•Sinusoidal cavity for variable wavelengths and solid volume fraction using Cu-water nanofluid is developed.•Four different solid volume fractions (ϕ = 0.02, 0.04, 0.06, 0.08), three different wavelengths (λ = 2, 3, 4), three different Rayleigh numbers (Ra=103,104, 105) have been considered.•When no magnetic field is applied, the kinetic energy is 1.89 times higher, and the Nusselt number is 7% higher for ϕ = 0.02 than for ϕ = 0.08 for λ = 3.•For no magnetic field intensity, the total entropy generation is 18% higher in ϕ = 0.02 than ϕ = 0.08 for λ = 2.•The highest values of entropy generation are found for the case of Ra=105, while the lowest values are found for the case of Ra=103. The use of thermofluids in a cavity under magnetohydrodynamics (MHD) with the natural convective flow has been extensively researched, with numerous attempts made by researchers. However, none have examined the role of nanofluid concentrations on a sinusoidal cavity with variable wavelengths at two walls using Cu-water nanofluid with MHD. Therefore, the present study evaluated nanofluid concentration impact in a sinusoidal cavity for three different wavelengths under magnetic field conditions and Rayleigh numbers using Cu-water nanofluid. Four different solid volume fractions (ϕ = 0.02, 0.04, 0.06, 0.08), three different wavelengths (λ = 2, 3, 4), three different Rayleigh numbers (Ra=103,104, 105) have been considered for the study using the finite element method. The LBB-stable element is used to approximate velocities, pressures, and temperatures. The numerical results have been validated from the past study. The variation of kinetic energy, average Nusselt number, viscous, thermal, magnetic, and total entropy generation for various wavelengths has been shown for varying the concentration of nanofluid and Hartmann number. Additionally, the effect of Rayleigh numbers on the entropy generations for three different wavelengths has been presented. The results revealed that the heat transfer rate, kinetic energy, and entropy generation rise with the decreased value of volume concentrations for most of the cases. It is also noticed that these parameters also negatively effect due to the magnetic field. Therefore, with the rise of the Hartmann number, the fluid flow, thermal parameters, and entropy generation decrease. Moreover, Ra has a notable role in entropy generation. Entropy generations have been found higher for Ra=105, and the lowest is found for the case of Ra=103.