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Mechanics
Streamlined Flow in Hydrodynamics
Streamlined 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 respect to time.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of streamline, laminar vs turbulent flow, equation of continuity for incompressible fluids, and basic applications of Bernoulli's theorem.
PGT extension
Velocity potential and stream function in two-dimensional flows, Euler's equation of motion along a streamline, and irrotational conditions linked to angular momentum and central forces.
Prerequisites
Centre of mass and translational kinematics
Angular momentum conservation in central forces
Pressure, density, and basic Newtonian mechanics
Definition and Characteristics of Streamlines
A streamline is an imaginary curve drawn in a fluid such that the tangent at any point on it gives the direction of the velocity vector of the fluid particle at that point.
Two streamlines can never intersect, as an intersection would imply two different velocity vectors at a single point, violating the steady-state assumption.
In steady flow, streamlines coincide identically with particle pathlines and streaklines.
Equation of Continuity
Based on the principle of conservation of mass applied to an incompressible fluid flowing through a tube of varying cross-sectional area.
The product of the cross-sectional area (A) and the fluid velocity (v) remains constant along any flow tube: Av = constant.
Physical interpretation: Where the pipe constricts, the velocity of the fluid increases; where the pipe widens, the velocity decreases.
Condition for Streamlined to Turbulent Transition
Determined by the dimensionless Reynolds number (Re = ρvd/η).
For flow in a cylindrical pipe, streamlined flow is strictly maintained when Re is less than the critical value (typically ~2000).
Higher velocities, larger tube diameters, or lower fluid viscosities destabilize streamlines, transitioning the flow into chaotic turbulence.
Irrotational vs Rotational Flow
In an ideal streamlined flow, if fluid elements possess no net angular velocity about their center of mass ($\omega = 0)$, the flow is termed irrotational.
Irrotational flow permits the definition of a scalar velocity potential function (φ), simplifying mathematical analysis.
Viscous boundary layers introduce local rotation (vorticity), modifying pure potential streamlined flow near solid boundaries.
Derivations
Equation of Continuity Derivation
Conditions: Steady, one-dimensional flow of an incompressible fluid through a rigid pipe
Consider a non-uniform pipe segment with cross-sectional areas A1 and A2 at entry and exit.
Let the fluid densities be ρ1 and ρ2, and velocities be v1 and v2 respectively.
In a small time interval Δt, the mass entering the left cross-section is m1 = A1 v1 ρ1 Δt.
Similarly, the mass leaving the right cross-section is m2 = A2 v2 ρ2 Δt.
By conservation of mass for a steady flow, m1 = m2, giving A1 v1 ρ1 = A2 v2 ρ2.
For an incompressible fluid (ρ1 = ρ2 = ρ), this simplifies to A1 v1 = A2 v2].
Result: A v = \text{constant}
Euler's Equation of Motion Along a Streamline
Conditions: Inviscid, steady flow along a streamline under conservative body forces
Consider a fluid element of mass d$m = A ds \rho along a streamline$, where ds is the length element.
Forces acting on the element in the direction of flow are pressure forces and component of gravity: (P)A - (P + \frac{\partial P}{\partial s} ds)A - \rho A ds g \cos\theta = dm \cdot a.
Substitute d$m = \rho A ds$ and acceleration $a = v \frac{\dv}{ds}$.
Simplify and divide throughout by ρ A ds, utilizing \cos\theta = \frac{dz}{ds} where z is vertical height.
Rearrange terms to group pressure, kinetic energy, and potential energy gradients together.
