Chapter 4 Couette-Poiseuille flow.pdf
- 1. Fluid Mechanics Chapter 4: Analytical solutions to the Navier-Stokes equations Dr. Naiyer Razmara
- 2. Some illustrative Examples A. Couette flow B. Poiseuille flow C. Flow in a circular pipe D. Flow near an Oscillating Plate Analytic solutions to the Navier-Stokes equations 1
- 3. Journal Bearing A. Flow between a Fixed and a Moving Plate: Plane Couette Flow 2
- 4. • Two-dimensional • Incompressible • Plane • Viscous flow • Between parallel plates a distance 2h apart • Assume that the plates are very wide and very long, so that the flow is essentially axial • The upper plate moves at velocity V but there is no pressure gradient • Neglect gravity effects The continuity equation A. Flow between a Fixed and a Moving Plate: Plane Couette Flow 3
- 5. • Thus there is a single nonzero axial-velocity component which varies only across the channel. • The flow is said to be fully developed (far downstream of the entrance). • The Navier-Stokes momentum equation for two-dimensional (x, y) flow: The no-slip condition A. Flow between a Fixed and a Moving Plate: Plane Couette Flow 4
- 6. The solution for the flow between plates with a moving upper wall, is: This is Couette flow due to a moving wall: a linear velocity profile with no-slip at each wall, as anticipated. A. Flow between a Fixed and a Moving Plate: Plane Couette Flow 5
- 7. B. Flow due to Pressure Gradient between Two Fixed Plate: Plane Poiseuille Flow (Channel Flow) Assumptions: • Both plates are fixed. • The pressure varies in the x direction. • The gravity is neglected. • The x-momentum equation changes only because the pressure is variable: • Poiseuille flows are driven by pumps that forces the fluid to flow by modifying the pressure. • Fluids flow naturally from regions of high pressure to regions of low pressure. • Typical examples are cylindrical pipe flow and other duct flows. • Figure below illustrates a fully developed plane channel flow. • Fully developed Poiseuille flows exists only far from the entrances and exits of the ducts, where the flow is aligned parallel to the duct walls. 6
- 8. The y- and z-momentum equations lead to: Thus the pressure gradient is the total and only gradient: The solution is accomplished by double integration: The constants are found from the no-slip condition at each wall: B. Flow due to Pressure Gradient between Two Fixed Plate: Plane Poiseuille Flow (Channel Flow) 7
- 9. Thus the solution to the flow in a channel due to pressure gradient, is: The flow forms a Poiseuille parabola of constant negative curvature. The maximum velocity occurs at the centerline: B. Flow due to Pressure Gradient between Two Fixed Plate: Plane Poiseuille Flow (Channel Flow) 8
- 10. If we assume the flow is in the x-direction, then the velocity vector is u(y, z, t) = (u(y, z, t), 0, 0) and Navier- Stokes equations can be simplified to C. Flow in circular pipe For a steady cylindrical pipe flow with radius r0 the solution is found simply by integrating twice and applying boundary conditions: 9
- 11. C. Flow in circular pipe 10
- 12. D. Flow near an Oscillating Plate Consider that special case of a viscous fluid near a wall that is set suddenly in motion as shown in Figure 1. The unsteady Navier-Stokes reduces to: 11
- 13. D. Flow near an Oscillating Plate 12
- 14. Substituting the last two equations into the reduced Navier-stokes equation, it follows that D. Flow near an Oscillating Plate 13