Question

Tutorial 2. Incompressible Navier-Stokes equations 18 September, 4-5 pm in FN2 In Lecture Notes 1 the Navier-Stokes equations (momentum balance) for incompressible flow were derived. They were eventually written in the following form dr 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 thex directionA. Gravity is pointing in the negative y-direction. Эк liquid H--A configuration 1 Config. 2. The start-up of the flow of liquid betw H of each other. There is no pressure gradient. Before time t-0 everything is standing still. At I-0, the upper plate starts moving with a constant velocity U. In this configuration we do not consider gravity een two very large parallel plates at distance liquid configuration 2 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 u, 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-oo), the simplified form of the NS equations as found under Item c. leads to a linear velocity profile between the plates

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