The kinetic theory of adsorption of polymers under shear flow near a plane wall leads to the

Question:

The kinetic theory of adsorption of polymers under shear flow near a plane wall leads to the following expression for the steady state concentration C(z) in the wallnormal direction (z-axis) 6 12 Ld 00-[- + ) - ()]] 1/1/4/ C(z) = exp 2s Z

where L_d is a characteristic length scale of polymer migration under shear flow, ε_s is the strength of polymer–wall attraction, and σ_s is a constant. L_d is some function of z, but it is only known numerically. In the far-field limit, you can use treat Ld as a constant, say L. Then the integral inside the square brackets can be calculated analytically, and you obtain the much simpler expression $6 12 L + [ - + ^ { ( 9 ) - ( 9 ) } ] Z C(z) = exp$

We want to calculate the amount of adsorbed Γ using the relation = [ [ C () - 1] dz Zi r =

where zi is the distance from the wall set by the polymer size and z_c is some distance from the wall.

Write aMATLAB program to calculate an approximation to the integral in Eq (8.4.15) using the multiple trapezoidal rule, with ε_s = 4.0, σ_s = 2.0, L = 0.5, z_i = 0.5, and z_c = 10.0. Plot the value you obtain with the number of segments, n = 1, . . . , 60.

Your program should automatically generate the plot. If your program is right, your plot will show oscillatory behavior. Answer in words the possible cause for this. Usually, the trapezoidal rule is not accurate for the above type of function. Explain in words why it is so.