2. Mathematical Modelling of Fluid Flow Flashcards
Define a Fluid
A fluid is a continuous continuum substance that cannot withstand sheer stress without continually deforming
What is sheer stress?
A stress that is applied parallel or tangential to a face of a material
What is the continuum approximation?
The assumption that matter is infinitely divisible
What is the mean free path?
The length a molecule of the medium travels between collisions with other molecules
When is the continnum approxmation valid?
When the mean free path (l) is much smaller than the length scale of flow (L)
What is the Knudsen number?
l/L
- Continuum approximation is valid when this is much less than 1
Describe when the continuum approximation fails (don’t say less than 1)
- The length scale over which the flow varies is too small
- The mean free path of the molecules is too large
What are the 5 main properties of fluids?
Density, Pressure, Dynamic Viscosity, Kinematic Viscosity, Surf. Tension
When is a fluid incompressible?
When the density of the fluid does not change in the flow
- Density is a const.
Define the density of a fluid
Mass of the fluid per unit volume
Define the pressure of a fluid
The force a fluid exerts on a wall per unit area
Define the dynamic viscosity of a fluid
The resistance of a fluid to shear
- Proportionality constant between the sheer stress σ and sheer rate γ dot
Define the kinematic viscosity of a fluid
The dynamic viscosity divided by the fluid density
Define the surface tension of a fluid
Tension in the liquid interface expressed as a force per unit length
When can we call a fluid ideal/inviscid?
When there is zero viscosity
Describe the Lagrangian description of fluid flow
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
Give an analogy which represents Lagrangian fluid flow
Someone on a boat flowing down a river
- Attention is on material fluid not location
Describe Eulerian flow
Fluid particles are represented by their current description
- Fluid particle occupies location x and moves with a speed u at time t
Give an analogy of Eulerian fluid flow
Standing on a riverbank and watching it flow
- Attention is on regions of space rather than material
State the notation and components for the material derivative
D/Dt
- d/dt is the local rate of change at a fixed Eulerian pos. x
- (u . ∇) is the advection/convection term
What are the 4 methods of fluid visualisation?
Particle Paths, Streaklines, Timelines, Streamlines
What are particle paths and describe the experimental technique
Paths followed by fluid particles
- Dye a small volume of fluid and follow it. Points visited = pathline
- Use long exposures/overlap video frames
What are streaklines and describe the experimental technique
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
What are timelines and describe the experimental technique
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
How do we model the deformation of finite volumes?
It is a collection of the infinitesimal deforming volumes
What is rate of change in relation to sources and sinks?
Rate of change = sources - sinks
How do we solve the dimensional inconsistency of the surface source term?
Divide the conservation by L^2 and take the limit L -> 0
- For any infinitesimal volume, the total surface contribution must vanish
Describe why we use Cauchy’s tetrahedron argument
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)
Describe what the notation of T_ij represents
T_ij is the Cauchy Stress tensor
- It describes the stress on the surface normal to j along i
What is the rate of change of the angular momentum equal to?
The net torque applied to the body
Describe what is meant by an inviscid fluid (qualitatively)
There is no friction between fluid elements
- Viscosity does not play a role
Explain how the pressure acts for an inviscid fluid element
It acts normal inward
- Pressure is a force that acts perpendicular to a surface
What do we assume about a Newtonian fluid?
It is incompressible and has a constant viscosity
What does it mean if a tensor is isotropic?
The components do not depend on the coordinate system used to represent the tensor
Describe the no slip boundary condition, and what type of fluids it applies to
The fluid velocity at the boundary is equal to the boundary veloctity
- Applies to viscous fluids
Describe the free slip boundary condition, and what type of fluids it applies to
The normal of the fluid velocity at the boundary is equal to the normal of the boundary veloctity (no penetration)
- Applies to inviscid fluids
Describe the kinematic boundary condition, and what type of fluids it applies to
The free surface follows the fluid
- Applies when there are moving material boundaries
