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Physics

Fluid Mechanics

Pascal's Law and Hydrostatic Pressure

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⚡ Quick Summary
Pascal's Law states that pressure changes in a liquid are transmitted equally throughout the liquid. Hydrostatic pressure increases with depth.
['dP = -ρg dz', 'P2 = P1 + ρgz']
  • Hydrostatic Pressure: The pressure at a point within a fluid at rest.
  • Derivation of Pressure Variation with Depth:
    • Consider a fluid element of area ΔS and height dz.
    • Forces on the element are due to pressure and weight.
    • Vertical equilibrium gives dP = -ρg dz, where ρ is density, g is acceleration due to gravity, and dz is the change in height.
    • Integrating from z=0 to z=h: P2 - P1 = -∫ρg dz (from 0 to h).
    • If density is constant: P2 = P1 + ρgh.
  • Pressure at the Same Horizontal Level: Pressure is the same at two points in the same horizontal level in a fluid.
  • Pascal's Law: If the pressure in a liquid is changed at a particular point, the change is transmitted to the entire liquid without being diminished in magnitude.
  • Application: Hydraulic Lift - Uses Pascal's Law to multiply force.

Hydraulic Lift

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⚡ Quick Summary
A hydraulic lift uses a liquid to transmit pressure from a small area to a larger area, creating a force multiplication effect.
F2 = A2 * (F1 / A1)
A hydraulic lift consists of two vertical cylinders A and B of different cross-sectional areas A1 and A2 connected by a horizontal tube. Pistons are fitted in both the cylinder. The load is kept on a platform fixed with the piston of larger area. A liquid is filled in the equipment. A valve V is fitted in the horizontal tube which allows the liquid to go from A to B when pressed from the A-side. The piston A is pushed by a force F1. The pressure in the liquid increases everywhere by an amount F1/A1. The valve V is open and the liquid flows into the cylinder B. It exerts an extra force F2 = A2(F1/A1) on the larger piston in the upward direction which raises the load upward. The advantage of this method is that if A2 is much larger than A1, even a small force F1 is able to generate a large force F2 which can raise the load. It may be noted that there is no gain in terms of work. The work done by F1 is same as that by F2. The piston A has to traverse a larger downward distance as compared to the height raised by B.

Atmospheric Pressure

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Atmospheric pressure is the force exerted by the air on a surface. It can be measured using a barometer.
P = Lim (∆S→0) F/∆S
The atmosphere of the earth is spread up to a height of about 200 km. This atmosphere presses the bodies on the surface of the earth. The force exerted by the air on any body is perpendicular to the surface of the body. We define atmospheric pressure as follows. Consider a small surface ∆S in contact with air. If the force exerted by the air on this part is F, the atmospheric pressure is P = Lim (∆S→0) F/∆S. Atmospheric pressure at the top of the atmosphere is zero as there is nothing above it to exert the force. The pressure at a distance z below the top will be ∫(0 to z) ρg dz.

Barometer

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A barometer measures atmospheric pressure by balancing the weight of a mercury column against the atmospheric pressure.
P0 = ρgH
Torricelli devised an ingenious way to measure the atmospheric pressure. The instrument is known as barometer. In this, a glass tube open at one end and having a length of about a meter is filled with mercury. The open end is temporarily closed (by a thumb or otherwise) and the tube is inverted in a cup of mercury. With the open end dipped into the cup, the temporary closure is removed. The mercury column in the tube falls down a little and finally stays there. The upper part of the tube contains vacuum as the mercury goes down and no air is allowed in. Thus, the pressure at the upper end A of the mercury column inside the tube is PA = zero. Let us consider a point C on the mercury surface in the cup and another point B in the tube at the same horizontal level. The pressure at C is equal to the atmospheric pressure. As B and C are in the same horizontal level, the pressures at B and C are equal. Thus, the pressure at B is equal to the atmospheric pressure P0 in the lab. Suppose the point B is at a depth H below A. If ρ be the density of mercury, PB = PA + ρgH or, P0 = ρgH. The height H of the mercury column in the tube above the surface in the cup is measured. Knowing the density of mercury and the acceleration due to gravity, the atmospheric pressure can be calculated using equation (13.4). The atmospheric pressure is often given as the length of mercury column in a barometer.

Pressure in a Fluid

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⚡ Quick Summary
The pressure at a point in a fluid increases with depth.
P = P0 + hρg
The pressure at a depth 'h' in a fluid with density 'ρ' is given by P = P0 + hρg, where P0 is the atmospheric pressure and g is the acceleration due to gravity.

