1. Mathematical Modelling of Fluids Flashcards
continuum hypothesis, velocity field, time derivatives, conservation of mass, incompressibility, viscosity, cartesian tensors, strain rate and vorticity tensors, polar tensors
Fluids
Definition
-substances tending to flow or conform to the outline of their container
Mechanics
Definition
-science studying the behaviour of physical objects subject to forces and displacement
Statics, Kinematics and Dynamics
Definitions
- mechanics can be split into three fields:
i) statics is the study of forces and torques when there is no motion
ii) kinematics is the study of motion regardless of causes (i.e. forces)
iii) dynamics is the study of forces and torques where there is motion (i.e the combination of statics and kinematics)
Fluid Dynamics
Definition
-the study of forces and torques on a moving fluid
Why is the continuum hypothesis necessary?
- we can not attempt to calculate the motion of individual molecules as there are far too many of them
- and their individual motion is dominated by high frequency fluctuations caused by collisions with neighbouring molecules
- instead we average the motion of a fluid ‘blob’
Fluid Particle
Definition
-instead of attempting to calculate the motion of individual particles, we represent the average motion of a ‘blob’ of fluid, a fluid particle of length d and volume 𝛿V
Continuum Hypothesis
Description
- we represent the average motion of a fluid particle of length d and volume 𝛿V
- we choose d so that:
i) d<
Continuum Hypothesis
Definition
-molecular details can be smoothed out by assigning to the velocity at a point P the average velocity in a fluid element 𝛿V centred on P
Fluid Molecule Size
-of order of magnitude ~10^-10
How many water molecules in a cubic centimeter?
10^23 molecules
What does a graph of average velocity u_ against fluid particle diameter d look like?
- lm at the small end of the x axis, a at the large end
- large fluctuations close to lm, graph appears as a zig-zag
- this smooths out as d->a
Velocity Field
Definition
- we define the velocity field u(x,t) as a smooth function of time and position, i.e. u is differentiable and integrable
- note that shock waves break this assumption
Local Density of Mass
-similar to the velocity field, local density of mass is defined:
ρ(x,t) = mass in 𝛿V / 𝛿V
Velocity Field
Shear Flow
u_ = (Uy/d, 0, 0)
Velocity Field
Stagnation Point Flow
u_ = (Ex, -Ey, 0)
Particle Paths
-a way to analyse the flow by injecting ‘tracers’ (i.e. passive particles that don’t alter/effect the flow in anyway) that follow the flow
-tracer particles follow the equation:
dx_/dy = u_(x_,t)
-with initial condition x_=xo_ = (xo,yo,zo) at t=to
Streamlines
-lines that are everywhere tangent to the velocity vector
-parameterise streamlines by a distance s:
dx_/ds = u_(x_(s), t)
-so:
dx/ds = u, dy/ds=v, dz/ds=w
=> dx/u = dy/v = dz/w
Particle Paths, Streamlines and Steady Flow
-in the context of a steady flow (∂u_/∂t=0), the particle paths and streamlines coincide
Partial/Eulerian Derivative
-rate of change of velocity at a fixed position in space, x_
∂u_/∂t
Material/Total/Lagrangian Derivative
-rate of change of velocity of a particle passing through _x
Df/Dt = ∂f/∂t + (u_ . ∇)f
Acceleration of Fluid Particles
-given by the Lagrangian derivative of velocity:
Du_/Dt = ∂u_/∂t + (u_ . ∇)u_
Conservation of Mass
- in all systems, mass is conserved
- consider a volume V of a fluid fixed in space, the mass Mv contained in the volume can only change if mass is carried into or out of V by the fluid
- mass flux is the mass flowing through S, the surface of V, per unit time
Continuity Equation
∂ρ/∂t + ∇.(ρu_) = 0
Lagrangian Form of the Continuity Equation
Dρ/Dt + ρ∇.u_ = 0
When does the density of a fluid particle moving within the fluid change?
