1.26. Let u(r, y,z,t) be a mass density in all of 3D space governed by the diffu

Question

1.26. Let u(r, y,z,t) be a mass density in all of 3D space governed by the diffusion flux rule. Assume there are no sources. Let R be a cube test volume with unit length sides and let E be another test volume that is outside R but inside a cube that completely encloses R with sides of two unit lengths. Suppose at time t there is a positive net flux of mass out of R. We know that diffusion flux goes from high density to low density, Determine the following true, false, or indeterminate and explain your answer: l. The mass inside R must be greater than the mass inside E but outside R at time t.

Explanation / Answer

1. False : The density of R should be more than density of S for flux out of R. Since volume of S is more than that of R, it's density can be less than that of R for masses less, greater or equal to S, depending on conditions. Hence it is not always true.

2. True. According to the diffusion flux eqn, the flux -> 0 for t-> infinity. Thus it's density and hence mass has to diminish with time.

3. False. As mass of R is also enclosed inside S, as mass of R decreases with time, so does mass of S.

4. True. From diffusion flux eqn.

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