELDERHOF
Keyword(s):  
Reynolds Number ◽  
Kinetic Energy ◽  
Relative Motion ◽  
Hydrodynamic Interaction ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Small Reynolds Number ◽  
Navier Stokes Equations ◽  
Small Constant ◽  
Distance Vector

The flow pattern generated by a sphere accelerated from rest by a small constant applied forceshows scaling behaviour at long times, as can be shown from the solution of the linearized Navier–Stokes equations. In the scaling regime the kinetic energy of the flow grows with thesquare root of time. For two distant settling spheres starting from rest the kinetic energy ofthe flow depends on the distance vector between centres; owing to interference of the flowpatterns. It is argued that this leads to relative motion of the two spheres. Thecorresponding interaction energy is calculated explicitly in the scaling regime.

scholarly journals Numerical studies of viscous incompressible flow between two rotating concentric spheres

2004 ◽  
Vol 2004 (2) ◽  
pp. 91-106 ◽  
Author(s):  
E. O. Ifidon
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Outer Sphere ◽  
Navier Stokes ◽  
Large Reynolds Number ◽  
Stable Configuration ◽  
Axially Symmetric ◽  
Closed Curves ◽  
Concentric Spheres ◽  
Flow Variables

The problem of determining the induced steady axially symmetric motion of an incompressible viscous fluid confined between two concentric spheres, with the outer sphere rotating with constant angular velocity and the inner sphere fixed, is numerically investigated for large Reynolds number. The governing Navier-Stokes equations expressed in terms of a stream function-vorticity formulation are reduced to a set of nonlinear ordinary differential equations in the radial variable, one of second order and the other of fourth order, by expanding the flow variables as an infinite series of orthogonal Gegenbauer functions. The numerical investigation is based on a finite-difference technique which does not involve iterations and which is valid for arbitrary large Reynolds number. Present calculations are performed for Reynolds numbers as large as 5000. The resulting flow patterns are displayed in the form of level curves. The results show a stable configuration consistent with experimental results with no evidence of any disjoint closed curves.

A paradox of hovering insects in two-dimensional space

2008 ◽  
Vol 617 ◽  
pp. 207-229 ◽  
Author(s):  
MAKOTO IIMA
Keyword(s):  
External Force ◽  
Stokes Equations ◽  
Hydrodynamic Force ◽  
Dimensional Space ◽  
Exact Formula ◽  
Uniform Flow ◽  
Far Field ◽  
Two Dimensional ◽  
Field Representation ◽  
Whole Space

A paradox concerning the flight of insects in two-dimensional space is identified: insects maintaining their bodies in a particular position (hovering) cannot, on average, generate hydrodynamic force if the induced flow is temporally periodic and converges to rest at infinity. This paradox is derived by using the far-field representation of periodic flow and the generalized Blasius formula, an exact formula for a force that acts on a moving body, based on the incompressible Navier–Stokes equations. Using this formula, the time-averaged force can be calculated solely in terms of the time-averaged far-field flow. A straightforward calculation represents the averaged force acting on an insect under a uniform flow, −〈V〉, determined by the balance between the hydrodynamic force and an external force such as gravity. The averaged force converges to zero in the limit 〈V〉 → 0, which implies that insects in two-dimensional space cannot hover under any finite external force if the direction of the uniform flow has a component parallel to the external force. This paradox provides insight into the effect of the singular behaviour of the flow around hovering insects: the far-field wake covers the whole space. On the basis of these assumptions, the relationship between this paradox and real insects that actually achieve hovering is discussed.

Viscous Flows

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Exact Solution ◽  
Reynolds Number ◽  
Stokes Equations ◽  
Viscous Flows ◽  
Navier Stokes ◽  
Small Reynolds Number ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
The Core ◽  
Flow Past A Sphere

Here some solutions of Navier–Stokes equations are found.The flow of a fluid along a pipe (Poisseuille flow) and that between two rotating cylinders (Couette flow) are the simplest. In the limit of large viscosity (small Reynolds number) the equations become linear: Stokes equations. Flow past a sphere is solved in detail. It is used to calculate the drag on a sphere, a classic formula of Stokes. An exact solution of the Navier–Stokes equation describing a dissipating vortex is also found. It is seen that viscosity cannot be ignored at the boundary or at the core of vortices.

The Navier–Stokes Equations

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Reynolds Number ◽  
Scale Invariance ◽  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Incompressible Limit ◽  
Small Reynolds Number ◽  
Navier Stokes Equations ◽  
Dimensionless Ratio ◽  
Energy And Momentum

When different layers of a fluid move at different velocities, there is some friction which results in loss of energy and momentum to molecular degrees of freedom. This dissipation is measured by a property of the fluid called viscosity. The Navier–Stokes (NS) equations are the modification of Euler’s equations that include this effect. In the incompressible limit, the NS equations have a residual scale invariance. The flow depends only on a dimensionless ratio (the Reynolds number). In the limit of small Reynolds number, the NS equations become linear, equivalent to the diffusion equation. Ideal flow is the limit of infinite Reynolds number. In general, the larger the Reynolds number, the more nonlinear (complicated, turbulent) the flow.

scholarly journals Angular velocity of a sphere in a simple shear at small Reynolds number

2016 ◽  
Vol 1 (8) ◽  
Author(s):  
J. Meibohm ◽  
F. Candelier ◽  
T. Rosén ◽  
J. Einarsson ◽  
F. Lundell ◽  
...  
Keyword(s):  
Reynolds Number ◽  
Angular Velocity ◽  
Simple Shear ◽  
Small Reynolds Number

A note on the solution of the Navier-Stokes equations for a spherically symmetric expansion into a very low pressure

1973 ◽  
Vol 59 (2) ◽  
pp. 391-396 ◽  
Author(s):  
N. C. Freeman ◽  
S. Kumar
Keyword(s):  
Shock Wave ◽  
Reynolds Number ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Low Pressure ◽  
Navier Stokes ◽  
Spherically Symmetric ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Type Flow

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

Sign in / Sign up

Export Citation Format

Share Document