PHYS 515: Plasma Physics & MHD Grade Assignments for 10/17/00 Note: assigments are due on Tuesday, Oct. 24. You can put them in my mailbox, or hand them in in class. 1. Shear viscosity: Consider a pipe with radius a, length L, and a pressure gradient Delta p between the two ends. The viscous force due to the shear in velocity that we derived is given by F_viscous = mu (grad)^2 v, where v is the velocity vector, and mu the shear viscosity coefficient. In the pipe the velocity is along the axis, i.e. v = v_z, and the velocity is zero at the boundary (r=a) and maximum (and finite) at the center r=0. There is axial symmetry. a) Write down the force-balance for a fluid (no magnetic field) with constant density in a stationary situation b) Solve for the velocity v_z as a function of radius c) Solve for the mass flux rate through the pipe d) How would you measure the shear viscosity in an experimental setup like that? 2. Consider, once more, a current sheet of the form B_z = B_0 tanh(y/d), with, to be specific, d = 1 km, and B_0 = 100 Gauss, values that one might think occur in the Solar corona. Assume n_e = 10^9 cm^-3. a) What is the drift velocity as a function of y? What is its maximum value (a number)? b) Let the resistivity be the Spitzer resistivity, e.g. as given in Choudhuri, eq. (13.27). The temperature is 10^6 K. The center of the current sheet is the location of the reversal of the plasma inflow in the y direction. What is the electric field strength in the center of the sheet, given Ohm's law (Choudhuri eq. 13.34), with all terms on the RHS ignored? c) Assume the electric field found in b) applies everywhere, i.e. it is constant. What is the equation for the inflow velocity, and what is it's maximum value? d) Is it justified to ignore the Hall term in Ohm's law? Why (not)?