A Mathematical Frame Work to Create Fluid Flow Devices……

P M V SubbaraoProfessor

Mechanical Engineering DepartmentI I T Delhi

Development of Conservation Laws for Fluid Flows

Reynolds Analogy of First Kind• This establishes analogy between

variation of field variable of a system considered Control Mass and Control Volume.

• Also called as Reynolds Transport Theorem (RTT)

m

j ts

n

i tstV jinletiexit

dsnvtxfdsnvtxfdVt

txfDt

tDF11

,,

ˆ.,ˆ.,,

RTT describes balance equation for field variable flowing through Control surfaces of a control volume.

tV

dVvtxft

txfDt

tDF

,.,

tStV

dsnvtxfdVt

txfDt

tDF ˆ.,,

Two Dimensional Description of A Control Volume

14

ˆ.,ˆ.,,

sstV

dsnvtxfdsnvtxfdVt

txfDt

tDF

Thermodynamic Version of RTT

n

iiinlet

n

iiexit

CVCM

FFdt

tdFdt

tdF1

,1

,

V

CV dVtxfF ,

inletinlet A

inA

inlet AdvfAdvfF ..

exitexit A

exitA

exit AdvfAdvfF ..

m

j ts

n

i tstV jinletiexit

dsnvtxfdsnvtxfdVt

txfDt

tDF11

,,

ˆ.,ˆ.,,

Steady State Steady Flow Thermodynamic Model

constant1

,1

,

n

iiinlet

n

iiexit

CM

FFdtdF

V

CM dVxfF

inletinlet A

inA

inlet AdvfAdvfF ..

exitexit A

exitA

exit AdvfAdvfF ..

Uniform State Uniform Flow Thermodynamic Model

n

iiinlet

n

iiexit

CVCM

FFdt

tdFdt

tdF1

,1

,

V

CV dVtfF

inletinlet A

inA

inlet AdvfAdvfF ..

exitexit A

exitA

exit AdvfAdvfF ..

Mass Flow Balance in Stationary Frame of Reference

• The conservation law of mass requires that the mass contained in a material volume V=V(t), must be constant:

tV

dVm

Consequently, above equation requires that the substantial changes of the above mass must disappear:

Mass contained in a material volume

0

tV

dVDtD

DtDm

Using the Reynolds transport theorem , the conservation of mass, results in:

0.

tVtV

dVvt

dVDtD

This integral is zero for any size and shape of material volume. Implies that the integrand in the bracket must vanish identically. The continuity equation for unsteady and compressible flow is written as:

0. v

t

This Equation is a coordinate invariant equation. Its index notation in the Cartesian coordinate system given is:

0

i

i

xv

t 0

3

3

2

2

1

1

xv

xv

xv

t

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

xy1tan

The gradient operator is given by,

zz

rrr

ˆˆ1ˆ

As a consequence the conservation of mass equation becomes,

011

zvv

rrvr

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

xy1tan

rz1cos

or

cossinsinsincos

zryrx

The gradient is

ˆ1ˆsin1ˆ

rrrr

As a consequence the conservation of mass equation becomes,

0

sin1sin

sin11 2

2

v

rv

rrvr

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, 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 Fluid 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 rate 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 rate 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.

0DtDm 0

DtDmv

GSmESED FFFFFDtDmtxv

DttxvDm ,,

GSmESED FFFFF

DttxvmD

,

GMSmESED FFFFF

DttxvDm

,

• Applying the Reynolds transport theorem to the left-hand side of Equation

tV

dVvvtv

DttxvmD

.,

tVtV

dVvvdVtv

DttxvmD

.,

Replace the second volume integral by a surface integral using the Gauss conversion theorem

tStV

dSvvndVtv

DttxvmD

.ˆ,

GSmESEDtStV

FFFFFdSvvndVtv

.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)

Conservation Momentum for A CV

MGMSmagESEDtStV

FFFFFdSvvndVtv

.ˆ

tVMGMSmagESED FFFFFdVvv

tv

.

tVSB FFdVvv

tv

.

The Body Forces on A Fluid Flow

• The body forces refer to long-ranged forces such as gravity or that arising from an electromagnetic field, and so on.

• Such forces, as the name implies, vary slowly with the distance between interacting elements.

• For instance, the gravitational force per unit mass acting on any object is very well represented by the constant vector g (|g| = 9.806m/s2) for distances from the earth’s surface that are less than the order of the earth’s radius ( 6.4 ×106m).

• A consequence of this slow variation is that such forces act equally on all the matter within an infinitesimal element of fluid continuum.

• Thus, long-ranged forces manifest as body forces.

Extensive Nature of Body Forces

• Assuming the body force per unit mass at a point x in the fluid, and at time t, to be F(x, t).

• The total force on an infinitesimal element around this point with volume dV is given by ρF(x, t)dV .

• ρF is now the force per unit volume. • For gravity, this force is simply the weight of the element,

ρgdV .• For a conducting fluid in an magnetic field (B), the Lorentz

force: FL i×B • i is the current density.

The System of Body Forces in Fluid Flows

• Gravitational Forces• Magnetic Forces (Magneto Hydrodynamic)

– Lorentz force• Electrical forces.

– Coulomb force– Dielectric force – Electrostriction