A thin capillary tube of length contains still water

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Temperature, heat and the macroscopic behaviour of matter

Mathematical Methods for Physics, Ch The classical partial differential equations of physics and their boundary conditions

Problem

A thin capillary tube of length L=4.0×10−2 mL = 4.0\times10^{-2}\,\mathrm{m} contains still water. At time t=0t=0 a dye is released at one end and its concentration c(x,t)c(x,t) (moles per unit volume) is uniform across the tube's cross-section, varying only along the axial coordinate xx. Conservation of the dye's moles in a short slice between xx and x+Δxx+\Delta x gives, in the limit Δx→0\Delta x\to0, ∂c∂t=−∂J∂x,\frac{\partial c}{\partial t} = -\frac{\partial J}{\partial x}, where J(x,t)J(x,t) is the molar flux (moles per unit area per unit time) in the +x+x direction. Fick's first law, an experimental fit rather than an exact law, states J=−Ds ∂c∂x,J = -D_{\mathrm{s}}\,\frac{\partial c}{\partial x}, with DsD_{\mathrm{s}} the (constant) diffusion coefficient of the dye in water. ((a)) Combine the two stated relations to derive the partial differential equation obeyed by c(x,t)c(x,t). State which of the two relations used is exact and which is an empirical closure. ((b)) State the SI units of DsD_{\mathrm{s}}, and show that L2/DsL^2/D_{\mathrm{s}} is the only combination of LL and DsD_{\mathrm{s}} with the units of time. ((c)) For this dye, Ds=5.0×10−10 m2 s−1D_{\mathrm{s}} = 5.0\times10^{-10}\,\mathrm{m^2\,s^{-1}}. Evaluate L2/DsL^2/D_{\mathrm{s}} and give it in hours.

Browse the problem library