Home › Forums › Student Help Forum › How do I ...? with Maple (Student)

confused about a differentiation

Posted on 2008-12-05 06:14 By mjason (Maple Rating 1 38)
Categories:

I am having a problem differentiating a function. I have a fluid in a channel with moving walls corresponding to y=a(t) (upper wall) and y=-a(t) (lower wall). The fluid is driven by suction out of the walls. The speed of the fluid being sucked out of the walls is the constant v_w. I am using the variable eta=y/a(t) to model the fluid. y is the normal direction to teh channel and x is the streamwise direction of the channel. So I have

x=streamwise direction of the channel

y=normal direction of the channel

a(t)=height of the channel

v_w=constant

t=time

eta=y/a(t) is the similarity variable

u=streamwise velocity

Xi=-v_w*x*F(eta,t)/(a(t)^2)

I want top partially differentiate Xi with respect to t.

I am confused as a(t) depends on t, eta depends on a(t) which in turn depends on t and F depends on t. What would i write on maple to get the correct differentiation?

Thanks

‹ changing variable in integral Integration Tutor: determine u-sub ›
Doug Meade's picture

clarification, and probable answer

Comment on 2008-12-05 13:49 from Doug Meade (Maple Rating 4 739)

There appear to be some inconsistencies in your description of the dependencies. Here's what I read:

a(t) depends on t,
eta depends on a(t) which in turn depends on t
and F depends on t

In your sample Maple, you have F depending on both eta and t. If F depends only on t, how does eta enter your equation? What is x?

While I do not know if the following is exactly the form you have in mind, it does show how to enter the dependencies and find the derivative wrt t.

Xi := v_w*x*eta(a(t))*F(t)/a(t)^2;
                            v_w x eta(a(t)) F(t)
                            --------------------
                                       2        
                                   a(t)         
diff( Xi, t );
                          / d      \                        / d      \
       v_w x D(eta)(a(t)) |--- a(t)| F(t)   v_w x eta(a(t)) |--- F(t)|
                          \ dt     /                        \ dt     /
       ---------------------------------- + --------------------------
                         2                                2           
                     a(t)                             a(t)            

                                   / d      \
            2 v_w x eta(a(t)) F(t) |--- a(t)|
                                   \ dt     /
          - ---------------------------------
                              3              
                          a(t)              

Here is, I hope, the derivative in a typeset form.

v_w*x*(D(eta))(a(t))*(diff(a(t), t))*F(t)/a(t)^2+v_w*x*eta(a(t))*(diff(F(t), t))/a(t)^2-2*v_w*x*eta(a(t))*F(t)*(diff(a(t), t))/a(t)^3

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed
Doug Meade's picture

try PDEtools[declare]

Comment on 2008-12-05 14:39 from Doug Meade (Maple Rating 4 739)

You can get slightly more efficient output by using PDEtools[declare]:

restart;
PDEtools[declare]([a(t),F(t),eta(a)]);
                       F(t) will now be displayed as F
                       a(t) will now be displayed as a
                     eta(a) will now be displayed as eta

Xi := v_w*x*eta(a(t))*F(t)/a(t)^2:

diff( Xi, t );

v_w*x*(D(eta))(a)*a[t]*F/a^2+v_w*x*eta*F[t]/a^2-2*v_w*x*eta*F*a[t]/a^3
 

Are we getting closer to what you want to do?

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.ed

 

thanks so much for all your

Comment on 2008-12-06 08:07 from mjason (Maple Rating 1 38)

thanks so much for all your help. sorry my description wasnt very good. I should've written F(eta,t) depends on eta and t but eta=y/a(t) so it also depends on t from the a(t) as well. x is unimportant. it can be considered as a constant as the differentiation is with respect to t.

thanks again

mj

andXi:=-v_w*x*F(eta,t)/a(t)^

Comment on 2008-12-06 08:35 from mjason (Maple Rating 1 38)

so i guess all i really need differentiating with respect to t is

Xi:=F(eta,t)/(a(t)^2);

where

eta=y/a(t)

differentiating with respect to t

Comment on 2008-12-06 12:25 from jakubi (Maple Rating 4 943)

Using 'PDEtools[declare]', for a more compact output, you may do it like:

Xi:=F(eta,t)/(a(t)^2);
Xi1:=subs(eta=y/a(t),Xi);
PDEtools[declare]([a(t)],quiet,prime=t);
expand(diff(Xi1,t));


        D[1](F)(y/a, t) y a'   D[2](F)(y/a, t)   2 F(y/a, t) a'
      - -------------------- + --------------- - --------------
                  4                   2                 3
                 a                   a                 a

thank you so much to both of

Comment on 2008-12-06 15:07 from mjason (Maple Rating 1 38)

thank you so much to both of you.

Comment viewing options

Select your preferred way to display the comments and click "Save settings" to activate your changes.
}