Manometer

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Manometer is a device used to measure pressure in a closed vessel.
P = P0 + hρg
A manometer is a U-tube containing a liquid. One end is open to the atmosphere, and the other is connected to the vessel. The pressure of the gas is equal to the atmospheric pressure plus hρg, where h is the difference in liquid levels and ρ is the density of the liquid.

Archimedes' Principle

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A body submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the body.
B = mg (where B is the buoyant force, m is the mass of the displaced fluid, and g is the acceleration due to gravity)
When a body is partially or fully immersed in a fluid, the fluid exerts an upward force of buoyancy equal to the weight of the displaced fluid. This principle can be derived from Newton's laws of motion. The buoyant force is the resultant of all contact forces exerted by the fluid on the body.

Floatation

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A body floats when the buoyant force equals its weight.
Weight of solid = Weight of displaced fluid (for floatation)
When a solid body is dipped into a fluid, it experiences an upward buoyant force. If this force equals the weight of the solid, the body will float in equilibrium. When the overall density of the solid is smaller than the density of the fluid, the solid floats with a part of it in the fluid. The fraction dipped is such that the weight of the displaced fluid equals the weight of the solid.

Accelerated Liquid in a Cylinder

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When a liquid in a cylinder is accelerated horizontally, the pressure difference between two points is proportional to the acceleration, density, and distance between the points. The pressure at two points on the same horizontal line will be different if the liquid is accelerating. The free surface of the liquid will be inclined at an angle θ with the horizontal, where tanθ = a/g.
['P₁ - P₂ = lρ₀a', 'tanθ = a/g']
Under the action of forces, a liquid contained in a cylinder accelerates towards the right. Newton's second law is applied to relate the pressure difference to the acceleration: * P₁ΔS - P₂ΔS = ma * (P₁ - P₂)ΔS = (ΔS)lρ₀a * P₁ - P₂ = lρ₀a Where: * P₁ and P₂ are the pressures at two points. * ΔS is the area. * m is the mass. * a is the acceleration. * l is the distance between the two points. * ρ₀ is the density of the liquid. In the absence of vertical acceleration, P₁ = P₀ + h₁ρg and P₂ = P₀ + h₂ρg, where h₁ and h₂ are the depths of points A and B from the free surface, respectively, and P₀ is atmospheric pressure. Substituting these expressions into the pressure difference equation, we get: * h₁ρg - h₂ρg = lρa * (h₁ - h₂)/l = a/g * tanθ = a/g, where θ is the inclination of the free surface with the horizontal.

Steady and Turbulent Flow

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Steady flow (or streamline flow) occurs when fluid particles passing a specific point have the same velocity. Turbulent flow is erratic, with varying velocities at the same point. A line of flow is the path of a particle, and in steady flow, it's called a streamline. Streamlines cannot intersect. A tube of flow is a collection of streamlines forming a tube, and fluid within different tubes cannot intermix.
[]
* **Steady Flow:** In steady flow, the velocity of fluid particles reaching a particular point is the same at all times. Each particle follows the same path as a previous particle. It's also called streamline flow. * **Turbulent Flow:** If a liquid is pushed rapidly, the flow may become turbulent. The velocities of different particles passing through the same point may be different and change erratically with time. * **Line of Flow (Streamline):** The path taken by a particle in flowing fluid. In steady flow, the line of flow is also called a streamline. The tangent to the streamline at any point gives the direction of all the particles passing through that point. Two streamlines cannot intersect. * **Tube of Flow:** Consider an area S in a fluid in steady flow. Draw streamlines from all the points of the periphery of S. These streamlines enclose a tube, of which S is a cross-section. Fluid flowing through different tubes of flow cannot intermix.

Irrotational Flow of an Incompressible and Nonviscous Fluid

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Simplified fluid flow analysis considering incompressibility, non-viscosity, and irrotational flow.
N/A
  • Incompressibility: Fluid density remains constant at all points and over time.
  • Nonviscous: Neglecting internal friction between fluid layers.
  • Irrotational flow: No net angular velocity of fluid particles.

Equation of Continuity

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The mass flow rate of a fluid is constant throughout a closed tube.
A₁v₁ = A₂v₂
The equation of continuity expresses the law of conservation of mass in fluid dynamics. It states that for an incompressible fluid, the product of the area of cross-section and the speed remains the same at all points of a tube of flow.

Bernoulli's Equation

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Relates speed, pressure, and height of a fluid at a point.
N/A
Bernoulli's equation is the application of the work-energy theorem in the case of irrotational and steady flow of an incompressible and nonviscous liquid. It relates the speed of a fluid at a point, the pressure at that point, and the height of that point above a reference level.

