## Fluid Kinematics

**Fluid Kinematics**deals with the motion of fluids such as displacement, velocity, acceleration, and other aspects. This topic is useful in terms of exam and knowledge of the candidate.- Kinematics is the branch of classical mechanics that describes the motion of bodies and systems without consideration of the forces the cause the motion.

### Types of Fluid Flows

Fluid flow may be classified under the following headings;

**Steady & Unsteady Flow**

**Uniform & Non-uniform Flow**

**Laminar & Turbulent Flow**

**Rotational & Irrotational Flow**

Combining these, the most common flow types are:

**Steady uniform flow**- Conditions do not change with position in the stream or with time.
- E.g. flow of water in a pipe of constant diameter at a constant velocity.

**Steady non-uniform flow**- Conditions change from point to point in the stream but do not change with time.
- E.g. Flow in a tapering pipe with constant velocity at the inlet.

**Unsteady uniform flow**- At a given instant in time the conditions at every point are the same but will change with time.
- E.g. A pipe of constant diameter connected to a pump pumping at a constant rate which is then switched off.

**Unsteady non-uniform flow**- Every condition of the flow may change from point to point and with time at every point.
- E.g. Waves in a channel

**Flow Pattern**

Three types of fluid element trajectories are defined: **Streamlines, Pathlines, **and** Streaklines**.

**Pathline**is the actual path travelled by an individual fluid particle over some time period. The pathline of a fluid element A is simply the path it takes through space as a function of time. An example of a pathline is the trajectory taken by one puff of smoke which is carried by the steady or unsteady wind.**Timeline**is a set of fluid particles that form a line at a given instant.**Streamline**is a line that is everywhere tangent to the velocity field. Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity field as illustrated in the figure below:

where u,v, and w are the velocity components in x, y and z directions respectively as sketched

**Streakline**is the locus of particles that have earlier passed through a prescribed point.A streakline is associated with a particular point P in space which has the fluid moving past it. All points which pass through this point are said to form the streakline of point P. An example of a streakline is the continuous line of smoke emitted by a chimney at point P, which will have some curved shape if the wind has a time-varying direction**Streamtube**: The streamlines passing through all these points form the surface of a stream-tube. Because there is no flow across the surface, each cross-section of the streamtube carries the same mass flow. So the streamtube is equivalent to a channel flow embedded in the rest of the flow field.

**Note:**

- The figure below illustrates
**streamlines, pathlines**, and**streaklines**for the case of a smoke being continuously emitted by a chimney at point P, in the presence of a shifting wind. - In a steady flow, streamlines, pathlines, and streaklines all coincide.
- In this example, they would all be marked by the smoke line.

**Velocity of Fluid Particle**

- Velocity of a fluid along any direction can be defined as the rate of change of displacement of the fluid along that direction
- Let
**V**be the resultant velocity of a fluid along any direction and,*u***v**and**w**be the velocity components inand*x, y*directions respectively.*z* - Mathematically the velocity components can be written as

**u = f ( x, y, z, t )**

**w = f ( x, y, z, t )**

**v = f ( x, y, z, t )**

- Let
*V*is resultant velocity at any point in a fluid flow._{R} - Resultant velocity VR
**=**u**i +**v**j +**w**k**

Where **u=dx/dt, v=dy/**dt** and w=dz/dt** are the resultant vectors in X, Y and Z directions, respectively.

### **Acceleration of Fluid Particle**

- Acceleration of a fluid element along any direction can be defined as the rate of change of velocity of the fluid along that direction.
- If
*a*_{x}*, a*and_{y}*a*are the components of acceleration along_{z}and*x, y*directions respectively, they can be mathematically written as*z**a*_{x}= du/ dt.

**Stream Function**

- The partial derivative of stream function with respect to any direction gives the velocity component at right angles to that direction. It is denoted by
**ψ**.

- Continuity equation for two-dimensional flow is

**Equations of Rotational Flow**

- As ψ satisfies the continuity equation hence if ψ exists then it is a possible case of fluid flow.
- Rotational components of fluid particles are:

**Equation of Irrotational Flow**

- If
then, flow is irrotational.*ω*_{x}= ω_{y}= ω_{z} - For irrotational flow,
*ω*= 0_{z}

- This is
**Laplace equation**for ψ.

**Note: **It can be concluded that if stream function (ψ) exits, it is a possible case of fluid flow. But we can’t decide whether flow is rotational or irrotational. But if stream function ψ satisfies Laplace equation then, it is a possible case of irrotational flow otherwise it is rotational flow.

**Velocity Potential Function**

- It is a scalar function of space and time such that its negative derivative with respect to any direction gives the fluid velocity in that direction. It is denoted b φ

We know that continuity equation for steady flow is

- If φ satisfies the Laplace equation, then it is a possible case of fluid flow.

**Rotational component ω_{z}** can be given by

- It shows that φ exits then, flow will be irrotational.

**Relation between Stream Function and Velocity Potential**

We know,

and

**Stream** versus **Velocity Function**

**Equipotential Line** versus **Stream Line**

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