Tutorial EG501V Computational Fluid Dynamics (AY 2023/24)

Tutorial 2. Incompressible Navier-Stokes equations

In Lecture Notes 1 the Navier-Stokes equations (momentum balance) for incompressible flow were derived. They were eventually written in the following form.

In this equation, the viscosity μ and the density ρ are constants. We now consider two simple flow configurations.

Config. 1. The steady state flow of a liquid in the space between two very large static parallel plates at distance H of each other in the presence of a constant pressure gradient in the x-direction Gravity is pointing in the negative y-direction.

Config. 2. The start-up of the flow of liquid between two very large parallel plates at distance H of each other. There is no pressure gradient. Before time t=0 everything is standing still. At t=0, the upper plate starts moving with a constant velocity U. In this configuration we do not   consider gravity.

Your assignments

For Configuration 1:

a.  Begin with the full, two-dimensional Navier-Stokes equations and determine which of the terms are zero and which are not.

b. Derive an expression for the velocity ux as a function of y by applying the simplified NS  equations (as found under Item a.) and by applying the no-slip condition at the two plates.

For Configuration 2:

c.  Begin with the full, two-dimensional Navier-Stokes equations and determine which of the terms are zero and which are not.

d. Make sure that when steady state has been reached ( t → ∞ ), the simplified form of theNS equations as found under Item c. leads to a linear velocity profile between the plates.