Momentum Balance

1. Momentum Balance Steven A. Jones BIEN 501/CMEN 513 Monday, March 19, 2007 Louisiana Tech University Ruston, LA 71272

2. Momentum Equation Note 1: The left hand side is the material derivative and will ultimately give rise to the nonlinear, convective acceleration terms. Note 2: The stresses themselves do not necessarily cause a change in momentum (acceleration of the fluid). Spatial variations of these stresses do. Note 3: Remember that this equation is just a restatement of Newton’s second law, with “ma” on the left hand side and “F” on the right hand side. Note 4: Remember that u is a vector and t is a 9-component tensor. The tensor has 3 rows (one per surface) and 3 columns (one per each velocity direction). Louisiana Tech University Ruston, LA 71272

3. Differential Form – Conservation of Momentum Select one component of velocity dx dy We can get a differential form if we convert the last integral to a volume integral. The divergence theorem says: dz Louisiana Tech University Ruston, LA 71272

4. Differential Form – Conservation of Momentum dx dy As with conservation of mass: dz Combine the two volume integrals: Louisiana Tech University Ruston, LA 71272

5. Differential Form – Conservation of Momentum In The left hand term (time derivative of momentum) is equal to the external force (by Newton’s 2nd law), so: Also, is the Eulerian derivative (a fixed location in space), so: Louisiana Tech University Ruston, LA 71272

6. Differential Form for Momentum In fluid mechanics, we write Newton’s second law backwards, by convention, so instead of: We write: Louisiana Tech University Ruston, LA 71272

7. External Forces on the Differential Cube There are three types of external forces on the differential cube: • Viscous forces • Pressure forces (always normal) • Body forces (e.g. gravity) In the momentum equation, these are expressed as force per unit volume. Louisiana Tech University Ruston, LA 71272

8. Newtonian Fluid Louisiana Tech University Ruston, LA 71272

9. Newtonian Fluid Typically one of the coordinates will be aligned with the boundary surface. E.g., for Poiseuille flow, the boundary is parallel to both the z and q directions. Since velocity is zero at the wall, there is no change with respect to the z and q directions. Louisiana Tech University Ruston, LA 71272

10. Poiseuille Flow Louisiana Tech University Ruston, LA 71272

11. Stokes Flow Not because uris 0, but because ur is 0 for all q. Louisiana Tech University Ruston, LA 71272