Lesson 3, part 3: linking stress to rate of strain

Understanding Stress Balance in Fluid Mechanics

Overview of Stress and Strain Tensors

• The lecture introduces the stress tensor and rate of strain tensor, emphasizing their symmetry properties.

• It discusses contributions to overall stress in a fluid, including static stresses like hydrostatic pressure and dynamic stresses related to strain rates.

Rules for Assembling a Stress Balance

• Three essential rules for forming a stress balance are outlined:

• Rule 1: Dimensional consistency is crucial; all terms must have compatible units.

• Rule 2: Rank consistency is necessary; tensors of different ranks cannot be added together.

• Rule 3: Symmetry must be maintained; any addition to the stress tensor should also be symmetric.

Contributions to Total Stress

• The total stress (Sigma) combines contributions from hydrostatic pressure (P) and shear stress tensor (tau).

• Pressure is treated as a scalar that needs manipulation to fit into the tensor framework, requiring it to be expressed correctly for summation with other tensors.

Manipulating Pressure for Tensor Consistency

• To integrate pressure into the total stress equation, it is represented as negative P multiplied by an identity tensor, ensuring proper rank and dimensionality.

• The identity tensor conveys that pressure acts equally in all normal directions without contributing shear stresses.

Final Formulation of Total Stress

• The final expression for total stress includes both hydrostatic pressure and shear components while adhering to all three rules established earlier.

• This formulation leads into force and momentum balances, ultimately connecting back to the Navier-Stokes equations.

Key Takeaways on Stress Summation Rules

• Recap of the three critical rules:

• All terms must be dimensionally consistent.

• Terms must maintain rank consistency when summed.