Transcription of Chapter 7 The Diffusion Equation - uni-muenster.de
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Chapter 7 The Diffusion Equation - uni-muenster.de
Chapter 7 The Diffusion EquationThe Diffusion Equation is a partial differential Equation which describes density fluc-tuations in a material undergoing Diffusion . The Equation can be written as: u(r,t) t= D(u(r,t),r) u(r,t) ,( )whereu(r,t)is the density of the diffusing material at locationr= (x,y,z)and (u(r,t),r)denotes the collectivediffusion coefficientfor densityuat the Diffusion coefficient doesn t depend on the density, ,Dis constant, thenEq. ( ) reduces to the following linear Equation : u(r,t) t=D 2u(r,t).( ) Equation ( ) is also calledthe heat equationand also describes the distribution ofa heat in a given region over ( ) can be derived in a straightforward way from thecontinuity equa -tion, which states that a change in density in any part of the system is due to inflowand outflow of material into and out of that part of the , no mate-rial is created or destroyed: u t+ =0,where is the flux of the diffusing material.
Equation (7.2) is also called the heat equation and also describes the distribution of a heat in a given region over time. Equation (7.2) can be derived in a straightforward way from the continuity equa-tion, which states that a change in density in any part of the system is due to inflow
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