A film of glycerine is sheared between two horizontal plates: the lower plate is fixed
PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Fluid statics and the continuum picture · Fluid statics and the continuum picture
Fluids and Continuum Physics, Ch The continuum hypothesis, pressure and the stress tensor
Problem
A film of glycerine is sheared between two horizontal plates: the lower plate is fixed, the upper plate slides in the +x direction, and y is the vertical. At a point in the film the stress tensor is σij=−1.00×10514014−1.00×105000−1.00×105Pa. Consider an imaginary cube of side 2.0 mm centred on that point, its faces normal to the coordinate axes. Recall the convention: t(n)=σn is the force per unit area exerted on the surface element by the material on the side into which the unit normal n points, and on each face of the cube n points outward.
((a)) Write the traction vector on the top face of the cube, whose outward normal is y^=(0,1,0).
((b)) Find the magnitude of the tangential (in-plane) force exerted on that face.
((c)) The upper plate is now brought to rest and the glycerine settles to rest at the same pressure. Write the traction vector on the top face then, and say in one sentence why every face of the cube now feels a traction along its own normal.