Fluid Kinematics | Velocity Potential Function and Stream Function

Fluid kinematics deals with the behavior of fluid motion without considering the forces causing the motion. Total fluid mass is considered instead of a single fluid particle for analysis i.e., the Eulerian method is used.

Types of Fluid Flow

We have already covered various types of fluid flow such as steady flow, unsteady flow, uniform flow, and nonuniform flow in our types of open channel flow blog. Here, we will discuss combinations of these types of flow along with examples.

Steady Flow

At a given section, fluid characteristics like velocity and density don't change with time. Flow through a constant diameter pipe is a good example.

∂V/∂t = 0, ∂Q/∂t = 0

Uniform Flow

At a given time, fluid characteristics like velocity and density don't change with space.

∂y/∂S = 0, ∂V/∂S = 0

Steady Uniform FLow

Example: Flow through a constant diameter pipe with constant discharge

Reason: As the discharge is constant, the velocity remains the same at a given section over time i.e., steady flow. Also, as the area of flow is constant, at a given time, the velocity between any two sections will remain constant i.e., uniform flow.

Steady Non-Uniform Flow

Example: Flow through a tapering diameter pipe at a constant rate.

Reason: As the discharge is constant, the velocity remains the same at a given section over time i.e., steady flow. But, as the pipe is tapered, the cross-sectional area decreases thereby increasing the velocity over distance. Therefore, at a given time, the velocity between two sections changes ie., non-uniform flow.

Unsteady Uniform Flow

Example: Fluid flow through a constant diameter pipe with varying discharge.

Reason: As the discharge changes, the velocity at a given section changes over time i.e., unsteady flow. But, as the area of flow is constant, at a given time, the velocity between any two sections will remain constant i.e., uniform flow.

Unsteady Non-Unifrom Flow

Example: Fluid flow through tampering diameter pipe with varying discharge.

Reason: As the discharge changes, the velocity at a given section changes over time i.e., unsteady flow. Also, as the pipe is tapered, the cross-sectional area decreases thereby increasing the velocity over distance. Therefore, at a given time, the velocity between two sections changes ie., non-uniform flow.

Continuity Equation

The flow of fluids can be expressed mathematically using the continuity equation. It is based on the law of conservation of mass. The general form of the continuity equation for a three-dimensional flow having steady, unsteady, uniform, non-uniform, compressible, and incompressible flow, is given as,

∂ρ/∂t + ∂(ρu)/∂x + ∂(ρv)/∂y + ∂(ρw)/∂z = 0

where,

ρ - density of fluid i.e., constant

u, v, w - velocity component in x, y, and z direction

For steady flow, ∂ρ/∂t = 0

For incompressible flow, ρ = constant

For uniform flow, ∂u/∂x = 0

Therefore, the continuity equation for a steady, uniform, and incompressible flow is given as,

∂u/∂x + ∂v/∂y + ∂w/∂z = 0

Rotational and Irrotational flow

In rotational flow, the fluid particles rotate about their mass center due to tangential stress caused by the viscosity of the fluid.

In such cases, the angular velocity of rotation is given by,

wz = (1/2) * ((∂v/∂x) - (∂u/∂y))

In the case of irrotational flow, the fluid particles don't rotate, and hence wz = 0.

∂v/∂x = ∂u/∂y

Acceleration of Fluid Flow

Acceleration in general is given as,

a = dV/dt = (∂V/∂s) * (ds/dt) = V * (∂V/∂s)

In the case of fluid flow, each velocity component is a function of x, y, z, and time (t). Therefore, the acceleration component for each velocity component should be found.

ax = (u * (∂u/∂x)) + (v * (∂u/∂y)) + (w * (∂u/∂z)) + (∂u/∂t)

ay = (u * (∂v/∂x)) + (v * (∂v/∂y)) + (w * (∂v/∂z)) + (∂v/∂t)

az = (u * (∂w/∂x)) + (v * (∂w/∂y)) + (w * (∂w/∂z)) + (∂w/∂t)

a = axi + ayj + azk, in vector format.

The terms (∂u/∂t), (∂v/∂t), and (∂w/∂t), are called temporal acceleration and they become zero in the case of steady flows.

The terms (u * (∂u/∂x)) + (v * (∂u/∂y)) + (w * (∂u/∂z)), (u * (∂v/∂x)) + (v * (∂v/∂y)) + (w * (∂v/∂z)), and (u * (∂w/∂x)) + (v * (∂w/∂y)) + (w * (∂w/∂z)), are called convective acceleration and they become zero in the case of uniform flow.

Therefore, in the case of a steady uniform flow, the acceleration is always zero.

Velocity Potential Function φ

The velocity potential function is a scalar function such that its negative derivative along any direction will give the velocity component in that direction.

u = -∂φ/∂x

v = -∂φ/∂y

The continuity equation in terms of the velocity potential function is called the Laplace function. For a steady, uniform, and incompressible flow, the Laplace equation should be satisfied.

Irrotational flow in terms of velocity potential function

Irrotational Flow by Velocity Potential Function

Stream Function ψ

The stream function is a scalar function such that its derivative along any direction gives the velocity component in the perpendicular direction, in the clockwise or anti-clockwise direction.

u = ∂ψ/∂y, v = -∂ψ/∂x

Irrotational flow in terms of stream function

Irrotational Flow by Stream Function

Note:

• Discharge between two points can be found as the difference in stream function between the two points.

• Vorticity for a two-dimensional fluid flow is given as, ∂v/∂x - ∂u/∂y

• Equation of streamline is given as, dx/u = dy/v

• ∂ψ/∂y = ∂φ/∂x

Fluid Flow Pattern

Streamline - curve line obtained during the flow of fluid particles such that a tangential drawn represents the resultant velocity of flow

Streamtube - formed by a number of streamlines

Pathline - line traced by fluid particles during a period of tie

Streak line - Locus of all fluid particles at any time instant which passes through a fixed point

We hope we have covered all the important details related to fluid kinematics. Test your knowledge by solving a GATE:2005 problem given below.