Fuel injector performance has a direct effect on the combustion efficiency, pollutant emissions and combustion instability of internal combustion engines. Liquid fuels are normally accelerated into an elevated combustion-chamber temperature to maintain a desirable homogeneous combustible mixture - liquid vapour and air. The accelerated jet breakup may be induced by cavitation, turbulent, hydrodynamic and aerodynamic forces interactions and variation in fluid properties. The absolute majority of studies have been devoted to the extensive study on some of the effects that cause jet instability and breakup, while others are still at their infant study. In particular, relatively few researchers have studied the combined effects of jet acceleration and non-isothermal condition on jet instability and breakup, despite its practical relevance in liquid fuel spray and combustion. A new analytical hydrodynamic instability and breakup model, which captured both jet acceleration and non-isothermal condition, for liquid jet is presented. The analytical model investigates the impact of two important phenomena on liquid jet instability and breakup; jet acceleration and non-isothermal conditions. These effects are naturally difficult to address by both experimental and CFD investigations. The new model analysis combined both hydrodynamic model and heat transfer equations, coupled through the surface-tension gradient, and is a novel address of this conjugated problem. Continued modelling in that area may benefit the development of the next generation of liquid fuel injectors and combustors, as simplified models have a quantitative agreement with experimental results.