Possibility of stable thin-shell around wormholes within the string cloud and quintessential field via the van der Waals and polytropic EOS in General relativity

In this work, we present the Einstein field equations (EFE) in the framework of a modified matter source and thus find out the solutions for the Wormhole. To obtain characteristic solutions set, we employ two kinds of equations of states (EOS), i.e., the van der Waals and polytropic EOS to the EFE. We adopt embedding class as a general technique for the system and discuss several physical attributes of the stellar system under embedded wormhole solutions. Detailed discussion about the matter contents of the thin-shell developed around the obtained wormhole solutions in the background of Schwarzschild BH surrounded by cloud and quintessence-type fluid distribution by using a well-known cut and paste approach. We have also performed stability analysis using the linearized radial perturbation method for the van der Waals EOS and polytropic EOS. Several interesting points have evolved from the entire investigation along with the features of the thin shell around the wormhole.