Assignment/Reading 9, Physics 426 Fluid Mechanics
Reading (Participation only, very short answer)
- Boundary Layers
- Instability
R1: Growth of a boundary layer…
What parameters set how thick a boundary layer will be? Consider a boat moving at 2 m/s (4 kts) through the water. How thick will the boundary layer be 1 m back from the bow with this scaling?
R2: Instability:
From the reading, qualitatively describe how we determine if a wavelength is unstable in a flow.
Assignment
A1: Shear Parallel Flow (/15)
Consider a flow between two parallel infinite plates a distance $H$ apart. The upper plate is moving with speed $U$ parallel to the other plate, which is motionless. The flow starts at rest and the plate is impulsively moved to speed $U$. [Assume that the flow is laminar (i.e. no turbulence develops) and that the viscosity of the fluid is given by $\nu/(HU) = 0.1$. Assume that there are no net pressure forces.]
- What is the steady-state solution of the flow between the two plates (i.e. $u_0(z) \equiv u(z,t=\infty)$)
- Derive a differential equation for the transient flow ($w(z,t) = u(z,t)-u_0(z)$) and state the spatial and temporal boundary conditions for $w(z,t)$.
- Determine an appropriate form for the solution to $w(z,t)$ (Hint: The solution to $w$ is separable in time and space. In order to match the initial condition, a discrete Fourier series in $z$ is required. )
- Using a graphics software, plot the solution at time $t/(H^2/\nu) = 0.01,\ 0.05,\ 0.2,\ 0.5,\ 1.,\ 10.$
- argue that the rate of work by the upper plate is equal to the rate of viscous dissipation in the fluid
Last Modified: 28 April 2024 Licence: Creative Commons Attribution required, non-commercial uses (CC BY-NC 4.0)