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Mechanics

Streamline Flow in Fluids

Streamline flow (or laminar flow) is a regime of fluid motion in which each particle of the fluid follows a smooth path, called a streamline, such that the paths of different particles never cross each other. At any given point in the fluid, the velocity vector of the fluid particles remains constant with time in both magnitude and direction.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Definition of streamline, laminar vs turbulent flow, equation of continuity (A1v1 = A2v2), Bernoulli's theorem and its direct applications like Venturi meter and Torricelli's law.

PGT extension

Euler's equation of motion for inviscid fluids, derivation of Bernoulli's equation from work-energy theorem, velocity potential and stream function for two-dimensional irrotational flow, vorticity.

Prerequisites

Equation of continuity and conservation of mass
Conservation of mechanical energy
Concept of pressure, density, and viscosity in fluids

Definition and Characteristics of Streamlines

A streamline is an imaginary curve drawn in a fluid flow such that the tangent at any point on the curve gives the direction of the velocity vector of the fluid particle at that point.
Two streamlines can never intersect, because if they did, the fluid particle at the point of intersection would have two different velocity vectors simultaneously, which is physically impossible.
In a steady flow, streamlines coincide with the actual path (trajectory) of individual fluid particles.

Equation of Continuity

Based on the law of conservation of mass for an incompressible fluid flowing through a tube of varying cross-section.
The product of the cross-sectional area (A) and the fluid velocity (v) remains constant along any streamline: Av = constant.
Physical interpretation: Where the pipe is narrower, the fluid speed is greater; where the pipe is wider, the fluid speed is smaller.

Bernoulli's Theorem

Derived from the work-energy theorem applied to a non-viscous, incompressible, and streamline fluid flow.
States that the sum of pressure energy per unit volume, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline.
Mathematical expression: P + (1/2)ρv² + ρgh = constant.

Conditions for Streamline Flow

The flow velocity at any fixed point must not change with time (steady flow condition).
The Reynolds number (R_e) must remain well below the critical value for the given geometry (typically R_e < 2000 in pipes).
Fluid viscosity and internal friction must not cause turbulent eddies under the operating flow speed.

Derivations

Equation of Continuity Derivation

Conditions: Incompressible fluid, steady flow through a rigid pipe of varying cross-section

Consider a section of a pipe with cross-sectional areas A₁ and A₂ at two points 1 and 2.
Let the fluid enter point 1 with speed v₁ and density ρ, and leave point 2 with speed v₂ and density ρ.
Mass entering in time interval dt is dm₁ = ρ A₁ v₁ dt.
Mass leaving in time interval dt is dm₂ = ρ A₂ v₂ dt.
By conservation of mass for an incompressible fluid (ρ₁ = ρ₂ = ρ), dm₁ = dm₂.

Result: A₁ v₁ = A₂ v₂

Bernoulli's Equation Derivation

Conditions: Streamline, steady, irrotational flow of an ideal (inviscid and incompressible) fluid

Consider a fluid element moving from a lower cross-section to a higher cross-section under pressure forces and gravity.
Work done by the pressure forces per unit volume during displacement is W_pressure = P₁ - P₂.
Work done by gravity against the weight of the fluid element per unit volume is W_gravity = ρ g h₁ - ρ g h₂.
By work-energy theorem, total net work done per unit volume equals the change in kinetic energy per unit volume: (&frac{1}{2}ρ v₂² - &frac{1}{2}ρ v₁²).
Rearranging terms by grouping parameters of state 1 on one side and state 2 on the other yields the constant sum.

Result: P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Equation of Continuity	$$A_1 v_1 = A_2 v_2$$	$A = Cross-sectional area$, v = Flow velocity m², m s^-1	Incompressible fluid in steady flow
Bernoulli's Theorem	$$P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$$	P = Static pressure, $\rho = Density$, v = Velocity, g = Acceleration due to gravity, h = Height Pa (N m^-2)	Inviscid, incompressible, steady, streamline flow
Torricelli's Law (Velocity of Efflux)	$$v = \sqrt{2gh}$$	v = Velocity of efflux, g = Acceleration due to gravity, h = Depth of orifice below free surface m s^-1	Large tank cross-section compared to orifice
Venturi Meter Flow Speed	$$v_1 = A_2 \sqrt{\frac{2(P_1 - P_2)}{\rho(A_1^2 - A_2^2)}}$$	A₁, A₂ = Areas, P₁ - P₂ = Pressure difference, $\rho = Density$ m s^-1	Horizontal pipe configuration
Reynolds Number	$$R_e = \frac{\rho v d}{\eta}$$	$\rho = Fluid density$, v = Flow speed, d = Characteristic dimension (diameter), $\eta = Dynamic viscosity$ Dimensionless	Determines transition from streamline to turbulent flow
Dynamic Pressure	$$q = \frac{1}{2}\rho v^2$$	q = Dynamic pressure, $\rho = Density$, v = Flow speed Pa	Incompressible streamline flow
Rate of Volume Flow (Discharge)	$$Q = A v = \frac{dV}{dt}$$	Q = Volume flow rate, A = Area, v = Velocity, V = Volume, t = Time m³ s^-1	Steady flow
Work-Energy Form of Bernoulli's Equation per unit mass	$$\frac{P}{\rho} + \frac{1}{2}v^2 + gh = \text{constant}$$	P = Pressure, $\rho = Density$, v = Velocity, g = Acceleration due to gravity, h = Height J kg^-1	Ideal fluid flow

