Abstract
In the present technical note, drag on axially symmetric body for conducting fluid in the presence of a uniform magnetic field is considered under the no-slip condition along with the matching condition(ρ2U2=H02μ3σ) involving Hartmans number and Reynolds number to define this drag as Oseen’s resistance or Oseen’s correction to Stokes drag is presented. Oseen’s resistance on sphere, spheroid, flat circular disk (broadside) are found as an application under the specified condition. These expressions of Oseen’s drag are seems to be new in magneto-hydrodynamics. Author claims that by this idea, the results of Oseen’s drag on axially symmetric bodies in low Reynolds number hydrodynamics can be utilized for finding the Oseen’s drag in magneto hydrodynamics just by replacing Reynolds number by Hartmann number under the proposed condition.
Keywords
- stokes drag
- Oseen’s resistance
- conducting fluid
- magnetic field
- Hartman number
- Reynolds number
1. Introduction
There are many fluids like plasmas, liquid metals, salt water, and electrolytes etc. lies under the class of magneto hydrodynamics and attracted the attention of mechanical engineers, scientists and chemists for a longer period of time. The main significant quantity of magneto hydrodynamic fluid past an axially symmetric particle or object is the drag experienced by the stationary body or moving through the fluid.
It was
where Ds is the classical Stokes drag and ‘R’ is the Reynolds number.
Where Ds is the classical Stokes drag and ‘M’ is the Hartmann number. He also proved that when the magnetic Reynolds number Rm, is small the magnetic field is essentially independent of the fluid motion.
For in depth information regarding the classical Stokes drag and Oseen’s drag on axi-symmetric bodies in relativistic fluid mechanics and magneto hydrodynamics, the books of Oseen [4] (in German Language), Happel and Brenner [26], Childress [27], Ferraro and Plumpton [28], Milne-Thompson [29], Cabannes [30], Mirela and Pop [31], Kim and Karrila [32] are referred by author.
2. Formulation of problem
We consider the equation of low Reynolds number flow of an incompressible conducting fluid past an axi-symmetric body in a magnetic field which is uniform at infinity.
In Eqs. (3–5), all entities are non-dimensional and their abbreviations are as follows;
U = free-stream velocity,
a = characteristic length of body,
Other symbols have their usual meanings in electro-hydrodynamics and magneto-hydrodynamics. Primed entities are in physical units (as per [5, 10]).
Following the perturbation method given by
where Ds is the Stokes drag for flow without magnetic field.
Now, in the section-4, we prove that the solution of drag given in Eq. (7) is Oseen’s drag or Oseen’s correction to Stokes drag by utilizing the idea of Oseen’s resistance given by
3. Oseen’s equations and Oseen’s drag
Let us consider the axially symmetric arbitrary body of characteristic length ‘a’ placed along principal axis (x-axis, say) in a uniform stream U of viscous [4, 13] fluid of density ρ and kinematic viscosity ν. When particle Reynolds number Ua/ν is low, the steady motion of incompressible fluid around this axially symmetric body is governed by Stokes equations [26],
subject to the no-slip boundary condition. It was
known as Oseen’s equation.
where Re = ρUa/μ is bodies Reynolds number.
Based on Oseen’s above idea and
where ‘a’ is any characteristic particle or body dimension and Re = ρUa/μ is the particle Reynolds number.
4. Matching condition for Hartmann number and Reynolds number
In the expression of drag (Eq. 7) given by
Under this condition, drag on axially symmetric body in the presence of a uniform magnetic field described by
5. Flow past sphere
We consider the sphere generated due to the revolution of circle of radius ‘a’ about axis of symmetry. The Oseen’s drag on sphere of radius ‘a’ placed under conducting fluid of uniform velocity U and uniform magnetic field H0 is given by (7) as
but for sphere, the classical Stokes drag Ds = 6πμUa, then, we have
which is in confirmation with Oseen’s drag (Eq. 10) on sphere given by
6. Flow past spheroid
6.1 Prolate spheroid
We consider the prolate spheroid generated by revolution of ellipse having semi-major axis length ‘a’ and semi-minor axis length ‘b’ about axis of symmetry. Stokes drag on prolate spheroid placed in uniform axial flow, with velocity U, parallel to axis of symmetry (x-axis) is given as (by utilizing DS conjecture given in [18])
Now, the Oseen’s correction as well as the solution of Oseen’s equation (Eq. 9) may be obtained for same prolate spheroid by substituting the value of Stokes drag (Eq. 14) in Brenner’s formula (Eq. 11) under the matching condition (Eq. 10) as
where
Equations (Eq. 16) and (Eq. 18) immediately reduces to the case of sphere (given in Eq. 13) in the limiting case as e → 0. On the other hand, the closed form expressions (Eq. 15) and (Eq. 17) due to Oseen for prolate spheroid appears to be new for magneto hydrodynamics as no such type of expressions are available in the literature for comparison.
6.2 Oblate spheroid
We consider the oblate spheroid generated by revolution of ellipse having semi-major axis length ‘b’ and semi-minor axis length ‘a’ about axis of symmetry. Stokes drag on oblate spheroid placed in uniform axial flow, with velocity U, parallel to axis of symmetry (x-axis) is given as (by utilizing DS conjecture given in
Now, the Oseen’s correction as well as the solution of Oseen’s equation (Eq. 19) may be obtained for same oblate spheroid by substituting the value of Stokes drag (6.6) in Brenner’s formula (Eq. 11) under the matching condition (Eq. 10) as
where
Equations (Eq. 21) and (Eq. 23) immediately reduces to the case of sphere (given in Eq. 13) in the limiting case as e → 0. On the other hand, the closed form expressions (Eq. 20) and (Eq. 22) due to Oseen for oblate spheroid appears to be new as no such type of expressions are available in the literature for comparison.
7. Flat circular disk (broadside on)
Lamb [20] provided the Stokes drag on flat circular disk of radius ‘a’ placed broadside on facing towards the uniform stream of velocity U as.
Now, under the matching conditions (Eq. 10), the Oseen’s drag on circular disk placed under the effect of magnetic field is given by Chang’s rule (Eq. 7) in terms of Hartmann number as
or
where
8. Conclusion
The problem of Oseen flow of an incompressible conducting fluid past axially symmetric body in the presence of a uniform magnetic field is tackled. The matching conditions are obtained by equating the small dimensionless Hartmann number and Reynolds number ensuring the
Acknowledgments
Author conveys his sincere thanks to the authorities of B.S.N.V. Post Graduate College, Lucknow, UP, India, to provide basic infrastructure facilities and moral support throughout the preparation of this paper.
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