Question

Perform a dimensional analysis to obtain an equation for rising air bubble velocity, v. What variables would v depend on; i.e. what variables would you include in your analysis?

Assume that v is a function of d (air bubble diameter). p_w (density of water), g (gravitational acceleration), and u_w (viscosity of water), and perform the dimensional analysis. How many dimensionless groups do you get (and why)? Look up the defintion of Reynolds number... Is Reynolds number of the dimensionless groups resulting from your analysis?

Answer 1

Given,

V = ( d, $\rho$ , , $\mu$ )

Let us apply Buckingham's $\pi$ - theorem for the above dimensional analysis.

$\therefore$ (V, D, $\rho$ , , $\mu$ ) = 0

Here, number of variables (n) = 5

We know that,

Dimension for length = L

Dimension for mass = M

Dimension for time = T

L, M, T are called fundamental dimensions.

Unit of velicity (V) = m/s = Length / time

$\therefore$ Dimension of V = L T^-1

Unit of diameter (D) = m = Length

$\therefore$ Dimension of D = L

Unit of gravity (g) = m/s² = Length / Time²

$\therefore$ Dimension of 'g' = LT^-2

Unit of mass density ( $\rho$ ) = kg/m³  = Mass / Length³

$\therefore$ Dimension of ' $\rho$ ' = ML^-3

Unit of dynamic viscosity ( $\mu$ ) = kg/m-s = Mass / (Length * Time)

$\therefore$ Dimension of ' $\mu$ ' = ML^-1T^-1

From the above observations it is clear that all three fundamental dimensions are present here.

Therefore,

Number of fundamental dimensions (m) = 3

As number of fundamental dimensions is less than number of variables present, there must be dimensionless group present

in excess of number of fundamental dimensions.

Therefore,

Number of dimension less groups = n -m = 5 - 3 = 2

Let us designate the dimensionless groups as $\pi$ ₁ and $\pi$ ₂

Dimension of $\pi$ -terms = M⁰L⁰T⁰

$\pi$ -terms consist of (m+1) = (3 + 1) = 4 variables.

'm' is also called as repeating variables i.e., these variables will be present in both the $\pi$ -terms here.

Let us select repeating variables as D, g, $\mu$ . [One geometric property i.e., D, One flow property i.e., 'g' and one fluid property i.e., $\mu$ ]

Therefore $\pi$ -terms can be written as,

$\pi$ ₁ = D^a₁. g^b₁. _$\mu$ ^c₁ . V ............................................................(1)

$\pi$ ₂ = D^a₂. g^b₂. _$\mu$ ^c₂ . $\rho$ ............................................................(2)

Where a₁, b₁, c₁, a₂, b₂ and c₂, are arbitrary powers values of which are to be found.

Substituting dimension of each term in equation (1) we get,

M⁰L⁰T⁰ = L^a₁. (L T^-2)^b₁. (ML^-1T^-1)^c₁ . LT^-1

$\Rightarrow$ M⁰L⁰T⁰ = L^{a₁ + b₁ - c₁ + 1}. T^{-2b₁ - c₁ - 1}. M^c₁ .

Equating the powers of each terms on both side,

0 = a₁ + b₁ - c₁ + 1 .......................................................(3) [equating powers of L on both side]

0 = - 2 b₁ - c₁ - 1 ............................................................(4)   [equating powers of T on both side]

0 = c₁ [equating powers of T on both side]

Therefore from equation (4),

0 = - 2 b₁ - 0 - 1

$\Rightarrow$ b₁ = - (1/2)

From equation (3),

0 = a₁ - (1/2) - 0 + 1

$\Rightarrow$ a₁ = - 1/2

Therefore from equation (1),

$\pi$ ₁ = D^(-1/2). g^(-1/2). _$\mu$ ⁰ . V

$\Rightarrow$ $\pi$ ₁ = V / (g * D)^(1/2) This is one of the two dimensionless groups.

Similarly,

Substituting dimension of each terms in equation (2) we get,

M⁰L⁰T⁰ = L^a₂. (L T^-2)^b₂. (ML^-1T^-1)^c₂ . ML^-3

$\Rightarrow$ M⁰L⁰T⁰ = L^{a₂ + b₂ - c₂ -3}. T^{-2b₁ - c₂}. M^{c₂ + 1} .

Equating the powers of each terms on both side,

0 = a₂ + b₂ - c₂ - 3   .......................................................(5) [equating powers of L on both side]

0 = - 2 b₂ - c₂   ............................................................(6)   [equating powers of T on both side]

0 = c₂ + 1   [equating powers of T on both side]

$\Rightarrow$ c₂ = -1

Therefore from equation (5),

0 = - 2 b₂ - 0 - (-1)

$\Rightarrow$ b₂ = (1/2)

From equation (3),

0 = a₂ + (1/2) - (-1) - 3

$\Rightarrow$ a₁ = 3/2

Therefore from equation (1),

$\pi$ ₂ = D^(3/2). g^(1/2). _$\mu$ ^-1 . $\rho$

$\Rightarrow$ $\pi$ ₁ = ( $\rho$ * g^(1/2) * D^(3/2)) / _$\mu$ This is another dimensionless groups.

We know that Reynolds number is given by,

R_e = ( $\rho$ * V * D) / _$\mu$

Comparing with the obtained dimensionless groups it is clear that Reynolds number is not a part of the above dimensionless groups.

Equation of air bubble velocity can be obtained by using the first dimensionless group as follows:

$\pi$ ₁ = V / (g * D)^(1/2)

i.e.,

V / (g * D)^(1/2) = Constant [As $\pi$ ₁ is a dimensionless parameter]

Therefore if model and prototype is considered above equation can be modified as,

{V / (g * D)^(1/2)}_model = {V / (g * D)^(1/2)}_prototype

Here, prototype is the rising air bubble.

Solving the above equation velocity of prototype can be obtained if model parameters are given.