2017.01.02 Phys 2425 Notes Section 9 - Fluid Dynamics (1).pdf

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Fluid Mechanics I. Density and Pressure The density 𝜌 (more properly called mass density) of any substance is defined to be equal to the mass of the substance divided by the volume of the substance. 𝜌 ≡ 𝑚 ? Density is an intensive quantity, meaning that it remains the same for a given substance no matter how much of the substance there is. The density of water, for instance, is 1000 kg/m 3 . The pressure of a fluid at a given location is defined to be the infinitesimal perpendicular force per infinitesimal unit area acting on a surface placed at that location in the fluid. 𝑝 ≡ 𝑑𝐹 𝑑? The average pressure is defined as the perpendicular force per finite unit area acting on a surface placed in a region in the fluid. The pressure acts in all directions in the fluid. See picture. 𝑝 ??? 𝐹 ? The SI unit of pressure the Pascal, where 1 Pa = 1 N/m 2 . The average air pressure at sea level has been measured to be 𝑝 ??? = 1.013 × 10 5 Pa.
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II. Energy Density Conservation in Fluids To study some aspects of fluids at rest (statics) and in motion (dynamics), we will be interested in the amount of energy contained in an infinitesimal volume of that fluid. The amount of energy per unit volume in a substance is called the energy density ( similarly, the mass density of a substance is the amount of mass in a unit volume of the substance ). Types of Energy Density A volume of a substance can contain various types of energy. We will now look at the possible types of energy we will be considering. Gravitational Potential Energy Density Certainly, the particles in the volume may have a non-zero vertical position. Therefore that volume of substance will contain gravitational potential energy. The total gravitational potential energy in a volume is, as we know, equal to ? ???? = 𝑀𝑔𝑦 ?? , where 𝑀 is the total mass of the particles in that volume and 𝑦 ?? is the center of mass vertical position of the volume (usually we abbreviate 𝑦 ?? to simply 𝑦 , where it is understood that we mean the height of the center of mass. Therefore the gravitational potential energy density, which is the gravitational potential energy per volume, is equal to ? ???? ? = 𝑀𝑔𝑦 ? = 𝜌𝑔𝑦 The gravitational potential energy density is proportional to the mass density of the substance and the vertical position of the volume of fluid of interest.
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