ME3560 – Fluid Mechanics
Summer 2016
ME 3560 Fluid Mechanics
Chapter III. Elementary Fluid
Dynamics – The Bernoulli Equation
1
Chapter III. Bernoulli Equation
Summer 2016
ME3560 – Fluid Mechanics
3.1 Newton’s Second Law
• A fluid particle can experience acceleration or deceleration as it moves
from one location to another. This motion is ruled by Newton's second
r
law of motion:
r
ΣF = ma
•Assuming that the flow is inviscid under a steady – state process, the
governing equations for a particle of fluid immersed in the flow will be
deducted next.
3.2 F = ma along a Streamline
•Streamlines are lines that
are tangent to the velocity
vector everywhere in the
flow.
2
Chapter III. Bernoulli Equation
ME3560 – Fluid Mechanics
r r
V × ds = 0
Summer 2016
(uiˆ + vˆj + wkˆ) × (dxiˆ + dyˆj + dzkˆ) = 0
(vdz − wdy)iˆ + ( wdx − udz) ˆj + (udy − vdx)kˆ = 0
dx dy dz
=
=
u
v
w
•Differential Equation Representing
Streamlines in 3D.
•Consider an infinitesimally small
fluid particle of size δs×δn×δy.
•Unit vectors along and normal to
the streamline are ŝ and ň.
•Steady flow, Newton's second law
along the streamline direction is
∂V
δFs = δmas = δmV
∂s
∂V
δFs = ρδV
∂s
Chapter III. Bernoulli Equation
ME3560 – Fluid Mechanics
Summer 2016
•The weight on the particle is δW = γδV,
(γ= ρg).
•The component of the weight in the
direction of the streamline is
δWs = −δW sin θ = −γδV sin θ
•In general, for steady flow, p=p(s, n).
•If the pressure at the center of the
particle is p.
•Its average value on the two end faces
perpendicular to the streamline are
•p + δps and p − δps.
∂pδs
δps ≈
st
•The 1 term in a Taylor series expansion for p is:
∂s 2
•If δFps is the net pressure force on the particle in s–direction:
δFps = ( p − δ p s )δ nδ y − ( p + δ p s )δ nδ y = −2δ p sδ nδ y
δFps = −
∂ ps
∂ ps
δ sδ nδ y = −
δV
∂s
∂s
Chapter III. Bernoulli Equation
Summer 2016
ME3560 – Fluid Mechanics
•The net force acting in the streamline direction on the particle is
∑ δF
s
= δ W s + δ F ps

∂ p
δ V
=  − γ sin θ −
∂s 

•Thus, from the equation obtained previously:
•And the previous expression:
∂p
∂V
− γ sin θ −
= ρV
= ρ as
∂s
∂s
•This equation can be rearranged and
integrated as follows.
•Notice that sin θ = dz/ds and that
VdV/ds = ½ d(V2)/ds.
•On the streamline n is const. (dn = 0):
dp=(∂p/∂s)ds+(∂p/∂n)dn = (∂p/∂s) ds.
•Hence, along a given streamline:
•p(s, n) = p(s) and ∂p/∂s = dp/ds.
Chapter III. Bernoulli Equation
∂V
δFs = ρδV
∂s
ME3560 – Fluid Mechanics
∂p
∂V
•Thus the equation:
− γ sin θ −
= ρV
= ρ as
•Can be presented as:
∂s
∂s
Summer 2016
1
dz d p 1 d (V 2 )
; dp + ρd (V 2 ) + γdz = 0 (on a streamline)
−γ
−
= ρ
2
ds d s 2
ds
dp V 2
∫ ρ + 2 + gz = C
(along a streamline)
•In general it is not possible to integrate the pressure term because the
density may not be constant.
•To complete the integral the relation between ρ and p must be known.
•For steady, inviscid, incompressible flow, the Bernoulli Equation is
obtained:
1
2
p+
2
ρ V + γ z = constant along streamline
•This equation is valid when: (1) viscous effects are assumed negligible,
(2) the flow is assumed to be steady, (3) the flow is assumed to be
incompressible, (4) the equation is applicable along a streamline.
Chapter III. Bernoulli Equation
Summer 2016
ME3560 – Fluid Mechanics
3.3 F = ma Normal to a Streamline
•In this section we will consider
application of Newton's second law
in a direction normal to the
streamline.
•Consider the force balance on the
fluid particle shown:
•Considering components in the
normal direction, n, and writing
Newton's second law in this
direction as
∑δ F
n
=
δ mV
R
2
=
Chapter III. Bernoulli Equation
ρδ VV
R
2
Summer 2016
ME3560 – Fluid Mechanics
•Assuming steady state flow, normal acceleration an = V2/R, where R is
the local radius of curvature of the streamlines.
•an is due to the change in direction of the particle's velocity as it moves
along a curved path.
•Assume that the only forces of importance are pressure and gravity. The
component of the weight (gravity force) in the normal direction is
δ Wn = −δ W cosθ = −γδ V cosθ
•If the pressure at the center of the
particle is p, then using a Taylor series
expansion: 
∂ p δ n
δ Fpn =  p −

δ sδ y
∂n 2 

∂ p δ n
δ sδ y
−  p +
∂ n 2Equation
 III. Bernoulli

Chapter
∂p
δ Fpn = −
δV
∂n
ME3560 – Fluid Mechanics
Summer 2016
•Thus, the net force acting in the normal direction on the particle is:

