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Example text
These equations can be generalized to the case in which the system shows helical symmetry. Omitting the calculations, we write the result [26] [see Eq. £. ~ ~ + ~ . £. £. 33) Here ~ is a function of the two coordinates r and e, e = cp- az, a = 21r /L. f3 = 1 + a 2r 2, and Lis the pitch of the helix. § 2. Axisymmetric Flow across an Azimuthal Magnetic Field An important case of MHD flow is the flow across an azimuthal magnetic field with velocity v = v 11 and field H = HfP. The equations describing flow of this type can be found from Eqs.
37) must be solved together for p and ~; the functions U(0 and B(O and the equations of the boundary surfaces must be given. If the field Hr,o is known we can write the current density j in terms of H r,o by means of . _ (rH In the first approximationp we can replace p in Eq. 115) by p0 (z). Equating the flow velocity v (z) to the critical velocity c s. we find the equation of the surface at which the critical velocity is reached: v 2 (z) (l+y) ---(2--y) B 2 r 2 p0 (z)=(y-l)U. 116) Since v(z) is an increasing function, while p 0(z) is a decreasing function, the critical surface is located at progressively larger values of z as the radius increases (Fig. lOa). B. In the other limiting case, (3 « 1, a cold plasma, in the zeroth approximation in the parameter (3, the density is p 0(r, z) = (U - v 2/2) jr2B 2• We now write c~ as c~= U- v2 /2 + ('\'- 2)W (p).