Mach-Zehnder interferometer formed by liquid-filled linear defect waveguides in a two-dimensional phononic crystal is numerically realized for sensing low concentrations of an analyte. The waveguides in the square phononic crystal of void cylinders in steel, as well as their T branches and sharp bends are utilized to construct interferometer arms. Sensing low concentrations of ethanol on the order of 0.1% in a binary mixture with water is achieved by replacing the contents of a number of waveguide core cells on one arm of the interferometer with the analyte. Computations are carried out through the finite-element method in an approach that takes the solid-liquid interaction at the waveguide core cells into account. Band analyses reveal linear variation of the central frequency of the transmission band within a band gap for ethanol concentrations up to 3.0%. Phase difference due to the imbalance of the sample and reference arms of the interferometer also varies linearly with ethanol concentration, leading in turn to a cosine variation of the Fourier component of the temporal interferometer response at the central input-pulse frequency. The induced phase difference in the investigated configuration becomes a -0.78 pi and -0.65 pi per percent increase of ethanol concentration as calculated from the band-structure and transient data, respectively. This is confirmed by transient finite-element simulations where totally destructive interference occurs for a concentration of approximately 1.5%. The proposed scheme, which can easily be adopted to other binary mixtures, offers a compact implementation requiring small amounts of analyte.