Gaya – Gaya Fiskositas ( Viscous Forces )

Mathematical Expression for the viscous forces

By reviewing the two-dimensional fluid element below:

Diketahui:

The friction forces on the side AB of length dx is : τ dx = μ (∂u/∂y)dx

Since the velocity at C is (u + (∂u/∂y ) dy)

The friction force on the side CD is:

dyy

dyy

uu

y

dydxy

udx

y

u2

2

dx = μ

= μ (1)

• Friction force on the side CD act in opposite diretions.

• If the forces due to the particle GHCD act in the OX direction on the side CD of ABCD,

• The forces due to the particle ABCD will act in the same direction of the AB side of a ABFE.

• And then, by reaction ABFE will cause a force in the –OX direction on ABCD.

The total shear force thus become: dxdyy

udxdy

y 2

2

(2)

• The friction force per unit of area is:

dxdyy

udxdy

y 2

2

dxdyy

udxdy

y 2

2

2

2

y

u

y

(3)

For a 3D incompressible fluide, The friction force components per unit of volume are:

They are written vectorially:

uz

u

y

u

x

u 2

2

2

2

2

2

2

V

vz

v

y

v

x

v 2

2

2

2

2

2

2

V

wz

w

y

w

x

w 2

2

2

2

2

2

2

V

V VVVV 2

2

2

2

2

2

2

zyx

Approximations Made on Viscous Forces

It has been shown experimentally that it is sometimes possible to consider friction effects as negligible. The preceding expression for the friction forces show that they can be neglected when the Laplacian of the velocity components ( ) is small. V 2

Theoretical Considerations of Surface Forces

1. A general Expression for Surface forces

2. The Nine Components of the External forces : Components of Lame

3. Value of the Lame Components in Some Particular Cases

4. Disipassion Function

A general Expression for Surface forces

Surfaces forces, as previously seen, consist of pressure force and friction force. These surface forces may be introduced without consideration of their physical nature. The advantage of so expressing the surface forces lies in its applicability to any kind of motion, e.g., viscous or turbulent, compressible or incompressible.

The Nine Components of the External forces : Components of

Lame Consider an elementary mass of fluid in the form of cube; its edges are parallel to the three coordinate axes OX, OY and OZ, shown in this figure:

The pairs of parallel forces acting on two opposite sides of the cube act in opposite directions, and their difference is obtained by a simple partial derivative with respect to the distance between the two considered sides.

Hence, the external forces may be defined by a tensor of rank two:

The addition of all the forces per unit volume acting in the same direction yields:

In the OX direction

In the OY direction

In the OZ direction

zzyzxz

zyyyxy

zxyxxx

zyxzxyxxx

zyxzyyyxy

zyxzzyzxz

Value of the Lame Components in Some Particular Cases

1. In the case of a perfect fluid, the shearing stress are zero and the normal forces become simply the pressure forces:

σxx = σyy = σzz = - p

τxy = τyz = τzx = 0

2. In a viscous incompressible fluid, it is possible to demonstrate that the normal forces (σ) are the sum of the pressure force and a viscous force proporsional to the coefficients of linear deformation:

3. In the case of a viscous compressible fluid, the shearing stresses are the same as in the above case, but the normal forces have to take into account the change of volume of the fluid particle. It may be seen that :

x

u

z

w

y

v

x

upxx

2

x

upxx

2

Disipassion Function

The energy transformed into heat either by change of volume or by friction may be obtained by adding the work done by all the external forces. This work is equal to the external forces times their displacement.

For instance, in the OX direction, the work of pressure forces is :

P dy dz u dt dz

dydxx

pp

dtdxx

uu

The work of all the forces in the OX direction is:

The total work per unit volume changed into heat in a unit period of time is called the “dissipation function” ф. It is a function of the liniear and angular rates of deformation and is found by substituting the valus of σ and τ.

222222

2 222y

u

x

v

x

w

z

u

z

v

y

w

z

w

w

v

x

udivV

dydxx

xxxx

dtdxx

uu

dydxxxy

xy

dtdx

x

uu

dydxxxz

xz

dtdxx

uu

σxx dy dz u dt - + τxy dy dz u dt -

+ τxz dy dz u dt - dz

dz