Lecture 4 Flashcards
(25 cards)
What assumptions are made for pressure-driven flow in a circular pipe?
Incompressible, Newtonian fluid
No body forces
Constant viscosity
Flow is axisymmetric and steady
Velocity only varies in the radial direction
How is the velocity profile for steady pipe flow derived?
Start from the Navier-Stokes equation.
Assume steady flow and no variation in the axial direction except pressure.
Solve for velocity by integrating the equation.
Why does flow take time to establish from rest in a pipe?
Initially, pressure accelerates the fluid in the center.
Viscous effects at the wall gradually propagate inward.
Flow is fully developed once vorticity diffuses across the pipe.
What is the governing equation for transient pipe flow from rest?
The unsteady Navier-Stokes equation simplifies to a diffusion equation for velocity.
What is the characteristic time scale for velocity diffusion in a pipe?
t_c=R^2/ν, meaning flow reaches steady state over time proportional to pipe radius squared.
What is Stokes’ First Problem and what does it describe?
It describes how fluid starts moving when a plate is suddenly set in motion.
The problem models vorticity diffusion from the plate.
Why is there no steady-state solution for Stokes’ First Problem?
Because the velocity profile continually changes over time.
Unlike pipe flow, there is no physical boundary that would enforce a final state.
How does velocity propagate into the fluid over time?
It spreads as [sqrt(𝜈𝑡)], meaning that vorticity diffuses outward over time.
What is the solution to the velocity field in Stokes’ First Problem?
The solution involves the error function and shows how velocity transitions smoothly from the plate into the fluid.
How does shear stress on the plate change over time?
It decreases as 𝑡^−1/2, meaning the force required to keep the plate moving decreases over time.
What does vorticity represent in fluid mechanics?
Vorticity measures the local rotation of a fluid element.
It is defined as 𝜔=∇×𝑢
How does vorticity evolve in a viscous fluid?
It diffuses according to ∂𝜔/∂𝑡=𝜈∇^2𝜔.
Why is vorticity important in boundary layers?
It controls the rate at which momentum diffuses away from a moving boundary.
How is vorticity related to shear stress?
Shear stress at the wall is proportional to the vorticity near the surface.
Why do we use non-dimensionalization in fluid mechanics?
To identify key dimensionless parameters.
To simplify equations and reveal dominant effects.
What is the Reynolds number and what does it indicate?
Re=ρUL/μ
It determines whether flow is laminar or turbulent.
What is the Prandtl number and what does it represent?
Pr=ν/α, where 𝛼 is thermal diffusivity.
It compares momentum and heat diffusion rates.
How do we non-dimensionalize the equation of motion in a spring-mass system?
Introduce characteristic scales for length and time.
Express the system in terms of dimensionless variables.
What is Darcy’s Law and when is it used?
It models flow through porous media.
Relates pressure gradient to flow rate via permeability.
Why does pressure drop increase when flow becomes turbulent?
Because turbulent flow introduces additional resistance due to chaotic motion.
What is the Moody Diagram and what does it show?
It provides the friction factor for pipe flow as a function of Reynolds number and surface roughness.
What happens to the velocity profile when flow transitions to turbulence?
It becomes flatter in the center and has a sharper gradient near the walls.
What is a gravity current and how does it evolve over time?
A gravity current is a flow driven by density differences.
The radius increases as 𝑡^1/8 due to balance between pressure forces and viscosity.
How does surface tension affect small-scale fluid motion?
Surface tension dominates at small scales where capillary forces exceed inertia.