Graphs and experiments

Velocity Profile in Laminar vs Turbulent Flow

Draw: A parabolic velocity profile across the diameter of a pipe for laminar flow, contrasted with a flattened (blunt) profile for turbulent flow.

Interpret: In streamline flow, maximum velocity occurs at the center of the pipe and zero velocity occurs at the walls due to no-slip condition.

Exam use: Tested conceptually in TGT/PGT to identify parabolic distribution with maximum velocity at axis.

Venturi Tube Pressure Head Variation

Draw: A horizontal pipe constriction connected with vertical manometric tubes showing liquid column heights dropping at the throat.

Interpret: At the narrower throat, velocity is higher, causing static pressure to drop according to Bernoulli's principle.

Exam use: Numerical and theoretical questions regarding pressure difference and manometer height differences.

Reynolds Dye Experiment

Draw: A schematic of a glass tank with a central filament of colored dye injected into flowing water.

Interpret: At low velocities, the dye filament remains a straight, distinct line (streamline flow). Above critical velocity, the filament breaks up into eddies (turbulent flow).

Exam use: Questions concerning the criterion for transition from streamline to turbulent flow.

Solved numericals

TGT core

Water flows through a horizontal pipe whose internal diameter changes from 4 cm at point A to 2 cm at point B. If the speed of water at point A is 2 m/s, find the speed of water at point B.

Identify given parameters: D₁ = 4 cm, D₂ = 2 cm, v₁ = 2 m/s.
Apply the equation of continuity: A₁v₁ = A₂v₂.
Since area is proportional to the square of diameter: (D₁²)v₁ = (D₂²)v₂.
Substitute values: (4)² × $2 = (2)^{2} \times v_{2}$.
Calculate v₂: 16 × $2 = 4 \times v_{2} = \rangle v_{2} = 32 / 4 = 8 m/s$.

Answer: 8 m s^-1

PGT extension

A horizontal venturi meter with a throat-to-main-area ratio of 1:2 is used to measure the flow rate of water ($\rho = 1000 kg/m^{3})$. If the manometer fluid indicates a pressure difference of 24 kPa between the main section and the throat, calculate the volume flow rate Q.

Let main area be A₁ and throat area be A₂. Given A₂/A₁ = 1/2, so A₁ = 2A₂.
Pressure difference Δ$P = P_{1} - P_{2} = 24000 Pa$.
Use Venturi meter velocity formula for v₁: v₁ = A₂ √[ 2ΔP / (ρ(A₁² - A_2²)) ].
Substitute A₁ = 2A₂: v₁ = A₂ √[ 2ΔP / (ρ(4A₂² - A₂²)) ] = A₂ √[ 2ΔP / (3ρA₂²) ] = √[ 2ΔP / (3ρ) ].
Calculate v₁: √[ 2 × 24000 / (3 × 1000) ] = √[ 48000 / 3000 ] = √$16 = 4 m/s$.
Volume flow rate $Q = A_{1} v_{1}$. Alternatively, compute using $Q = A_{2} v_{2} = A_{2} A_{1} \sqrt{}[$...] / A₂. Let us find v₂ from continuity: A₁v₁ = A₂v₂ => 2A₂(4) = A₂v₂ => v₂ = 8 m/s. Thus $Q = A_{2}v_{2} = A_{2} \times 8$. (Assuming unit area or expressing symbolically if area not given; let's assume A₂ = 0.01 m² for numerical completion: $Q = 0.01 \times 8 = 0.08 m^{3}/s)$.

