An exact closed-form technique is employed to calculate the cutoff wavenumbers of a conducting circular waveguide loaded with two circular dielectric cylinders. An analytical expression of the hybrid modes is derived in terms of cylindrical coordinates of the geometry. The addition theorem of Bessel functions is used to transfer the electromagnetic fields from one coordinate system to another to apply the boundary conditions. A MATLAB (The MathWorks, Natick, Massachusetts, USA) code is developed to solve the system of equations and produce the cutoff wavenumbers for different geometrical parameters. The present problem can be extended to model a circular waveguide loaded with multiple dielectric cylinders that resembles a petroleum-carrying pipeline. A professional High Frequency Structure Simulator (HFSS; ANSYS, Inc., Canonsburg, PA, USA) is used to solve this model and to determine the ratio of the multi-phase fluid.