In this article, thermal–hydraulic performance and entropy generation (EG) characteristics for pressure-driven flow in a wavy channel with linearly varying amplitude (LVA) at the entrance region are computationally investigated. The computational simulations have been conducted for a wide range of Reynolds number 5 ≤ Re ≤ 1000 and normalized entrance length (EL) of LVA 0 ≤ EL ≤ 25.5. The results reveal that the flow field and heat transfer rate for the wavy channel with varying amplitude are remarkably different from those for a wavy channel (WC) with uniform amplitude, and the characteristics can be modulated by varying EL. The reversal of flow takes place in the wavy passages beyond a threshold value of Re, and the number of recirculating zones and the strength of the flow reversal strongly depend on EL. The average Nusselt number for the present WC is more than that of the plane channel (PC) after a critical value of Re only and at Re = 1000, the enhancements in average Nusselt number as compared to the plane channel are 6.91%, 20.67%, 26.37%, and 36.54%, for EL = 25.5, 11.5, 5.5, and 0, respectively. The combined influences of the augmentation in the average Nusselt number and the frictional pressure drop are presented in terms of performance factor (PF), which consistently decreases with the increase in Re for all non-zero EL, and the decrement is steeper for lower Re values. The average total entropy generation (EG) for WC is lower than PC at higher Re values, and the maximum percentage decrease in average total EG for WC compared to PC is achieved for EL = 11.5 at Re = 1000.