Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Topics covered in this course include fluid properties, fluid statics, fluid kinematics, control volume analysis, dimensional analysis, internal flows, differential analysis, external flows CFD, compressible flow and turbomachinery. Key words for this lecture are: Stream Function, Differential Equation, Momentum in Fluid Flow, Navier-Stokes Equation, Linear Momentum, Shear Stress, Shear Strain Rate, Cauchy's Equation, Stress Tensor, Derivation of the Navier-Stokes Equation
Typology: Exercises
1 / 7
This page cannot be seen from the preview
Don't miss anything!
Today, we will :
Derivation of the Navier-Stokes Equation (Section 9-5, Çengel and Cimbala)
We begin with the general differential equation for conservation of linear momentum, i.e., Cauchy’s equation , which is valid for any kind of fluid,
The problem is that the stress tensor σ ij needs to be written in terms of the primary unknowns in the problem in order for Cauchy’s equation to be useful to us. The equations that relate σ ij to other variables in the problem – velocity, pressure, and fluid properties – are called constitutive equations. There are different constitutive equations for different kinds of fluids.
Types of fluids:
Some examples of non-Newtonian fluids:
We consider only Newtonian fluids in this course.
Cauchy’s equation:
For Newtonian fluids , the shear stress is linearly proportional to the shear strain rate.
Examples of Newtonian fluids: water, air, oil, gasoline, most other common fluids.
Stress tensor
For Newtonian fluids (see text for derivation), it turns out that
Now we plug this expression for the stress tensor σ ij into Cauchy’s equation. The result is the famous Navier-Stokes equation , shown here for incompressible flow.
To solve fluid flow problems, we need both the continuity equation and the Navier-Stokes equation. Since it is a vector equation, the Navier-Stokes equation is usually split into three components in order to solve fluid flow problems. In Cartesian coordinates,
We have achieved our goal of writing σ ij in terms of pressure P , velocity components u , v , and w , and fluid viscosity μ.
Navier-Stokes equation:
