Non-dimensionalization of Equations
Manus AI
March 26, 2025
1 Non-dimensionalization of Allen-Cahn Equation
The Allen-Cahn equation is given by:
∂ϕ 1
= −M µ, where µ = W ′ (ϕ) − ϵ∇2 ϕ (1)
∂t ϵ
1.1 Step 1: Identify the dimensional parameters and variables
• ϕ: phase field variable (dimensionless)
• t: time [T]
• M : mobility coefficient [L2 /T]
• ϵ: interface width parameter [L]
• W (ϕ): double-well potential (dimensionless)
• µ: chemical potential [1/L2 ]
1.2 Step 2: Define characteristic scales
We need to introduce:
• Characteristic length: Lc
• Characteristic time: Tc
1.3 Step 3: Define dimensionless variables
• x∗ = x
Lc (dimensionless space)
• t∗ = t
Tc (dimensionless time)
• ϕ is already dimensionless
1.4 Step 4: Rewrite the equation in terms of dimensionless variables
For the time derivative:
∂ϕ 1 ∂ϕ
= (2)
∂t Tc ∂t∗
For the Laplacian:
1 ∗2
∇2 ϕ = ∇ ϕ (3)
L2c
1
Substituting into the original equation:
1 ∂ϕ 1 ′ 1 ∗2
= −M W (ϕ) − ϵ 2 ∇ ϕ (4)
Tc ∂t∗ ϵ Lc
Multiplying both sides by Tc :
1 ′ 1 ∗2
∂ϕ
= −M T c W (ϕ) − ϵ ∇ ϕ (5)
∂t∗ ϵ L2c
Rearranging:
∂ϕ 1 1
= −M Tc W ′ (ϕ) + M Tc ϵ 2 ∇∗2 ϕ (6)
∂t∗ ϵ Lc
1.5 Step 5: Choose appropriate scales to simplify the equation
We want to choose Lc and Tc to make the coefficients simple, ideally equal to 1.
Let’s set:
1. Lc = ϵ (natural length scale of the interface)
ϵ2
2. Tc = M (diffusive time scale)
With these choices:
∂ϕ ϵ2 1 ′ ϵ2 1
∗
= − · M · W (ϕ) + · M · ϵ · 2 ∇∗2 ϕ (7)
∂t M ϵ M ϵ
= −ϵW ′ (ϕ) + ∇∗2 ϕ (8)
For a typical double-well potential W (ϕ) = 14 (1 − ϕ2 )2 , we have W ′ (ϕ) = ϕ3 − ϕ.
Substituting:
∂ϕ
= −ϵ(ϕ3 − ϕ) + ∇∗2 ϕ (9)
∂t∗
= −ϵϕ3 + ϵϕ + ∇∗2 ϕ (10)
If we further define W̃ (ϕ) = 1ϵ W (ϕ) so that W̃ ′ (ϕ) = ϕ3 − ϕ, then:
∂ϕ
= −W̃ ′ (ϕ) + ∇∗2 ϕ (11)
∂t∗
1.6 Step 6: Final non-dimensionalized form
Dropping the asterisks for simplicity, the non-dimensionalized Allen-Cahn equation is:
∂ϕ
= ∇2 ϕ − W̃ ′ (ϕ) (12)
∂t
or with the specific double-well potential:
∂ϕ
= ∇2 ϕ + ϕ − ϕ 3 (13)
∂t
2 Non-dimensionalization of Cahn-Hilliard Equation
The Cahn-Hilliard equation is given by:
∂ϕ 1
= M ∇2 µ, where µ = W ′ (ϕ) − ϵ∇2 ϕ (14)
∂t ϵ
2
2.1 Step 1: Identify the dimensional parameters and variables
• ϕ: phase field variable (dimensionless)
• t: time [T]
• M : mobility coefficient [L4 /T]
• ϵ: interface width parameter [L]
• W (ϕ): double-well potential (dimensionless)
• µ: chemical potential [1/L2 ]
2.2 Step 2: Define characteristic scales
We need to introduce:
• Characteristic length: Lc
• Characteristic time: Tc
2.3 Step 3: Define dimensionless variables
• x∗ = x
Lc (dimensionless space)
• t∗ = t
Tc (dimensionless time)
• ϕ is already dimensionless
2.4 Step 4: Rewrite the equation in terms of dimensionless variables
For the time derivative:
∂ϕ 1 ∂ϕ
= (15)
∂t Tc ∂t∗
For the Laplacian:
1 ∗2
∇2 ϕ = ∇ ϕ (16)
L2c
For the chemical potential:
1 ′ 1
µ= W (ϕ) − ϵ 2 ∇∗2 ϕ (17)
ϵ Lc
For the Laplacian of the chemical potential:
1 ∗2
∇2 µ = ∇ µ (18)
L2c
Substituting into the original equation:
1 ∂ϕ 1 ∗2 1 ′ 1 ∗2
= M ∇ W (ϕ) − ϵ ∇ ϕ (19)
Tc ∂t∗ L2c ϵ L2c
Multiplying both sides by Tc :
1 ∗2 1 ′ 1 ∗2
∂ϕ
= M T c ∇ W (ϕ) − ϵ ∇ ϕ (20)
∂t∗ L2c ϵ L2c
3
2.5 Step 5: Choose appropriate scales to simplify the equation
We want to choose Lc and Tc to make the coefficients simple, ideally equal to 1.
