What units will this equation have?

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SUMMARY

The discussion focuses on the units associated with the diffusion equation, specifically ∂u/∂t = D (∂²u/∂x²). It establishes that the variable u represents the density of a substance, such as heat or smell, and that diffusivities (D) have dimensions of L²T⁻¹. The participants confirm that both sides of the equation maintain consistent dimensions of density over time (T⁻¹), leading to the conclusion that the units for the diffusion coefficient D are indeed m²/s.

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RyanUSF
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Let u be density of something (heat, smell, etc.). Assume the something only diffuses; there’s no convection or ballistic transport. Let’s work in one spatial dimension (x). Then u satisfies the diffusion equation,

∂u/∂t = D ((∂^2)u/(dx^2))
 
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would separating the equation such that D = (∂u/∂t)((∂x^2)/(d^2u)) be of any help? Would the spatial dimensions cancel eacher out just giving dx^2/dt so units of m^2/s?
 
RyanUSF said:
Let u be density of something (heat, smell, etc.). Assume the something only diffuses; there’s no convection or ballistic transport. Let’s work in one spatial dimension (x). Then u satisfies the diffusion equation,

∂u/∂t = D ((∂^2)u/(dx^2))

Both sides have dimensions of (\mbox{something density}) T^{-1}. Diffusivities always have dimensions of L^2 T^{-1}.
 

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