Result: \frac{\partial P}{\rho} + v \, dv + g \, dz = 0
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, steady fluid flow
Volumetric Flow Rate (Discharge)
$$Q = A v = \frac{V}{t}$$
Q = Discharge rate, V = Volume, t = Time m3 s-1
Steady flow
Reynolds Number
$$R_e = \frac{\rho v d}{\eta}$$
$\rho = Density$, v = Velocity, d = Characteristic dimension (diameter), $\eta = Coefficient of viscosity$ Dimensionless
Cylindrical pipe flow
Bernoulli's Equation (Energy Conservation per unit volume)
$$P + \frac{1}{2}\rho v^2 + \rho g z = \text{constant}$$
P = Static pressure, $\rho = Density$, v = Velocity, g = Acceleration due to gravity, z = Elevation N m-2 (Pa)
$\rho = Density$, &vec;v = Velocity, P = Pressure, &vec;g = Acceleration due to gravity N m-3
Non-viscous fluid
Mass Flux Rate
$$\frac{dm}{dt} = \rho A v$$
m = Mass, t = Time, $\rho = Density$, A = Area, v = Velocity kg s-1
Steady state flow
Graphs and experiments
Velocity Profile in Laminar Pipe Flow
Draw: A parabolic velocity distribution curve across the diameter of a circular pipe, showing maximum velocity at the center and zero velocity at the inner walls (no-slip condition).
Interpret: Viscous drag retards fluid layers near the wall, whereas the central core experiences minimal shear constraint, forming a parabolic profile.
Exam use: Tested conceptually in TGT/PGT matching questions regarding parabolic distribution vs plug flow.
Reynolds Dye Experiment Setup
Draw: A glass tank containing water with a central fine nozzle injecting colored dye into a horizontal glass tube where flow rate can be controlled via a valve.
Interpret: At low velocities, the dye forms a distinct, straight, unblended filament (streamlined). As velocity increases past critical, the filament wavers and disperses completely (turbulent).
Exam use: Direct theoretical foundation questions regarding the experimental criteria of critical velocity and Reynolds number.
Streamlines Around an Obstacle (Airfoil/Sphere)
Draw: Smooth, continuous flow lines diverging and converging symmetrically around a streamlined body without detachment points.
Interpret: Demonstrates irrotational flow conservation where flow lines crowd over the upper curved surface of an airfoil, indicating localized velocity increase.
Exam use: Used in PGT questions analyzing pressure differentials and lift generation.
Solved numericals
TGT core
Water flows through a horizontal pipe whose internal diameter changes from 4 cm to 2 cm. If the speed of water at the wider section is 2 m s-1, find its speed at the narrower section.
Identify given values: d1 = 4 cm, v1 = 2 m s-1, d2 = 2 cm.
Area is proportional to diameter squared: A ∝ d2.
Apply the equation of continuity: A1 v1 = A2 v2 ⇒ d12 v1 = d22 v2.
Substitute the values: (4)2 × $2 = (2)^{2} \times v_{2}$.
Solve for v2: 16 × $2 = 4 \times v_{2} \Rightarrow v_{2} = 32 / 4 = 8 m s^{-1}$.
Answer: 8 m s-1
PGT extension
A two-dimensional incompressible flow field is given by the velocity components u = 2xy and $v = x^{2} - y^{2}$. Determine whether the flow is possible (satisfies continuity) and if it is irrotational.
Substitute: ωz = 1/2 (2x - 2x) = 0. Thus, the flow is irrotational.
Answer: Flow is possible (incompressible) and irrotational
PGT extension
Calculate the maximum velocity for steady streamlined flow of water (viscosity $\eta = 10^{-3} Pa s$, density $\rho = 1000 kg m^{-3}) through a pipe of diameter 5 cm$, assuming the critical Reynolds number is 2000.
Write down the Reynolds number formula: $R_e = \rho v d / \eta$.
Rearrange to solve for velocity v: $v = (R_e \times \eta) / (\rho \times d)$.
Substitute given values: Re = 2000, $\eta = 10^{-3} Pa s$, $\rho = 1000 kg m^{-3}$, d = 0.05 m.
$v = (2000 \times 10^{-3}) / (1000 \times 0.05)$.
$v = 2 / 50 = 0.04 m s^{-1}$.
