## 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**

**Fluid Dynamics and Flow Measurements**

**Fluid Dynamics**

Fluid Dynamics is the beginning of the determination forces which cause motion in fluids. This section includes various forces such as Inertia, Viscous, etc., Bernoulli's theorems, Vortex motion, forced motion etc.

Include momentum correction factor, impact of jets etc.

Dynamics is that branch of mechanics which treats the motion of bodies and the action of forces in producing or changing their motion.

**Flow rate**

**Mass flow rate**

**Volume flow rate - Discharge**- More commonly we use volume flow rate Also know as discharge. The symbol normally used for discharge is Q.

**Continuity**

This principle of conservation of mass says matter cannot be created or destroyed. This is applied in fluids to fixed volumes, known as control volumes (or surfaces).

- For any control volume, the principle of conservation of mass defines,

**Mass entering per unit time = Mass leaving per unit time + Increase of mass in control vol per unit time **

- For steady flow there is no increase in the mass within the control volume,

**Mass entering per unit time = Mass leaving per unit time**

**Applying to a stream-tube**

Mass enters and leaves only through the two ends (it cannot cross the stream tube wall).

for steady flow,

**ρ _{1}∂A_{1}u_{1 }= ρ_{2}∂A_{2}u_{2}= Constant= Mass flow rate**

**This is the continuity equation.**

**Some example applications of Continuity**

A liquid is flowing from left to right. By the continuity, **ρ _{1}A_{1}u_{1}_{ }= ρ_{2}A_{2}u_{2}**

**As we are considering a liquid, **

**ρ**

_{1}=ρ_{2}**Q**

_{1 }= Q_{2}**A**

_{1}u_{1}=A_{2}u_{2}### Velocities in pipes coming from a junction

**mass flow into the junction = mass flow out**

**ρ _{1}Q_{1}= ρ_{2}Q_{2 }+ ρ_{3}Q_{3}**

When incompressible,

**Q _{1}_{ }= Q_{2}_{ }+ Q_{3}**

**A _{1}u_{1}=A_{2}u_{2}_{ }+ A_{3}u_{3}**

**Vortex flow**

- This is the flow of rotating mass of fluid or flow of fluid along curved path.

### Free vortex flow

- No external torque or energy required. The fluid rotating under certain energy previously given to them. In a free vortex mechanics, overall energy flow remains constant. There is no energy interaction between an external source and a flow or any dissipation of mechanical energy in the flow.
- Fluid mass rotates due to the conservation of angular momentum.
- Velocity inversely proportional to the radius.
- For a free vortex flow

**vr= constant **

**v= c/r**

**At the center (r = 0) of rotation, velocity approaches to infinite, that point is called singular point.**- The free vortex flow is
**irrotational**, and therefore, also known as the irrotational vortex. - In free vortex flow, Bernoulli’s equation can be applied.

*Examples* include a whirlpool in a river, water flows out of a bathtub or a sink, flow in centrifugal pump casing and flow around the circular bend in a pipe.

### Forced vortex flow

- To maintain a forced vortex flow, it required a continuous supply of energy or external torque.
- All fluid particles rotate at the constant angular velocity ω as a solid body. Therefore, a flow of forced vortex is called as a solid body rotation.
- Tangential velocity is directly proportional to the radius.
- v = r ω
- ω = Angular velocity.
- r = Radius of fluid particle from the axis of rotation.

- The surface profile of vortex flow is parabolic.

- In forced vortex total energy per unit weight increases with an increase in radius.
- Forced vortex is not irrotational; rather it is a
**rotational flow**with constant vorticity 2ω.

** Examples **of forced vortex flow is rotating a vessel containing a liquid with constant angular velocity, flow inside the centrifugal pump.

**Energy Equations**

- This is the equation of motion in which the forces due to gravity and pressure are taken into consideration. The common fluid mechanics equations used in fluid dynamics are given below
- Let,
**Gravity force F**,_{g}**Pressure force F**_{p, }**Viscous force**F,_{v}**Compressibility force**F, and_{c}**Turbulent force F**_{t.}

**F _{net} = F_{g} + F_{p} + F_{v }+ F_{c} + F_{t}**

- If fluid is
**incompressible**, then**F**_{c}= 0

**∴ F_{net} = F_{g} + F_{p} + F_{v} + F_{t}**

This is known as **Reynolds equation of motion**.

- If fluid is
**incompressible and turbulence**is negligible, then,*F*_{c}= 0, F_{t}= 0

*∴* *F _{net} = F_{g} + F_{p} + F_{v}*

This equation is called as **Navier-Stokes equation**.

- If fluid flow is considered
**ideal**then, a**viscous effect**will also be negligible. Then

*F _{net} = F_{g} + F_{p}*

This equation is known as **Euler’s equation**.

- Euler’s equation can be written as:

**Bernoulli’s Equation**

It is based on law of conservation of energy. This equation is applicable when it is assumed that

- Flow is steady and irrotational
- Fluid is ideal (non-viscous)
- Fluid is incompressible

It states in a steady, ideal flow of an incompressible fluid, the total energy at any point of the fluid is constant.

The total energy consists of pressure energy, kinetic energy and potential energy or datum energy. These energies per unit weight of the fluid are:

- Pressure energy

- Kinetic energy

- Datum energy = z

Bernoulli’s theorem is written as:

- Bernoulli’s equation can be obtained by
**Euler’s equation**

As fluid is incompressible, ρ = constant

where,

**Restrictions inthe application of Bernoulli’s equation**- Flow is steady
- Density is constant (incompressible)
- Friction losses are negligible
- It relates the states at two points along a single streamline, (not conditions on two different streamlines)

**The Bernoulli equation is applied along streamlines like that joining points 1 and 2 **

Total head at 1 = Total head at 2

This equation assumes no energy losses (e.g. from friction) or energy gains (e.g. from a pump) along the streamline. It can be expanded to include these simply, by adding the appropriate energy terms

**Note Point:**

The Bernoulli equation is often combined with the continuity equation to find velocities and pressures at points in the flow connected by a streamline.

### Kinetic Energy Correction Factor (α)

In a real fluid flowing through a pipe or over a solid surface, the velocity will be zero at the solid boundary and will increase as the distance from the boundary increases. The kinetic energy per unit weight of the fluid will increase in a similar manner.

The kinetic energy in terms of **average velocity V** at the section and a kinetic energy correction factor α can be determined as:

In which m = ρAVdt is the total mass of the fluid flowing across the cross-section during dt. By comparing the two expressions for kinetic energy, it is obvious that,

The numerical value of α will always be greater than 1, taking kinetic energy correction factor, α, as

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