, which is the ratio of reaction to diffusion coefficients.
The boundary conditions are
(a,t)=O and y(1,t)=1 for all time >0
Here a is the internal boundary, a parameter that corresponds in spherical geometries to the ratio of the radius of a support media to the total radius of the bioparticle, in cylindrical geometries to the radius of a cylinder without biofilm to the total radius of the cylinder with biofilm and in slab geometry to be the ratio of the inactive region of diffusion to the thickness of biofilm measured from the centre of the slab. This may vary but for our purposes we shall take it to be a constant.
The major part of this thesis is concerned with the solution to the steady state associated with problem (P).
Using the maximum principle and, methods of upper and lower solutions and standard topologocal results from non-linear analysis, existence , uniqueness and monotonicity results are obtained.
In particular it is shown that the steady state problem has a unique solution
for all values
,.,n,.,f C "\ ,-,.f -1-h'"' .f,-,,.,..,..,
1
above in the geometries slab, cylinder and sphere.
Liiv ,v1111
It is also shown that if F(0) = 0, F(y) 0, the unique solution of the steady state problem associated with (P) is strictly greater than zero. This is indeed true for nth order and Michaelis-Menten kinetics.
For the zero order case F(0) * 0 implies that our solution to the steady state problem associated with (P) could become negative. Having a negative concentration is not a physical reality and we impose a third boundary
condition that redefines a in terms of Thiele modulus,