Fluid Mechanics PDF
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Uploaded by ElatedChrysoprase1998
St. John's Public School
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These notes cover fundamental concepts and formulas in fluid mechanics, ideal for students preparing for competitive examinations like JEE. Topics include density, pressure, buoyancy, and fluid dynamics.
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# Fluid Dynamics ## Density - **Uniform Substance:** ρ = M/V - **Non-Uniform Substance:** ρ = dm/dV - 1 g/cc = 10^3 kg/m^3 - **Water** ρ(water) = 1g/cc = 10^3 kg/m^3 - **Mercury** ρ Hq = 13.6 g/cc - 1000 L = 1 m^3 ## Mixtures - **Two liquids of equal volume and different densities:** - ρ<su...
# Fluid Dynamics ## Density - **Uniform Substance:** ρ = M/V - **Non-Uniform Substance:** ρ = dm/dV - 1 g/cc = 10^3 kg/m^3 - **Water** ρ(water) = 1g/cc = 10^3 kg/m^3 - **Mercury** ρ Hq = 13.6 g/cc - 1000 L = 1 m^3 ## Mixtures - **Two liquids of equal volume and different densities:** - ρ<sub>mix</sub> = (ρ<sub>1</sub> + ρ<sub>2</sub>)/2 - **Two liquids of equal mass and different densities:** - ρ<sub>mix</sub> = 2ρ<sub>1</sub>ρ<sub>2</sub>/(ρ<sub>1</sub>+ρ<sub>2</sub>) - **n Equal mass liquids:** - ρ<sub>mix</sub> = 1/ρ<sub>1</sub> + 1/ρ<sub>2</sub> + … + 1/ρ<sub>n</sub> - ρ<sub>mix</sub> = (ρ<sub>1</sub> + ρ<sub>2</sub> + ρ<sub>3</sub> + …+ ρ<sub>n</sub>)/n ## Pressure - **Definition:** - P = F/A - P = dF/A - **Units:** - N/m<sup>2</sup> (or) Pascal (or) atm - 1 N/m<sup>2</sup> = 1 Pascal - 1 atm = 1.013 x 10<sup>5</sup> N/m<sup>2</sup> ## Atmospheric Pressure - **Formula:** P<sub>o</sub> = ρgh - **Standard Atmospheric Pressure:** P<sub>o</sub> = 1 atm - **Mercury** ρm = 13.6 x 10<sup>3</sup> kg/m<sup>3</sup> - **Gravity** g = 9.8m/s<sup>2</sup> - **Height (Mercury column)** h = 76cm = 0.76m ## Pressure Due to Liquid Column - **Total Pressure** P<sub>T</sub> = P = P<sub>o</sub> + ρgh - **Absolute Pressure** P = P<sub>o</sub> + ρgh - **Gauge Pressure:** P<sub>liquid</sub> = gauge pressure = (P - P<sub>o</sub>) - **Accelerating Container** P = P<sub>o</sub> + ρg<sub>eff</sub>h - g<sub>eff</sub> = g + a ## Accelerating Fluid - **Case 1:** - P<sub>B</sub> - P<sub>A</sub> = ρg<sub>eff</sub>h - P<sub>B</sub> - P<sub>A</sub> = ρ(g + a)sh - **Case 2:** - Tan θ = ma/mg = a/g = ρg<sub>eff</sub>h/mg - Tanθ = h/L = a/g - h = (g/g)L ## Hydrostatic Paradox - P<sub>A</sub> = P<sub>B</sub> = P<sub>C</sub> = P<sub>D</sub> ## Pascal's Law - **Formula:** F<sub>1</sub>/A<sub>1</sub> = F<sub>2</sub>/A<sub>2</sub> ## Rotating Fluid in Container (Cylindrical) - **Position Formula:** y = w<sup>2</sup>x<sup>2</sup>/2g - **Height Formula using position formula:** h = w<sup>2</sup>R<sup>2</sup>/2g ## Rotating Density (RD) (Specific Gravity) - **Formula:** RD = P<sub>body</sub>/P<sub>water</sub> - **Alternative formula:** RD = weight<sub>body</sub>/weight<sub>equal volume of water</sub> - **Another formula:** RD = M<sub>body</sub>/M<sub>equal volume of water</sub> - **Note:** RD