1. Mathematical Modelling of Fluids Flashcards
What is fluid dynamics?
-the study of the motion of liquids, gases and plasmas which have no large scale structure and can be deformed to an unlimited extent (in contrast with solids)
Continuum Hypothesis
Individual Molecules
- one cubic centimetre of water contains of the order 10^23 molecules typically of size lm ≈ 10^(-10) m
- they are in continuous motion, even in still water
- it is impossible to calculate the motion (velocity and position) of individual particles
Continuum Hypothesis
Fluid Particle
- as it is impossible to track the motion of every individual molecule we instead concentrate on ‘bulk properties’ of fluids
- i.e. to lool at the motion, mass, etc. of a ‘blob’ of fluid called a fluid particle
Continuum Hypothesis
Pipe Example - Description
-in a section of a pipe of radius a≈1cm, we calculate the average velocity of all molecules in a test volume 𝛿V of length d
Mesoscopic
Definition
-between macroscopic and atomic/molecular scales
Continuum Hypothesis
Pipe Example - Scales
d = length of test volume i.e. 𝛿V = d^3 a = pipe radius lm = molecular length -if d>>lm , then 𝛿V contains many molecules and the fluctuations due to individual molecules are averaged out -if d<<a></a>
Continuum Hypothesis
-molecular detail can be smoothed out by assigning the velocity at a point P to be the average velocity in a fluid element 𝛿V centred in P
-thus we can define the velocity field |u(|x,t) as a smooth function, differentiable and integrable
-similarly:
ρ(|x,t) = mass in 𝛿V/𝛿V
is the local density of mass
Velocity Field
Definition
-the fluid velocity is defined, within the continuum hypothesis, as the vector field |u(|x,t) function of space and time
Shear Flow
Definition
-flow between two parallel plates when one is moved relative to the other with a constant velocity U
Shear Flow
Velocity Field
|u = U*y/d ^ex
Stagnation-Point Flow
Definition
-flow with a point at which |u=0
Stagnation-Point Flow
Velocity Field
|u = (Ex , -Ey , 0)
- where E is a constant
- the point |x=0 where |u=0 is a stagnation point
Vortex Flow
Definition
-flow in rotation about a central point
Vortex Flow
Velocity Field
|u = ( y/(x²+y²) , -x/(x²+y²),0)
or in cylindrical polars;
|u = ( sinθ/r , -cosθ , 0)
-the flow is singular at |x=0 : || |u || ~ 1/r which tends to infinity as r->0
What are the three methods of visualising flow?
1) Pathlines / Particle Paths
2) Streamlines
3) Streaklines
Pathlines / Particle Path
Description
- drop a tracer particle into the flow
- the particle is imaginary, it is so small/light that it has no effect on the flow
- follow the path it takes through the flow as time passes
Pathlines / Particle Path
Equations
d|x/dt = |u(|x,t)
|x=|xo at t=to
Streamlines
Description
- streamlines are everywhere tangent to the local velocity at some instant
- it is like taking a snapshot at some time t and drawing curves tangent to the flow at every point
- if the flow changes with time t then you will get a different snapshot at each time t you choose to look at the streamlines BUT t is treated as a constant since for any given snapshot t does not change
Streamlines
Equations
d|x/ds = |u(|x,t)
|x=|xo at t=to
-where s parameterises the curve:
|x = (x(s;t) , y(s;t) , z(s;t))
Streaklines
Description
- like releasing a continuous stream of dye from a point xo at time to
- the dye will move around and define a curve, the position of fluid elements that passed through |xo at some time to prior to the current time t
Strealines
Equations
-streakline |x(to,t) satisfies:
∂|x/∂t = |u(|x,t)
|x=|xo at t=to
Pathlines, Streamlines, Streaklines and Steady Flow
- for a steady flow there is no explicit time dependence, ∂|u/∂t=0
- therefore the pathlines, streamlines and streaklines are identical
Pathlines, Streamlines, Streaklines and Unsteady Flow
- an unsteady flow does have explicit time dependence
- this means that the equations for pathlines, streamlines and streaklines will all be different
Time Derivatives
Eulerian Description of Fluids
-at a fixed point in space
-let |f(|x,t) be some quantity of interest
-the partial derivative
(∂f/∂t)|x = ∂/∂t |f(|x,t)
-is the rate of change of |f at fixed position |x
-e.g. ∂|f/∂t = 0 means that f remains constant at that point in the field
BUT ∂|u/∂t is NOT the acceleration of a fluid particle
Time Derivatives
Lagrangian Description of Fluids
-following fluid particles
-let |f(|x,t) be some quantity of interest
-the convective derivative:
D/Dt |f(|x,t)
-is the rate of change of |f when |x is the position of a fluid particle (i.e. |x travels with the fluid along particle paths
-e.g. D|f/Dt = 0 implies that |f remains constant along particle paths
Convective Derivative
-also known as the Lagrangian derivative or material derivative
-defined as:
Df/Dt = d/dt f(|x(t),t)
= ∂f/∂t + ∂f/∂xdx/dt + ∂f/∂ydy/dt + ∂f/∂zdz/dt
= ∂f/∂t + u∂f/∂x + v∂f/∂y + z∂f/∂z
-so:
D|f/Dt = ∂|f/∂t + (|u . |∇)|f
-where |u = (u,v,w)
-or as an operator
D/Dt = ∂/∂t + (|u . |∇)
Time Derivatives
Lagrangian - Steady Flow
-for a steady flow ∂|u/∂t=0
-in steady flows, the rate of change of f following a fluid particle becomes:
Df/Dt = (|u . |∇) f
-so in steady flows,
(|u . |∇) f=0
implies that f is constant along streamlines, though that constant may be different for each streamline
-also since particle paths and streamlines are identical for time-independent flows:
Df/Dt = ||u|| (^e . ∇) f
= ||u|| df/ds
-where ^e is a unit vector in the s direction, along the streamline