aml715-17

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)