Understand the foremost Economic Theory of Engineering ….

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Study of Navier-Stokes Equations

Differential Momentum Conservation Equations for Fluid Flows

ijgvvt

v

.

vfij

0.

vt

It is a must to explore the most important microscopic relation.

jiijjijiijjiij vveevveef ,,,, 2

1,

2

1

Philosophy of Science

• The goal which physical science has set itself is the simplest and most economical abstract expression of facts.

• The human mind, with its limited powers, attempts to mirror in itself the rich life of the world, of which it itself is only a small part…….

• In reality, the law always contains less than the fact itself.• A Law does not reproduce the fact as a whole but only in

that aspect of it which is important for us, the rest being intentionally or from necessity omitted.

Newtonian (linear) Viscous Fluid• "Newton's law of viscosity" is a theory in physics named after English

physicist Sir Isaac Newton.

• This was published in Philosophiae Naturalis Pricipia Mathematica in 1687.

dy

duxy

• The original law was modified as, under conditions of steady streamline flow, the shearing stress needed to maintain the flow of the fluid is proportional to the velocity gradient in a direction to the direction of flow

• Newton’s Law of viscosity is the most Economic solution to highly complex truth.

Relation between Stress & Strain on A General Plane Axes

These stress and strain components on a general plane must obey stokes laws, and hence

vCKp xxxx

.2

xyxy K

Newtonian (linear) Viscous Fluid:

• Compare stokes equations with Newton’s Law of viscosity.

dy

duxy

The linear coefficient K is equal to twice the ordinary coefficient of viscosity, K = 2.

xyxy K

vCKp xxxx

.2 xyxy K

A virtual Viscosity

• The coefficient C2, is new and independent of and may be called the second coefficient of viscosity.

• In linear elasticity, C2, is called Lame's constant and is given the symbol , which is adopted here also.

• Since is associated only with volume expansion, it is customary to call it as the coefficient of bulk viscosity .

General deformation law for a Newtonian (linear)viscous fluid:

j

i

i

jijij x

v

x

vvp .

This deformation law was first given by Stokes (1845).

vp xxxx

.2

xyxy 2

Thermodynamic Pressure Vs Mechanical Pressure

• Stokes (1845) pointed out an interesting consequence of this general Equation.

• By analogy with the strain relation, the sum of the three normal stresses xx , yy and zz is a tensor invariant.

• Define the mechanical pressure as the negative one-third of this sum.

• Mechanical pressure is the average compression stress on the element.

zzyyxxp 3

1vp xxxx

.2

vp yyyy

.2

vp zzzz

.

vpp

.3

2

Stokes Hypothesis

• The mean pressure in a deforming viscous fluid is not equal to the thermodynamic property called pressure.

• This distinction is rarely important, since v is usually very small in typical flow problems.

• But the exact meaning of mechanical pressure has been a controversial subject for more than a century.

• Stokes himself simplified and resolved the issue by an assumption:

03

2

This relation, frequently called the Stokes’ relation,.This is truly valid for monoatomic gases

Above equation leads to 3

2

The Controversy

• Stokes hypothesis simply assumes away the problem.

• This is essentially what we do in this course.

• The available experimental evidence from the measurement of sound wave attenuation, indicates that for most liquids is actually positive.

is not equal to -2/3, and often is much larger than .

• The experiments themselves are a matter of some controversy.

3

2

Thus spake : Ernst Mach

• In mentally separating a body from the changeable environment in which it moves, what we really do is to extricate a group of sensations on which our thoughts are fastened and which is of relatively greater stability than the others, from the stream of all our sensations.

• It is highly an economical reason to think that the fastness of a flying machine is described in terms of velocity (km/hr) !!!!

Incompressible Flows

• Again this merely assumes away the problem.

• The bulk viscosity cannot affect a truly incompressible fluid.

• In fact it does affect certain phenomena occurring in nearly incompressible fluids, e.g., sound absorption in liquids.

• Meanwhile, if .v0, that is, compressible flow, we may still be able to avoid the problem if viscous normal stresses are negligible.

• This is the case in boundary-layer flows of compressible fluids, for which only the first coefficient of viscosity is important.

• However, the normal shock wave is a case where the coefficient cannot be neglected.

• The second case is the above-mentioned problem of sound-wave absorption and attenuation.

Bulk Viscosity Coefficient

• The second viscosity coefficient is still a controversial quantity.

• Truly saying, may not even be a thermodynamic property, since it is found to be frequency-dependent.

• The disputed term, divv, is almost always so very small that it is entirely proper simply to ignore the effect of altogether.

• Collect more discussions on This topic and submit as an assignment: Date of submission: 22nd September 2015.

The Navier-Stokes Equations

• The desired momentum equation for a general linear (newtonian) viscous fluid is now obtained by substituting the stress relations, into Newton's law.

• The result is the famous equation of motion which bears the names of Navier (1823) and Stokes (1845).

• In scalar form, we obtain

z

u

x

w

zx

v

y

u

yv

x

u

xx

pg

Dt

Dux

2

y

w

z

v

zv

y

v

yy

u

x

v

xy

pg

Dt

Dvy

2

vz

w

zy

w

z

v

yz

u

x

w

xz

pg

Dt

Dwz

2

These are the Navier-Stokes equations, fundamental to the subject of viscousfluid flow. Considerable economy is achieved by rewriting them as a single vector equation, using the indicia1 notation:

vx

v

x

v

xpg

Dt

vDij

i

j

j

i

j

Incompressible Flow

• If the fluid is assumed to be of constant density, divv vanishes due to the continuity equation.

• The vexing coefficient disappears from Newton's law.

• NS Equations are not greatly simplified, if the first viscosity is allowed to vary with temperature and pressure.

i

j

j

i

j x

v

x

v

xpg

Dt

vD

vpgDt

vD

v

pg

Dt

vD

2

• This leads to assumption of is constant, many terms vanish.

• A much simpler Navier-Stokes equation for constant viscosity is

ijgvvt

v

.

Incompressible NS Equations in Cylindrical Coordinate system

222

2

2

2

22

2

2

211

r

v

rr

v

r

v

z

vv

rr

v

r

pg

r

v

z

vv

v

r

v

r

vv

t

v

rrrrrr

rz

rrr

r

Navier- Stokes equation in r-direction:

222

2

2

2

22

2

211

r

v

rr

v

r

v

z

vv

rr

v

r

pg

r

vv

z

vv

v

r

v

r

vv

t

v

r

rzr

Navier- Stokes equation in -direction:

Navier- Stokes equation in z-direction:

rr

v

z

vv

rr

v

z

pg

z

vv

v

r

v

r

vv

t

v

zzzzz

zz

zzr

z

2

2

2

2

22

2 11

Fluid Mechanics Made Easy

• Incompressibility is an excellent point of departure in the theory of incompressible viscous flow.

• It is essential to remember that it assumes constant viscosity.

• For non-isothermal flows, it may be a rather poor approximation.

• This approximation is highly objectionable, particularly for liquids, whose viscosity is often highly temperature-dependent.

• For gases, whose viscosity is only moderately temperature-dependent, this is a good approximation

• This fails only when compressibility becomes important, i.e., when v 0.