scholarly journals On the Perturbation of the Three-Dimensional Stokes Flow of Micropolar Fluids by a Constant Uniform Magnetic Field in a Circular Cylinder

2011 ◽  
Vol 2011 ◽  
pp. 1-41 ◽  
Author(s):  
Panayiotis Vafeas ◽  
Polycarpos K. Papadopoulos ◽  
Pavlos M. Hatzikonstantinou
Keyword(s):  
Magnetic Field ◽  
Circular Cylinder ◽  
Stokes Flow ◽  
Uniform Magnetic Field ◽  
Colloidal Suspension ◽  
Three Dimensional ◽  
Flow Fields ◽  
Creeping Flow ◽  
Spherical Particles ◽  
Circular Duct

Modern engineering technology involves the micropolar magnetohydrodynamic flow of magnetic fluids. Here, we consider a colloidal suspension of non-conductive ferromagnetic material, which consists of small spherical particles that behave as rigid magnetic dipoles, in a carrier liquid of approximately zero conductivity and low-Reynolds number properties. The interaction of a 3D constant uniform magnetic field with the three-dimensional steady creeping motion (Stokes flow) of a viscous incompressible micropolar fluid in a circular cylinder is investigated, where the magnetization of the ferrofluid has been taken into account and the magnetic Stokes partial differential equations have been presented. Our goal is to apply the proper boundary conditions, so as to obtain the flow fields in a closed analytical form via the potential representation theory, and to study several characteristics of the flow. In view of this aim, we make use of an improved new complete and unique differential representation of magnetic Stokes flow, valid for non-axisymmetric geometries, which provides the velocity and total pressure fields in terms of easy-to-find potentials. We use these results to simulate the creeping flow of a magnetic fluid inside a circular duct and to obtain the flow fields associated with this kind of flow.

Optimum profiles in two-dimensional Stokes flow

1995 ◽  
Vol 450 (1940) ◽  
pp. 603-622 ◽  
Keyword(s):  
Circular Cylinder ◽  
Cross Section ◽  
Stokes Flow ◽  
Unit Length ◽  
Three Dimensional ◽  
Variational Formula ◽  
Effective Radius ◽  
Two Dimensional ◽  
Equivalent Radius ◽  
Optimum Profile

We consider the problem of designing the section of a cylinder to minimize the drag per unit length it experiences when placed perpendicular to a uniform stream at low Reynolds number; we suppose the area of the cross-section to be given, and the flow to be two-dimensional. The relevant properties of a cylinder of general cross-section in a particular orientation can conveniently be expressed in terms of its equivalent radius; when the drag and flow at infinity are parallel, this equivalent radius is the radius of the circular cylinder giving rise to the same drag per unit length. We obtain a variational formula for this equivalent radius when the surface of the cylinder is perturbed; this shows that the optimum profile we seek must be such that the flow past it has a vorticity of constant magnitude at its surface, and this fact enables the optimum to be determined analytically. The efficacy of a particular section may be measured by its effective radius, this being the equivalent radius when the length scale is chosen to give the section an area π ; thus a circular cylinder has an effective radius of 1. The minimum possible effective radius, achieved by the optimum profile, is 0.88876. To illustrate some of the arguments we exploit in a more familiar setting, we also obtain a variational formula for the drag on a three-dimensional body in Stokes flow when its surface is perturbed.

scholarly journals The 3D Happel model for complete isotropic Stokes flow

2004 ◽  
Vol 2004 (46) ◽  
pp. 2429-2441 ◽  
Author(s):  
George Dassios ◽  
Panayiotis Vafeas
Keyword(s):  
Angular Velocity ◽  
Stokes Flow ◽  
Total Pressure ◽  
Mechanical Energy ◽  
Creeping Flow ◽  
Spherical Particles ◽  
Mathematical Treatment ◽  
Normal Velocity ◽  
Tensor Fields ◽  
Concentric Spheres

