Thispaperaddressesanonstationaryflowofheat-conductiveincompressibleNewtonianfluidwithtemperature-dependentviscositycoupledwithlinearheattransferwithadvectionandaviscousheatsourceterm,underNavier/Dirichletboundaryconditions.Thepartialregularityforthevelocityofthefluidisprovedforeachproperweaksolution,thatis,forsuchweaksolutionswhichsatisfysomelocalenergyestimatesinasimilarwaytothesuitableweaksolutionsoftheNavier-Stokessystem.Finally,westudythenatureofthesetofpointsinspaceandtimeuponwhichproperweaksolutionscouldbesingular.