Answer: 0.04 m s-1
Assumptions and limitations
The fluid is assumed to be completely incompressible ($\rho = constant)$, which is valid for liquids but breaks down for high-speed gases where compressibility effects dominate.
Internal friction or fluid viscosity is neglected in ideal streamlined flow derivations like Bernoulli's theorem, requiring correction via Navier-Stokes equations for real viscous fluids near boundaries.
Common misconceptions
Common error: Streamlines represent the physical trajectory of a single moving fluid particle under all conditions.
Correct: Streamlines represent instantaneous velocity field tangents; they match particle pathlines only when the flow is strictly steady.
In unsteady flow fields, the velocity vector at a spatial point changes with time, causing pathlines and streamlines to diverge.
Common error: An increase in fluid velocity inside a constricted pipe implies an increase in pressure due to crowding.
Correct: An increase in velocity results in a decrease in static pressure, as dictated by Bernoulli's energy conservation principle.
Kinetic energy per unit volume increases at constrictions, requiring a corresponding drop in static pressure energy term.
Quick revision
Streamlines never intersect and their tangent gives the instantaneous flow direction.
The continuity equation (A1v1 = A2v2) stems directly from conservation of mass for incompressible fluids.
Critical Reynolds number (~2000 in pipes) marks the boundary between laminar (streamlined) and turbulent flow regimes.
Must remember
A streamline is an imaginary curve whose tangent at any point indicates the direction of the fluid velocity vector.
Two streamlines can never intersect, as an intersection would imply two different velocity vectors at a single spatial point.
The Equation of Continuity $A_1 v_1 = A_2 v_2$ is based on the conservation of mass for an incompressible fluid.
Reynolds number $R_e = \frac{\rho v d}{\eta}$ determines the transition from streamlined to turbulent flow; critical value for pipes is roughly 2000.
In irrotational flow, fluid elements possess no net angular velocity about their center of mass ($\omega = 0$), allowing the use of velocity potential $\phi$.
Glossary
StreamlineAn imaginary curve drawn in a fluid showing the instantaneous direction of fluid velocity.
Equation of ContinuityStatement of mass conservation for steady incompressible flow: $A v = \text{constant}$.
Reynolds Number ($R_e$)A dimensionless number used to predict flow transition from laminar/streamlined to turbulent.
Velocity Potential ($\phi$)A scalar function whose negative gradient yields the velocity vector in irrotational flows.
Stream Function ($\psi$)A scalar function used in 2D incompressible flow where lines of constant $\psi$ represent streamlines.
Streamlined vs Turbulent Flow
Parameter
Streamlined (Laminar) Flow
Turbulent Flow
Reynolds Number ($R_e$)
Less than critical (~2000)
Greater than critical (~2000-4000)
Fluid Paths
Smooth, orderly, non-intersecting
Chaotic, eddying, mixing
Mathematical Analysis
Relatively simple (potential/stream functions)
Extremely complex, empirical methods required
Practice Quiz
Choose an option to reveal the explanation.
1. What does the tangent at any point on a streamline represent in a fluid flow?
Explanation: A streamline is defined as an imaginary curve drawn in a fluid such that the tangent at any point gives the direction of the velocity vector of the fluid particle at that point.
2. Why can two streamlines never intersect each other?
Explanation: An intersection of two streamlines would imply two distinct tangent directions at the same point, hence two different velocity vectors for a single fluid particle, which is impossible in steady flow.
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 formulated by applying the principle of conservation of mass to an incompressible fluid flowing through a tube of varying cross-section.
4. In a constricted pipe, how does the velocity of an incompressible fluid change?
Explanation: According to the equation of continuity ($A v = \text{constant}$), where the cross-sectional area $A$ decreases (pipe constricts), the velocity $v$ must increase.
5. What is the dimensionless parameter used to determine whether a fluid flow in a cylindrical pipe is streamlined or turbulent?
Explanation: The Reynolds number ($R_e = \frac{\rho v d}{\eta}$) is the dimensionless parameter determining the transition between streamlined and turbulent flow regimes.