-the density of a fluid particle moving with the field only changes if there is an expansion or contraction in the flow
Incompressibility Constraint
-an incompressible fluid is a fluid whose density does not vary (in space or time:
Dρ/Dt + ρ∇.u_ = 0
=>
∇.u_ = 0
-also referred to as an alternative form of the continuity equation for incompressible fluids
Streamfunction
-the incompressibility constraint places a restriction on the velocity field:
u_ = ∇ x ψ
-due to the vectorial equality:
∇ . (∇xf) = 0
Streamfunction for 2D Cartesian Incompressible Flows
-consider the incompressible flow :
u_ = u(x,y)ex^ + v(x,y)ey^
-if this flow is incompressible, we can introduce the streamfunction:
u = ∂ψ/∂x, v = - ∂v/∂y
Streamline
Definition
- a streamline is a line on which the streamfunction is constant, dψ=0
- the gradient of the streamfunction is orthogonal to the velocity vector
Flux Between Two Lines
-consider the two streamlines ψt and ψp
-the flux between the two lines is given by:
Q = ψt - ψp
-where ψt is the top line and ψp is the bottom line
Stokes Streamfunction
-consider:
u_ = u(r,z)er^ + w(r,z)ez^
-in this case, Ψ is in the eθ^ direction, and we can define:
Ψ_ = (1/r Ψ(r,z)) eθ^
-and the fluid velocity is given by u_ = ∇ x Ψ_
What are the properties of Stokes streamfunctions?
-Stokes streamfunctions have properties analagous to planar streamfunctions
i) Ψ is constant on streamlines
ii) relation between volume flux and streamtubes:
Q = 2π(Ψo - Ψi)
Dynamic Viscosity
µ, related to force
-units are pascal seconds
Kinematic Viscosity
ν, does not arise from considering force:
ν = µ/ρ
-units are metres squared per second
Stress
Definition
-the density of force
σ = F/A
Shear Rate
Definition
γ’ = U/d
-where U is the velocity of the wall and d is the distance between the walls
What is the relationship between stress and shear rate?
σ = µ*γ’
Scalar Product in Suffix Notation
a_ . b_ = ai bi = ai bi 𝛿ij
-where 𝛿ij is the kronecker delta which =1 for i=j and =0 otherwise
Vector Product in Suffix Notation
a_ x b_ = εijk aj bk
-where εijk is the alternating tensor which =1 for ijk=123,312,231 =-1 for ijk=321,132,213 and 0 otherwise
Velocity Gradient
Kij = ∂ui/∂xj
Double Dot Product
D = A : B = Aij B kl ∂jk ∂il
-where A and B are matrices
Vector Product Rule
a_ x (b_ x c_) = b_(a_.c_) + - c_(a_.b_)
Grad in Suffix Notation
-gradient of a scalar:
∇p = ∂p/∂xi
-gradient of a vector:
∇u_ = ∂ui/∂xj
Divergence in Suffix Notation
-for vectors
∇ . u_ = ∂ui/∂xj 𝛿ij = ∂ui/∂xi
-for matrices:
∇ . A|j = ∂Aij/∂xi
Curl in Suffix Notation
-for vectors:
∇ x u_ = εijk ∂/∂xj uk
Why do we use strain rate and vorticity tensors?
- consider the velocity gradient vector ∂ui/∂xj, this is a nine components
- for an incompressible flow, ∇.u_=0, we know that the trace is 0 so we can write one of the components on the main diagonal in terms of the other two
- this reduces the number of unknown components to 8, which is still a lot!
- a useful simplification is to decompose the velocity gradient into the sum of a symmetric and an antisymmetric tensor, the strain rate and vorticity tensors
Decomposing the Velocity Gradient
∂ui/∂xj = Eij + Ωij
- where Eij is the strain-rate tensor which is symmetric
- and Ωij is the vorticity tensor which is antisymmetric
Strain Rate Tensor
Eij = 1/2 * (∂ui/∂xj + ∂uj/∂xi)
-symmetric
Vorticity Tensor
Ωij = 1/2 * (∂ui/∂xj - ∂uj/∂xi)
-anti-symmetric
Relationship Between Vorticity and the Vorticity Tensor
Ωij = 1/2 εkji ωk
= -1/2 εkij ωk
