kn[n_,R_]:=(k/.FindRoot[k Sin[k]==R Cos[k]/3,{k,(n-1)Pi,(n-1/2)Pi}][[1]])

Integrate[Cos[k z],{z,0,1}]/Integrate[Cos[k z]^2,{z,0,1}]//FullSimplify

A[R_,m_,z_,t_]:=Sum[((2 Sin[kn[n,R]])/(kn[n,R]+Cos[kn[n,R]] Sin[kn[n,R]]))Cos[kn[n,R]z]Exp[-3kn[n,R]^2 t/R],{n,1,m}]

pl[Re_]:=Plot[Evaluate[Join[{1},Table[Evaluate[A[Re,20,z,t]],{t,0.5,3,0.5}]]],{z,0,1},PlotLabel->R==Re,AxesLabel->{"z","A"},PlotLegends->Join[{t==0},Table[t,{t,0.5,3,0.5}]],PlotRange->{0,1}]

pl[1.0]

pl[0.1]

(* When R is small, A is approximately spatially uniform.*)

pl[100]

(* When R is large, there is a boundary layer near z=1. *)