General Viscosity Law - 2 | Fluid Mechanics for Mechanical Engineering PDF Download

Nine Scalar Quantities of Stress System - Stress Tensor

The set of nine components of stress tensor can be described as

                                                   (24.12)     

• The  stress tensor is symmetric,

• This means that two shearing stresses with subscripts which differ only in their sequence are equal. For example τ_2x = τ_x2

• Considering the equation of motion for instantaneous rotation of the fluid element (Fig. 24.1) about y axis, we can write

where  =dxdydz is the volume of the element,  is the angular acceleration 

  dl_y  is the moment of inertia of the element about y-axis

• Since dl_y is proportional to fifth power of the linear dimensions and    is proportional to the third power of the linear dimensions, the left hand side of the above equation and the second term on the right hand side vanishes faster than the first term on the right hand side on contracting the element to a point.
• Hence, the result is

τ_2x = τ_x2

From the similar considerations about other two remaining axes, we can write

τ_xy = τ_yx
τ_yz = τ_zy

which has already been observed in Eqs (24.2a), (24.2b) and (24.2c) earlier.

• Invoking these conditions into Eq. (24.12), the stress tensor becomes

                                                      (24.13)                        

• Combining Eqs (24.10), (24.11) and (24.13), the resultant surface force per unit volume becomes

                                          (24.14)

• As per the velocity field,

                                                  (24.15)

By Newton's law of motion applied to the differential element, we can write

 

Substituting Eqs (24.15), (24.14) and (24.6) into the above expression, we obtain

                                        (24.16c)

Since    

Similarly others follow.

• So we can express    in terms of field derivatives,

• These differential equations are known as Navier-Stokes equations.
• At this juncture, discuss the equation of continuity as well, which has a general (conservative) form

• In case of incompressible flow ρ = constant. Therefore, equation of continuity for incompressible flow becomes

                                                                                (24.19)                   

• Invoking Eq. (24.19) into Eqs (24.17a), (24.17b) and (24.17c), we get
   

Similarly others follow

Thus, 

                  (24.20c)

Vector Notation & derivation in Cylindrical Coordinates - Navier-Stokes equation

• Using, vector notation to write Navier-Stokes and continuity equations for incompressible flow we have

and

                                                                                                              (24.22)              

• we have four unknown quantities, u, v, w and p ,
• we  also have four equations, - equations of motion in three directions and the continuity equation.
• In principle, these equations are solvable but to date generalized solution is not available due to the complex nature of the set of these equations.
• The highest order terms, which come from the viscous forces, are linear and of second order
•  The first order convective terms are non-linear and hence, the set is termed as quasi-linear.

FIG 24.2 Cylindrical polar coordinate and the velocity components 

                                        (24.23a)

                                       (24.23b)

                                                                       (24.23b)

                                                                                                            (24.23c)