If u = ax, v = ay and w = -2az are the velocity components for a fluid flow, check whether they satisfy the continuity equation. If they do, is the flow rotational or irrotational? Also obtain equation of streamlines passing through the point (2, 2, 4).

Three-Dimensional Flow Analysis

Problem Statement

The velocity components in a three-dimensional flow are:

u = ax
v = ay
w = -2az

Check whether they satisfy the continuity equation. If they do, determine if the flow is rotational or irrotational. Also obtain the equation of streamlines passing through the point (2, 2, 4).

1. Verify the Continuity Equation

For a three-dimensional flow to be physically possible, it must satisfy the continuity equation:

∂u/∂x + ∂v/∂y + ∂w/∂z = 0

Given:
u = ax
v = ay
w = -2az

Step 1.1: Calculate the partial derivatives
∂u/∂x = a
∂v/∂y = a
∂w/∂z = -2a

Step 1.2: Verify the continuity equation
∂u/∂x + ∂v/∂y + ∂w/∂z = a + a + (-2a) = 0

The continuity equation is satisfied, confirming this is a possible case of fluid flow.

2. Check for Irrotationality

A flow is irrotational if all vorticity components are zero. In a three-dimensional flow, we need to check:

ω_x = ½(∂w/∂y – ∂v/∂z)
ω_y = ½(∂u/∂z – ∂w/∂x)
ω_z = ½(∂v/∂x – ∂u/∂y)

Step 2.1: Calculate ω_x
∂w/∂y = ∂(-2az)/∂y = 0
∂v/∂z = ∂(ay)/∂z = 0
ω_x = ½(0 – 0) = 0

Step 2.2: Calculate ω_y
∂u/∂z = ∂(ax)/∂z = 0
∂w/∂x = ∂(-2az)/∂x = 0
ω_y = ½(0 – 0) = 0

Step 2.3: Calculate ω_z
∂v/∂x = ∂(ay)/∂x = 0
∂u/∂y = ∂(ax)/∂y = 0
ω_z = ½(0 – 0) = 0

Since all components of vorticity are zero (ω_x = ω_y = ω_z = 0), the flow is irrotational.

3. Find Equation of Streamlines

The differential equation of streamlines is given by:

dx/u = dy/v = dz/w

Substituting the velocity components:
dx/ax = dy/ay = dz/(-2az)

Step 3.1: From the first two terms (dx/ax = dy/ay)
dx/x = dy/y
Integrating:
ln x = ln y + ln C₁
x/y = C₁ … (a)

Step 3.2: From the first and third terms (dx/ax = dz/(-2az))
dx/x = dz/(-2z)
dx/x = -dz/(2z)
Integrating:
ln x = -½ ln z + ln C₂
ln x + ½ ln z = ln C₂
x·z^1/2 = C₂ … (b)

Step 3.3: Apply the condition that streamline passes through (2, 2, 4)
From equation (a): C₁ = 2/2 = 1
From equation (b): C₂ = 2·4^1/2 = 2·2 = 4

Therefore, the equations of the streamline are:
x/y = 1 ⟹ x = y
x·z^1/2 = 4

The streamline passing through (2, 2, 4) is defined by the equations x = y and x·z^1/2 = 4.

Conclusion

We have shown that the given velocity field:

u = ax
v = ay
w = -2az

1. Satisfies the continuity equation (∂u/∂x + ∂v/∂y + ∂w/∂z = 0), confirming it represents a possible fluid flow.

2. Has zero vorticity in all directions (ω_x = ω_y = ω_z = 0), confirming the flow is irrotational.

3. The streamline passing through the point (2, 2, 4) is described by the equations:
x = y (or x/y = 1)
x·z^1/2 = 4

Physical Interpretation

This three-dimensional, irrotational flow has the following characteristics:

It represents a linear flow field where velocity components vary linearly with position
Since the flow is irrotational, a velocity potential function φ exists
The streamlines form a family of curves in 3D space, with any particular streamline defined by two equations
The flow exhibits expansion in the x and y directions (positive u and v) and compression in the z direction (negative w)
This pattern corresponds to fluid elements being stretched horizontally while being compressed vertically
Due to its irrotational nature, Bernoulli’s equation can be applied between any two points in the flow field