Streamlines, Streaklines and Particle Paths Flashcards
Define a streamline
An imaginary line in the flow which is everywhere tangential to the fluid velocity u.
There is only one streamline through any point, streamlines cannot cross.
Three possible equations from steamlines
dx/u = dy/v
dx/u = dz/w
dy/v = dz/w
.
What is a stagnation point?
A point such that u = 0.
Define a particle path
The path of a fluid particle, in general not the same as a stream line.
Equations of particle paths
dx/dt = u
dy/dt = v
dz/dt = w
What is a streakline?
Where pollutant is inserted into a flow continuously from some fixed point, and this produces a trail of the pollutant leading off downstream.
What is steady flow?
The flow is the same for all time. All three visualizations of flow are the same for steady flow.
What is the material derivative?
δρ/δt = (ρ(r + δr, t + δt) − ρ(r, t)) / δt
Dρ/Dt = ∂ρ/∂t + u ∂ρ/∂x + v ∂ρ/∂y + w ∂ρ/∂z
Dρ/Dt = ∂ρ/∂t + u.∇ρ
Equation for any fluid property
DA/Dt = ∂A/∂t + u.∇A
Acceleration of a fluid particle
Du/Dt = ∂u/∂t + (u.∇)u
Mass conservation integral
M = triple integral (V) ρ dV, V is the region of space inside a net.
Continuity equation
Dρ/Dt + ρ∇.u = 0
Incompressibility condition
∇.u = 0
Wave vector u is said to be solenoidal.
What is a uniform fluid?
No density variations.
How is summation implied in the summation convention?
By the occurrences of the same suffix.
Define an external force
Long range forces that act on every particle of matter, examples are gravity, coriolis force, electric and magnetic fields.
Also referred to as “body forces”.
Define an internal force
Short-range effect; one fluid particle feels a force from another fluid particle where they come into contact.
What is hydrostatic equilibrium?
Forces are in balance, fluid is at rest.
Total force acting on the fluid particle is zero.
triple integral (V) ρ F dV − double closed integral (S) p n dS = 0
ρ F − ∇p = 0
What is a free surface?
Liquid in an open container, measure z upwards from the surface of the liquid.
p0 is the value of p just below the free surface.
Divergence theorem in suffix notation
triple integral (V) ∂A/∂xq dV = double closed integral (S) A nˆq dS
Stokes’ theorem in suffix notation
double integral (S)εq β α ∂A/∂xαnˆβ dS
= closed integral (C) A dxq
Total rate of flow of momentum into V across S
- double integral (S) u.n ρ u dS
Euler’s Equation of Motion
ρ Du/Dt = ρ F − ∇p
Pressure for liquids and gases
ρ = constant, for liquids
p = kρ^γ for gases.
Boundary conditions for two media occupying regions 1 and 2.
i) p1 = p2 on S
ii) n.u1 = n.u2 on S.
