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Conservation_of_Mass_for_a_Differential_

Conservation_of_Mass_for_a_Differential_ - Conservation of...

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Conservation of Mass for a Differential Open System (Control Volume) [Corrected] Consider a fluid flowing unsteadily such that the only important spatial variables are x and y . This is a two-dimensional, unsteady flow field as shown in the figure below. Every intensive property of the flowing fluid is a function of only three variables: time and two spatial coordinates, x and y . For example in this two-dimensional, unsteady flow field the following variables can be measured: Scalar Field Vector Field Pressure, Temperature, Density P = P ( x, y, t ); T = T ( x, y, t ); ρ = ρ ( x, y, t ) x- and y-velocity V x = V x ( x, y, t ) and V y = V y ( x, y, t ) Velocity V = V ( x, y, t ) = V x i + V y j Linear momentum density ρ V = ρ ( x, y, t ) V ( x, y, t ) = ( ρ V x ) i + ( ρ V y ) j In each case, the measured variables depend on x , y , and t and are referred to as either scalar or vector fields. For purposes of analysis, we will also assume that the flow field has a constant and uniform depth z in the direction of the z - axis. Consider writing the conservation of mass for a small, finite-sized open system shown by dashed lines in the figure. We get the standard relationship between the rate of change of the mass inside the system and
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