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Slide 1 - Kinematics I Slide 2 - Many Particles make up a Continuum Slide 3 - Material Coordinates Slide 4 - Material Coordinates (continued) Slide 5 - Spatial Coordinates -- why? Slide 6 - Material (Langrangian) and Spatial (Eularian) Coordinates Slide 7 - Material and Spatial Coordinates (continued) Slide 8 - Material and Spatial Coordinates (continued) Slide 9 - The Material Derivative Slide 10 - The Material Derivative (continued) Slide 11 - Kinematics I summary
"Kinematic" refers to relationships between spatial position, orientation, and time.
[D1]
Mass-point mechanics: trajectory capital X as a function of lowercase t - position history of particle.
The extension to a countable number of particles is obvious: Capital X superscript lowecase n equals capital X subscript lowercase n, as a function of lowercase t; capital V superscipt lowercase n equals , capital V superscipt lowercase n, as a function of lowercase t, lowercase n is an integer.
Or, we may write instead, capital X equals capital X as a function of lowercase t and lowercase n; capital V equals capital V as a function of lowercase t and lowercase n.
For a continuum, the integers are insufficient; we require continuous identification variables.
Capital X = capital X as a function of lowercase t, lowercase a, lowercase b, and lowercase c.
The triple, lowercase a lowercase b lowercase c, must "span" the space.
The continuous variables, the triple lowercase a lowercase b lowercase c, are known as material coordinates since they identify particular pieces of the continuum. How should they be chosen?
[D2]
One convenient choice for the triple lowercase a lowercase b lowercase c is to let them designate the positions of the material particles at some arbitrary reference time, call it lowercase t equals 0, i.e.,
The triple lowercase a lowercase b lowercase c equals capital X subscript 0 equals capital X as a function of zero and capital X subscript 0.
Therefore, for the use of the traditional material coordinates of mass-point mechanics, we have the position field
Bold capital X equals bold capital X as a function of lowercase t and bold capital X subscript 0
along with the velocity and acceleration easily determined from this by differentiation with respect to lowercase t.
In fluid mechanics, however, it is usually much more convenient to work with spatial coordinates that are fixed (specified) in space rather than attached to individual parcels of fluid.
We make measurements in a spatial frame of reference.
[D3]
When performing an analysis of flow past an object, some calculations are facilitated by the use of a spatial fram (e.g., a control-volume momentum balance).
[D4]
Since our fundamental principles (e.g., Newton's 2nd Law of Motion) are expressed in terms of material coordinates and we would prefer to work in more convenient spatial ones, we must be able to transition between the two.
Assume that capital theta as a function of capital X subscript 0 and lowercase t, and lowercase theta as a function of lowercase x and lowercase t are the material (capital) and spatial (lower-case) designations of a variable of interest. (In general, in this module, we shall use upper and lower case letters to denote material and spatial variables, respectively.) We wish to determine:
(the derivative of capital theta with respect to lowercase t) subscript capital X subscript 0
the derivative of capital theta holding the material identity fixed.
Using the chain rule for differentiation,
(The derivative of capital theta with respect to lowercase t) subscript capital X subscript 0 equals (The derivative of lowercase theta with respect to lowercase t) subscript lowercase x times (the derivative of lowercase t with respect to lowercase t) subscript capital X subscript 0 plus (the derivative of lowercase theta with respect to lowercase x subscript lowercase i)subscript lowercase t times (the derivative of lowercase x subscript lowercase i with respect to lowercase t) subscript capital X subscript 0.
where we have used Cartesian tensor notation and the summation convention.
But the derivative of t with respect to t equals one
and
(the derivative of lowercase x subscript lowercase i with respect to lowercase t) subscript capital X subscript 0 equals (the derivative of capital X subscript lowercase i with respect to lowercase t) subscript X subscript 0 equals capital V subscript lowercase i as function of bold capital X subscript 0 and lowercase t,
the material velocity.
Since we desire to express the right-hand side of our equation using only spatial variables, we'll use lower-case notation to write (The derivative of capital theta with respect to lowercase t) subscript bold X subscript 0 equals (the derivative of lowercase theta with respect to lowercase t) subscript bold lowercase x, plus lowercase v subscript lowercase i, as a function of lowercase bold x and lowercase t, times (the derivative of lowercase theta with respect to lowercase x subscript i) subscript t.
We may drop the subscripts x and t since there is no confusion regarding what is held fixed. Finally, in tensor or vector form, we have,
The derivative of capital theta with respect to lowercase t = the derivative of lowercase theta with respect to lowercase t, plus lowercase v subscript lowercase i, times the derivative of lowercase theta with respect to lowercase x subscript i.
The derivative of capital theta with respect to lowercase t equals the derivative of lowercase theta with respect to lowercase t, plus bold lowercase v dot del lowercase theta.
The spatial quantity
The derivative of lowercase theta with respect to lowercase t plus bold lowercase v dot del lowercase theta
is known as the material (substantial, Stokes) derivative and is usually indicated by either
lowercase d lowercase theta with respect to lowercase t, or capital D lowercase theta with respect to lowercase t.
(In red) Capital D lowercase theta over capital D lowercase t equals (in light purple) the partial derivative of lowercase theta with respect to lowercase t, plus (in dark purple) bold lowercase v dot del lowercase theta.
(In red) Time rate of change of lowercase theta following the material;
(In light purple) Time rate of change of lowercase theta at a fixed point in space (the local derivative).
(In dark purple) Time rate of change of lowercase theta due to movement of fluid from one location to another (the convective rate of change).
We have discussed the notions of material and spatial frames of reference and their value to the study of fluid mechanics.
We have further found how to represent time derivatives following the material in the (usually!) more convenient spatial frame of reference.
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