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at 10:30 am. Under suitable conditions, the rate of oxygen metabolism by vascular tissue is very nearly teel and and metabolism by cardiovascular tissue with respect to the oxygen concentration. Consider the metabolism of oxygen by a long cylindrical strand of cardiac tissue with radius R. Treat the system as pseudo-binary, composed only of diffusing oxygen ("A") and stationary tissue ("B"). Oxygen diffuses through the tissue with a diffusion coefficient DAB The outside of the tissue (r-R) is exposed to air, where the equilibrium concentration of oxygen within the tissue is CAO- The tissue metabolizes the oxygen everywhere in the tissue according to a zeroth-order homogeneous reaction with rate R, -k. Because of the zeroth-order reaction, part of the central tissue may be starved of oxygen at steady-state; i.e. within Oer<R, the oxygen concentration is zero, where RR is the critical radius at which the oxygen concentration is zero. Assume constant physical properties. Postulate CA CA(r) only. Assume that the mole fraction of oxygen is small (r, <<1) and thus any convective correction to mass transport is negligible. The aim of this problem is to calculate the concentration profile of oxygen within the tissue and determine under what conditions a starved, oxygen- free core exists. (6) Using principles of conservation of mass, derive the differential equation that governs the concentration of oxygen (c) within the cylindrical cardiovascular tissue. (ii) Non-dimensionalize the governing differential equation in (i) using appropriate scales, letting I and be the dimensionless concentration and r-coordinate, respectively. A dimensionless parameter emerges-label it a. What is its physical meaning? Solve for I in terms of two unknown constants of integration. (iii) (iv) What is the boundary condition at the outside of the tissue (r= R)? Non-dimensionalize it. Use this condition to solve for a constant of integration. (v) What is the boundary condition at the interface between oxygenated and un-oxygenated tissue (i.e., where the tissue is starved of oxygen, r=R)? Non-dimensionalize it, labeling the non-dimensional critical radius. Use this condition to solve for a constant of integration. Write down the final expression for I(E). Box it. It should contain (vi) (vii) At steady state, there is no mass transport of oxygen across the interface between oxygenated and un-oxygenated tissue (r= R), i.e., the diffusive mass flux is zero. Non-dimensionalize this additional boundary condition. (viii) (ix) A starved, oxygen-free core only exists only when a is above a certain threshold value. First, be qualitative: explain why this makes physical sense based on your definition of a; i.e., why would you expect an oxygen-free core when a>>1 and why you would not expect one when a<<1? Examining how a scales with the different physical parameters may be helpful. Second, be qu ative: use the additional condition from (vii) as a constraint to derive this threshold value of a. Hint: enforce the condition of zero diffusive flux, then gather all terms containing on one side of the equation and all other terms not containing on the other side of the equation. Notice that the equality can only be satisfied when a is above a critical threshold value.