Expanding complex expression and replacing variables in the result

Question

When I'm finished expanding my expression I would like the result to be expressed again in terms of the original variables (like α, S1, T1, T2). I've tried replace, but can't seem to have good results. Any suggestions?

c = ((1 + I) Pi)/D1;
τ = I τy;
u0 = ((1 - I) π τ)/(D1 *ρ0* f);
ζ[h_] := (π h)/D1;
α[h_] := (Cosh[ζ[h]]*
      Cos[ζ[h]])^2 + (Sinh[ζ[h]]*Sin[ζ[h]])^2;

S1[h_] := (Cosh[ζ[h]]*Cos[ζ[h]])/α[h];
T1[h_] := (Cosh[ζ[h]]*Sinh[ζ[h]])/α[h];
T2[h_] := (Cos[ζ[h]]*Sin[ζ[h]])/α[h];

vg[h_] := (
    2*π)/(ρ0*f*D1)*((1 - S1[h])/(T1[h] - T2[h])*τy) - 
   ug*(T1[h] + T2[h] - 2*π*h/D1)/(T1[h] - T2[h]);
ugc[h_] := ug + I vg[h];
uec[z_, h_] := 
  u0*Sinh[c*(z + h)]/Cosh[c*h] - ugc[h]*Cosh[c*z]/Cosh[c*h];
uc[z_, h_] := uec[z, h] + ugc[h];

expr = ComplexExpand[uc[z, h], TargetFunctions -> {Re, Im}]