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17 2.19afh 2.19e Classification

17 2.20 Classification

17 2.21ab 2.21c Classification: assume u=u(x,y,z)

18 2.24 2.26 Canonical form

18 2.25 Canonical form

18 2.22cef 2.22d Characteristics

18 2.27ab 2.27d 2D Canonical form

18 2.28dejo 2.28nml 2D Canonical form

98 7.20 7.19 Unsteady heat conduction in a bar

98 7.22 7.21 Unsteady heat conduction in a bar

98 7.24 7.26 Unidirectional viscous flow

98 7.26 Unidirectional viscous flow

99 7.27 7.28 Acoustics in a pipe

99 7.31 7.29 Vibrations of a string

99 7.35 7.36 Acoustics above a plate

99 7.37 7.37 Steady heat conduction in a plate

99 7.38 7.39 Steady heat conduction in a disk Solve the problem below

1. Solve the unsteady flow in a circular pipe of unit radius. The nonzero axial velocity component satisfies the heat equation

   

   The boundary condition on the pipe wall is

   

   For initial condition, assume that is a delta function at r=0.5 and , in other words that

   

   for any function . Sketch the velocity profile u(r,0,t) at for a small time t. Determine and draw the shape of the velocity profile (i.e. u(r,0,t)/u(0,0,t)) for a large time t. Does u become independent of for large times?

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