MAE 130C - Midterm review questions Viscous Stress Navier Stokes

Define a Newtonian fluid in terms of the relationship between stress and deformation

A Newtonian fluid is that where viscous shear stress is directly proportional to the rate of shear strain/deformation at all times

t= du/dy (mu)

Why is the "Rate of Rotation" tensor not included in the stress-strain relationship for a fluid?

In a Newtonian fluid, the rotation rate cannot generate viscous stresses (as only the shear rate generates viscous stresses)

Rigid rotation does not deform fluids meaning there are no stresses

What physical property of the fluid does μ (dynamic viscosity) represent at the molecular level?

The inertial friction caused by the transfer of momentum and cohesive forces between moving molecules

molecular momentum transfer, higher mu = higher resistance to motion

In the incompressible Navier-Stokes equation, what happens to the ∇ · V term?

It goes to zero as the divergence of velocity represents continuity/mass conservation, and we assume this to be true

represents true incompressible flow, there is no volume change, density is constant

Explain the physical role of the convective acceleration term (V · ∇)V.

This represents the acceleration a fluid particle experiences due to a change in its position within a non-uniform velocity field, rather than a change of velocity over time at a fixed point

acceleration of fluid particle due to the movement through velocity field

In the x-momentum equation, what does the term μ∇2u physically represent?

Forces due to shear stresses (namely viscous force per unit volume)

viscous diffusion of momentum

What physical condition at a solid boundary is fundamentally responsible for the generation of a boundary layer, and what property of the fluid enables this?

The no-slip condition; viscosity or viscous shear stresses

wall slows the fluid, viscosity slows upward flow which creates the boundary layer

Why can we often neglect the ∂2u/∂x2 term in a boundary layer compared to ∂2u/∂y2?

By OoM analysis, ∂2u/∂x2~U/l^2 and ∂2u/∂y2~U/δ^2. Because Lis much larger than delta, we say that U/δ^2 is much larger and thus neglect U/l^2 or ∂2u/∂x2

What is the physical interpretation of the "Stokes Hypothesis" for the bulk viscosity of a fluid?

The bulk viscosity of a Newtonian fluid is zero, meaning that the mechanical pressure (average normal stress during the flow) equals thermodynamic equilibrium pressure

How does the "No-Slip Condition" generate vorticity at a wall?

The wall forces the fluid immediately adjacent to it match its velocity, creating a large velocity profile and thus shearing stress

no slip matches velocity →large velocity profile →shearing stress created vorticity

In a fully developed duct flow, why does the convective acceleration vanish?

The velocity profile does not change along the direction of the flow