## Fluid Dynamics and the Navier-Stokes Equations

### Magnetohydrodynamics (MHD) and the Formation of Jet(s) Nobody has seen a black hole up till 2014. Nevertheless, schematic diagram such as the one shown in figure 23 invariably presents a system composed with a central object (the black hole), an accretion disk, and a pair of jets moving along twisted magnetic field lines. This picture of the black hole is based theoretically on the combination of three branches of physics - Fluid Dynamics, Electromagnetism, and Gravitation.

The effect of gravity from the black hole is characterized by the escape velocity vesc(r) = (2GM/r)1/2, where M is the mass of the black hole, and r = (R2 + Z2)1/2 provides a link between the spherical and cylindrical coordinates (see upper left corner insert in Figure 23).

With the assumptions of infinite conductivity (for the plasma in the system), isotropic pressure, local charge neutrality, non-relativistic inter-particle speeds), the suite of MHD equations are :

#### Figure 23 Black Hole Schematic [view large image] These set of equations can be solved only through numerical computation. In general, the magnetic field has only the z component Bz initially. When the accretion disk is set into rotation, the field lines is wrapped around the rotation axis creating the helical field in the z direction. As ionized matter cannot cross field lines, it is obliged to follow the lines and thus form a collimated jet (Figure 23). The following is a scenario of the numerical computation according to a letter in Nature with the title "A magnetic switch that determines the speed of astrophysical jets". • A dense accretion disk (shown in dark) orbits a compact object of mass M.
• Acceleration takes place mostly at R0 ~ 3.6 Rs, where Rs = 2GM/c2 is the Schwartzschild radius.
• The dense disk is sandwiched by two layers of tenuous corona (shown stippled) whose temperature is hotter than the disk but still colder than that in the halo. The coronal density is assumed to be 10 times the asymptotic halo density, and the disk density is essentially infinite in the simulations.
• #### Figure 24 MHD Computation [view large image]

• Open magnetic field lines, making an angle = 30o with the rotation axis, protrude from the disk and through the corona initially. The lines are dragged along by rotation creating a B component.
• • The corona is replenished continually from the disk. The coronal flows are decidedly jet-like. The flow speed is a strong function of the ratio = vA/vesc. The jet speed is low for < 1, i.e., for weak magnetic field (relative to gravity). The speed rises sharply when > 1. The ratio is therefore referred to as "magnetic switch" to turn on a pair of jets from a state with no jet.
• Figure 25 shows the jet speed as a function of coronal magnetic field strength for both stellar disk (lower and left axes) and black hole's (upper and right axes). The vc for the proto-star is the velocity at rc = 2x1012 cm
• #### Figure 25 Jet Switch ~ 30 Rsun, while = (1 - v2/c2)-1/2 is the Lorentz factor and = 11(vA/c)2 for the black hole. The two dash lines are taken from the "Relativistic Wind Theory" which excludes the effect of gravity.

The simulation demonstrates that the jet(s) provides a mechanism to remove the angular momentum of the system. Otherwise, the formation of galaxy and star would not be possible as the in-falling material reaches the so called centrifugal barrier - where nothing can move toward the rotation axis. This kind of model can also be applied to proto-stellar object before the onset of thermonuclear reaction.

By comparing other studies on the same subject but taken into account the relativistic effects, the above-mentioned scenario is still valid with some modifications such as restricting vA < c, replacing v by v, changing vesc to jet[2(1+vA2/c2)GM/r]1/2 etc. The quantitative difference is in the range of a factor of two or so. The qualitative conclusion remains the same.

N.B.   MHD and specifically the Alfven velocity vA was developed by the Swedish physicist Hannes Alfven (1908-1995) almost single-handedly. The Alfven velocity is associated with the transverse wave motion of the magnetic field lines. It seems that the speed of the Alfven wave vA can be greater than the velocity of light for large B or small . However, it can be shown that if the displacement current term is retained in the Ampere's Law, the Alfven wave velocity becomes uA = vA/[1+(vA/c)2]1/2, which is reduced to uA ~ c for vA >> c, while uA ~ vA for vA << c. His work was not taken seriously until such wave was detected in the lab in the late 1950s. Eventually, he was awarded the Nobel prize for the efforts in 1970.

.