of body = W<sub>air</sub>/W<sub>loss</sub> = W<sub>air</sub>/(W<sub>air</sub> - W<sub>water</sub>) - **Note:** P<sub>liquid</sub>/P<sub>water</sub> = (W<sub>air</sub> - W<sub>liquid</sub>)/(W<sub>air</sub> - W<sub>water</sub>) = RD ## Archimedes Principle - **Buoyancy Force formula:** F<sub>B</sub> = ρ<sub>c</sub>(V<sub>i</sub>)g ## Principle of Rotation - **Velocity formula:** V<sub>i</sub> = P<sub>b</sub>A/P<sub>L</sub> - **Force formula:** F<sub>in</sub> = P<sub>b</sub>/P<sub>L</sub> ## Force Acting on Wall of Container Due To Liquid - **Force formula:** F<sub>wall</sub> = ρ<sub>q</sub>wh<sup>2</sup>/2 ## Torque on Wall of Container Due To Liquid - **Torque formula:** T = ρ<sub>q</sub>wh<sup>3</sup>/3 - **Torque is unstable:** T = F x R (unit = m/s<sup>3</sup>) ## Fluid Dynamics - **Streamline flow (Laminar flow):** the fluid particles move in parallel, smooth, and continuous paths - **Turbulent flow:** characterized by chaotic and irregular motion, creating eddies and vortices - **Ideal Flow:** Incompressible, Non-viscous, streamlined flow - **Rate of Flow of Fluid:** R<sub>F</sub> = AV = PAV/t - **Equation of Continuity:** AV = const; A<sub>1</sub>V<sub>1</sub> = A<sub>2</sub>V<sub>2</sub>; A<sub>1</sub>V<sub>1</sub> = A<sub>2</sub>V<sub>2</sub> + A<sub>3</sub>V<sub>3</sub> - **K.E of fluid:** KE = 1/2pV<sup>2</sup> = 1/2W<sub>v</sub><sup>2</sup>/g ## Bernoullis Principal - **Formula:** P + ρgh + 1/2ρV<sup>2</sup> = const - **Note 1:** P<sub>1</sub> + ρgh<sub>1</sub> + 1/2ρV<sub>1</sub><sup>2</sup> = P<sub>2</sub> + ρgh<sub>2</sub> + 1/2 ρV<sub>2</sub><sup>2</sup> - **Note 2:** P<sub>1</sub> + 1/2ρV<sub>1</sub><sup>2</sup> = P<sub>2</sub> + 1/2 ρV<sub>2</sub><sup>2</sup> - **Application:** P - P<sub>o</sub> + ρgh = 1/2ρ(V<sub>2</sub><sup>2</sup> - V<sub>1</sub><sup>2</sup>) - **Velocity Formula using Application formula:** V<sub>2</sub> = √[(2(P - P<sub>o</sub> + ρgh))/(ρ(1 - (A<sub>2</sub>/A<sub>1</sub>)<sup>2</sup>))] - **Speed of efflux formula:** V = √2gh ## Range of Liquid Ejected From Base of Large Container - **Range formula:** R = √4h(H - h) - **Maximum Range formula:** h = H/2 - **Maximum Range result:** R<sub>max</sub> = H ## Time Taken to Decrease Liquid Level in Open Container - **Formula:** t = A√[2(h<sub>i</sub> - h<sub>f</sub>)/g] ## Venturimeter - **Formula:** Rate of flow = A<sub>1</sub>A<sub>2</sub>√[2ρg<sub>h</sub>/(ρ(A<sub>2</sub><sup>2</sup> - A<sub>1</sub><sup>2</sup>))] ## Buoyancy Force - **Formula:** F<sub>B</sub> = ½ρAg ## Important Formulas for JEE - **Hydrostatic Pressure:** P = P<sub>o</sub> + ρgh - **Buoyancy Force:** F<sub>B</sub> = ρVg - **Continuity Equation:** AV<sub>1</sub> = AV<sub>2</sub> - **Bernoullis Equation:** P + 0.5ρV<sup>2</sup> + ρgh = constant - **Rate of Flow (Venturi Meter):** Q = AV - **Surface Tension: ** F = L - **Capillary Rise:** h = 2cos θ/ (ρgr) - **Viscosity:** F = A(du/dy) These formulas provide a concise summary of key principles in fluid mechanics that are essential for solving problems in JEE examinations.
