Problem 5.10
Difference between inflow and outflow in orifice flow
A wall in which there is a small nozzle, or hole, of radius R, separates two large compartments. The pressure p1 in the left-hand compartment far from the nozzle is greater than the pressu
Problem 5.20
Solid disc spinning up
Consider a solid disc with radius R, thickness h, and density . If the disc is free to rotate about its axis without friction, and if a torque T is applied to it about that axis, show by applying the angular momentum th
Problem 5.19
Spark-ignited spherical combustion See also Problem 3.7
Fig. 1 A combustible mixture of air and fuel is initially at rest at a density c and uniform pressure p . At t = 0, the mixture is ignited at the origin by a spark and a flame front begi
Problem 5.18
Rocket firing on test bed
The figure shows a rocket burning solid propellant, the system being mounted on a stationary test bed. The rocket body has an inside area A1 where the fuel is, and tapers to a smaller area A2 at the exit plane. The c
Problem 5.17
Rocket accelerating against gravity
After its second booster has been fired, a space vehicle finds itself outside the earth's r atmosphere moving vertically upward against gravity g at speed V0. Its total mass at that point is M0. At t = 0, t
Problem 5.16
Rocket assisted braking
A high-speed test vehicle with initial velocity V0 is to be brought quickly to rest on level ground by firing a rocket in the direction of motion. The rocket burns solid propellant at a mass rate mR (kg/s) and ejects g
Problem 5.15
Hinged flat plate swung against wall
A flat plate is hinged at one side to s smooth floor, as shown, and held at a small angle o ( o < 1) relative to the floor. The entire system is submerged in a liquid of constant density . At t = 0, a vert
Problem 5.14
Sand thrown onto rolling plate
A steel plate of mass M is supported by frictionless rollers. At t<0 the plate is moving at speed Vo to the right. Beginning at t = 0, sand is dumped onto the plate from above at a mass flow rate ms and with hor
Problem 5.13
Rain on roof
A violent rainstorm hits a roof inclined at an angle from the horizontal. The rain pours down at a mass flow rate m per unit horizontal area, each drop falling vertically at a speed V. Soon a steady state is established where the
Problem 5.12
Hovering platform
The sketch shows a hovering platform of half-width b and length L, the latter being large compared with b. The platform has a perforated bottom through which air is pumped at a steady total volume flow rate Q, the air leavin
Problem 5.11
Air cushion vehicle
The sketch (a) shows the cross-section of an air cushion vehicle of the "peripheral jet" type, first developed by Christopher Cockerell in the mid-1950s. A fan draws air from the ambient atmosphere at pressure pa and compr
Problem 5.9
Scrubber
A scrubber is a device that removes pollutants from air by allowing the air to come in contact with a spray of water droplets. The pollutants are absorbed by the droplets, and the air that passes through is clean. The polluted droplet
Problem 5.8
Jet pump
The device connected between compartments A and B is a simplified version of a jet pump. A jet (or ejector) pump is a device which uses a small, very high-speed jet with relatively low volume flow rate to move fluid at much larger vol
Problem 5.7
Pressure drop in a reaction zone
The sketch shows a liquid emulsiona finely-divided mixture of two immiscible, incompressible liquidswith mean density 1 entering a reaction zone of a constant-area reactor with speed V1. The components of the e
Problem 5.6
Narrowing of water stream from tap
An incompressible viscous liquid flows out of a long, circular tube with a small radius R. At the exit plane (1), the velocity profile is the parabolic one characteristic of fully developed, laminar flow in a
Problem 5.5
Sudden expansion in pipe
2 An incompressible fluid passes through a sudden expansion in a pipe, from area A1 = R1 2 to A2 = R2 . The flow just downstream of the expansion looks like a jet of radius R1 that emerges into an almost stagnant, re-c
Problem 5.4
Hydraulic jump
The top figure shows a "hydraulic jump" in a steady, two-dimensional water flow with a free surface. The "jump" is a relatively sudden increase in liquid depth in the direction of the flow, and a decrease in velocity, which is i
Problem 5.3
Force on sluice gate
Sluice gates are used to regulate water level (or flow rate) in open channels. The figure shows a gate that is adjusted so that the upstream depth is maintained at a depth h1. The density of water is , and the acceleration
Problem 5.2
Force on firemans hose
We are interested in how the shape of the nozzle and the direction of the stream affects the force required to hold a firemans hose. The sketch shows a test rig where a hosenozzle combination is held firmly by a support.
Problem 5.1
Force on nozzle Ain Sonin
A circular pipe with radius R1 carries a fluid with density to a converging nozzle with exit radius R2, mounted at its end. Before the flow is started, the bolts that compress the nozzle against pipe flange (see the f
The Centrifugal Turbine Ain Sonin, MIT
2.25 Fall 2004
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The Centrifugal Turbine Ain Sonin, MIT
2.25 Fall 2004
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The Centrifugal Turbine Ain Sonin, MIT
2.25 Fall 2004
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The Centrifugal Turbine Ain Sonin, MIT
2.25 Fall 2004
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2.25
Fall 2004 Ain A. Sonin, Gareth H. McKinley, MIT
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5.1
Control Volume Theorems
The Reynolds transformation, which allows the laws for material volumes to be reexpressed in terms of an arbitrarily specified control volume (an open system with an arbitr
1 Equation of Motion in Streamline Coordinates Ain A. Sonin, MIT 2.25 Advanced Fluid Mechanics Euler's equation expresses the relationship between the velocity and the pressure fields in inviscid flow. Written in terms of streamline coordinates, this equa
2.25 Fall 2004 G.H. McKinley
Navier Stokes Equation Dv v = + v v
= p + 2v + g Dt t Euler Equation; inviscid flow v + v v
= p + g t
v v =
Barotropic flow; = f ( p ) only
(
1 vv 2
)
v v
scalar potential for conservative body force (e.g. = gz )
v
2.25
Fall 2004; Gareth H. McKinley, Ain A. Sonin
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4.1 4.2 4.3
Inviscid Flow I: Euler's Equation of Motion, Bernoulli's Integral, and the Effects of Streamline Curvature
Euler's equation for inviscid motion: a relationship between fluid acceleration (conv
1
On Choosing and Using Control Volumes: Six Ways of Applying the Integral Mass Conservation Theorem to a Simple Problem
Ain A. Sonin, MIT 2001, 6 pages
Reference: Ain A. Sonin, Fundamental Laws of Motion for Particles, Material Volumes, and Control Volum
2.25
Ain A. Sonin, Gareth McKinley MIT
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3.1
Mass Conservation in Flowing Media
Mass conservation law in differential form. The physical significance of the v divergence of the velocity: V = the rate of increase of the material's volume, per unit volume.
2.25 PROBLEM 1.9: DICUSSION AND SOLUTION Ain Sonin, MIT The net upward lift force on a helium-filled balloon under quasi-static conditions is
L=(
air He
)gV
mg
(1)
where air is the density of the ambient air around the balloon, He is the density of the he
2.25
Ain A. Sonin, Gareth H. McKinley; MIT
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2.1
Static Fluids
Equation for pressure distribution in a fluid which is static in some reference frame (its "rest-frame"). Boundary condition on pressure at a density discontinuity in the absence of surface te