Fluid Machanics

Control-Volume Method ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Control-Volume Selection ,[object Object],[object Object],[object Object],[object Object], No trigonometric functions appear in u rel • n Trigonometric functions appear in u rel • n   Unknown p in the injection slot is irrelevant Unknown p in the injection slot is part of solution 

Solution Strategy ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Useful Control-Volume Theorem-1 ,[object Object],[object Object],[object Object],[object Object]

Useful Control-Volume Theorem-2 ,[object Object],[object Object],[object Object],[object Object]

Fluid Statics Revisited-1 ,[object Object],[object Object],What we know about the solution… F = F x i + F z k F x = force on the projection of arc AB on a vertical plane (viz., surface OA) F z = weight of the column of fluid above arc AB

Fluid Statics Revisited-2 ,[object Object],[object Object],[object Object],[object Object]

Fluid Statics Revisited-3 ,[object Object],[object Object],[object Object],[object Object],Zero because p = p a at the free surface  This is F  ,[object Object],[object Object]

Fluid Statics Revisited-4 ,[object Object],[object Object], Force on the projection of arc AB on a vertical plane  Weight of the column of fluid above arc AB

Steady Flow in a Pipe-1 ,[object Object],[object Object],  Zero for steady flow Net flux of mass out of CV is zero…what flows in flows out  u rel = u for a stationary CV

Steady Flow in a Pipe-2 ,[object Object],[object Object],[object Object],[object Object], Pipe inlet  Pipe outlet  Pipe walls u 1 , n 1 opposite direction  u 2 , n 2 same  direction

Steady Flow in a Pipe-3 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Sphere Falling in a Cylinder-1 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object], A Galilean transformation puts the sphere at rest

Sphere Falling in a Cylinder-2 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object], < 0 … fluid entering CV   > 0… fluid leaving CV

Sphere Falling in a Cylinder-3 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],A 2 = area  between sphere and cylinder A 1 = area of the cylinder 

Deforming Control Volume-1 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Deforming Control Volume-2 ,[object Object],[object Object],[object Object],[object Object]

Deforming Control Volume-3 ,[object Object],[object Object],[object Object],[object Object],The net flux of mass out of the CV is positive 

Channel Flow with Suction-1 ,[object Object],[object Object]

Channel Flow with Suction-2 ,[object Object],[object Object],[object Object], Fluid  entering CV at inlet  Fluid leaving CV at outlet  Fluid leaving CV at upper wall

Channel Flow with Suction-3 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object], Dot product with i extracts the x component

Channel Flow with Suction-4 ,[object Object],[object Object],[object Object],x momentum is carried out at y = H in the y direction – true only for an inviscid fluid

Channel Flow with Suction-5 ,[object Object],[object Object],[object Object],[object Object],[object Object]

Indirect Force Computation-1 ,[object Object],[object Object]

Indirect Force Computation-2 ,[object Object],[object Object],[object Object],[object Object],

Indirect Force Computation-3 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object], Reference pressure to the atmospheric value, p a 0

Indirect Force Computation-4 ,[object Object],[object Object],[object Object],[object Object], Upper outlet  Jet from leak  Lower outlet

Indirect Force Computation-5 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Fluid pressure on a channel wall segment dS is p n dS , and the atmosphere contributes –p a n dS

Indirect Force Computation-6 ,[object Object],[object Object],[object Object], Upper outlet  Jet from leak  Lower outlet  Inlet

Using Bernoulli’s Equation-1 ,[object Object],[object Object],[object Object]

Using Bernoulli’s Equation-2 ,[object Object],[object Object],[object Object],[object Object],

Using Bernoulli’s Equation-3 ,[object Object],[object Object],[object Object],x momentum  from left inlet  Force on the “device” x momentum exiting  at outlet  Pressure force at left inlet

Using Bernoulli’s Equation-4 ,[object Object],[object Object],[object Object],y momentum  from bottom inlet  Force on the “device” y momentum exiting  at outlet  Pressure force at bottom inlet

Using Bernoulli’s Equation-5 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Using Bernoulli’s Equation-6 ,[object Object],[object Object],[object Object],[object Object],  

Reaction Forces-1 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],To find the force on an object, we have thus far computed it from the pressure integral…there is an alternative Part of this integral is F disk  Reaction force, R , equals – F disk 

Reaction Forces-2 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],0 

Reaction Forces-3 ,[object Object],[object Object],[object Object],[object Object]

Nonuniform Velocity Profiles-1 ,[object Object],[object Object],[object Object]

Nonuniform Velocity Profiles-2 ,[object Object],[object Object],[object Object],[object Object]

Nonuniform Velocity Profiles-3 ,[object Object],[object Object],Momentum  entering CV  Rate at which it enters Momentum  leaving CV Rate at which  it leaves Pressure  at inlet

Nonuniform Velocity Profiles-4 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],As expected, Fx is positive so that the drag is in the correct direction

Accelerating Control Volume-1 ,[object Object],[object Object],[object Object],[object Object],[object Object]

Accelerating Control Volume-2 ,[object Object],[object Object],[object Object],[object Object]

Accelerating Control Volume-3 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Accelerating Control Volume-4 ,[object Object],[object Object],[object Object],[object Object],Since ρ e w e A e > 0, the rocket’s t otal mass is decreasing

Accelerating Control Volume-5 ,[object Object],[object Object],[object Object],Net pressure force  on the rocket, D p Weight of the  rocket, M g  Friction drag on the rocket

Accelerating Control Volume-7 ,[object Object],[object Object],[object Object], Mass conservation tells us this is zero

Fluid Machanics

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Fluid Machanics