What is Kinematics of Flow?
Kinematics of flow deals with the motion of fluids without considering the forces causing the motion.
It describes:
How fluids move
Velocity of flow
Acceleration of flow
Flow patterns
(Not concerned with pressure and forces — that is Dynamics)
2. Types of Fluid Flow
(A) Steady and Unsteady Flow
Steady Flow
Fluid properties at a point do not change with time
∂
𝑉
∂
𝑡
=
0
∂t
∂V
=0
Example: Water flowing in a long straight pipe at constant discharge
Unsteady Flow
Fluid properties change with time
∂
𝑉
∂
𝑡
≠
0
∂t
∂V
=0
Example: Filling of a tank
(B) Uniform and Non-Uniform Flow
Uniform Flow
Velocity does not change with space
∂
𝑉
∂
𝑠
=
0
∂s
∂V
=0
Example: Flow in a long straight channel of constant section
Non-Uniform Flow
Velocity changes from point to point
Example: Flow near pipe entrance
(C) Laminar and Turbulent Flow
Laminar Flow
Fluid flows in smooth parallel layers
Occurs when:
𝑅
𝑒
<
2000
Re<2000
Turbulent Flow
Irregular, chaotic motion
Occurs when:
𝑅
𝑒
>
4000
Re>4000
(D) Compressible and Incompressible Flow
Type Description
Incompressible Density constant (liquids)
Compressible Density changes (gases)
(E) Rotational and Irrotational Flow
Type Description
Rotational Fluid particles rotate
Irrotational No rotation of particles
3. Flow Descriptions
(A) Lagrangian Method
Tracks individual fluid particles
Example: Tracking a floating object
(B) Eulerian Method
Studies flow at fixed points in space
Used in engineering analysis
4. Velocity and Acceleration in Fluid Flow
Velocity is a vector:
𝑉
⃗
=
𝑢
𝑖
^
+
𝑣
𝑗
^
+
𝑤
𝑘
^
V
=u
i
^
+v
j
^
+w
k
^
Where:
u, v, w = velocity components in x, y, z directions
Types of Acceleration
Total acceleration has two parts:
(A) Local Acceleration
Due to change of velocity with time
𝑎
𝑙
=
∂
𝑉
∂
𝑡
a
l
=
∂t
∂V
(B) Convective Acceleration
Due to change of velocity with position
𝑎
𝑐
=
𝑢
∂
𝑢
∂
𝑥
+
𝑣
∂
𝑢
∂
𝑦
+
𝑤
∂
𝑢
∂
𝑧
a
c
=u
∂x
∂u
+v
∂y
∂u
+w
∂z
∂u
5. Streamline, Pathline, Streakline
Streamline
Line drawn such that velocity vector is tangent at every point
No two streamlines can intersect
Pathline
Actual path followed by a fluid particle
Streakline
Locus of particles that passed through a fixed point
Example: Smoke from chimney
6. Continuity Equation
Based on conservation of mass
For incompressible flow:
𝐴
1
𝑉
1
=
𝐴
2
𝑉
2
=
𝑄
A
1
V
1
=A
2
V
2
=Q
Where:
A = Area
V = Velocity
Q = Discharge
7. One-Dimensional, Two-Dimensional, Three-Dimensional Flow
Type Velocity varies in
1-D One direction only
2-D Two directions
3-D All directions
8. Velocity Potential and Stream Function
Velocity Potential (ϕ)
Scalar function where:
𝑢
=
∂
𝜙
∂
𝑥
,
𝑣
=
∂
𝜙
∂
𝑦
u=
∂x
∂ϕ
,v=
∂y
∂ϕ
Exists for irrotational flow
Stream Function (ψ)
Defined so that:
𝑢
=
∂
𝜓
∂
𝑦
,
𝑣
=
−
∂
𝜓
∂
𝑥
u=
∂y
∂ψ
,v=−
∂x
∂ψ
Streamlines are given by:
𝜓
=
constant
ψ=constant
9. Example
Water flows through a pipe reducing from 0.04 m² to 0.01 m². If velocity at larger section is 2 m/s, find velocity at smaller section.
𝐴
1
𝑉
1
=
𝐴
2
𝑉
2
A
1
V
1
=A
2
V
2
0.04
×
2
=
0.01
×
𝑉
2
0.04×2=0.01×V
2
𝑉
2
=
8
𝑚
/
𝑠
V
2
=8m/s
10. Importance in Civil Engineering
Used in:
Pipe flow analysis
Open channel flow
Groundwater movement
Hydraulic machinery
River engineering
Tags:
Fluid Mechanics