Fluid Flow and Bernoulli's Equation

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This section discusses fluid flow, focusing on the application of the work-energy theorem and the derivation of Bernoulli's equation. It relates pressure, kinetic energy, and potential energy in a fluid system.
<ul><li>Equation of Continuity: A1v1 = A2v2</li><li>Bernoulli's Equation: P1 + ρgh1 + 1/2ρv1^2 = P2 + ρgh2 + 1/2ρv2^2</li><li>Bernoulli's Equation (General Form): P + ρgh + 1/2ρv^2 = constant</li></ul>
  • Consider a liquid flowing through a tube with varying cross-sectional areas A1 and A2, velocities v1 and v2, and pressures P1 and P2 at points A and B respectively. The density of the liquid is ρ.
  • AA' = v1∆t and BB' = v2∆t are distances traveled by the liquid in time ∆t at points A and B.
  • Equation of Continuity: A1v1∆t = A2v2∆t
  • Mass of the liquid: ∆m = ρA1v1∆t = ρA2v2∆t
  • Forces acting on the liquid:
    • P1A1 by the liquid on the left
    • P2A2 by the liquid on the right
    • (∆m)g, the weight of the liquid
    • N, contact forces by the walls of the tube
  • Work done by P1A1: W1 = (P1A1)(v1∆t) = P1(∆m/ρ)
  • Work done by P2A2: W2 = -(P2A2)(v2∆t) = -P2(∆m/ρ)
  • Work done by the weight: W3 = (∆m)gh1 − (∆m)gh2, where h1 and h2 are heights at A and B respectively.
  • Change in Kinetic Energy: KE_BB' - KE_AA' = 1/2(∆m)v2^2 - 1/2(∆m)v1^2
  • Total Work Done: W = W1 + W2 + W3 = P1(∆m/ρ) - P2(∆m/ρ) + (∆m)gh1 - (∆m)gh2
  • By Work-Energy Theorem: P1(∆m/ρ) - P2(∆m/ρ) + (∆m)gh1 - (∆m)gh2 = 1/2(∆m)v2^2 - 1/2(∆m)v1^2

Hydrostatics

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When fluid is at rest, pressure at different heights is related as P1 - P2 = rho * g * (h2 - h1).
P1 - P2 = rho * g * (h2 - h1)
If the speed of the fluid is zero everywhere, we get the situation of hydrostatics. Putting v1 = v2 = 0 in the Bernoulli equation (13.10) P1 + ρgh1 = P2 + ρgh2 or, P1 − P2 = ρ g(h2 − h1) as expected from hydrostatics.

Speed of Efflux (Torricelli's Theorem)

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The speed of liquid coming out of a hole at depth h is the same as a particle falling freely through height h: v = sqrt(2gh).
v = sqrt(2gh)
Consider a liquid of density ρ filled in a tank of large cross-sectional area A1. There is a hole of cross-sectional area A2 at the bottom and the liquid flows out of the tank through the hole. Suppose A2 << A1. Bernoulli equation gives P0 + 1/2 ρv1^2 + ρgh = P0 + 1/2 ρv2^2. By the equation of continuity A1v1 = A2v2. If A2 << A1, this equation reduces to v2 = sqrt(2gh). The speed of liquid coming out is called the speed of efflux. This is known as Torricelli’s theorem.

Ventury Tube

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Ventury tube is a device to measure fluid speed. By using continuity equation and Bernoulli's principle, we can relate pressure difference to speeds: (P1 - P2) = 1/2 * rho * (v2^2 - v1^2) and P1-P2 = rho*g*h.
['(P1 - P2) = 1/2 * rho * (v2^2 - v1^2)', 'P1 - P2 = rho*g*h', 'A1v1 = A2v2']
A ventury tube is used to measure the flow speed of a fluid in a tube. It consists of a constriction or a throat in the tube. As the fluid passes through the constriction, its speed increases in accordance with the equation of continuity. The pressure thus decreases as required by Bernoulli equation. Let v1 and v2 be the speeds of the liquid at A1 and A2. As both the cross sections are open to the atmosphere and by Bernoulli equation, (P1 − P2 ) = 1/2 * ρ * (v2^2 − v1^2). Figure (13.18) also shows two vertical tubes connected to the ventury tube at A1 and A2. If the difference in heights of the liquid levels in these tubes is h, we have P1 − P2 = ρgh.