6. For fluid flow in a cylindrical pipe, streamlined flow is typically maintained as long as the Reynolds number remains below:
Explanation: In cylindrical pipe flow, streamlined flow is strictly maintained when the Reynolds number is less than the critical value, which is approximately 2000.
7. What characterizes an irrotational flow?
Explanation: In an irrotational flow, fluid elements possess no net angular velocity about their center of mass, meaning the vorticity or curl of the velocity vector is zero.
8. The existence of a scalar velocity potential function $\phi$ is valid under which condition?
Explanation: Irrotational flow ($\nabla \times \vec{v} = 0$) mathematically permits the definition of a scalar velocity potential function $\phi$ such that $\vec{v} = -\nabla \phi$.
9. What is the SI unit of the volumetric flow rate (discharge $Q = A v$)?
Explanation: Volumetric flow rate is volume per unit time, having the SI unit of cubic meters per second ($\text{m}^3 \text{s}^{-1}$).
10. In a two-dimensional incompressible flow, the velocity components $u$ and $v$ can be expressed in terms of the stream function $\psi$ as:
Explanation: For two-dimensional incompressible flow, the stream function relations are defined as $u = \frac{\partial \psi}{\partial y}$ and $v = -\frac{\partial \psi}{\partial x}$ to automatically satisfy the continuity equation.
11. What happens to pathlines and streamlines in a steady flow?
Explanation: In steady flow, the velocity field does not change with time, causing streamlines, pathlines, and streaklines to coincide identically.
12. Which of the following factors promotes the transition from streamlined to turbulent flow?
Explanation: Higher velocities increase the Reynolds number, destabilizing streamlines and transitioning the flow into turbulence.
13. What does the mass flux rate formula $\frac{dm}{dt} = \rho A v$ represent?
Explanation: The mass flux rate $\frac{dm}{dt}$ measures the mass of fluid crossing a given cross-sectional area per unit time, expressed as $\rho A v$.
14. What is the physical implication of viscous boundary layers near solid boundaries in an otherwise ideal fluid?
Explanation: Viscous boundary layers introduce velocity gradients near solid walls, causing local rotation (vorticity) which modifies pure potential streamlined flow.
15. What is the condition for Bernoulli's equation regarding the nature of the fluid flow?
Explanation: Bernoulli's standard equation requires the fluid to be inviscid, incompressible, irrotational, and flowing in a steady state.
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 of varying cross-section. At a point where the cross-sectional area is $4 \text{ cm}^2$, the velocity of water is $2 \text{ m s}^{-1}$. What is the velocity of water at a constriction where the area becomes $1 \text{ cm}^2$?
Explanation: Using the equation of continuity $A_1 v_1 = A_2 v_2$: $(4 \text{ cm}^2)(2 \text{ m s}^{-1}) = (1 \text{ cm}^2)(v_2)$, which gives $v_2 = 8 \text{ m s}^{-1}$.
2. If the velocity potential function is given by $\phi = -xy$, what are the velocity components $u$ and $v$ along the $x$ and $y$ axes?
Explanation: Since $\vec{v} = -\nabla \phi$, the components are $u = -\frac{\partial \phi}{\partial x} = -(-y) = y$ and $v = -\frac{\partial \phi}{\partial y} = -(-x) = x$.
3. A fluid of density $\rho$ and coefficient of viscosity $\eta$ flows through a pipe of diameter $d$ with velocity $v$. The dimensions of the Reynolds number $R_e = \frac{\rho v d}{\eta}$ are:
Explanation: Reynolds number is a dimensionless quantity, so its dimensional formula is $[M^0 L^0 T^0]$.
4. Which of the following statements regarding streamlines is incorrect?
Explanation: Streamlines represent instantaneous velocity field tangents and match particle pathlines only in steady flow, making statement C incorrect for general unsteady flow.
