2. Mathematical Modelling of Fluid Flow

Question 1

Define a Fluid

Answer 1

A fluid is a continuous continuum substance that cannot withstand sheer stress without continually deforming

Question 2

What is sheer stress?

Answer 2

A stress that is applied parallel or tangential to a face of a material

Question 3

What is the continuum approximation?

Answer 3

The assumption that matter is infinitely divisible

Question 4

What is the mean free path?

Answer 4

The length a molecule of the medium travels between collisions with other molecules

Question 5

When is the continnum approxmation valid?

Answer 5

When the mean free path (l) is much smaller than the length scale of flow (L)

Question 6

What is the Knudsen number?

Answer 6

l/L
- Continuum approximation is valid when this is much less than 1

Question 7

Describe when the continuum approximation fails (don’t say less than 1)

Answer 7

1. The length scale over which the flow varies is too small
2. The mean free path of the molecules is too large

Question 8

What are the 5 main properties of fluids?

Answer 8

Density, Pressure, Dynamic Viscosity, Kinematic Viscosity, Surf. Tension

Question 9

When is a fluid incompressible?

Answer 9

When the density of the fluid does not change in the flow
- Density is a const.

Question 10

Define the density of a fluid

Answer 10

Mass of the fluid per unit volume

Question 11

Define the pressure of a fluid

Answer 11

The force a fluid exerts on a wall per unit area

Question 12

Define the dynamic viscosity of a fluid

Answer 12

The resistance of a fluid to shear
- Proportionality constant between the sheer stress σ and sheer rate γ dot

Question 13

Define the kinematic viscosity of a fluid

Answer 13

The dynamic viscosity divided by the fluid density

Question 14

Define the surface tension of a fluid

Answer 14

Tension in the liquid interface expressed as a force per unit length

Question 15

When can we call a fluid ideal/inviscid?

Answer 15

When there is zero viscosity

Question 16

Describe the Lagrangian description of fluid flow

Answer 16

A label is applied to an infinitesimal volume of fluid
- These volumes move with the velocity of the fluid
- A finite Lagrangian volume evolves as a collection of infinitesimal Lagrangian volumes

Question 17

Give an analogy which represents Lagrangian fluid flow

Answer 17

Someone on a boat flowing down a river
- Attention is on material fluid not location

Question 18

Describe Eulerian flow

Answer 18

Fluid particles are represented by their current description
- Fluid particle occupies location x and moves with a speed u at time t

Question 19

Give an analogy of Eulerian fluid flow

Answer 19

Standing on a riverbank and watching it flow
- Attention is on regions of space rather than material

Question 20

State the notation and components for the material derivative

Answer 20

D/Dt
- d/dt is the local rate of change at a fixed Eulerian pos. x
- (u . ∇) is the advection/convection term

Question 21

What are the 4 methods of fluid visualisation?

Answer 21

Particle Paths, Streaklines, Timelines, Streamlines

Question 22

What are particle paths and describe the experimental technique

Answer 22

Paths followed by fluid particles
- Dye a small volume of fluid and follow it. Points visited = pathline
- Use long exposures/overlap video frames

Question 23

What are streaklines and describe the experimental technique

Answer 23

The locus of Lagrangian fluid particles that passed through a fixed Eulerian point
- Continuolusly introduce dye form a point in the fluid. Streak formed is the streakline

Question 24

What are timelines and describe the experimental technique

Answer 24

The locus of Lagrangian points that coincided with a given curve at source reference time t_0 in the past
- Release a puff of dye in a curve and follow the direction

Question 25

How do we model the deformation of finite volumes?

Answer 25

It is a collection of the infinitesimal deforming volumes

Question 26

What is rate of change in relation to sources and sinks?

Answer 26

Rate of change = sources - sinks

Question 27

How do we solve the dimensional inconsistency of the surface source term?

Answer 27

Divide the conservation by L^2 and take the limit L -> 0
- For any infinitesimal volume, the total surface contribution must vanish

Question 28

Describe why we use Cauchy’s tetrahedron argument

Answer 28

Can calculate the surface source term for the tetrahedron
- Tetrahedron is infinitely small so h^B does not depend on x, but the normal dependece remains (depends on the orientation)

Question 29

Describe what the notation of T_ij represents

Answer 29

T_ij is the Cauchy Stress tensor
- It describes the stress on the surface normal to j along i

Question 30

What is the rate of change of the angular momentum equal to?

Answer 30

The net torque applied to the body

Question 31

Describe what is meant by an inviscid fluid (qualitatively)

Answer 31

There is no friction between fluid elements
- Viscosity does not play a role

Question 32

Explain how the pressure acts for an inviscid fluid element

Answer 32

It acts normal inward
- Pressure is a force that acts perpendicular to a surface

Question 33

What do we assume about a Newtonian fluid?

Answer 33

It is incompressible and has a constant viscosity

Question 34

What does it mean if a tensor is isotropic?

Answer 34

The components do not depend on the coordinate system used to represent the tensor

Question 35

Describe the no slip boundary condition, and what type of fluids it applies to

Answer 35

The fluid velocity at the boundary is equal to the boundary veloctity
- Applies to viscous fluids

Question 36

Describe the free slip boundary condition, and what type of fluids it applies to

Answer 36

The normal of the fluid velocity at the boundary is equal to the normal of the boundary veloctity (no penetration)
- Applies to inviscid fluids

Question 37

Describe the kinematic boundary condition, and what type of fluids it applies to

Answer 37

The free surface follows the fluid
- Applies when there are moving material boundaries