Bernoulli's Equation Applications

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This section discusses applications of fluid dynamics principles, including aspirator pumps and the Magnus effect on spinning balls.
2 gh = v 2 − v 2
Aspirator Pump: When a fluid flows through a constricted region, its speed increases, and the pressure decreases. This principle is used in aspirator pumps to spray liquids. A barrel with a constriction is connected to a liquid vessel. Air is forced through the barrel, increasing speed at the constriction, which reduces pressure. This draws liquid up from the vessel and sprays it out with the air.
Change of Plane of Motion of a Spinning Ball: Spinning balls can change their direction in flight due to the Magnus effect. As a spinning ball moves through the air, the air on one side moves faster (due to being dragged by the spin), while the air on the other side moves slower. According to Bernoulli's principle, faster-moving air has lower pressure. The pressure difference creates a net force on the ball, causing it to deviate from its original path.

Pressure and Force in Fluids

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The pressure at a depth in a fluid is equal to the atmospheric pressure plus the pressure due to the height of the fluid column.
p = ρgh
The pressure at the surface of a fluid is equal to the atmospheric pressure. The pressure at a depth h in a fluid is equal to the atmospheric pressure plus the pressure due to the height of the fluid column, which is given by ρgh, where ρ is the density of the fluid and g is the acceleration due to gravity.

Equilibrium of Floating Objects

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When an object floats, the buoyant force equals its weight. Torque equilibrium is also required if the object is free to rotate.
['Weight of plank: mg = 2lρg', 'Buoyant force: F = (2lρg)/cosθ', 'Equilibrium condition (torque): mg(OB)sinθ = F(OA)sinθ']
For an object of length `2l` and mass per unit length `ρ` floating at an angle θ, the following concepts are used: * **Weight of the plank:** `mg = 2lρg` * **Mass of the part OC of the plank:** `(l/cosθ)ρ` * **Mass of water displaced:** `0.5 * (l/cosθ) * ρ = (lρ)/cosθ` * **Buoyant force F:** `F = (2lρg)/cosθ` For equilibrium: * Torque of weight about O = Torque of buoyant force about O * `mg(OB)sinθ = F(OA)sinθ`

Simple Harmonic Motion of Floating Cylinder

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A cylinder floating vertically undergoes SHM when displaced and released. The frequency depends on the water displaced, the radius, and the mass of the block.
['Equilibrium: πr^2 h ρg = W', 'Restoring force: F = -πr^2 ρgx', 'Frequency: ν = (1/2π) * √(πr^2 ρg / M)']
For a cylindrical block of wood with mass M and radius r floating in water: 1. **Equilibrium condition:** `πr^2 h ρg = W`, where h is the height submerged, ρ is the density of water, and W is the weight of the block. 2. **Net Force at displacement x:** `F = -πr^2 ρg x = -kx`, where `k = πr^2 ρg` 3. **The frequency of SHM:** `ν = (1/2π) * √(k/M) = (1/2π) * √(πr^2 ρg / M)`

Pressure in Fluids

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Pressure at a depth in a fluid is given by P = P0 + ρgz, where P0 is the pressure at the surface, ρ is the density of the fluid, g is the acceleration due to gravity, and z is the depth.
P = Lim (∆s→0) F/∆S, P1 - P2 = ρgz
  • Pressure in a fluid increases with depth.
  • The equation P = Lim (∆s→0) F/∆S defines pressure as the force per unit area.
  • The equation P1 - P2 = ρgz relates the pressure difference between two points in a fluid to the density of the fluid, the acceleration due to gravity, and the vertical distance between the points.

Archimedes’ Principle

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Archimedes' principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
Buoyant Force = Weight of displaced fluid
Archimedes' Principle describes buoyancy, stating that the upward buoyant force on an object immersed in a fluid is equal to the weight of the fluid the object displaces.

Equation of Continuity

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The equation of continuity states that for an incompressible fluid, the product of the area of cross-section and the velocity of the fluid remains constant along a flow tube.
A1V1 = A2V2
The equation of continuity, A1V1 = A2V2, where A is the cross-sectional area and V is the velocity of the fluid. This equation is based on the principle of conservation of mass.

Buoyancy and Archimedes' Principle

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Objects immersed in fluids experience an upward force (buoyant force) equal to the weight of the fluid displaced by the object. This principle helps determine if an object will float or sink.
Buoyant Force (F_b) = Weight of fluid displaced = V_displaced * density_fluid * g
The buoyant force is the upward force exerted on an object submerged in a fluid. It is equal to the weight of the fluid displaced by the object. Whether an object floats or sinks depends on the balance between its weight and the buoyant force. If the buoyant force is greater than or equal to the weight of the object, it floats. If the buoyant force is less than the weight of the object, it sinks.

