Water flowing from a tap appears smooth when the flow rate is low, but as the flow rate increases beyond a certain value, disturbances and irregularities begin to appear. If a stream of ink is introduced when the flow is smooth, the ink moves in a straight line without mixing with other layers. However, when introduced during turbulent flow, the ink mixes with the surrounding water. In this section, we will study the first type, known as streamline or laminar flow.
Streamline flow in fluids is described as a flow in which the fluids flow in parallel layers with no interruption or mixing of the layers, and the velocity of each fluid particle passing by remains constant over time at a particular position.
There are no turbulent velocity variations at low fluid velocities, and the fluid tends to flow without lateral mixing. The fluid particles move in a specific order with regard to the particles travelling in a straight line parallel to the pipe wall so that the contiguous layers slide past each other like playing cards.
Streamline
The path traveled by fluid particles under steady flow conditions is defined as a streamline. The tangent at any point on the curve reveals the direction of the fluid velocity at that moment if the flow lines are represented as curves. The curves, as shown in the figure below, indicate how the fluid particles move in relation to time.
The curve represents the flow of the fluid particles in a steady state. This map is time-stationary, which means that every particle passing a point behaves precisely like the particle that came before it.
In a laminar flow, the streamlines follow the equation of continuity. i.e.,
Av = Constant,
where A is the fluid flow's cross-sectional area, and v is the fluid's velocity at that location.
The term Av is defined as the fluid's volume flux or flow rate, which is constant in a steady flow.
Let us consider the cross-sectional area at Point 1 to be A1 and at Point 2 to be A2,
Then, by the equation of continuity:
A1v1 = A2v2
This represents the inverse relationship between area and velocity. The velocity of the liquid decreases as the cross-section area increases, and vice versa.
Moreover, the flow of the water can also be determined by the quantity named Reynolds number, which determines whether a flow pattern is laminar or turbulent, which is discussed below:
Reynold's Number
The nature of flow in a pipe depends on the Reynolds number. If the Reynolds number is less than 2000, the flow is laminar, and if it is greater than 2000, the flow is turbulent. Values around 2000 indicate the transition between the two types of flow.
The formula for the Reynolds number is given as:
\boxed{Re = \frac{\rho V D}{\mu}} where:
- Re is the Reynolds number,
- p is the density of the fluid,
- V is the velocity of flow,
- D is the diameter of the pipe and
- u is the viscosity of the fluid.
Solved Problems
Question 1: On which theorem is the continuity equation based?
Solution: The flow of the fluid is assumed to be incompressible, laminar, and non-viscous for using the continuity equation. The continuity equation is based on the conservation of mass. The product of area and velocity gives the volume flow rate of fluid. Since, the density of fluid is the same at all points, the mass of fluid passing through two different regions is the same at the same time.
Question 2: A tube having a uniform cross-section is kept vertical in such a way that water enters from the top and leaves from the bottom. If the speed at a point Below the opening is ‘v.' What will be the speed at a point B vertically below A such that the distance between A & B is ‘2h’?
Solution: According to the equation of continuity, the velocity at B will be the same as that at A since the area of the cross-section is the same at both points. Although gravity will increase its velocity, pressure will also decrease downwards, and therefore, the net effect will be zero change in velocity.
Question 3: A horizontal pipeline carries water in streamlined flow. At a point along the pipe, where the cross-sectional area is 25 cm², the water velocity is 0.5 m/s. Find the velocity of the water at a point where the cross-sectional area is 2 cm².
Solution: Given
Area at point 1, A1 = 25 cm2
Area at point 2, A2 = 2 cm2
Velocity of point 1, v1 = 0.5 m/s
Let us consider the unknown velocity be v2.
According to the continuity equation,
A1v1 = A2v2
Rearrange the formula in terms of v2.
v2 = A1v1 / A2
Substitute all the values in the above expression.
v2 = (25 cm2) × (0.5 m/s) / (2 cm2)
= 6.25 m/s
Thus, the velocity of water at a point of the second cross-section is 6.25 m/s.
Question 4: Determine the flow of fluid having a relative density of 100 kg/m3 and a, the viscosity of 0.5 N s/m2, with a velocity of 5 m/s through a pipe of 0.2 m.
Solution: The type of flow can be determined by the value of the Reynolds number.
Given:
Velocity of fluid, V = 5 m/s
Diameter of pipe, D = 0.2 m
Relative density of fluid, p =100 kg/m3
Viscosity of fluid = 0.5 N s/m2
The formula of the Reynolds number is given as:
Re = (p × V × D) / u
Substitute all the values in the formula to calculate the Reynolds number.
Re = (100 kg/m3) × (5 m/s) × (0.2 m) / (0.5 N s/m2)
= 200
Since, the Reynolds number is less than 2000, the flow of liquid is laminar.
Question 5: Why do two streamlines never intersect?
Solution: The direction of the net velocity of the flow is given by the tangent at a point to the streamline.
If the two streamlines cross, it denotes two different directions of velocity at a point, which are not feasible. As a result, two streamlines cannot intersect.
Unsolved Problems
Question 1: Water flows steadily through a horizontal pipe of varying diameter. At a narrow section, the cross-sectional area is 0.01 m², and the velocity is 2 m/s. What is the velocity at a wider section where the area is 0.05 m²?
Question 2: A fluid with a density of 800 kg/m³ flows in a pipe of diameter 0.1 m at a velocity of 1.5 m/s. The viscosity of the fluid is 0.002 Pa·s. Determine whether the flow is laminar or turbulent.
Question 3: A small pipe carries oil with a velocity of 0.8 m/s. The pipe’s diameter is 0.05 m, and the oil has a density of 900 kg/m³ and a viscosity of 0.01 N·s/m². Calculate the Reynolds number and identify the type of flow.
Question 4: In a vertical pipe, water flows downward with a velocity of 1 m/s. If the pipe’s diameter suddenly reduces by half, what will be the new velocity of the water at the narrow section?
Question 5: A dye is introduced in a pipe to observe laminar flow. Explain qualitatively how the dye behaves in laminar flow and how it differs if the flow becomes turbulent.