The creeping flow through a swarm of spherical particles that move with constant velocity in an arbitrary direction and rotate with an arbitrary constant angular velocity in a quiescent Newtonian fluid is analyzed with a 3D sphere-in-cell model. The mathematical treatment is based on the two-concentric-spheres model. The inner sphere comprises one of the particles in the swarm and the outer sphere consists of a fluid envelope. The appropriate boundary conditions of this non-axisymmetric formulation are similar to those of the 2D sphere-in-cell Happel model, namely, nonslip flow condition on the surface of the solid sphere and nil normal velocity component and shear stress on the external spherical surface. The boundary value problem is solved with the aim of the complete Papkovich-Neuber differential representation of the solutions for Stokes flow, which is valid in non-axisymmetric geometries and provides us with the velocity and total pressure fields in terms of harmonic spherical eigenfunctions. The solution of this 3D model, which is self-sufficient in mechanical energy, is obtained in closed form and analytical expressions for the velocity, the total pressure, the angular velocity, and the stress tensor fields are provided.

The distortion of a magnetic field by the flow of a conducting fluid past a circular cylinder

1965 ◽  
Vol 22 (3) ◽  
pp. 561-578 ◽  
Author(s):  
R. Seebass ◽  
K. Tamada
Keyword(s):  
Magnetic Field ◽  
Integral Equation ◽  
Circular Cylinder ◽  
Potential Flow ◽  
Uniform Magnetic Field ◽  
The Body ◽  
Flow Potential ◽  
Field Lines ◽  
The Magnetic Field ◽  
Rear Stagnation Point

The distortion of a uniform magnetic field, aligned with the flow at infinity, by the potential flow of an inviscid conductor about a circular cylinder is determined. Potential flow of the fluid occurs when the interaction parameter is small; this is the case studied here. In the flow-potential and stream-function plane the problem may be formulated as a singular integral equation. Solutions of this equation show that for small fluid conductivities the magnetic field lines are distorted in the sense of being dragged along by the motion of the fluid. This process continues as the conductivity increases, with fewer and fewer of the magnetic field lines entering the body. For large conductivity this reduced flux of field lines enters over most of the body surface and exits in the neighbourhood of the rear stagnation point; behind the body there is a jet-like structure of magnetic field lines.

On angles of separation in Stokes flows

1983 ◽  
Vol 133 ◽  
pp. 427-442 ◽  
Author(s):  
M. E. O'Neill
Keyword(s):  
Circular Cylinder ◽  
Stokes Flow ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Stokes Flows ◽  
Eddy Structure ◽  
Toroidal Vortices ◽  
Mathematical Equivalence ◽  
Infinite Sets

A two-dimensional Stokes flow close to the line of contact of two touching cylinders or three-dimensional axisymmetric Stokes flow close to the point of contact of two touching bodies is shown in general to separate into infinite sets of eddies with angles of separation from the bodies which tend to 58.61° as the line or point of contact is approached. The flow near the vertex of a conical cusp is shown to be a system of nested toroidal vortices and the separation angles tend to 45.25° as the vertex is approached. Stokes flow between parallel planes or within a circular cylinder is shown in general to separate far from the generating disturbances with cellular eddy structure and separation angles which tend to 58.61° and 45.25° respectively. The mathematical equivalence of the various problems is established.

Liquid-metal flow in a system of electrically coupled U-bends in a strong uniform magnetic field

1995 ◽  
Vol 299 ◽  
pp. 73-95 ◽  
Author(s):  
Sergei Molokov ◽  
Robert Stieglitz
Keyword(s):  
Magnetic Field ◽  
Pressure Drop ◽  
Liquid Metal ◽  
Uniform Magnetic Field ◽  
Flow Direction ◽  
Metal Flow ◽  
Three Dimensional ◽  
Recirculatory Flow ◽  
The Magnetic Field ◽  
Very High

Liquid-metal magnetohydrodynamic flow in a system of electrically coupled U-bends in a strong uniform magnetic field is studied. The ducts composing the bends are electrically conducting and have rectangular cross-sections. It has been anticipated that very strong global electric currents are induced in the system, which modify the flow pattern and produce a very high pressure drop compared to the flow in a single U-bend. A detailed asymptotic analysis of flow for high values of the Harmann number (in fusion blanket applications of the order of 103−104) shows that circulation of global currents results in several types of peculiar flow patterns. In ducts parallel to the magnetic field a combination of helical and recirculatory flow types may be present and vary from one bend to another. The magnitude of the recirculatory motion may become very high depending on the flow-rate distribution between the bends in the system. The recirculatory flow may account for about 50% of the flow in all bends. In addition there are equal and opposite jets at the walls parallel to the magnetic field, which are common to any two bends. The pressure drop due to three-dimensional effects linearly increases with the number of bends in a system and may significantly affect the total pressure drop. To suppress this and some other unwelcome tendencies either the ducts perpendicular to the magnetic field should be electrically separated, or the flow direction in the neighbouring ducts should be made opposite, so that leakage currents cancel each other.

