or, using j=σ(E-uB) 4 Combining(2)and (4) d (E The flow velocity u evolves along x according to the momentum equation(ignoring pressure forces neglect for now Substitute(2)into(6) u wh d B2 μodx Integrate mu+wH B WH B2-B2 Putting this in Equation(5), B(B:-B2 If we approximate the conductivity o as a constant this can be integrated as B(B2-B2) This integral can actually be calculated analytically but the resulting expression is not very transparent. It is more useful to examine its behavior qualitatively. The 16.522, Space Propulsion Lecture 22 Prof. Manuel martinez-Sanchez Page 2 of 816.522, Space Propulsion Lecture 22 Prof. Manuel Martinez-Sanchez Page 2 of 8 or, using z yx V E E = , B -B , u = u H ≡ ≡ j = E - uB σ ( ) (4) Combining (2) and (4), 0 ( ) dB = - E - uB dx σµ (5) The flow velocity u evolves along x according to the momentum equation (ignoring pressure forces) ( )x du dP m + A = j B A = jBwH dx dx × i G JG (6) neglect for now Substitute (2) into (6): 2 0 0 du 1 dB wH d B m = - B wH = - dx dx dx 2 ⎛ ⎞ ⎜ ⎟ µ µ ⎝ ⎠ i (7) Integrate: 2 2 0 0 0 0 B B mu + wH = mu + wH 2 2 µ µ i i neglect 2 2 0 0 wH B -B u = 2 m µ i (8) Putting this in Equation (5), ( ) 2 2 0 0 0 dB wH =- E- B B -B dx 2 m ⎡ ⎤ σµ ⎢ ⎥ ⎣ ⎦ µ i (9) If we approximate the conductivity σ as a constant, this can be integrated as B0 0 2 2 B 0 0 dB x = wH E - B(B - B ) 2 m σµ µ ∫ i (10) This integral can actually be calculated analytically, but the resulting expression is not very transparent. It is more useful to examine its behavior qualitatively. The