The brain, suspended in cerebrospinal fluid (CSF) and not attached to skeletal muscles, moves within the skull. The origin and physiological impact of this movement are yet to be understood, but we suspect that it may substantially contribute to CSF circulation and clearing of toxic metabolites from the central nervous system (CNS). Here we investigate how mechanical stimuli transmitted directly to the CNS promote fluid exchange between brain and the subarachnoid space (SAS). A mixture-theory-based poroelasticity model is adopted to represent both brain and SAS, each with its own constitutive properties. The model's numerical implementation is through an arbitrary Lagrangian–Eulerian finite element method (FEM). Both pressure and fluid velocity can be discontinuous across the brain/SAS interface. We simulated how the CNS moves in response to mechanical stimuli provided by the vertebral venous plexus (VVP), a network of veins that can communicate pressure changes from the thoracic cavity directly to the dura of the spinal cord. While our geometry is highly simplified, our results are encouragingly in line with experiments. Among the numerical challenges we encountered we discuss the enforcement of the constituents' incompressibility condition and the enforcement of the jump conditions at the brain/SAS interface. Here, after reviewing our modeling choices, we will first present our numerical implementation, in which we used integration by parts for the mass balance equation but not for the full momentum equation. We used second order Lagrange polynomials for the displacement and velocity fields, and first order for the pressure. Again, velocity and pressure can be discontinuos at the brain/SAS interface. Finally, we will present results indicating that the VVP-CNS interaction can induce CNS motions promoting appreciable brain/SAS fluid exchange. Some pertinent experimental evidence will also be discussed.