Flexographic printing is widely used in the packaging field, but there are still some problems in the printing of flexographic ink on non-absorbent substrates, such as low precision and unstable quality. In this paper, the printing process of flexographic ink is simulated. The interaction of fluid flow, temperature change, and solid deformation in flexographic printing is studied systematically by using the method of fluid–solid thermal coupling for the first time. The process of ink channel formation under static extrusion and fluid–solid thermal coupling was analyzed. The influences of printing pressure, printing speed, ink layer thickness, and ink viscosity on the ink channel were explored. The results show that the printing speed increases and the temperature in the stamping area increases. The printing speed is nonlinear related to the ink flow channel, the influence on the channel is slow at a low speed, the channel increases sharply at a medium and high speed, and tends to be stable at a high speed. When the printing speed is 200 m/min, the ink temperature in the stamping area is 1.5 °C higher than that at the entrance. With an increase in printing pressure, the ink flow channel width showed a trend of decreasing first and then stabilizing, and the pressure was about 0.4 MPa, showing a small fluctuation; the greater the pressure, the higher the temperature of the ink, which will change the performance of the ink and plate, causing adverse effects on the printing belt. The channel width showed obvious nonlinear characteristics with an increase and decrease in ink thickness. When the ink thickness is 30 μm, the deformation of the plate reaches the maximum, and the width of the ink circulation channel is correspondingly the widest. The change in ink viscosity has little influence on the stability of the ink’s internal flow rate and temperature field. The research results provide theoretical support for the transfer of ink printing from gravure to flexo printing.