Fluid Mechanics

Question 1

What is a fluid?

Answer 1

A substance which fills up a vessel with no fixed shape

Question 2

What is the equation for density?

Answer 2

ρ=m/V

Question 3

What is the equation for the Knudsen number?

Answer 3

K_n=λ/L
Where λ is the mean free path
L is the size of the system

Question 4

What is the equation for stress?

Answer 4

τ=F/A=μ ∂u_x/∂y

Question 5

What is the equation for strain?

Answer 5

e=∂X/∂y
Where X is the deformation caused by the stress

Question 6

How do stress and strain link for a fluid?

Answer 6

τ_fluid= μ ∂e/∂t
Where μ is the viscosity

Question 7

What is steady motion?

Answer 7

When it is independent of t and so ∂/∂t = 0
No acceleration

Question 8

What are Newtonian fluids?

Answer 8

When μ does not depend on the velocity u
air, water, plasma

Question 9

What are non Newtonian fluids?

Answer 9

When μ does depend on the velocity u
oil, blood, emulsions

Question 10

What is the difference between a Lagrangian and Eulerian fluid element?

Answer 10

Lagrangian: Moves with the fluid in such a way into or out of the element (constant mass)
Eulerian: fixed in space, and does not move with the flow

Question 11

What is a streamline?

Answer 11

A curve which at a fixed time at each location is tangential to the fluid velocity u

Question 12

What differential equation describes streamlines?

Answer 12

dλ = dx/μ_x= dy/μ_y = dz/μ_z

Question 13

What is a laminar flow?

Answer 13

A flow in 1D, parallel to a single direction
No vortices

Question 14

What is the equation for the viscous force?

Answer 14

F_μ=μA ∂u_x/∂y
Where A=ΔxΔz

Question 15

What is the equation for the Reynolds number?

Answer 15

Re=ρ₀u₀L₀/μ
Where ρ₀ is the density
u₀ is the characteristic speed
L₀ is the characteristic spatial scale
μ is the viscosity

Question 16

What does the Reynolds number mean?

Answer 16

If Re is small the flow is viscous dominant (laminar)
If Re is large then the flow is turbulent (μ can be neglected)

Question 17

What is the mass of a fluid element?

Answer 17

m=ρu⋅ndS
Where n is the surface normal vector

Question 18

What is the condition of incompressibility?

Answer 18

∇⋅u=0

Question 19

How is the condition of incompressibility obtained?

Answer 19

Mass is conserved
rate of mass change of fluid in volume V = the total mass flow through S per second

Question 20

What is the continuity equation for fluids?

Answer 20

∂ρ/∂t+∇⋅(ρu)=0

Question 21

What is the general equation for steady, viscous flow?

Answer 21

∇P = μ∇²u

Question 22

What is the Navier-Stokes equation?

Answer 22

ρ∂u/∂t + ρ(u⋅∇)u = -∇p + μ∇²u + ρgk
Where k is the negative z direction

Question 23

What is the adiabatic equation for pressure?

Answer 23

d/dt (P/ρ^γ) = 0
Where P is the pressure
γ is the adiabatic index

Question 24

What is an inviscid fluid?

Answer 24

Where λ and μ are both 0

Question 25

When is the Navier-Stokes equation applicable?

Answer 25

When the Reynolds number is small

Question 26

How does the velocity behave for NS near a boundary?

Answer 26

It experiebces a sharp gradient

Question 27

How do the Euler and NS solutions differ near boundary?

Answer 27

NS solutions have a transition region, δ

Question 28

What is the estimate for the transition region width for the NS solutions?

Answer 28

δ ~= sqrt(μL₀/ρ₀u₀)

Question 29

How does the Reynolds number and transition region width relate?

Answer 29

δ/L₀ ~= (Re)^-1/2

Question 30

What is the equation for Bernoulli's principle?

Answer 30

ρu²/2 + ρgz + P = constant along a streamline

Question 31

What are the conditions for Bernoulli's principle?

Answer 31

Incompressible Inviscous Steady

Question 32

When is the BP applicable?

Answer 32

When the flow speed is slow

Question 33

What is a stagnation point?

Answer 33

Where the velocity is equal to 0

Question 34

Are the arrival times for air above and below and aerofoil the same?

Answer 34

No

Question 35

What is the equation for vorticity?

Answer 35

ω = ∇ × u

Question 36

What is irrotational flow?

Answer 36

When the vorticity, ω, is equal to 0 everywhere

Question 37

What is the equation for circulation?

Answer 37

K = ∮ u⋅dl Over a path Γ

Question 38

What is the circulation for irrotational flow?

Answer 38

0

Question 39

What is Kelvin's circulation theorem?

Answer 39

The circulation, K, around any closed loop Γ moving with the fluid is constant for inviscid flows

Question 40

What is the equation for the circulation theorem?

Answer 40

dK/dt=d/dt [∮ u⋅dl]

Question 41

What is a barotropic fluid?

Answer 41

A fluid for which the density is a function of pressure ρ = ρ(P)

Question 42

What is the Laplace equation?

Answer 42

∇²Φ = 0

Question 43

What are the conditions for a potential flow?

Answer 43

Circulation and Vorticity are 0, ∇ × u = 0 Incompressible, ∇⋅u = 0

Question 44

When is the circulation not zero for a potential flow?

Answer 44

If the path Γ is around r = 0

Question 45

What is the generalised B.P.?

Answer 45

dΦ/dt + u²/2 + gz + P/ρ = constant Φ is the potential flow, u = ∇Φ

Question 46

What is the deep water approximation?

Answer 46

d >> λ

Question 47

What is the equation for phase velocity?

Answer 47

V_phase = ω/k

Question 48

What is the equation for group velocity?

Answer 48

V_group = dω/dk

Question 49

What is the dispersion relation for surface waves?

Answer 49

ω² = gk tanh(kd) Small angle approximation leads to ω = sqrt(gh)k

Question 50

What is the condition for dispersion less waves?

Answer 50

The phase and group velocities must be equal ω/k = dω/dk

Question 51

At what angle is the wave vector to the shore?

Answer 51

Always perpendicular

Question 52

What is the equation for the Magnus force?

Answer 52

L = ρU₀×K

Question 53

How does the lift force work?

Answer 53

The velocity is higher above the object than below By the BP higher velocity means lower pressure So higher pressure below than above and so a lift force

Question 54

What are the physics of the lift?

Answer 54

In the vicinity of the boundary, the no slip boundary condition Flow moves with the cylinder

Question 55

When is the circulation equal to zero?

Answer 55

When the integration path does not enclose the rotation axis

Question 56

How do two vortices rotating oppositely interact?

Answer 56

They are pushed towards one another by the Magnus effect And drift in the vertical direction

Question 57

How do two vortices rotating in the same direction interact?

Answer 57

They repel one another

Question 58

What does an adverse pressure gradient do?

Answer 58

Decelerates the flow

Question 59

What does a favourable pressure gradient do?

Answer 59

Accelerates the flow

Question 60

What is the Struhal number?

Answer 60

St = D/Pu D is the diameter of cylinder/vortex u if the flow speed upstream P is the time period of vortex shedding

Question 61

What is required for a lift force on an aerofoil?

Answer 61

A circulation around it

Question 62

What is the drag force?

Answer 62

F_drag ~ ρu²A