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Introduction to Fluid Mechanics

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Viscosity

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Viscosity is a measure of a fluid's resistance to flow. It quantifies the internal friction in the fluid as layers move past one another. The formula for dynamic viscosity is given by Newton's law of viscosity, τ=μdudy\tau = \mu \frac{du}{dy}, where τ\tau is the shear stress, μ\mu is the dynamic viscosity, dudu is the change in velocity across a distance dydy between two fluid layers.

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Bernoulli's Equation

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Bernoulli's Equation relates the pressure, velocity, and elevation of a fluid in steady, incompressible flow along a streamline. The equation is given as P+12ρv2+ρgh=constantP + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}, where PP is the fluid pressure, ρ\rho is the fluid density, vv is the fluid velocity, and hh is the elevation height above a reference plane, with gg being the acceleration due to gravity.

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Reynolds Number

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The Reynolds Number is a dimensionless quantity used to predict the flow regime in a fluid. It is calculated using the formula Re=ρULμRe = \frac{\rho \cdot U \cdot L}{\mu}, where ρ\rho is the fluid density, UU is the velocity of the fluid, LL is a characteristic linear dimension (like diameter of a pipe), and μ\mu is the dynamic viscosity. Laminar flow occurs at low Reynolds numbers, while turbulent flow occurs at high Reynolds numbers.

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Pressure

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Pressure is defined as the force exerted per unit area. In fluid mechanics, pressure at a point in a fluid is isotropic and can be calculated using the formula P=FAP = \frac{F}{A}, where PP is the pressure, FF is the normal force, and AA is the area over which the force is distributed.

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Buoyancy

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Buoyancy is the upward force exerted by a fluid that opposes the weight of an object immersed in it. It is determined by Archimedes' Principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object. The formula for buoyant force (FbF_b) is Fb=ρgVF_b = \rho \cdot g \cdot V, where ρ\rho is the fluid density, gg is the acceleration due to gravity, and VV is the volume of fluid displaced.

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