Stokes Law of Cause & Effect in VFF…

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Deformation Law for Navier-Stokes Equations

General Deformation Law along General Axes

• The transformation rule between a normal stress or strain rate in the new system and the principal stresses or strain rates is given by,

2133

2122

2111 nmlxx

2133

2122

2111 nmlxx

0.121

21

21 nmlUsing the equation

Shear Stresses along General Axes

• Similarly, the shear stresses (strain rates) are related to the principal stresses (strain rates) by the following transformation law:

213321222111 nnmmllxy

213321222111 nnmmllxy These stress and strain components must obey stokes law, and hence

vCKp xxxx

.2

xyxy K

Note that the all direction cosines have all politely vanished.

Similarly, a simplification is done to give

In search of true Parameters Responsible for Deformation

• Euler and his colleagues, contributed valuable analytical research on frictionless flows.

• The next significant analytical advance was the addition of frictional-resistance terms to Euler's in viscid (frictionless) equations.

• This was done, with varying degrees of elegance by

• Navier in 1827

• Cauchy in 1828

• Poisson in 1829

• St. Venant in 1843 and

• Stokes in 1845

• The first four wrote their equations in terms of an unknown molecular function.

• Stokes was the first to use the first coefficient of viscosity p

Newtonian (linear) Viscous Fluid:

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

dy

duxy

We see that the linear coefficient K is equalto twice the ordinary coefficient of viscosity, K = 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 simply resolved the issue by an assumption:

03

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 -2p/3, and often is much larger than p.

• The experiments themselves are a matter of some controversy.

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 divv # 0, 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.