| Summary: | This paper addresses the solution of the incompressible second-grade fluid models. Fundamental qualitative properties of the solution are primarily studied for proving the adequacy of the physical interpretations of the proposed model. We use the Liouville-Caputo fractional derivative with its generalized version that gives more comprehensive physical results in the analysis and investigations. In this work, both the <inline-formula><math display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Laplace homotopy transform method (<inline-formula><math display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-LHTM) and the heat balance integral method (HBIM) are successfully combined to solve the fractional incompressible second-grade fluid differential equations. Numerical simulations and their physical interpretations of the mentioned incompressible second-grade fluid model are ensured to illustrate the main findings. It is also proposed that one can recognize the differences in physical analysis of diffusions such as ballistic diffusion, super diffusion, and subdiffusion cases by considering the impact of the orders <inline-formula><math display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mi>φ</mi></semantics></math></inline-formula>.
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