1. UNSTEADY VISCOUS FLOW Viscous effects confined to within some finite area near the boundary → boundary layer In unsteady viscous flows at low Re the boundary layer thickness δ grows with time; but in periodic flows, it remains constant If the pressure gradient is zero, Navier-Stokes equation (in x) reduces to: Assume linear, horizontal motion

2. Heat Equation– parabolic partial differential equation - linear Requires one initial condition and two boundary conditions U y Total of three conditions Impulsively started plate – Stokes first problem

3. Heat Equation– parabolic partial differential equation Can be solved by “Separation of Variables” Suppose we have a solution: Substituting in the diff eq: May also be written as: Moving variables to same side:The two sides have to be equal for any choice of y and t, The minus sign in front of k is for convenience

4. This equation contains a pair of ordinary differential equations:

5. increasing time Applying B.C., B = 0; C =1;

6. New independent variable: η is used to transform heat equation: Substituting into heat equation: Alternative solution to“Separation of Variables” – “Similarity Solution” from:

7. To transform second order into first order: 2 BC turn into 1 With solution: Integrating to obtain u: Or in terms of the error function: For η > 2 the error function is nearly 1, so that u → 0

8. Then, viscous effects are confined to the region η < 2 This is the boundary layer δ δ grows as the squared root of time increasing time

10. UNSTEADY VISCOUS FLOW Oscillating Plate – Stokes’ second problem Ucos(ωt) y Look for a solution of the form: Euler’s formula

11. Fourier’s transform in the time domain: B.C. in Y Substitution into:

12. Most of the motion is confined to region within: Ucos(ωt) y

14. UNSTEADY VISCOUS FLOW Oscillating Plate Look for a solution of the form: Euler’s formula Ucos(ωt) y W

15. Fourier’s transform in the time domain: B.C. in Y Substitution into:

17. Same solution as for unbounded oscillating plate