The current work seeks to examine numerical heat transfer by using a complicated channel with a trapezoid shape hanging in the channel. This channel demonstrates two-dimensional laminar flow, forced convective flow, and incompressible flow. To explore the behavior of heat transfer in complex channels, several parameters, such as the constant Prandtl number (Pr = 6.9), volume fraction ( $\mathrm{\phi }$) equal to (0.02 to 0.04), Cauchy number (Ca) equal to (10⁻⁴ to 10⁻⁸), and Reynolds number equal to (60 to 160) were utilized. At the complex channel, different elastic walls are used in different locations, with case A being devoid of an elastic wall, cases B and C each having three elastic walls before and after the trapezoid shape, respectively, and case D having six elastic walls. The geometry of a complicated channel with varying L2/H2 and B/H2 ratios is investigated. The trouble was solved using the FEM with the ALE technique. The results showed that the best case with an elastic wall is reached for B/H2 = 0.8 and L2/H2 = 3. When compared to the channel without a flexible wall in case A, the highest reading for Nusselt was recorded at case C with a percentage of 34.5 percent, followed by case B (31.4 percent) and then case D (21.5 percent). It also has the highest Nusselt number reading at Ca = 10⁻⁴ and Re = 160, or about 6.4 when compared to Ca = 10⁻⁵ and Ca = 10⁻⁸. In case A, ∆P increases as the Re grows; however, in cases B and C, the ∆P reduces as the Re increases, but in case D, the ∆P increases with increasing Re.