[問題] 四道PDE的應用題

看板Physics作者victor0410 (zero)時間19年前 (2007/05/01 01:00)推噓5(5推 0噓 11→)

留言16則, 2人參與討論串1/1

恩以下四題其實是PDE課本的習題不過都是牽扯到物理所以PO在這裡發問我相信對這個板的強者大大應該是輕而易舉吧^^" 就麻煩板上大大指教了謝謝 m(_ _)m 1.A flexible chain of length l is hanging from one end x=0 but oscillates horizontally.Let the x axis point downward and the u axis point to the right. Assume that the force of gravity at each point of the chain equals the weight of the part of the chain below the point and is directed tangentially along the chain.Assume the oscillations are small.Find the PDE satisfied by the chain. αu αu ANS: ----=αg[(l-x)----]/αx αt αx 2.Suppose that some particles which are suspended in a liquid medium would be pulled down at the constant velocity V>0 by gravity in the absence of diffusion.Taking account of the diffusion,find the equation for the concentration of particles.Assume homogeneity in the horizontal directions x and y.Let the z axis point upwards. αu α^2u αu ANS: ----=k ----- + V ---- αt αz^2 αz 3.Derive the equation of one-dimensional diffusion in a medium that is moving along the x axis to the right at constant speed V. αu α^2u αu ANS: ----=k ----- - V ---- αt αx^2 αx 4.(a)Show that the temperature of a metal rod,insulated at the end x=0, satisfies the boundary condition αu/αx = 0. (b)Do the same for the diffusion of gas along a tube that is closed off at the end x=0.(Use the Fick's law.) (c)Show that the three-dimensional version of (a)(insulated solid) or (b) (impermeable container)leads to the boundary condition αu/αn = 0. -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 59.126.209.22

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