Many fluids exhibit a linear relationship between the stress components and the velocity gradients. Such fluids are called Newtonian fluids and include common fluids such as water, oil and air. (Potter & Wiggert, 2002) This set of notes is a development of the basic Navier-Stokes equations for Newtonian fluids. It is similar to developments in Viscous Fluid Flows, with the possible exception of the terminology for the shear stresses.(Munson, Young & Okiishi, 2006)
Normally, this set of equations includes conservation of mass, momentum, and frequently energy. Species conservation equations are nearly identical to the energy equations except for the diffusion coefficients, so they are not usually explicitly given, unless there is reason to emphasize the interactions between heat and mass transfer.
The equation of incompressible fluid flow,
is the kinematic viscosity,
v is the velocity of the fluid parcel,
p is the pressure
is the fluid density.
B).Derivation of Newtonian Fluid: Navier-Stokes Equation
For Newtonian fluid the viscous stress is proportional to the rate of shearing strain (angular deformation rate). The stresses may be expressed in term of velocity gradients and fluid properties in constitutive equation as follows:
For most gases and for monatomic gases exactly, the second coefficient of viscosity is related to the viscosity by
a condition that is known as Stokes hypothesis. With this relationship the negative average of the three normal stress is equal to the pressure, that is,
So, using constitutive equation above, this can be shown to always be true for a liquid in which , and with Stokes hypothesis it is also true for gas.
Substitute the constitutive equation into the differential momentum equation, there result using Stokes hypothesis,
Where we assumed a homogeneous fluid, that is, fluid properties are independent of position.
For an incompressible flow the continuity equation allows the equation above to be reduce, these are the Navier-Stokes Equation
Where we have used the Laplacian
Combining the above, the Navier-Stokers equation takes the vector form
C).Examples of Navier-Stokers with explanation are expected
i) For example, the measurement of changes in wind velocity in the atmosphere can be obtained with the help of an anemometer in a weather station or by mounting it on a weather balloon. The anemometer in the first case is measuring the velocity of all the moving particles passing through a fixed point in space, whereas in the second case the instrument is measuring changes in velocity as it moves with the fluid.
ii) In fluid dynamics, the Taylor-Green vortex is a 2-dimensional, unsteady flow of a decaying vortex, which has the exact closed form solution of incompressible Navier-Stokes equations in Cartesian coordinates.
iii) One of the obvious examples is sound. Description of such phenomena requires more general presentation of the Navier-Stokes equation that takes into account fluid compressibility.
D).Example to understanding
Simplify the x-component Navier-Stokes equation for steady flow in a horizontal, rectangular channel assuming all streamlines parallel to the walls. Lets the x-direction be in the direction of flow.
If the streamlines are parallel the walls, only the x-component of velocity will be nonzero. Letting v = w =0 the continuity equation for an incompressible flow becomes,
Showing that u = u(y , z) . the acceleration is then
The x-component momentum equation then simplifies to