92.450 / 550 Math Modeling Homework 10 Due August 3 Brush re. Let u(s ) where s = x c t be the (nondimensionalized) temperature at position x and time t of a brush re propagating to the right with speed c . Then k u + c u h u = Q , 0, s <0 s>0 (1) together with the matching conditions u (0) = T0 (2), u(0+) = T0 (3) and u (0) = u (0+) (4) where T0 is the ignition temperature of the brush. Here k is the diffusivity, h the heat transfer coefcient and Q the rate at which the re generates heat (all positive constants). You will nd it useful to know the general solution of k u + c u h u = C is u (s ) = Ae r 1 s + B e r 2 s C /h where A, B are arbitrary constants, r 1 = 1 2k c 2 + 4k h + c and r 2 = 1 2k c 2 + 4k h c . Note r1 , r2 > 0. (a) s < 0. Use C = Q in (1). Find the constants A, B so the solution is bounded and satises the initial condition (2). (b) s > 0. Now use C = 0 in (1). Find A, B so the solution is bounded and satises (3). (c) Finally apply the matching condition (4) to obtain the formula for T0 in terms of k , h,Q and c . (d) Plot the graph of T0 as a function of c . What is the maximum value of T0 for which a re can propagate? 92.550 students (extra credit for 92.450 students) Fire break. Suppose now the region ahead of the re s > 0 is devoid of fuel. The traveling wave will no longer propagate into this region but instead approaches a steady state solution U obtained by setting c = 0 in u Q Q e h /k x , x < 0 h 2h U (x ) = Q e h /k x , x >0 2h (e) Sketch the graph of U . Make sure to show T0 on the graph. ( f ) Find the largest value of x for which U (x ) T0 . This gives the minimum width of the break which will halt the progress of the brush re. 92.450 / 550 Math Modeling Homework 10 Due August 3 Brush re. Let u(s ) where s = x c t be the (nondimensionalized) temperature at position x and time t of a brush re propagating to the right with speed c . Then k u + c u h u = Q , 0, s <0 s>0 together with the matching conditions u (0) = T0 (1), u(0+) = T0 (2) and u (0) = u (0+) (3) where T0 is the ignition temperature of the brush. Here k is the diffusivity, h the heat transfer coefcient and Q the rate at which the re generates heat (all positive constants). You will nd it useful to know the general solution of k u + c u h u = C (C constant) is u (s ) = Ae r 1 s + B e r 2 s C /h where A, B are arbitrary constants, r 1 = 1 2k c 2 + 4k h + c and r 2 = (4) 1 2k c 2 + 4k h c . Note r1 , r2 > 0. (a) s < 0. Use C = Q in (4). Find the constants A, B so the solution is bounded and satises the initial condition (1). (b) s > 0. Now use C = 0 in (4). Find A, B so the solution is bounded and satises (2). (c) Finally apply the matching condition (3) to obtain the formula for T0 in terms of k , h,Q and c . (d) Plot the graph of T0 as a function of c . What is the maximum value of T0 for which a re can propagate? 92.550 students (extra credit for 92.450 students) Fire break. Suppose now the region ahead of the re s > 0 is devoid of fuel. The traveling wave will no longer propagate into this region but instead approaches a steady state solution U obtained by setting c = 0 in u Q Q e h /k x , x < 0 h 2h U (x ) = Q e h /k x , x >0 2h (e) Sketch the graph of U . Make sure to show T0 on the graph. ( f ) Find the largest value of x for which U (x ) T0 . This gives the minimum width of the break which will halt the progress of the brush re