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Velocity and acceleration of a particle of a fluid

  1. Oct 7, 2016 #1
    1. The problem statement, all variables and given/known data
    Fluid mech.png
    2. Relevant equations
    x = (x1,x2,x3) are the coordinates of the current configuration and X = (X1,X2,X3) are the coordinates of the reference configuration where x = y(X,t) where y is a deformation of X.

    3. The attempt at a solution
    (a)
    taking the partial derivatives of xi with respect to t:
    ∂(x1)/∂t = αX2 + 2(α^2)X3*t
    ∂(x2)/∂t = 2(α^2)*X1*t + αX3
    ∂(x3)/∂t = αX1 +2(α^2)X2t

    so v(X,t) = (αX2 + 2(α^2)X3*t , 2(α^2)*X1*t + αX3 , αX1 +2(α^2)X2t )
    (1) v((d,d,d),0) = ( αd , αd , αd )

    a(X,t) = ∂v/∂t = ( 2(α^2)X3, 2(α^2)*X1 , 2(α^2)X2 )

    (2) a((d,d,d),0) = ( 2(α^2)d , 2(α^2)*d, 2(α^2)d )


    (b)
    for this part we know that x = (d,d,d) , so we wan't to find the X that corresponds to this point so we can plug it into the functions v(X,t) and a(X,t).

    rearranging the three equations of motion:
    X1 = x1 - αX2t - (α^2)X3t^2 (a)
    X2 = x1 - αX3t - (α^2)X1t^2 (b)
    X3 = x1 - αX1t - (α^2)X2t^2 (c)

    now, subbing (b) and (c) into (a) we have:

    X1 = x1 - α( x1 - αX3t - (α^2)X1t^2 )t - (α^2)( x1 - αX1t - (α^2)X2t^2 )t^2

    from the above it seems like like I won't be able to express X1 solely in terms of spatial coordinates because if I expand the second and third term I won't be able to get rid of X3 and X2.
    Any help would be appreciated.
     
    Last edited: Oct 7, 2016
  2. jcsd
  3. Oct 7, 2016 #2
    You have 3 linear algebraic equations in three unknowns, X1, X2, and X3 corresponding to (d,d,d) @ t.
     
  4. Oct 7, 2016 #3
    Thanks for the reply,
    looking at the system of equations:
    d = X1 + αX2t + (α^2)X3t^2
    d = X2 + αX3t + (α^2)X1t^2
    d = X3 + αX1t + (α^2)X2t^2

    if I begin subtracting the equations from one another to remove the d terms and then manipulating the resultant equations I will get:
    Xi(terms involving α and t ) = 0 , for i =1,2,3
    I'm not sure what I can say about this other than that either the Xi or () must be zero.
     
  5. Oct 7, 2016 #4
    Just use Gaussian elimination.
     
  6. Oct 7, 2016 #5
    Thanks again, using Gaussian elimination as you suggested I found X1,X2 and X3 all to be equal to d(1-αt)/(1-(α^3)(t^3)).
    I suspect this to be the correct answer as its easy to see why when t approaches α^-1 this motion is unrealistic.
     
    Last edited: Oct 7, 2016
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