Let’s set:
1. Lc = ϵ (natural length scale of the interface)
ϵ4 ϵ2
2. Tc = M ϵ2 = M (diffusive time scale)
With these choices:
M ϵ2 1 ∗2 1 ′ 1 ∗2
∂ϕ
= ∇ W (ϕ) − ϵ ∇ ϕ (21)
∂t∗ M ϵ2 ϵ ϵ2
ϵ2 1
ϵ
= 2 ∇∗2 W ′ (ϕ) − 2 ∇∗2 ϕ (22)
ϵ ϵ ϵ
1 1
= ∇∗2 W ′ (ϕ) − ∇∗2 ϕ (23)
ϵ ϵ
1 ∗2 ′
= ∇ W (ϕ) − ϵ2 ∇∗2 ϕ (24)
ϵ
If we define W̃ (ϕ) = 1ϵ W (ϕ) so that W̃ ′ (ϕ) = 1ϵ W ′ (ϕ), then:
∂ϕ
= ∇∗2 W̃ ′ (ϕ) − ∇∗2 ϕ (25)
∂t∗
2.6 Step 6: Final non-dimensionalized form
Dropping the asterisks for simplicity, the non-dimensionalized Cahn-Hilliard equation is:
∂ϕ
= ∇2 W̃ ′ (ϕ) − ∇2 ϕ (26)
∂t
For a typical double-well potential where W̃ ′ (ϕ) = ϕ3 − ϕ:
∂ϕ
= ∇2 ϕ 3 − ϕ − ∇ 2 ϕ (27)
∂t
3 Non-dimensionalization of Convection-Diffusion Equation
The convection-diffusion equation is given by:
∂u
+ ∇ · (vu) − D∇2 u = q (28)
∂t
3.1 Step 1: Identify the dimensional parameters and variables
• u: concentration or temperature [U]
• t: time [T]
• v: velocity field [L/T]
• D: diffusion coefficient [L2 /T]
• q: source/sink term [U/T]
• x: spatial coordinate [L]
4
3.2 Step 2: Define characteristic scales
We need to introduce:
• Characteristic length: Lc
• Characteristic time: Tc
• Characteristic velocity: Vc
• Characteristic concentration: Uc
• Characteristic source: Qc
3.3 Step 3: Define dimensionless variables
• x∗ = x
Lc (dimensionless space)
• t∗ = t
Tc (dimensionless time)
• u∗ = u
Uc (dimensionless concentration)
• v∗ = v
Vc (dimensionless velocity)
• q∗ = q
Qc (dimensionless source)
3.4 Step 4: Rewrite the equation in terms of dimensionless variables
For the time derivative:
∂u Uc ∂u∗
= (29)
∂t Tc ∂t∗
For the convection term:
Vc · Uc ∗
∇ · (vu) = ∇ · (v ∗ u∗ ) (30)
Lc
For the diffusion term:
Uc ∗2 ∗
D∇2 u = D ∇ u (31)
L2c
For the source term:
q = Qc · q ∗ (32)
Substituting into the original equation:
Uc ∂u∗ V c · Uc ∗ Uc
∗
+ ∇ · (v ∗ u∗ ) − D 2 ∇∗2 u∗ = Qc · q ∗ (33)
