Lecture 3 Flashcards
(21 cards)
What is the fundamental assumption of unidirectional flow?
The velocity vector has only one nonzero component, typically in the z-direction.
This assumption simplifies the continuity equation:
ββ
u=0
meaning that velocity does not change in the flow direction.
How do the Navier-Stokes equations simplify for fully developed pipe flow?
Since there is no acceleration in the flow direction, the momentum equation reduces to:
ππ/ππ§=π(1/π π/ππ(πππ’/ππ))
which is a balance between pressure gradient and viscous forces.
What boundary conditions apply for pipe flow?
No-slip condition: π’=0 at the pipe wall.
Maximum velocity at the centerline: ππ’/ππ=0 at π=0.
What is the velocity profile for laminar, pressure-driven pipe flow?
The velocity follows a parabolic profile:
π’(π)=1/4π(ππ/ππ§)(π
^2βπ^2)
How does the volumetric flow rate depend on pipe radius and pressure gradient?
he Hagen-Poiseuille equation gives:
π=ππ
^4/8π (Ξπ/πΏ)
meaning that flow rate is proportional to the fourth power of the pipe radius.
What are the dimensions of viscosity, and why are they important?
Dynamic viscosity π has dimensions:
[π]=π/πΏπ
which relate force, length, velocity, and time, determining a fluidβs resistance to flow.
What does Buckingham Pi theorem state?
Given n variables and r fundamental dimensions, there are πβπ independent dimensionless groups. Helps in reducing the number of variables in a system.
What is the significance of the Reynolds number?
It is the ratio of inertial forces to viscous forces: π
π=πππΏ/π
Determines whether a flow is laminar (π
π<2000) or turbulent (π
π>4000).
How does pressure drop relate to flow rate in laminar pipe flow?
From the Hagen-Poiseuille equation: Ξπ/πΏ=8ππ/ππ ^4 showing that pressure drop increases with increasing viscosity and decreasing pipe radius.
What does vorticity represent in fluid mechanics?
Vorticity π is the local rotational motion of the fluid:
π=βΓπ’
It measures the tendency of a fluid to rotate at a given point.
How does vorticity change in a flow?
It evolves according to the vorticity transport equation:
βπ/βπ‘=πβ^2π indicating that vorticity spreads over time due to viscosity, just like heat diffusion.
What is the characteristic time scale for vorticity diffusion?
The diffusion time scale follows: π‘βΌπΏ^2/π where πΏ is the characteristic length scale and π is kinematic viscosity.
What is the Moody Diagram used for?
It helps determine the friction factor π for turbulent pipe flow, which is needed to compute pressure losses.
At what Reynolds number does flow transition from laminar to turbulent in a pipe?
Laminar:
π
e<2000
Turbulent: π
π>4000
The transition region occurs between these values.
How does friction factor relate to Reynolds number in laminar flow?
For laminar flow, the Darcy-Weisbach friction factor is:
π=64/π
π which shows that friction decreases as flow becomes more inertial.
What does Darcyβs law describe?
he movement of fluid through porous media, given by:
π=π/π Ξπ/πΏ where π is the permeability, which depends on the size and structure of the pores
How does permeability relate to the smallest characteristic dimension of a porous medium?
In general, π scales with the square of the smallest flow dimension: πβπ ^2
What is Stokesβ first problem?
It describes the diffusion of momentum near a wall when a plate suddenly starts moving.
The velocity solution involves the error function:
π’(π¦,π‘)=πerf(π¦/sqrt(4ππ‘))
What is the key time scale for boundary layer development?
The boundary layer grows with time as:
πΏβΌsqrt(ππ‘) where π is kinematic viscosity.
What is the main experimental criterion for the transition to turbulence?
A significant increase in pressure drop at a critical Reynolds number, typically around π πβ2000
What is the friction factor for turbulent flow?
It is determined from empirical correlations or the Moody diagram.
