Recent experiments have revealed that a vascular structure called the vertebral venous plexus transfers pressure increases from the abdominal cavity to the central nervous system (CNS), and that such pressure increases cause movement of the brain. This movement, clearly observable through in vivo experiments, also generates a complex cerebrospinal fluid (CSF) flow and a concurrent exchange/mixing of CSF and interstitial fluid (IF, the fluid saturating brain parenchyma). The implications of this mechanical coupling between the abdomen and the CSF are significant and make it essential to quantify the fluid flow activated by brain motion. As direct flow measurements in the living brain are extremely challenging, mathematical models for this quantification become very appealing. We use Hamilton’s principle to model fluid flow through brain tissue and to quantify fluid exchange between brain parenchyma and the subarachnoid space (SAS; the outermost CSF-filled compartment in the CNS). We model both brain and fluid-filled compartments using mixture-theory-based poroelasticity. Our model accounts for large deformations and inertia. The constituent fluid and solid skeleton are assumed to be incompressible. The variational framework of Hamilton’s principle is also used to create a corresponding finite element scheme whereby flow velocity and pore pressure can be discontinuous across the brain/SAS interface, while some control of these discontinuities is built into the interfacial constitutive behavior. In addition to presenting our modeling and numerical framework, we will also show numerical results comparing brain surface displacements to those experimentally measured, as well as companion estimates of CSF/IF fluid exchange. Our flow estimates indicate that brain motion can indeed drive physiologically-relevant fluid exchange.