∂ p
∑ δ Fn = δ Wn + δ Fpn = − γ cosθ + ∂ n δ V

•Then, by combining the previous eqn. with 
2
2
δ mV
ρδ VV
∑ δ Fn = R = R
•The equation of motion along the normal direction is found (notice that
2
cos θ = dz/dn):
dz ∂ p ρV
−γ
−
=
dn ∂ n
R
•That is, a change in the direction of flow of a fluid particle (i.e., a
curved path, R< ∞) is accomplished by the appropriate combination of
pressure gradient and particle weight normal to the streamline. A larger
speed or density or a smaller radius of curvature of the motion requires a
larger force unbalance to produce the motion.
Chapter III. Bernoulli Equation
ME3560 – Fluid Mechanics
Summer 2016
•Neglecting gravity (commonly done for gas flows) or if the flow is in a
horizontal (dz/dn = 0) plane, the previous eqn. becomes ∂ p
ρV 2
=∂n
R
•This indicates that the pressure increases with distance away from the
center of curvature (∂p/∂n is negative since ρV2/R is positive—the
positive n direction points toward the “inside” of the curved streamline).
•Multiply the previous eqn. by dn, and noticing that ∂p/∂n = dp/dn if s is
constant, the integral 2across the streamline (in the n direction) is:
∫
dp
V
+∫
dn + gz = const. across the streamline
R
ρ
•To complete the indicated integrations, it is necessary to know how the
density varies with pressure and how the fluid speed and radius of
curvature vary with n. For incompressible flow ρ = const. Still it is
necessary to know the relation between V and R with n[V = V(s, n) and
R= R(s, n)]. Thus, the final form of Newton's second law applied across
the streamlines for steady, inviscid, incompressible flow is
V2
p + ρ∫
dn + γ z = constant across the streamline
Chapter III. Bernoulli Equation
R
ME3560 – Fluid Mechanics
Summer 2016
3.5 Static, Stagnation, Dynamic, and Total Pressure
•From Bernoulli’s Equation:
1
p + ρ V 2 + γ z = constant along streamline
2
•p, is the actual thermodynamic
pressure (static pressure) of the
fluid as it flows. To measure its
value, one could move along with
the fluid, thus being “static” relative
to the moving fluid.
•Or by drilling a hole in a flat surface and fasten a piezometer tube as
indicated by the location of point (3).
•The pressure in the flowing fluid at (1) is p1 = γh3-1 + p3, the same as if
the fluid were static. From the manometer relations: p3 = γh4-3. Thus,
since h3-1+ h4-3= h it follows that p1 = γh.
Chapter III. Bernoulli Equation
ME3560 – Fluid Mechanics
Summer 2016
•From Bernoulli’s Equation:
1
p + ρ V 2 + γ z = constant along streamline
2
•The term γ z, is the hydrostatic
pressure. It is not actually a pressure
but does represent the change in
pressure possible due to potential
energy variations of the fluid as a
result of elevation changes.
•The term ρV2/2, is the dynamic pressure.
•Its interpretation can be seen in the figure by considering the pressure at
the end of a small tube inserted into the flow and pointing upstream.
After the initial transient motion has died out, the liquid will fill the tube
to a height of H as shown. The fluid in the tube, including that at its tip,
(2), will be stationary. That is, V2= 0, or point (2) is a stagnation point.
Chapter III. Bernoulli Equation
Summer 2016
ME3560 – Fluid Mechanics
•Applying the Bernoulli equation
between points (1) and (2), (V2= 0)
and z1 = z2:
1
2
p2 = p1 +
2
ρ V1
•The pressure at the stagnation point
is greater than the static pressure, p1,
by an amount ρV2/2, the dynamic
pressure.
•There is a stagnation point on any
stationary body that is placed into a
flowing fluid.
•Some of the fluid flows “over” and some
“under” the object. The dividing line (or
surface for two-dimensional flows) is termed
the stagnation streamline and terminates at the
stagnation point on the body. For symmetrical
objects (such as a baseball) the stagnation point
is clearly at the tip or front of the object.
Chapter III. Bernoulli Equation
Summer 2016
ME3560 – Fluid Mechanics
3.6 Examples of Use of the Bernoulli Equation
•Between any two points, (1) and (2), on a streamline in steady, inviscid,
incompressible flow the Bernoulli equation can be applied in the form
1
1
2
p1 + ρ V1 + γ z1 = p 2 + ρ V22 + γ z 2
2
2
•If 5 of the 6 variables are known, the remaining one can be determined.
• Frequently, it is necessary to introduce other equations (continuity, etc.)
3.6.1 Free Jets
•When a jet of liquid of diameter
d flows from the nozzle with
velocity V. Bernoulli’s equation
gives:
1
2
γh=
2
ρV
•The fluid leaves as a “free jet”
(p2 = 0)
Chapter III. Bernoulli Equation
V = 2 gh
Summer 2016
ME3560 – Fluid Mechanics
3.6.3 Flowrate Measurement
•Several devices using principles involved in
the Bernoulli equation have been developed to
measure fluid velocities and flowrates.
•The Pitot-static tube is an example.
•Other examples are flowrate meters, for pipes
and channels.
•The flowrate in a pipe can be measured using:
the orifice meter, the nozzle meter, and the
Venturi meter.
1
1
2
•Application of Bernoulli’s equation results in: p1 + ρ V1 = p2 + ρ V22
2
2
•Which when combined with the mass
conservation equation:Q = V1 A1 = V2 A2
•Yields:
Q = A2
Chapter III. Bernoulli Equation
2( p1 − p2 )
ρ[1 − ( A2 / A1 ) 2 ]