Answer: 0.08 m^3 s^-1 (for $A_2 = 0.01 m^2)$

TGT core

A small orifice is punched at a depth of 2.0 m below the free surface of water in a large tank. Calculate the velocity of efflux of water from the orifice. (Take $g = 9.8 m/s^{2})$

Identify depth h = 2.0 m and $g = 9.8 m/s^{2}$.
Apply Torricelli's law: $v = \sqrt{2gh}$.
Substitute the values: $v = \sqrt{2 \times 9.8 \times 2.0}$.
Calculate: $v = \sqrt{39.2} \approx 6.26 m/s$.

Answer: 6.26 m s^-1

Assumptions and limitations

The fluid is assumed to be ideal: non-viscous (zero internal friction) and incompressible (constant density).
The flow must be steady, meaning fluid properties at any point do not vary with time.
Real fluids possess viscosity; hence Bernoulli's equation and streamline relations hold strictly only as approximations away from boundary walls where viscous shear stresses are negligible.

Common misconceptions

Common error: Streamlines represent the physical walls or boundaries of a pipe.

Correct: Streamlines are purely mathematical curves drawn tangent to velocity vectors inside the fluid body.

Streamlines exist throughout the fluid continuum, not just at solid boundaries.

Common error: Higher fluid velocity implies higher static pressure according to Bernoulli's theorem.

Correct: Higher fluid velocity corresponds to lower static pressure along a streamline.

Since kinetic energy per unit volume increases with velocity, static pressure must drop to keep the total sum constant.

Common error: Streamlines can intersect if the fluid flow accelerates sharply.

Correct: Streamlines can never intersect under any steady flow condition.

Intersection would imply two distinct velocity vectors at a single spatial point for a single fluid particle.

Quick revision

In streamline flow, velocity at any point is constant over time and streamlines never cross.
Equation of Continuity: A_1 $v_1 = A_2 v_2$
Bernoulli's Theorem: P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}

Must remember

A streamline is an imaginary curve such that the tangent at any point gives the direction of the fluid velocity vector.
Two streamlines can never intersect, as it would imply two velocity vectors at a single point.
The equation of continuity $A_1 v_1 = A_2 v_2$ expresses the conservation of mass for an incompressible fluid.
Bernoulli's theorem states that $P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$ along a streamline for an inviscid, incompressible fluid.
The Reynolds number $R_e = \frac{\rho v d}{\eta}$ determines whether fluid flow remains streamline or becomes turbulent ($R_e < 2000$ typically indicates streamline flow in pipes).
Torricelli's law gives the velocity of efflux as $v = \sqrt{2gh}$ where $h$ is the depth of the orifice below the free surface.
Higher fluid velocity along a streamline corresponds to lower static pressure, derived directly from conservation of energy.
For two-dimensional irrotational flow, velocity potential and stream function can be defined to analyze streamline behavior.
Euler's equation of motion forms the foundational differential basis for inviscid fluid flow before integration yields Bernoulli's theorem.
Dynamic pressure is defined as $q = \frac{1}{2}\rho v^2$ and represents the kinetic energy per unit volume of the moving fluid.

Glossary

StreamlineAn imaginary curve whose tangent at any point indicates the instantaneous velocity vector of the fluid particle.

Equation of ContinuityThe mathematical statement $A_1 v_1 = A_2 v_2$ reflecting mass conservation for steady, incompressible fluid flow.

Bernoulli's TheoremThe conservation of mechanical energy principle applied per unit volume along a streamline in an ideal fluid: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$.

Reynolds Number ($R_e$)A dimensionless quantity used to predict flow transitions from laminar/streamline to turbulent flow regimes.

Torricelli's LawThe speed of efflux of a liquid through an orifice at a depth $h$ below the free surface is equal to that of a body falling freely through height $h$.

Venturi MeterA device used for measuring the rate of flow of a fluid through a pipe using differential pressure.

Dynamic PressureThe kinetic energy density of a flowing fluid, given by $q = \frac{1}{2}\rho v^2$.

Inviscid FluidA theoretical fluid with zero viscosity ($\eta = 0$), experiencing no internal friction during flow.

Velocity PotentialA scalar function whose spatial derivatives give the fluid velocity components in irrotational flow.

Stream FunctionA scalar function in two-dimensional flow whose constant values define streamlines.

Streamline Flow vs Turbulent Flow

Parameter	Streamline (Laminar) Flow	Turbulent Flow
Path of particles	Well-defined, non-intersecting trajectories	Erratic, crossing, and fluctuating paths with eddies
Reynolds number ($R_e$)	Typically $R_e < 2000$	Typically $R_e > 4000$
Fluid velocity	Constant or smoothly varying at any fixed point	Fluctuates erratically in magnitude and direction with time
Governing principles	Strictly obeys equation of continuity and Bernoulli's theorem	Complex viscous dissipation, heavily reliant on empirical friction factors

Practice Quiz

Choose an option to reveal the explanation.

