Stokes' First problem

[Ángel Martin]

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