AdvancedTraining_2_Governing_Equations.pdf
Governing Equations
Fluid Properties
-Properties required to describe flow:
-normal shear stress (pressure)
-viscosity, μ, (gives tangential shear stress)
-density, ρ
-plus, velocity of fluid flow, u,v,w
-temperature of fluid, T
Fluid Properties
-Properties required to describe flow:
-normal shear stress (pressure)
-viscosity, μ, (gives tangential shear stress)
-density, ρ
-plus, velocity of fluid flow, u,v,w
-temperature of fluid, T
Governing Equations
-Conservation of Mass
-Newton’s Second Law of Motion
-First Law of Thermodynamics Mass
-Mass of volume = density × volume
-Mass = ρ δxδyδz
-Assuming volume does not deform, rate of change of mass with time =
-Mass into control volume
-Volume flow rate = velocity × area
-Mass flow rate = density × velocity × area
-In x direction = ρ u δyδz
-Mass out of system
-Mass flow rate = density × velocity × area
-We need to account for change in velocity and density across the volume
-x velocity out = u + (?u/?x)δx
-density out = ρ + (?ρ/?x)δx
Mass
-In + Made = Out + Accumulated
-Mass cannot be created, so
-In = Out + Accumulated
-Accumulated + Out - In = 0
Momentum
-Use Newton’s Second Law to relate forces on a control volume to the acceleration of the fluid
-Forces are shear stresses and normal stresses plus body forces such as gravity
-Can express acceleration as rate of change of velocity
-Need to consider change of velocity in space and time
Momentum
-Now, use Newton’s Second Law to relate forces on a control volume to the acceleration of the fluid
-Force = Mass × Acceleration
-With Mass = ρ δxδyδz
-and body force per unit mass = fx
Energy
-First Law of Thermodynamics
-Energy is Conserved
-Rate of change of energy within the element = Net flux of heat into the element + rate of work done on element due to body and surface forces + Source terms
-Net energy change in fluid = sum of work done on fluid + net rate of heat addition +energy sources
-Include potential energy as a source term
-Can write equations for each direction and find the total rate of work done on the fluid particle by surface stresses
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