Motion of a single particle

Motion of a single particle

Mass × Acceleration = Sum of forces

u

F

Σ dt =

V

d ρ

For steady motion du_p/dt=0

Forces on a body

The most common forces on a body moving through a fluid are:

Weight: mg=ρ_pV_pg Buoyancy: ρ_fV_pg

Drag: Resistance to motion of a particle owing to interaction with the fluid.

Drag and lift

Lift L

Drag D U

In an inviscid flow D=0 always

L≠0 when there is circulation around the body

Force coefficients

The forces on a body can be non-dimensionalised by the lift and drag coefficients, C_L and C_D, where

A C_{D f r r}

D u u

F

ρ

2

= 1 F_L C_L

ρ

= 1

where u_r is the slip velocity, the relative velocity between the particle and the fluid u_r=u_f-u_p and A is the projected frontal area of the object.

Viscous drag

Generated by the presence of a laminar boundary layer over the surface of a body.

Can be calculated exactly to give the Stokes Law for a sphere r

d u

D = 3 πµ

Only true when the flow is viscously dominated i.e.

Re_p=ρ_fd_pu_r/µ<<1.

or

p

C

Re

= 24

Pressure drag

Caused by the asymmetrical pressure distribution generated

through separation of a flow around an object when fluid inertia is important.

When fully separated, the drag coefficient is approximately constant: for a sphere when Re>≈1000, C_D≈0.44 or

4 2

44 1 . 0

2 p r

r f

π

ρ

⋅

= u u

F_D for a sphere.

The value of the constant and the threshold Reynolds number is different for differently shaped objects.

Drag curve

Disk

Sphere

C

≈ 0.44

C_D = 24

Re_p

(

)

No wall effects

Other forces: Magnus effect

Lift L

Resultant flow

direction

Rotation of particle

Inviscid flow L=ρ^u_rΓ

Other forces: shear force

L

u_f y x

Saffman Lift Force

L = 6.46 µ ^d

4

∂ u

∂ y

1

ν ^u

ν=kinematic viscosity = µ/ρ

Other forces: thermophoresis

Temperature T

F

= −6 πµ ^d

2 K

∇T T

F_T

K_t is the thermophoretic coefficient which is a function of the particle material properties

Terminal velocity of a sphere

When a particle is travelling at a constant speed i.e. it is in equilibrium, then

4 0

1 ⎟ ⎟ − 3

=

⎠

⎞

⎜ ⎜

⎝

⎛ −

=

p D p

f p

f

p

u

d g C

dt du

ρ ρ ρ

ρ

When the fluid is quiescent i.e.u_f=0, u_r=u_p=u_t.

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −

= 1

3 4

f p D

p

t C

g d

u ρ

ρ

or when Re<<1.

(

)

p t

u gd

ρ ρ

µ

= 18

2

Example

What is the terminal velocity of a glass sphere ( ρ

=2500 kg/m

) falling in air ( ρ

=1.2 kg/m

,

µ =1.7×10

kg/ms) when (a)d

= 50 µm

(b) d

= 500 µm (c)d

= 5 mm

Always confirm you are in the right regime by checking the Reynolds number.

⇒ u_t=0.20 m/s, Re = 0.71 Viscously dominated

⇒ u_t=17.59 m/s, Re = 6209 Inertially dominated

Intermediate Reynolds numbers

D

t

C

u 3 . 69

= Re = 35 . 3 ^u

Guess high Re

C

u

(m/s) Re

0.44 5.56 196

0.70 4.41 156

0.80 4.12 145

0.90 3.89 137

Drag curve for example

Sphere

Unsteady motion of a particle under gravity

x

y

g

x &

α

y &

Fluid stagnant u_p

p p

u y

u x

/ sin

/ cos

&

&

=

=

α α

Initial conditions

v y

y

w x

x t

=

=

=

=

=

&

&

, 0

, 0

, 0

What are x(t) and y(t)?