Physical model exact solutions improve crucial industrial production processes by giving operatorsand designers a greater grasp of how systems operate in practice. The current casestudy aims to find exact momentum and pressure variation solutions during the unsteadystatefiltration process to advance fluid purification. Lie symmetry analysis is used to transforma system of PDEs representing the flow and pressure variation into solvable ODEswithout changing the dynamics of the case study. The velocity (momentum) and pressuresolutions are then determined by integrating the obtained solvable ODEs. The effects ofphysical parameters resulting from the process dynamics are examined using the exact solutionsacquired to identify the parameter combination that results in the maximum permeatesoutflow. Graphical representations of the flow velocity and pressure are presented and analysedas well as the skin friction tabular. The results indicate that as time evolves, permeateoutflow decreases because internal momentum, work done, and pressure diminish over timedue to the clustering of particles. In addition, low wave speed, small porosity, more chamberspace, high injection rate, and stronger magnetic effects are ideal for minimising the effectsof no-slip condition at the bottom wall during operation.