Control of Chaotic Flows and Fluid Forces (Previous Part)

**Perturbation Expansion**

The distribution function may be expanded as a multiscale perturbation expansion in terms of a small parameter as follows:

The second component of the multiscale perturbation expansion of is not a second order derivative; it simply indicates the expansion of the time scale in the second position.

and .

Using the above expansions in Eq. (??), we obtained

(1)

### Control of Chaotic Flows and Fluid Forces Around Two Offset Cylinders in Presence of Control Plate (Part 2)

Equating coefficient of

(2)

**O():**

(3)

**Moments and Recombination**

**Zeroth Moment**

To find zeroth moment of Eq. (2) of **O()** sum the equation over

(4)

(5)

Distribution function is written as

By adding the above equation over , we get

(6)

Since , and

By adding the above equation over , we get

Since the sum of weight factor is always equal to one i.e. . Therefore from above equation we get

Using above equation in Eq. (6), we get

From above equation, we get

Now since,

So Eq. (5) becomes

(7)

**First Moment**

To find zeroth moment of Eq. (2) of **O()** we multiply this equation with and sum over \textit{i} and index changing to

(8)

Since

So Eq. (8) becomes

(9)

**Second Moment**

To find zeroth moment of Eq. (2) of **O()** we multiply this equation with and sum over \textit{i} and index changing to

(10)

Since

So Eq. (10) becomes

(11)

Now we calculate the zeroth, and first moment of **O()**.

**Zeroth Moment of O()**

To find zeroth moment of Eq. (3) of **O()** sum the equation over

(12)

Now since,

So Eq. (12) becomes

(13)

**First Moment of O()**

To find zeroth moment of Eq. (3) of **O()** we multiply this equation with and sum over \textit{i} and index changing to

(14)

(15)

Now since,

So Eq. (15) becomes

(16)

Now by combining Eq. (7) and (13), we get

Control of Chaotic Flows and Fluid Forces (Previous Part)

Very knowledgeable