Pressure in Fluids

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Pressure in a fluid increases with depth. The force exerted by a fluid on a surface is the pressure multiplied by the area of the surface.
P = P_0 + ρgh; F = PA
Pressure at a depth 'h' in a fluid is given by P = P_0 + rho*g*h, where P_0 is the atmospheric pressure, rho is the density of the fluid, and g is the acceleration due to gravity. The force exerted by the fluid on a submerged surface is the integral of the pressure over the area of the surface. For a flat surface at a constant depth, the force is simply the pressure at that depth multiplied by the area.

Specific Gravity

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Specific gravity is the ratio of the density of a substance to the density of a reference substance, usually water.
SG = ρ_substance / ρ_water
Specific gravity (SG) is a dimensionless quantity that represents the ratio of the density of a substance to the density of a reference substance, typically water at 4°C (which is approximately 1000 kg/m³). SG = (density of substance) / (density of water)

Simple Harmonic Motion in Fluids

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When an object floating in a fluid is slightly displaced and released, it can undergo simple harmonic motion. The time period of oscillation depends on the object's mass, the fluid's density, and the geometry of the object.
T = 2π * sqrt(m / (Aρg))
For a cylindrical object floating vertically, the time period (T) of small vertical oscillations can be calculated using T = 2π * sqrt(m / (A * rho * g)), where m is the mass of the object, A is the cross-sectional area of the cylinder, rho is the density of the fluid, and g is the acceleration due to gravity.

Fluid Flow

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Deals with the properties and behavior of fluids (liquids and gases) in motion.
Continuity equation: A₁v₁ = A₂v₂ (where A is the cross-sectional area and v is the velocity of the fluid) Torricelli's theorem: v = √(2gh) (where v is the velocity of efflux, g is the acceleration due to gravity, and h is the height of the fluid above the orifice)
This section covers concepts related to fluid dynamics, including flow rate, continuity equation, pressure differences in fluid flow, and applications like Torricelli's theorem for fluid efflux.

Surface Tension

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Surface tension is the property of the surface of a liquid that allows it to resist an external force, due to the cohesive nature of its molecules.
Surface Tension = Force / Length
Surface Tension: Force per unit length acting on the surface of a liquid, tending to minimize the surface area. Contact Angle: The angle formed at the point where a liquid interface meets a solid surface.

Viscosity

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Viscosity is a measure of a fluid's resistance to flow.
N/A
Terminal Velocity: The constant speed that a freely falling object eventually reaches when the resistance of the fluid through which it is falling prevents further acceleration.

Flow of Fluids

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Describes the behavior of fluids in motion, including concepts like viscosity, pressure, and flow rate.
Equation of Continuity: A1V1 = A2V2; Bernoulli's Equation: P + (1/2)ρv^2 + ρgh = constant; Viscosity Force: F = ηA(dv/dx)
Includes steady and unsteady flow, streamlines, viscosity, Bernoulli's equation, and surface tension.

Pascal's Law

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Pressure applied to a confined fluid is transmitted equally throughout the fluid.
P = F/A
Explains Pascal's law and its relation to pressure.

Poiseuille Equation

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Describes the rate of flow of an incompressible fluid through a cylindrical pipe.
Q = (πr^4ΔP) / (8ηL)
Describes the rate of flow of an incompressible fluid through a cylindrical pipe.

Pressure

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Force per unit area.
P = F/A
Discusses pressure, atmospheric pressure, excess pressure inside drops and bubbles, and pressure variation with height.

Reynolds Number

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A dimensionless number that predicts whether flow will be laminar or turbulent.
Re = (ρvD) / μ
A dimensionless number that predicts whether flow will be laminar or turbulent.

Streamline

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A line whose tangent at any point is the direction of the fluid velocity at that point.
N/A
A streamline is a line whose tangent at any point is the direction of the fluid velocity at that point. Streamlines are used to visualize the flow of a fluid.

Turbulent Flow

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Irregular fluid motion with chaotic changes in pressure and velocity.
N/A
Turbulent flow is a type of fluid flow characterized by irregular motion, chaotic changes in pressure and velocity, and the formation of eddies and vortices.

Viscosity

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A measure of a fluid's resistance to flow.
N/A
Viscosity is a measure of a fluid's resistance to flow. It describes the internal friction within a fluid.

Terminal Velocity

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The constant speed that a freely falling object eventually reaches when the resistance of the medium prevents further acceleration.
N/A
Terminal velocity is the constant speed that a freely falling object eventually reaches when the force of air resistance equals the force of gravity.