Stokes' First problem
11-09-2016, 09:59 AM (This post was last modified: 11-09-2016 05:33 PM by Ángel Martin.)
Post: #1
 Ángel Martin Senior Member Posts: 1,071 Joined: Dec 2013
Stokes' First problem
Stokes’ First Problem. [ P1STOKE ]
From the author’s Engineering Collection, included in the ETSII4 module (ETI4 on the CL Library)

This program calculates the velocity at a point placed at a distance Y from the bottom and an instant t in an unsteady viscous boundary layer flow. The bottom is suddenly imposed at t=0 a constant velocity U0 and the fluid has a kinematic viscosity "nu". Vertical distances (y) are measured from the bottom (y=0) up.

The expression for the instant velocity at a distance y can be related to the cumulative probability function of a normal distribution as follows:

U(y,t) = 2 U0 [ 1- F( y / sqr(2 nu t) )

Example: for U0 = 1 m/s, nu = 10 m^2/s; Y = 0.5 m and t = 2 s
the result is: U(Y,T) = 0.911 m/s

The original version of this program used a polynomial approximation to calculate F, with an accuracy limited to 4 to 6 decimal places, depending on the value of the argument. A modern version based on the ERF implementation on the SandMath brings that to at least 8 decimal places and a much faster execution – thanks to MCODE and the improved algorithm used.

U(y,t) = U0 [ 1 – erf { [ y / 2 sqr( nu t) ] }

Below you can see the program listing using the new approach. Note that R00-R03 are used by ERF:

Code:
 01    LBL “P1STK” 02    “U0=?”  03    PROMPT 04    STO 04 05    “NU=?” 06    PROMPT 07    STO 05 08    LBL 00 09    “Y=?” 10    PROMPT 11    “T=?” 12    PROMPT 13    RCL 05 14    * 15    ST+ X 16    SQRT 17    / 18    ERF 19    CHS 20    1 21    + 22    RCL 04 23    * 24    “U=” 25    ARCL X 26    PROMPT 27    GTO 00 28    END
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