text: Consider a cylindrical duct with axis along  direction with radius  through which fuel is fed from the bottom and the tube mouth is located at . Oxidizer is fed along the same axis, but in the concentric tube of radius  outside the fuel tube. Let the mass fraction in the fuel tube be  and the mass fraction of the oxygen in the outside duct be . Fuel and oxygen mixing occurs in the region . The following assumptions were made in the analysis:
  * The average velocity is parallel to axis ( direction) of the ducts, 
  *The mass flux in the axial direction is constant, 
  *Axial diffusion is negligible compared to the transverse/radial diffusion
  * The flame occurs infinitely fast (Burke–Schumann limit), therefore flame appears as a reaction sheet across which properties of flow changes
  * Effects of gravity has been neglected
  Consider a one-step irreversible Arrhenius law, , where  is the mass of oxygen required to burn unit mass of fuel and  is the amount of heat released per unit mass of fuel burned. If  is the mass of fuel burned per unit volume per unit time and introducing the non-dimensional fuel and mass fraction and the Stoichiometry parameter,
  the governing equations for fuel and oxidizer mass fraction reduce to
  where Lewis number of both species is assumed to be unity and  is assumed to be constant, where  is the thermal diffusivity. The boundary conditions for the problem are
  The equation can be linearly combined to eliminate the non-linear reaction term  and solve for the new variable
  where  is known as the mixture fraction. The mixture fraction takes the value of unity in the fuel stream and zero in the oxidizer stream and it is a scalar field which is not affected by the reaction. The equation satisfied by  is
  (If the Lewis numbers of fuel and oxidizer are not equal to unity, then the equation satisfied by  is nonlinear as follows from Shvab–Zeldovich–Liñán formulation). Introducing the following coordinate transformation
  reduces the equation to
  The corresponding boundary conditions become
  The equation can be solved by separation of variables
  where  and  are the Bessel function of the first kind and  is the nth root of  Solution can also be obtained for the planar ducts instead of the axisymmetric ducts discussed here.