A population of micro-organisms lives in a thin surface layer

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow

Mathematical Methods for Physics, Ch The integral theorems: Gauss, Stokes and Green

Problem

A population of micro-organisms lives in a thin surface layer, of depth hh, on a still pond, so that motion across the layer's top and bottom is negligible. Let n(x,y,t)n(x,y,t) be the number of organisms per unit area at the point (x,y)(x,y) of the surface, and let J(x,y,t)=(Jx,Jy)\mathbf{J}(x,y,t) = (J_x,J_y) be the number of organisms crossing unit length of any line on the surface per second, in the direction of J\mathbf{J}. Each organism divides at a constant rate rr (units s−1\mathrm{s^{-1}}), so a patch with no organisms crossing its edge would grow its population at the rate rr times the number present. For any fixed region AA of the surface, of boundary curve ∂A\partial A, the global population balance reads ddt∫An dA=r∫An dA−∮∂AJ⋅n^ dℓ,\frac{\mathrm{d}}{\mathrm{d}t}\int_A n\,\mathrm{d}A = r\int_A n\,\mathrm{d}A - \oint_{\partial A}\mathbf{J}\cdot\hat{\mathbf{n}}\,\mathrm{d}\ell, where n^\hat{\mathbf{n}} is the outward normal to ∂A\partial A lying in the surface and dℓ\mathrm{d}\ell is arc length along ∂A\partial A. ((a)) Using only the three-dimensional divergence theorem, show that ∮∂AJ⋅n^ dℓ=∫A(∂Jx∂x+∂Jy∂y)dA.\oint_{\partial A}\mathbf{J}\cdot\hat{\mathbf{n}}\,\mathrm{d}\ell = \int_A\left(\frac{\partial J_x}{\partial x}+\frac{\partial J_y}{\partial y}\right)\mathrm{d}A. ((b)) Convert the global population balance into a local differential equation for nn and J\mathbf{J}.

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