Tc ∂t Lc Lc
Dividing by Uc :
1 ∂u∗ Vc ∗ 1 Qc ∗
∗
+ ∇ · (v ∗ u∗ ) − D 2 ∇∗2 u∗ = q (34)
Tc ∂t Lc Lc Uc
Multiplying by Tc :
∂u∗ Vc · Tc ∗ Tc Qc · Tc ∗
+ ∇ · (v ∗ u∗ ) − D 2 ∇∗2 u∗ = q (35)
∂t∗ Lc Lc Uc
3.5 Step 5: Choose appropriate scales to simplify the equation
We have several options for choosing scales. Let’s consider two common approaches:
5
3.5.1 Option 1: Diffusion-dominated scaling
If diffusion is the dominant process, we can set:
L2c
1. Tc = D (diffusive time scale)
2. Vc = Lc
Tc = D
Lc (consistent velocity scale)
3. Qc = Uc
Tc = Uc ·D
L2c (consistent source scale)
With these choices:
∂u∗ D L2c 1 ∗ L2c 1 ∗2 ∗ Uc · D L2c 1 ∗
+ · · ∇ · (v ∗ ∗
u ) − D · · ∇ u = · · q (36)
∂t∗ Lc D Lc D L2c L2c D Uc
∂u∗
= ∗ + ∇∗ · (v ∗ u∗ ) − ∇∗2 u∗ = q ∗ (37)
∂t
This gives us the Péclet number P e = 1, indicating that diffusion and convection are equally important.
3.5.2 Option 2: Convection-dominated scaling
If convection is the dominant process, we can set:
1. Tc = Lc
Vc (convective time scale)
2. Qc = Uc
Tc = Uc ·Vc
Lc (consistent source scale)
With these choices:
∂u∗ Vc · Tc ∗ Tc Qc · Tc ∗
+ ∇ · (v ∗ u∗ ) − D 2 ∇∗2 u∗ = q (38)
∂t∗ Lc Lc Uc
∂u∗ V c · Lc 1 ∗ Lc 1 ∗2 ∗ Uc · Vc Lc 1 ∗
= ∗ + · ∇ · (v ∗ u∗ ) − D · 2∇ u = · · q (39)
∂t Vc Lc Vc Lc Lc Vc Uc
∂u∗ D
= ∗ + ∇∗ · (v ∗ u∗ ) − ∇∗2 u∗ = q ∗ (40)
∂t Vc · Lc
This introduces the Péclet number P e = Vc ·Lc
D as a coefficient of the diffusion term:
∂u ∗
1 ∗2 ∗
∗
+ ∇∗ · (v ∗ u∗ ) − ∇ u = q∗ (41)
∂t Pe
3.6 Step 6: Final non-dimensionalized form
Dropping the asterisks for simplicity, the non-dimensionalized convection-diffusion equation is:
For diffusion-dominated scaling:
∂u
+ ∇ · (vu) − ∇2 u = q (42)
∂t
For convection-dominated scaling:
∂u 1 2
+ ∇ · (vu) − ∇ u=q (43)
∂t Pe
where P e = Vc ·Lc
D is the Péclet number, which represents the ratio of convective to diffusive transport.
