In this paper, the transport of flow and heat transfer through parallel plates arranged horizontally against each other is studied. The mechanics of fluid transport and heat transfer are formulated utilizing systems of the coupled higher-order numerical model. This governing transport model is investigated by applying the variation of the parameter's method. Result obtained from the analytical study is reported graphically. It is observed from the generated result that the velocity profile and thermal profile drop by increasing the squeeze parameter. The drop inflow is due to limitations in velocity as plates are close to each other. Also, thermal transfer due to flow pattern causes decreasing boundary layer thickness at the thermal layer, consequently drop in thermal profile. The analytical obtained result from this study is compared with the study in literature for simplified cases, this shows good agreement. The obtained results may therefore provide useful insight to practical applications including food processing, lubrication, and polymer processing industries amongst other relevant applications.