1. Mathematical Modelling of Fluids

Question 1

What is fluid dynamics?

Answer 1

-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)

Question 2

Continuum Hypothesis

Individual Molecules

Answer 2

• 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

Question 3

Continuum Hypothesis

Fluid Particle

Answer 3

• 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

Question 4

Continuum Hypothesis

Pipe Example - Description

Answer 4

-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

Question 5

Mesoscopic

Definition

Answer 5

-between macroscopic and atomic/molecular scales

Question 6

Continuum Hypothesis

Pipe Example - Scales

Answer 6

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>

Question 7

Continuum Hypothesis

Answer 7

-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

Question 8

Velocity Field

Definition

Answer 8

-the fluid velocity is defined, within the continuum hypothesis, as the vector field |u(|x,t) function of space and time

Question 9

Shear Flow

Definition

Answer 9

-flow between two parallel plates when one is moved relative to the other with a constant velocity U

Question 10

Shear Flow

Velocity Field

Answer 10

|u = U*y/d ^ex

Question 11

Stagnation-Point Flow

Definition

Answer 11

-flow with a point at which |u=0

Question 12

Stagnation-Point Flow

Velocity Field

Answer 12

|u = (Ex , -Ey , 0)

• where E is a constant
• the point |x=0 where |u=0 is a stagnation point

Question 13

Vortex Flow

Definition

Answer 13

-flow in rotation about a central point

Question 14

Vortex Flow

Velocity Field

Answer 14

|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

Question 15

What are the three methods of visualising flow?

Answer 15

1) Pathlines / Particle Paths
2) Streamlines
3) Streaklines

Question 16

Pathlines / Particle Path

Description

Answer 16

• 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

Question 17

Pathlines / Particle Path

Equations

Answer 17

d|x/dt = |u(|x,t)

|x=|xo at t=to

Question 18

Streamlines

Description

Answer 18

• 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

Question 19

Streamlines

Equations

Answer 19

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))

Question 20

Streaklines

Description

Answer 20

• 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

Question 21

Strealines

Equations

Answer 21

-streakline |x(to,t) satisfies:
∂|x/∂t = |u(|x,t)
|x=|xo at t=to

Question 22

Pathlines, Streamlines, Streaklines and Steady Flow

Answer 22

• for a steady flow there is no explicit time dependence, ∂|u/∂t=0
• therefore the pathlines, streamlines and streaklines are identical

Question 23

Pathlines, Streamlines, Streaklines and Unsteady Flow

Answer 23

• an unsteady flow does have explicit time dependence

- this means that the equations for pathlines, streamlines and streaklines will all be different

Question 24

Time Derivatives

Eulerian Description of Fluids

Answer 24

-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

Question 25

Time Derivatives

Lagrangian Description of Fluids

Answer 25

-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

Question 26

Convective Derivative

Answer 26

-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 . |∇)

Question 27

Time Derivatives

Lagrangian - Steady Flow

Answer 27

-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