4 Non-dimensionalization of Navier-Stokes Equation
The Navier-Stokes equations are given by:
∂v
ρ + v · ∇v − η∇2 v + ∇P = f (44)
∂t
∇·v =0 (45)
6
4.1 Step 1: Identify the dimensional parameters and variables
• v: velocity field [L/T]
• t: time [T]
• ρ: density [M/L3 ]
• η: dynamic viscosity [M/(L·T)]
• P : pressure [M/(L·T2 )]
• f : external force per unit volume [M/(L2 ·T2 )]
• x: spatial coordinate [L]
4.2 Step 2: Define characteristic scales
We need to introduce:
• Characteristic length: Lc
• Characteristic time: Tc
• Characteristic velocity: Vc
• Characteristic pressure: Pc
• Characteristic force: Fc
4.3 Step 3: Define dimensionless variables
• x∗ = x
Lc (dimensionless space)
• t∗ = t
Tc (dimensionless time)
• v∗ = v
Vc (dimensionless velocity)
• P∗ = P
Pc (dimensionless pressure)
• f∗ = f
Fc (dimensionless force)
4.4 Step 4: Rewrite the equation in terms of dimensionless variables
For the time derivative:
∂v Vc ∂v ∗
= (46)
∂t Tc ∂t∗
For the convective term:
Vc ∗ V2
v · ∇v = Vc · v · ∇∗ v ∗ = c v ∗ · ∇∗ v ∗ (47)
Lc Lc
For the viscous term:
Vc ∗2 ∗
η∇2 v = η ∇ v (48)
L2c
For the pressure gradient:
Pc ∗ ∗
∇P = ∇ P (49)
Lc
For the external force:
f = Fc f ∗ (50)
7
For the incompressibility constraint:
Vc ∗ ∗
∇·v = ∇ ·v =0 (51)
Lc
Substituting into the momentum equation:
Vc ∂v ∗ Vc2 ∗
Vc Pc ∗ ∗
ρ ∗
+ v · ∇ ∗ ∗
v − η 2 ∇∗2 v ∗ + ∇ P = Fc f ∗ (52)
Tc ∂t Lc Lc Lc
Dividing by ρVc2 /Lc (which has units of force per unit volume):
Lc ∂v ∗ η Pc ∗ ∗ Fc Lc ∗
∗
+ v ∗ · ∇∗ v ∗ − ∇∗2 v ∗ + 2
∇ P = f (53)
Vc Tc ∂t ρVc Lc ρVc ρVc2
4.5 Step 5: Choose appropriate scales to simplify the equation
We want to choose the scales to make the coefficients simple. Let’s set:
1. Tc = Lc
Vc (advective time scale)
2. Pc = ρVc2 (dynamic pressure scale)
ρVc2
3. Fc = Lc (consistent force scale)
With these choices:
Lc Vc ∂v ∗ η ρV 2 ρV 2 /Lc · Lc ∗
· ∗
+ v ∗ · ∇∗ v ∗ − ∇∗2 v ∗ + c2 ∇∗ P ∗ = c f (54)
Vc Lc ∂t ρVc Lc ρVc ρVc2
∂v ∗ η
∗
+ v ∗ · ∇∗ v ∗ − ∇∗2 v ∗ + ∇∗ P ∗ = f ∗ (55)
∂t ρVc Lc
The coefficient of the viscous term is the inverse of the Reynolds number:
ρVc Lc
Re = (56)
η
So the equation becomes:
∂v ∗ 1 ∗2 ∗
+ v ∗ · ∇∗ v ∗ − ∇ v + ∇∗ P ∗ = f ∗ (57)
∂t∗ Re
The incompressibility constraint remains:
∇∗ · v ∗ = 0 (58)
4.6 Step 6: Final non-dimensionalized form
Dropping the asterisks for simplicity, the non-dimensionalized Navier-Stokes equations are:
∂v 1 2
+ v · ∇v − ∇ v + ∇P = f (59)
∂t Re
∇·v =0 (60)
where Re = ρVc Lc
η is the Reynolds number, which represents the ratio of inertial forces to viscous forces.
8
4.7 Alternative scaling for pressure-driven flows
For pressure-driven flows, we might choose a different pressure scale:
ηVc
Pc = (61)
Lc
This would lead to:
∂v ∗ 1 ∗2 ∗ 1 ∗ ∗ Fc Lc ∗
∗
+ v ∗ · ∇∗ v ∗ − ∇ v + ∇ P = f (62)
∂t Re Re ρVc2
(63)
If we also set Fc = L2c ,
ηVc
then:
∂v ∗ 1 ∗2 ∗ 1 ∗ ∗ 1 ∗
+ v ∗ · ∇∗ v ∗ − ∇ v + ∇ P = f (64)
∂t∗ Re Re Re
(65)
Multiplying by Re:
∂v ∗
Re + Re(v ∗ · ∇∗ v ∗ ) − ∇∗2 v ∗ + ∇∗ P ∗ = f ∗ (66)
∂t∗
(67)
This form is useful for studying the limit of small Reynolds numbers (Stokes flow).
