Thermodynamic analysis for Non-linear system (Van-der-Waals EOS) with viscous cosmology

The following paper is motivated by the recent works of Kremer [Gen Relativ Gravit 36(6):1423–1432, 2004; Phys Rev D 68(12):123507, 2003], Vardiashvili (Inflationary constraints on the van Der Waals equation of state, arXiv:1701.00748, 2017), Jantsch (Int J Mod Phys D 25(03):1650031, 2016), Capozziello (Phys Lett A 299(5–6):494–498, 2002) on Van-Der-Waals EOS cosmology. The main aim of this paper is to analyze the thermodynamics of a Non-linear system which in this case is Van-Der-Waals fluid EOS (Capozziello et al., Quintessence without scalar fields, arXiv:astro-ph/0303041, 2003). We have investigated the Van-Der-Waals fluid system with the generalized EOS as $$p=w\left( \rho ,t \right) \rho +f\left( \rho \right) -3\eta \left( H,t \right) H$$ p = w ρ , t ρ + f ρ - 3 η H , t H (Brevik et al., Int J Geom Methods Mod Phys 15(09):1850150, 2018). The third term signifies viscosity which has been considered as an external parameter that only modifies pressure but not the density of the liquid. The $$w(\rho ,t)$$ w ( ρ , t ) and $$f(\rho )$$ f ( ρ ) are the two functions of energy density and time that are different for the 3 types of Vander Waal models namely one parameter model, two parameters model and three parameters model (Ivanov and Prodanov, Eur Phys J C 79(2):118, 2019; Elizalde and Khurshudyan, Int J Mod Phys D 27(04):1850037, 2018). The value of EOS parameter ($$w_{EOS})$$ w EOS ) (Capozziello et al., Quintessence without scalar fields, arXiv:astro-ph/0303041, 2003; Obukhov and Timoshkin, Russ Phys J 60(10):1705–1711, 2018) will showdifferent values for different models. We have studied the changes in the parameters for different cosmic phases [Kremer, Phys Rev D 68(12):123507, 2003; Capozziello et al., Phys Lett A 299(5–6):494–498, 2002; Capozziello et al., Quintessence without scalar fields, arXiv:astro-ph/0303041, 2003]. We have also studied the thermodynamics and the stability conditions for the three models in viscous condition [Obukhov and Timoshkin, Russ Phys J 60(10):1705–1711, 2018; Panigrahi and Chatterjee, Gen Relativ Gravit 49(3):35, 2017; Panigrahi and Chatterjee, J Cosmol Astropart Phys 2016(05):052, 2016; Chakraborty et al., Evolution of FRW Universe in Variable Modified Chaplygin Gas Model, arXiv:1906.12185, 2019]. We have discussed the importance of viscosity (Brevik and Grøn, Astrophys Space Sci 347(2):399–404, 2013) in explaining accelerating universe with negative pressure (Panigrahi and Chatterjee, Gen Relativ Gravit 49(3):35, 2017).Finally, we have resolved the finite time future singularity problems [Brevik et al., The effect of thermal radiation on singularities in the Dark Universe, arXiv:2103.08430, 2021; Odintsov and Oikonomou, Phys Rev D 98(2):024013, 2018; Odintsov and Oikonomou, Int J Mod Phys D 26(08):1750085, 2017; Frampton et al., Phys Rev D 85(8):083001, 2012; Frampton et al., Phys Lett B 708(1–2):204–211, 2012; Frampton et al., Phys Rev D 84(6):063003, 2011] and discussed the thermodynamics energy conditions [Visser and Barcelo, Energy conditions and their cosmological implications. In: Cosmo-99, pp 98–112, 2000; Chattopadhyay et al., Eur Phys J C 74(9):1–13, 2014; Arora et al., Phys. Dark Universe 31:100790, 2021; Sharma and Pradhan, Int J Geom Methods Mod Phys 15(01):1850014, 2018; Sahoo et al., AstronomischeNachrichten 342(1–2):89–95, 2021; Yadav et al., Mod Phys Lett A 34(19):1950145, 2019; Sharma et al., Int J Geom Methods Mod Phys 17(07):2050111, 2020, Moraes and Sahoo, Eur Phys J C 77(7):1–8, 2017; Hulke et al., New Astron 77:101357, 2020; Singla et al., Gravit Cosmol 26(2):144–152, 2020; Sharif et al., Eur Phys J Plus 128(10):1–11, 2013] with those models.