What units will this equation have?

[RyanUSF]

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))

 

[RyanUSF]

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?

 

[pasmith]

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 $L^2 T^{-1}$.