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A Mathematical Frame Work to Create Fluid Flow Devices……
P M V SubbaraoProfessor
Mechanical Engineering Department
I I T Delhi
Conservation Laws for Viscous Fluid Flows
Viscous Fluid Flow is A Control Volume Flow
0ˆ.,ˆ.,,
11,,
m
j ts
n
i tstV jinletiexit
dsnvtxdsnvtxdVt
tx
Dt
Dm
Steady Viscous Flow
If the density does not undergo a time change (steady flow), the above equation is reduced to:
0ˆ.,ˆ.,,
11,,
m
j ts
n
i tstV jinletiexit
dsnvtxdsnvtxdVt
tx
Dt
Dm
0ˆ.ˆ.11
,,
m
j ts
n
i ts jinletiexit
dsnvxdsnvxDt
Dm
0ˆ. ts
dsnvxDt
Dm
Continuity Equation in Cartesian Coordinates
• The continuity equation for unsteady and compressible flow is written as:
0.
vt
This Equation is a coordinate invariant equation. Its index notation in the Cartesian coordinate system given is:
0
i
i
x
v
t
0
3
3
2
2
1
1
x
v
x
v
x
v
t
0.
tVtV
dVvt
dVDt
D
Continuity Equation in Cylindrical Polar Coordinates
• Many flows which involve rotation or radial motion are best described in Cylindrical Polar Coordinates.
• In this system coordinates for a point P are r, and z.
The velocity components in these directions respectively are vr ,v and vz. Transformation between the Cartesian and the polar systems is provided by the relations,
22 yxr
x
y1tan
The gradient operator is given by,
zz
rrr
ˆˆ1ˆ
As a consequence the conservation of mass equation becomes,
0
11
z
vv
rr
vr
rtzr
Continuity Equation in Cylindrical Polar CoordinatesSpherical polar coordinates are a system of curvilinear coordinates
that are natural for describing atmospheric flows.Define to be the azimuthal angle in the x-y -plane from the x-axis with 0 < 2 . to be the zenith angle and colatitude, with 0 < r to be distance (radius) from a point to the origin.
The spherical coordinates (r,,) are related to the Cartesian coordinates (x,y,z) by
222 zyxr
x
y1tan
r
z1cos
or
cos
sinsin
sincos
z
ry
rx
The gradient is
ˆ1ˆsin
1ˆ
rrrr
As a consequence the conservation of mass equation becomes,
0
sin
1sin
sin
11 2
2
v
r
v
rr
vr
rtr
Balance of Linear Momentum
• The momentum equation in integral form applied to a control volume determines the integral flow quantities such as blade lift, drag forces, average pressure.
• The motion of a material volume is described by Newton’s second law of motion which states that mass times acceleration is the sum of all external forces acting on the system.
• These forces are identified as electrodynamic, electrostatic, and magnetic forces, viscous forces and the gravitational forces:
• For a control mass
GMSmESED FFFFFDt
txvDm
,
This equation is valid for a closed system with a system boundary that may undergo deformation, rotation, expansion or compression.
Balance of Momentum for Flow
• In a flow, there is no closed system with a defined system boundary.
• The mass is continuously flowing from one point to another point.
• Thus, in general, we deal with mass flow rather than mass.
• Consequently, the previous equation must be modified in such a way that it is applicable to a predefined control volume with mass flow passing through it.
• This requires applying the Reynolds transport theorem to a control volume.
The Preparation
• The momentum balance for a CM needs to be modified, before proceeding with the Reynolds transport theorem.
• As a first step, add a zero-term to CM Equation.
0Dt
Dm0
Dt
Dmv
GSmESED FFFFFDt
Dmtxv
Dt
txvDm ,
,
GSmESED FFFFF
Dt
txvmD
,
• Applying the Reynolds transport theorem to the left-hand side of Equation
tV
dVvvt
v
Dt
txvmD
.
,
tVtV
dVvvdVt
v
Dt
txvmD
.
,
Replace the second volume integral by a surface integral using the Gauss conversion theorem
tStV
dSvvndVt
v
Dt
txvmD
.ˆ
,
GSmESED
tStV
FFFFFdSvvndVt
v
.ˆ
Viscous Fluid Flows using a selected combination of Forces
• Systems only due to Body Forces.
• Systems due to only normal surface Forces.
• Systems due to both normal and tangential surface Forces.
– Thermo-dynamic Effects (Buoyancy forces/surface)…..
– Physico-Chemical/concentration based forces (Environmental /Bio Fluid Mechanics)
