Title: MECH 221 FLUID MECHANICS (Fall 06/07) Tutorial 5
1MECH 221 FLUID MECHANICS(Fall 06/07)Tutorial 5
2Outline
- Equations of motion for inviscid flow
- Conservation of mass
- Conservation of momentum
- Bernoulli Equation
- Bernoulli equation for steady flow
- Static, dynamic, stagnation and total pressure
- Example
31. Equations of Motion for Inviscid Flow
- Conservation of Mass
- Conservation of Momentum
-
41.1. Conservation of Mass
- Mass in fluid flows must conserve. The total mass
in V(t) is given by - Therefore, the conservation of mass requires that
- dm/dt 0.
- where the Leibniz rule was invoked.
51.1. Conservation of Mass
- Hence
-
-
- This is the Integral Form of mass conservation
equation.
61.1. Conservation of Mass
- As V(t)?0, the integrand is independent of V(t)
and therefore, -
- This is the Differential Form of mass
conservation and also called as continuity
equation.
71.2. Conservation of Momentum
- The Newtons second law,
- is Lagrangian in a description of momentum
conservation. For motion of fluid particles that
have no rotation, the flow is termed
irrotational. An irrotational flow does not
subject to shear force, i.e., pressure force
only. Because the shear force is only caused by
fluid viscosity, the irrotational flow is also
called as inviscid flow
81.2. Conservation of Momentum
- For fluid subjecting to earth gravitational
acceleration, the net force on fluids in the
control volume V enclosed by a control surface S
is -
-
- where s is out-normal to S from V and the
divergence theorem is applied for the second
equality. - This force applied on the fluid body will leads
to the acceleration which is described as the
rate of change in momentum.
91.2. Conservation of Momentum
-
-
- where the Leibniz rule was invoked.
-
101.2. Conservation of Momentum
- Hence
-
-
- This is the Integral Form of momentum
conservation equation. -
111.2. Conservation of Momentum
- As V?0, the integrands are independent of V.
Therefore, - This is the Differential Form of momentum
conservation equation for inviscid flows.
121.2. Conservation of Momentum
- By invoking the continuity equation,
-
- The momentum equation can take the following
alternative form - which is commonly referred to as Eulers
equation of motion.
132.1. Bernoulli Equation for Steady Flows
- From differential form of the momentum
conservation equation - g-gVz
- By vector identity,
- Therefore, we get,
142.1. Bernoulli Equation for Steady Flows
- Assumption,
- Steady flow
- v and t are independent
- Irrotational flow
- Vxv0
0 (Steady flow)
0 (irrotational flow)
152.1. Bernoulli Equation for Steady Flows
- Finally, we can get,
- Or
- where vmagnitude of velocity vector v,
- i.e. vv(u2v2w2)
162.1. Bernoulli Equation for Steady Flows
- Since, for dr in any direction,
we have - For anywhere of irrotational fluids
- For anywhere of incompressible fluids
172.1. Bernoulli Equation for Steady Flows
- Bernoulli Equation in different form
- Energy density
- Total head (H)
182.2. Static, Dynamic, Stagnation and Total
Pressure
- Consider the Bernoulli equation,
- The static pressure ps is defined as the pressure
associated with the gravitational force when the
fluid is not in motion. If the atmospheric
pressure is used as the reference for a gage
pressure at z0.
(for incompressible fluid)
192.2. Static, Dynamic, Stagnation and Total
Pressure
- Then we have as also from chapter
2. - The dynamic pressure pd is then the pressure
deviates from the static pressure, i.e., p
pdps. - The substitution of p pdps. into the
Bernoulli equation gives
202.2. Static, Dynamic, Stagnation and Total
Pressure
- The maximum dynamic pressure occurs at the
stagnation point where v0 and this maximum
pressure is called as the stagnation pressure p0.
Hence, - The total pressure pT is then the sum of the
stagnation pressure and the static pressure,
i.e., pT p0 - ?gz. For z -h, the static
pressure is ?gh and the total pressure is p0
?gh.
212.3.1. Example (1)
- Determine the flowrate through the pipe.
222.3.1. Example (1)
- Procedure
- Choose the reference point
- From the Bernoulli equation
- P, V, Z all are unknowns
- For same horizontal level, Z1Z2
- V V(P1, P2)
- From the balance of static pressure
- P ?gh
- ?h is given, ?m, ?water are known
- V V(?h, ?m, ?water)
- Q AV pD2V/4
232.3.1. Example (1)
- From the Bernoulli equation,
242.3.1. Example (1)
- From the balance of static pressure,
252.3.1. Example (1)
262.3.2. Example (2)
- A conical plug is used to regulate the air flow
from the pipe. The air leaves the edge of the
cone with a uniform thickness of 0.02m. If
viscous effects are negligible and the flowrate
is 0.05m3/s, determine the pressure within the
pipe.
272.3.2. Example (2)
- Procedure
- Choose the reference point
- From the Bernoulli equation
- P, V, Z all are unknowns
- For same horizontal level, Z1Z2
- Flowrate conservation
- QAV
282.3.2. Example (2)
- From the Bernoulli equation,
292.3.2. Example (2)
- From flowrate conservation,
302.3.2. Example (2)
- Sub. into the Bernoulli equation,
31The End