SOME GEOMETRIC PROPERTIES OF MAGNETO-FLUID FLOWS

By employing an anholonomic description of the governing equations, certain geometric results are obtained for a class of non-dissipative magnetofluid flows. The stream lines are geodesics on a normal congruence of the surfaces which are the Maxwellian surfaces.

This paper presents a kinematic study of the steady and non-dissipative magneto-fluid flows with the use of anholonomic geometric results.It has been shown that if at each point of the flow region the magnetic field line is in the plane of the stream line and its binormal, there exists a normal congruence of the surfaces which are the Maxwellian surfaces.The stream lines are the geodesics and their binormals are parallels on these surfaces.The speed and the fluid pressure are constant along the binormals of the stream lines.The Bernoulli equation in MHD has also been derived in a special case.

THE BASIC EQUATIONS
The basic equations governing the steady and non-dissipative magneto-fluid flows given by Lundquist ill are as follows: (),j 0, (2.1) PvJ vi, j + P'i vHj Hi,j -*v(H2)'i' where is the constant magnetic permeability, p the fluid pressure, p the fluid density, Q the entropy of the fluid, @ the velocity field vector, and H j the magnetic field vector.
Using the substitution h j /H (2.9) 3. ANHOLONOMIC GEOMETRIC RESULTS In this section we are going to state the results given by Marris and Passman 5].If we consider the orthonormal basis {t i, ni, b i} introduced along the stream lines, where t i i b i n and are the unit vectors along the tangents, the principal normals, and the binormals of the stream lines, we have the results t t.. k n.
(3.1) t j ni, j =-k t i + b i and t j bi, j ---n i which are the Serret Frenet formulae, where k and are the curvature and torsion of tJ-lines.The latter results are augmented by the Intrinsic deriva- rives along n3-1ines and b3-1+/-nes given by, in ters of the eight parameters i bj t i J n j t k, , t' n' n j, b,j, 8nt 'j ni' 8bt ,j bi' 3 where t and are abnormalities of tJ-lines and nJ-lines respectively given by n and a t t i eijk(tk),j (3.10) It should be noted that here we employ the parameter On instead of Ob-n as employed by lrris and Passan; b the abnormality of bJ-lines, is expressed in terms of other parameters by the relation Now we will derive some kinematic properties of magneto-fluid flows employing the results of earlier section.First we will decompose the basic equations governing the flow.
The velocity vector v j and magnetic field vector h j can be expressed as q t j (4.1) =ht n where q is the the magnitude of the velocity; and ht, h n, are the components of magnetic field vectors along tj, nj, b j respectively.
(4.10) (4.8) and ( 4.9a) imply b j (n q ),j + b j 0 if h b # 0 (4.11) If q 0, the velocity vanishes; that means the fluid is stationary.[Since the fluid is not stationary, q # 0].Therefore, (4.9b) implies 0 if # 0" (4.12) that is, the abnormality of the principal normal vanishes which is the necessary and sufficient condition for the existence of normal congruence of the surfaces containing tJ-lines and bJ-lines.Since the vector h j is in the plane of t j and b j, the magnetic field lines lie on these surfaces.Hence, the normal con- gruence of the surfaces is nothing but the family of the Maxwellian surfaces.
Since the principal normals of the stream lines are the normals of these surfaces the stream lines are the geodesics.Thus we have the theorem: THEOREM i.If at each point of the flow region the component of the magnetic field vector in the direction of the principal normal of the stream line vanishes there exists a normal congruence of the surfaces containing the stream lines and their binormals which are the geodesics and parallels respectively.These surfaces are the Maxwellian surfaces.
If the magnetic field lines coincide with the binormals of the stream lines and the density does not vary along the stream lines, the equations (4.3) (4.6c) will be put in the forms tJ q,j + q (nt + bt 0 q,j + q t j h,j + h q ent 0. (4.16c) Using (4.13) in (4.16c), we get t j h,j h 8bt 0. (4.17) From (4.14c) and (4.16a) we have the theorem: THEOREM 2. If the magnetic field lines coincide with the binormals of the stream lines, the fluid pressure and magnitude of the velocity are constant a- long the binormals.
Let us take the additional assumption that the magnetic intensity is con- stant along the stream lines, i.e. t j h 0, then (4.17This is the Bernoulli equation.
PROPERTIES OF THE FLOW.
geodesic curvature of bJ-lines vanishes.Thus we see that the mag- netic ield lines are the geodesics on the normal congruence of the surfaces, and hence we get the theorem: THEOREM 3. If the fluid density and the magnetic intensity do not vary along the stream lines, the family of the Maxwellian surfaces is the normal congruence of developables.The stream lines and the magnetic field lines are orthogonal geodesics on these developables.Employing (4.18) in (4.14a), we get + P + 1/2h 2 constant along the stream lines.
pq t j q j + t j p*.