Stokes flow of a conducting fluid past an axially symmetric body in the presence of a uniform magnetic field

1960 ◽  
Vol 9 (3) ◽  
pp. 473-477 ◽  
Author(s):  
I-Dee Chang
Keyword(s):  
Magnetic Field ◽  
Stokes Flow ◽  
Uniform Magnetic Field ◽  
Hartmann Number ◽  
Magnetic Effect ◽  
Symmetric Body ◽  
Axially Symmetric ◽  
Low Reynolds Number Flow ◽  
Reynolds Number Flow ◽  
Number Flow

Low Reynolds number flow of an incompressible fluid past an axially symmetric body in the presence of a uniform magnetic field is studied using a perturbation method. It is found that for small Hartmann number M an approximate drag formula is given by $ D^ \prime = D^\prime_0 \left(1 + \frac {D^\prime_0} {16\pi \rho vaU}M\right) + O(M^2),$ where D′0 is the Stokes drag for flow with no magnetic effect.

Inverse cascades in incompressible fluid and magnetofluid turbulence

1996 ◽  
Vol 56 (3) ◽  
pp. 467-491
Author(s):  
Murshed Hossain
Keyword(s):  
Magnetic Field ◽  
Incompressible Fluid ◽  
Uniform Magnetic Field ◽  
Spatial Scales ◽  
Three Dimensional ◽  
Self Organization ◽  
Anisotropic Fluid ◽  
Uniform Rotation ◽  
Fluid Turbulence ◽  
Turbulent Fluctuations

Absolute equilibrium statistical theory and numerical simulations are reviewed in the context of inverse cascades in two- and three-dimensional incompressible fluid and magnetofluid turbulence. Turbulent fluctuations of physically interesting quantities undergo inverse cascade to larger spatial scales, leading to self-organization under certain circumstances. In particular, most systems with more than one quadratic ideal invariant, or, having some kind of imposed anisotropy, exhibit inverse cascades. Anisotropic fluid turbulence in the presence of a uniform rotation and magnetofluid turbulence in the presence of a uniform magnetic field are considered.

Weak magnetohydrodynamic turbulence and intermittency

2015 ◽  
Vol 770 ◽  
Author(s):  
R. Meyrand ◽  
K. H. Kiyani ◽  
S. Galtier
Keyword(s):  
Magnetic Field ◽  
Uniform Magnetic Field ◽  
Three Dimensional ◽  
Leading Order ◽  
Magnetohydrodynamic Turbulence ◽  
Asymptotic Regime ◽  
Resonant Interactions ◽  
Vector Separation ◽  
Parallel Current

Three-dimensional numerical simulation is used to investigate intermittency in incompressible weak magnetohydrodynamic turbulence with a strong uniform magnetic field $\boldsymbol{b}_{\mathbf{0}}$ and zero cross-helicity. At leading order, this asymptotic regime is achieved via three-wave resonant interactions with the scattering of a wave on a 2D mode for which $k_{\Vert }=0$. When the interactions with the 2D modes are artificially reduced, we show numerically that the system exhibits an energy spectrum with $k_{\bot }^{-3/2}$, whereas the expected exact solution with $k_{\bot }^{-2}$ is recovered with the full nonlinear system. In the latter case, strong intermittency is found when the vector separation of structure functions is taken transverse to $\boldsymbol{b}_{\mathbf{0}}$. This result may be explained by the influence of the 2D modes whose regime belongs to strong turbulence. In addition to shedding light on the origin of this intermittency, we derive a log-Poisson law, ${\it\zeta}_{p}=p/8+1-(1/4)^{p/2}$, which fits the data perfectly and highlights the important role of parallel current sheets.

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