If a full wave rectification of an A.C. current flows through a closed circuit in series a coil of wire wound around a solenoidal core, what happens to the current in the overall circuit?
What is the overall circuit? We would need more details to answer this.
the current in the overall circuit would be like the curve of square of sine function
Assuming you've got an AC source (wall plug) hooked up to a bridge rectifier (with or without transformer), the output will be a rectified sinewave. If you apply a rectified sine to an inductor, coil, solenoid, etc, current will flow in one direction only, and the inductor will attempt to smooth out the drops in voltage, and oppose any change in current. This is similar to placing a capacitor in parallel, and basically filters out any AC harmonics, leaving a cleaner DC output. Bottom line: current gets filtered to produce cleaner DC.