Chapter 2, Problem 1P

For the velocity fields given below, determine:

1. (a) whether the flow field is one-, two-, or three-dimensional, and why.
2. (b) whether the flow is steady or unsteady, and why.

(The quantities a and b are constants.)

1. 1 $\stackrel{\to }{V}=[(ax+t)e^{by}]\hat{i}$
2. 2 $\stackrel{\to }{V}=(ax-by)\hat{i}$
3. 3 $\stackrel{\to }{V}=ax\hat{i}+[e^{bx}]\hat{j}$
4. 4 $\stackrel{\to }{V}=ax\hat{i}+b{x}^2\hat{j}+ax\hat{k}$
5. 5 $\stackrel{\to }{V}=ax\hat{i}+[e^{bt}]\hat{j}$
6. 6 $\stackrel{\to }{V}=ax\hat{i}+b{x}^2\hat{j}+ay\hat{k}$
7. 7 $\stackrel{\to }{V}=ax\hat{i}+[e^{bt}]\hat{j}+ay\hat{k}$
8. 8 $\stackrel{\to }{V}=ax\hat{i}+[e^{by}]\hat{j}+az\hat{k}$

To determine

(a) Whether the flow field is one-, two-, or three-dimensional, and why.

(b) Whether the flow is steady or unsteady, and why.

Explanation of Solution

Condition for checking the Flow dimensions:

The flow is dependent on a number of space coordinates such as x, y, and z. Depending on the number of coordinates, the flow is stated as one or two or three dimensional.

Condition for checking whether the flow is steady or unsteady:

When the flow is changing its velocity with respect to time, then the fluid is unsteady. If the flow is not changing its velocity with respect to time, then the fluid is steady.

Calculation:

The given velocity field is,

  $\overrightarrow{V}=\left[(ax+t)e^{by}\right]\widehat{i}$

(a)

Here, the flow is dependent on two space coordinates x and y. Thus, the flow is two dimensional.

(b).

Here, the velocity of flow is changing with respect to time $\left(t\right)$. Thus, the flow is unsteady flow.

To determine

(a) Whether the flow field is one-, two-, or three-dimensional, and why.

(b) Whether the flow is steady or unsteady, and why.

Explanation of Solution

The given velocity field is,

  $\overrightarrow{V}=(ax-by)\widehat{i}$

(a)

Here, the flow is dependent on one space coordinate x Thus, the flow is two dimensional.

(b).

Here, the velocity of flow is does not change with respect to time $\left(t\right)$. Thus, the flow is steady flow.

To determine

(a) Whether the flow field is one-, two-, or three-dimensional, and why.

(b) Whether the flow is steady or unsteady, and why.

Explanation of Solution

The given velocity field is,

  $\overrightarrow{V}=ax\widehat{i}+\left[e^{hx}\right]\widehat{j}$

(a)

Here, the flow is dependent on two space coordinates x only. Thus, the flow is one dimensional.

(b).

Here, the velocity of flow is does not change with respect to time $\left(t\right)$. Thus, the flow is steady flow.

To determine

(a) Whether the flow field is one-, two-, or three-dimensional, and why.

(b) Whether the flow is steady or unsteady, and why.

Explanation of Solution

The given velocity field is,

  $\overrightarrow{V}=ax\widehat{i}+b{x}^2\widehat{j}+ax\widehat{k}$

(a)

Here, the flow is dependent on space coordinates x only. Thus, the flow is one dimensional.

(b).

Here, the velocity of flow is does not change with respect to time $\left(t\right)$. Thus, the flow is steady flow.

To determine

(a) Whether the flow field is one-, two-, or three-dimensional, and why.

(b) Whether the flow is steady or unsteady, and why.

Explanation of Solution

The given velocity field is,

  $\overrightarrow{V}=ax\widehat{i}+\left[e^{bt}\right]\widehat{j}$

(a)

Here, the flow is dependent on space coordinates x only. Thus, the flow is one dimensional.

(b).

Here, the velocity of flow is changing with respect to time $\left(t\right)$. Thus, the flow is unsteady flow.

To determine

(a) Whether the flow field is one-, two-, or three-dimensional, and why.

(b) Whether the flow is steady or unsteady, and why.

Explanation of Solution

The given velocity field is,

  $\overrightarrow{V}=ax\widehat{i}+b{x}^2\widehat{j}+ay\widehat{k}$

(a)

Here, the flow is dependent on two space coordinates x and y. Thus, the flow is two dimensional.

(b).

Here, the velocity of flow is does not change with respect to time $\left(t\right)$. Thus, the flow is steady flow.

To determine

(a) Whether the flow field is one-, two-, or three-dimensional, and why.

(b) Whether the flow is steady or unsteady, and why.

Explanation of Solution

The given velocity field is,

  $\overrightarrow{V}=ax\widehat{i}+\left[e^{bt}\right]\widehat{j}+ay\widehat{k}$

(a)

Here, the flow is dependent on two space coordinates x and y. Thus, the flow is two dimensional.

(b).

Here, the velocity of flow is changing with respect to time $\left(t\right)$. Thus, the flow is unsteady flow.

To determine

(a) Whether the flow field is one-, two-, or three-dimensional, and why.

(b) Whether the flow is steady or unsteady, and why.

Explanation of Solution

The given velocity field is,

  $\overrightarrow{V}=ax\widehat{i}+\left[e^{by}\right]\widehat{j}+az\widehat{k}$

(a)

Here, the flow is dependent on two space coordinates x , y, and z. Thus, the flow is three dimensional.

(b).

Here, the velocity of flow is does not change with respect to time $\left(t\right)$. Thus, the flow is steady flow.