Vector Concepts for Fluid Mechanics
1. A vector has magnitude and direction and can be described as an arrow pointing in the assigned direction with a length equal to its magnitude (Figure 1).
2. Vectors can be described in Cartesian coordinates as:
3. The coefficients , , and †are the , , and †components of the vector.
4. †are themselves vectors.† †is a vector pointing in the †direction, †is a vector pointing in the †direction, and †is a vectory pointing in the †direction.† See Figure 2.
5. The magnitude of a vector is its length, which is given by:
which is simply a statement of the Pythagorean theorem.
Problem 1: Given that the vector †can be represented as , show that the magnitude of †is 1.
Problem 2: Show that the equation for the magnitude of a vector is simply a statement of the Pythagorean theorem as follows.† First note that †and †are perpendicular to one another because †is parallel to the †axis and †is parallel to the †axis.† Use the Pythagorean theorem to obtain †in terms of †and .† Now, since †and †are both perpendicular to the †axis, †is also perpendicular to the †axis (i.e., it lies in a plane that is perpendicular to the †axis.† Use this information to write the length of the vector †in terms of †and .† Then back substitute for †to obtain the length of †in terms of , †and .
6. The dot product between two vectors, †and †is
7. The dot product is a scalar value.
8. The value of the dot product is , where †is the angle between the two vectors.
9. The dot product of one vector with another is physically the projection of that vector on the other vector, multiplied by the length of the other vector.
10. A unit vector is a vector with magnitude of one.
11. Any vector can be normalized to become a unit vector by simply dividing by its own magnitude.
12. Recalling 9 above, the dot product of any vector with a unit vector is the projection of that vector in the direction of the unit vector.† (e.g., the shadow cast by the vector by a light source that is in the plane of the two vectors and that is perpendicular to the unit vector).
13. The cross product of two vectors is a vector and can be written as:
15. The magnitude of the cross-product has the value of , where †is the angle between the two vectors.
16. The direction of the cross-product is perpendicular to both of the two original vectors, or, equivalently, perpendicular to the plane in which the two original vectors lie.
17. The above concepts are ones that students should know from their statics course.† The students will need this much review, and they will need to be reminded that they already know these concepts from statics.† Next come some concepts that will be new in the sense of understanding vector applications.† In other words, while students may have seen the gradient operator in their math class, they may not yet understand what it means physically.
18. The gradient of a function †is defined as .
19. The gradient is a vector.
20. However, the function itself is not a vector in this case.
21. The gradient is a function of the spatial coordinates.
22. Each term of the gradient is the rate of change of the function as one moves in the given coordinate direction, assuming that there is no change in position with respect to the other two coordinate directions.† To illustrate the gradient, consider the situation shown in Figure 4.† A paper mill produces a sulfurous odor that diminishes in pungency with distance from the mill.† A person at point A experiences a relatively large rate of reduction of the odor as he moves in the †direction.† The same person experiences a relatively small rate of reduction of odor as he moves in the †direction.† Thus, the †component of the gradient is large and the †y component of the gradient is small.
Problem 3: Assume in the paper mill example that the odor drops off with distance from the mill according to the equation .† Give the equation for the gradient at any location †from the mill, assuming that the origin of the †and †axes is at the center of the mill.† Hint:† You should write †and then take the derivatives with respect to †and .† Answer: .
Problem 4: For the same paper mill, what are the †and †components of the gradient found in Problem 3 at a point ?† How do they agree with the above statement that the odor should drop off more quickly along the †direction than along the †direction?
23. Velocity is a vector quantity.† Itís magnitude is the speed of an object.† Itís direction is the direction in which the object is moving.
24. The gradient operator can be described as a vector, .† Thus, the gradient of a function is the gradient operator applied to the function.
25. The divergence of a vector †(such as velocity) is the dot product of the gradient operator with the vector.
26. The divergence is a scalar quantity because it is a dot product.
27. One may take the divergence of the gradient of a function.† This scalar quantity is known as the Laplacian operator.
28. One may also consider the divergence to be an operator †defined as:
30. The curl of a vector is the cross-product of the gradient operator with the vector.
31. The curl of a vector is another vector.
32. The curl of a vector can be written as:
and this form provides one with an easy way to remember where the ďĖď signs must go.
Problem 5: Which of these are not legitimate vector operations.† Why?† Assume †is a scalar function.†
Problem 6: For each legitimate operation in Problem 5, state what kind of entity (vector or scalar) is the result.