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Quiz1 - Quiz 1 1 Consider a concentration of a chemical c...

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Unformatted text preview: Quiz 1 1. Consider a concentration of a chemical c ( t,x ). We denote by φ ( t,x ) the flux of the chemical (amount of chemical per unit surface flowing to the right per unit time). 1.1. Write the conservation of chemical in a rod ( a,b ) of cross-sectional area A . 1.2. Assuming that the flux is proportional to the gradient of concentration (with a negative multiplicative constant), find a P.D.E. for c ( t,x ). 2. Consider the same problem in two space dimensions, where the chemical concentration is c ( t,x,y ) and the (vectorial) flux φ ( t,x,y ). To fix notation, you are asked to write the conservation of the chemical on a domain A with boundary ∂A and outward unit vector at the boundary n ( x,y ) for ( x,y ) ∈ ∂A . Solutions. 1.1. The total amount of chemical in the rod is given by Z b a c ( t,y ) Ady. Conservation of the amount of chemical means that the change of that amount per unit time (rate of change) equals the amount of chemical entering minus the amount of chemical exiting the rod per unit time, i.e.,amount of chemical exiting the rod per unit time, i....
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Quiz1 - Quiz 1 1 Consider a concentration of a chemical c...

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