42   const unsigned int N_index = chem_mixture.species_index(
"N");
 
   43   const unsigned int N2_index = chem_mixture.species_index(
"N2");
 
   45   const double gamma = 0.03;
 
   51   const double w_s = 0.2;
 
   53   const double rho = 1.0e-3;
 
   54   const double rho_s = rho*w_s;
 
   56   const double T = 620.1;
 
   57   const double R_N = chem_mixture.R( chem_mixture.species_index(
"N") );
 
   59   const double R = 30.1;
 
   63   const double drho_dws = -rho*rho_s/R;
 
   69   const double omega_dot_N = wall_N.omega_dot( rho_s, T );
 
   71   const double domega_dot_dT_N = wall_N.domega_dot_dT( rho_s, T );
 
   73   const double domega_dot_dws_N = wall_N.domega_dot_dws( rho_s, w_s, T, R );
 
   75   const double tol = 1.0e-15;
 
   79     double rel_error = std::fabs( (omega_dot_N - omega_dot_exact)/omega_dot_exact );
 
   83         std::cerr << std::setprecision(16) << std::scientific
 
   84                   << 
"Mismatch in omega_dot_N!" << std::endl
 
   85                   << 
"omega_dot_N = " << omega_dot_N << std::endl
 
   86                   << 
"omega_dot_exact = " << omega_dot_exact << std::endl
 
   87                   << 
"rel error = " << rel_error << std::endl;
 
   95     double rel_error = std::fabs( (domega_dot_dT_N - domega_dot_dT_exact)/domega_dot_dT_exact );
 
   99         std::cerr << std::setprecision(16) << std::scientific
 
  100                   << 
"Mismatch in domega_dot_dT_N!" << std::endl
 
  101                   << 
"domega_dot_dT_N = " << domega_dot_dT_N << std::endl
 
  102                   << 
"domega_dot_dT_exact = " << domega_dot_dT_exact << std::endl
 
  103                   << 
"rel error = " << rel_error << std::endl;
 
  111     double rel_error = std::fabs( (domega_dot_dws_N - domega_dot_dws_exact)/domega_dot_dws_exact );
 
  113     if( rel_error > tol )
 
  115         std::cerr << std::setprecision(16) << std::scientific
 
  116                   << 
"Mismatch in domega_dot_dws_N!" << std::endl
 
  117                   << 
"domega_dot_dws_N = " << domega_dot_dws_N << std::endl
 
  118                   << 
"domega_dot_dws_exact = " << domega_dot_dws_exact << std::endl
 
  119                   << 
"rel error = " << rel_error << std::endl;
 
const libMesh::Real two_pi