1. What does the tangent drawn at any point on a streamline represent?

Explanation: By definition, a streamline is an imaginary curve such that the tangent at any point gives the direction of the velocity vector of the fluid particle at that location.

2. Why can two streamlines never intersect each other in a steady fluid flow?

Explanation: If two streamlines intersected, a single fluid particle arriving at that point would have two distinct velocity vectors simultaneously, which is physically impossible.

3. The equation of continuity $A_1 v_1 = A_2 v_2$ is a direct consequence of which conservation law?

Explanation: The equation of continuity is derived directly from the law of conservation of mass applied to an incompressible fluid flowing through a tube of varying cross-section.

4. In a horizontal pipe of varying cross-section, where the cross-sectional area is smaller, the fluid speed is:

Explanation: From the equation of continuity $Av = \text{constant}$, area and velocity are inversely proportional. Thus, a smaller area results in a greater fluid speed.

5. Bernoulli's theorem is derived by applying which theorem to an inviscid, incompressible fluid flow?

Explanation: Bernoulli's theorem is derived from the work-energy theorem applied along a streamline for a non-viscous and incompressible fluid.

6. Which of the following represents the correct mathematical expression for Bernoulli's theorem per unit volume?

Explanation: Bernoulli's equation per unit volume states that the sum of static pressure ($P$), kinetic energy per unit volume ($\frac{1}{2}\rho v^2$), and potential energy per unit volume ($\rho gh$) remains constant along a streamline.

7. What is the dimensionless quantity used to determine the transition from streamline to turbulent flow in pipes?

Explanation: The Reynolds number ($R_e = \frac{\rho v d}{\eta}$) is the dimensionless parameter used to determine whether a fluid flow regime is laminar/streamline or turbulent.

8. For streamline flow in a cylindrical pipe, the Reynolds number should typically remain below what value?

Explanation: In smooth circular pipes, flow remains laminar or streamline when the Reynolds number $R_e < 2000$.

9. Torricelli's law gives the velocity of efflux of liquid from an orifice at a depth $h$ below the free surface as:

Explanation: Torricelli's law states that the velocity of efflux is given by $v = \sqrt{2gh}$, identical to the velocity of a freely falling body from height $h$.

10. According to Bernoulli's theorem, if the velocity of a fluid increases along a streamline, what happens to its static pressure?

Explanation: Because the total sum of pressure energy, kinetic energy, and potential energy per unit volume is constant along a streamline, an increase in velocity (kinetic energy term) requires a corresponding decrease in static pressure.

11. What is the unit of dynamic pressure $q = \frac{1}{2}\rho v^2$?

Explanation: Dynamic pressure represents energy per unit volume, which has SI units of Pascals ($\text{Pa}$) or $\text{N m}^{-2}$, identical to static pressure.

12. In PGT physics, for a two-dimensional irrotational flow, the fluid velocity can be expressed as the negative gradient of which scalar function?

Explanation: In irrotational flow, a velocity potential $\phi$ exists such that the velocity vector is given by $\vec{v} = -\nabla \phi$.

13. What is the vorticity of a fluid flow that is classified as irrotational?

Explanation: By definition, an irrotational flow has zero vorticity, meaning the curl of the velocity field is zero ($\nabla \times \vec{v} = 0$).

14. Which of the following conditions is NOT required for the validity of Bernoulli's theorem along a streamline?

Explanation: Bernoulli's theorem requires the fluid to be inviscid, incompressible, steady, and streamline. Thermal conductivity is not a primary condition for its mechanical derivation.

15. Euler's equation of motion for an ideal fluid relates fluid acceleration and pressure gradients assuming which property?

Explanation: Euler's equation of motion is formulated for inviscid (zero viscosity) fluids, equating inertial forces to pressure and body forces.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. Water flows through a horizontal pipe whose internal diameter is $4\text{ cm}$ at one end and $2\text{ cm}$ at the other end. If the velocity of water at the larger end is $2\text{ m s}^{-1}$, what is the velocity of water at the smaller end?

Explanation: Using the equation of continuity $A_1 v_1 = A_2 v_2$ or $\pi r_1^2 v_1 = \pi r_2^2 v_2$. Given $d_1 = 4\text{ cm}$ ($r_1 = 2$), $d_2 = 2\text{ cm}$ ($r_2 = 1$), and $v_1 = 2\text{ m s}^{-1}$. Thus, $(2)^2 \times 2 = (1)^2 \times v_2 \implies 4 \times 2 = 1 \times v_2 \implies v_2 = 8\text{ m s}^{-1}$.

