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UNSTEADY VISCOUS FLOW

UNSTEADY VISCOUS FLOW. Assume linear, horizontal motion. 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.

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UNSTEADY VISCOUS FLOW

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

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

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

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

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

  8. For η > 2 the error function is nearly 1, so that u → 0 Then, viscous effects are confined to the region η < 2 This is the boundary layer δ increasing time δgrows as the squared root of time

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

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

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

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

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

  14. Same solution as for unbounded oscillating plate

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