5. In a 2D incompressible flow, if the stream function is $\psi = 2xy$, what is the magnitude of the velocity vector at point $(1, 1)$?
Explanation: Velocity components are $u = \frac{\partial \psi}{\partial y} = 2x$ and $v = -\frac{\partial \psi}{\partial x} = -2y$. At $(1,1)$, $u = 2$ and $v = -2$. Magnitude is $\sqrt{2^2 + (-2)^2} = \sqrt{8} = 2\sqrt{2}$.
6. What happens to the static pressure of an ideal fluid when it passes through a constriction in a pipe where its velocity increases?
Explanation: As velocity increases at a constriction, kinetic energy per unit volume increases, resulting in a corresponding decrease in static pressure according to Bernoulli's energy conservation principle.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If the radius of a pipe carrying water is halved, by what factor does the flow velocity change to maintain a constant volumetric discharge?
Explanation: From continuity $A_1 v_1 = A_2 v_2$, since $A = \pi r^2$, halving the radius reduces the area by a factor of 4 ($\pi (r/2)^2 = \pi r^2 / 4$). Thus, velocity must increase by a factor of 4.
2. What condition must be satisfied for a velocity potential $\phi$ to exist in a fluid flow field?
Explanation: A velocity potential $\phi$ exists if and only if the flow is irrotational, which mathematically means the curl of the velocity vector is zero ($\nabla \times \vec{v} = 0$).
3. In a steady, ideal fluid flow, which of the following remains constant along a streamline?
Explanation: Bernoulli's equation states that for an inviscid, incompressible, irrotational, steady flow, the sum of static pressure, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline.
4. If the Reynolds number for a fluid flowing in a pipe exceeds 4000, the flow is definitively classified as:
Explanation: Reynolds numbers exceeding 4000 (well above the critical threshold of ~2000) signify that inertial forces dominate over viscous forces, resulting in chaotic turbulent flow.
5. What is the physical significance of lines of constant stream function ($\psi = \text{constant}$) in a 2D flow?
Explanation: By definition, lines along which the stream function $\psi$ is constant correspond precisely to the streamlines of a two-dimensional incompressible flow.
6. A tube has cross-sectional areas of $A_1 = 20 \text{ cm}^2$ and $A_2 = 5 \text{ cm}^2$ at two points. If the mass flux rate is $100 \text{ g s}^{-1}$ and density is $1 \text{ g cm}^{-3}$, what is the velocity $v_1$?
Explanation: Mass flux $\frac{dm}{dt} = \rho A_1 v_1$. Substituting values: $100 = (1)(20)(v_1)$, which gives $v_1 = \frac{100}{20} = 5 \text{ cm s}^{-1}$.
7. Which term in Euler's force equation represents convective acceleration?
Explanation: In Euler's equation $\rho \left( \frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \nabla)\vec{v} \right) = -\nabla P + \rho \vec{g}$, the term $\frac{\partial \vec{v}}{\partial t}$ is local acceleration and $(\vec{v} \cdot \nabla)\vec{v}$ is convective acceleration.
8. Why do pathlines and streamlines diverge in an unsteady fluid flow?
Explanation: In unsteady flow, the velocity at any spatial point fluctuates over time. Consequently, the instantaneous tangent (streamline) differs from the historical trajectory of a moving particle (pathline).
9. If an incompressible fluid flows through a pipe whose diameter increases by a factor of 3, the flow velocity changes by what factor?
Explanation: From continuity, $A_1 v_1 = A_2 v_2$. Since area scales as $d^2$, increasing diameter by a factor of 3 increases area by 9 times, causing velocity to decrease by a factor of 9.
10. Which of the following functions are mutually orthogonal in a two-dimensional ideal (irrotational and incompressible) fluid flow?
Explanation: In 2D potential flow, equipotential lines ($\phi = \text{constant}$) and streamlines ($\psi = \text{constant}$) intersect each other orthogonally everywhere except at stagnation points.