2. An open tank filled with water has a small hole pierced at a depth of $5\text{ m}$ below the free water surface. Taking $g = 10\text{ m s}^{-2}$, the velocity of efflux of water through the hole is:

Explanation: According to Torricelli's law, velocity of efflux $v = \sqrt{2gh} = \sqrt{2 \times 10 \times 5} = \sqrt{100} = 10\text{ m s}^{-1}$.

3. If the Reynolds number for a fluid flowing through a circular tube exceeds $4000$, the flow is predominantly:

Explanation: When the Reynolds number exceeds 4000, inertial forces dominate over viscous forces, leading to chaotic, irregular, and turbulent flow.

4. At what point along a streamline in a horizontal Venturi meter is the static pressure minimum?

Explanation: At the narrowest cross-section, fluid velocity is maximum according to the equation of continuity. By Bernoulli's theorem, maximum velocity corresponds to minimum static pressure.

5. Which of the following represents the work-energy form of Bernoulli's equation expressed per unit mass?

Explanation: Dividing Bernoulli's equation per unit volume by density $\rho$ yields the work-energy form per unit mass: $\frac{P}{\rho} + \frac{1}{2}v^2 + gh = \text{constant}$.

6. Why do streamlines never cross each other?

Explanation: If streamlines crossed, it would mean a fluid particle at that intersection possesses two different instantaneous velocities simultaneously, which is impossible in steady streamline flow.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. If the radius of a pipe carrying fluid decreases to half of its original value while maintaining the volume flow rate constant, by what factor does the flow velocity increase?

Explanation: Since $Q = A v = \pi r^2 v = \text{constant}$, velocity $v \propto \frac{1}{r^2}$. If radius $r$ becomes $\frac{r}{2}$, velocity increases by a factor of $(2)^2 = 4$.

2. In Bernoulli's equation, what physical quantity does the term $\rho gh$ represent?

Explanation: The term $\rho gh$ represents gravitational potential energy per unit volume of the fluid at height $h$ above a reference level.

3. What is the dimensionless combination of variables that constitutes the Reynolds number $R_e = \frac{\rho v d}{\eta}$?

Explanation: The Reynolds number is fundamentally the ratio of inertial forces ($\rho v^2$) to viscous forces ($\eta v / d$) acting on fluid elements.

4. A liquid flows through a pipe of varying cross section. At a point where the cross-sectional area is $0.02\text{ m}^2$, the flow speed is $3\text{ m s}^{-1}$. What is the volume flow rate $Q$?

Explanation: Volume flow rate $Q = A v = 0.02\text{ m}^2 \times 3\text{ m s}^{-1} = 0.06\text{ m}^3\text{ s}^{-1}$.

5. For a two-dimensional fluid flow, if the stream function $\psi = \text{constant}$ along a curve, what does that curve represent?

Explanation: By definition in fluid mechanics, lines along which the stream function $\psi$ is constant correspond precisely to streamlines of the flow.

6. How does dynamic pressure $q$ change if the velocity of an incompressible fluid is tripled?

Explanation: Dynamic pressure $q = \frac{1}{2}\rho v^2$ is proportional to $v^2$. If velocity $v$ is tripled ($3v$), dynamic pressure increases by $3^2 = 9$ times.

7. Which of the following assumptions is fundamental to Euler's equation of motion for fluids?

Explanation: Euler's equation of motion is derived by applying Newton's second law to an ideal fluid element, specifically assuming zero viscosity (inviscid fluid).

8. If a fluid is irrotational, which mathematical condition must be satisfied by its velocity field $\vec{v}$?

Explanation: A flow is irrotational if the curl of the velocity vector is zero, i.e., $\nabla \times \vec{v} = 0$.

9. In a steady streamline flow, what is the relationship between streamlines and particle trajectories?

Explanation: In a steady flow condition, the velocity at any given point does not change with time, causing streamlines to coincide exactly with the actual paths (trajectories) traveled by fluid particles.

10. A horizontal tube has areas $A_1 = 4\text{ cm}^2$ and $A_2 = 2\text{ cm}^2$. If the pressure difference between the two sections is $\Delta P$, what causes this pressure difference according to Bernoulli's principle?

Explanation: In a horizontal pipe, height $h$ is constant, so potential energy terms cancel. The pressure difference arises entirely from the change in kinetic energy per unit volume necessitated by the equation of continuity across differing areas.