Physics
Fluids
Fluid Flow - Rate of Flow
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⚡ Quick Summary
The rate of flow of a fluid through a tube is determined by the cross-sectional area and the velocity of the fluid.
Equation of continuity: v_A / v_B = A_B / A_A
Bernoulli's equation: P_A + (1/2) * ρ * v_A^2 = P_B + (1/2) * ρ * v_B^2
The rate of flow is calculated using the equation of continuity and Bernoulli's equation. The equation of continuity relates the velocities and cross-sectional areas at two points in a tube. Bernoulli's equation relates the pressure, velocity, and height of a fluid at two points in a flow.
Bernoulli's Equation Application - Tank with Opening
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Bernoulli's equation can be used to find the velocity of water coming out of an opening in a tank, considering the pressure at the surface of the water and at the opening.
Bernoulli's equation: P_A + ρgh + (1/2) * ρ * v_A^2 = P_B + (1/2) * ρ * v_B^2
When a tank has a large cross-sectional area compared to the opening, the velocity of the water at the surface of the tank is negligible. Bernoulli's equation is applied between the surface of the water in the tank and the opening, considering the pressure at both points.
Buoyancy
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An object floats when the buoyant force (upward force exerted by the fluid) equals the object's weight. The weight of the displaced fluid equals the weight of the object.
Buoyant Force = Weight of displaced fluid
A solid floats in a liquid in a partially dipped position. The liquid exerts a force of buoyancy on the solid which is equal to the weight of the solid. The weight of the displaced liquid equals the weight of the solid. The weight of the dipped part of the solid is equal to the weight of the displaced liquid.
Pressure in Accelerating Fluids
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When a fluid-filled container accelerates, the pressure distribution changes. The resultant normal force on a surface is affected by the acceleration.
Pressure variation in accelerating fluids
A closed cubical box completely filled with water is accelerated horizontally. The resultant normal force by the water on the top of the box depends on the acceleration.
Fluid Dynamics - Continuity Equation
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For an incompressible fluid flowing through a tube, the volume flow rate is constant. If the cross-sectional area changes, the velocity must also change to maintain a constant flow rate.
A1V1 = A2V2 (Continuity Equation)
Water enters through one end of a cylindrical tube and leaves through the other. The relationship between the speeds at the entrance and exit depends on whether the tube is horizontal or vertical.
Bernoulli's Theorem
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Bernoulli's theorem states that for an ideal fluid, the total energy (pressure, kinetic, and potential) per unit volume remains constant along a streamline.
P + (1/2)ρv^2 + ρgh = constant
Bernoulli theorem is based on the conservation of energy.
Pressure in Fluids
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Pressure at a point in a fluid depends on depth and external pressure. In a horizontal tube, pressure can vary depending on fluid velocity.
P = P0 + ρgh
Water is flowing through a long horizontal tube. The pressure at two points A and B depends on the conditions.
Torricelli's Theorem
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The speed of efflux of a fluid from an opening in a container is the same as the speed a body would acquire falling freely from the surface of the fluid to the opening.
v = sqrt(2gh)
Water and mercury are filled in two cylindrical vessels up to same height. The velocity of water and mercury coming out of the holes are related to their densities.
Fluid Dynamics - Tank with Hole
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⚡ Quick Summary
The water level in a tank with a hole at the bottom will reach an equilibrium where the inflow equals the outflow. The height depends on inflow speed and gravity.
Outflow Rate = A * sqrt(2gh)
A large cylindrical tank has a hole of area A at its bottom. Water is poured in the tank by a tube of equal cross-sectional area A ejecting water at the speed v. The water level will reach a certain height.