A fluid element, often called a material element. The reason for this is that the unit vectors in cylindrical coordinates change direction when the particle is moving. It has been accepted for inclusion in Chemistry Education Materials by an authorized administrator of DigitalCommons@UConn. The Material Derivative The equations above apply to a fluid element which is a small “blob” of fluid that contains the same material at all times as the fluid moves. In the Lagrangian reference, the velocity is only a function of time. The Laplacian in Spherical Polar Coordinates Carl W. David University of Connecticut, Carl.David@uconn.edu This Article is brought to you for free and open access by the Department of Chemistry at DigitalCommons@UConn. The Material Derivative in Cylindrical Coodrinates This one a little bit more involved than the Cartesian derivation. Figure 1.

f is a function of stationary spacial coordinates, as indicated by the lower case letter x. The Material Derivative fluid element. Material Derivative Example: The heat along a one dimensional fluid is defined as f(x,t). The unit vectors in the spherical coordinate system are functions of position. "ˆ = z ˆ #r ˆ sin! r ˆ =! It is convenient to express them in terms of the spherical coordinates and the unit vectors of the rectangular coordinate system which are not themselves functions of position.

r r = xx ˆ + yy ˆ + zz ˆ r = x ˆ sin!cos"+ y ˆ sin!sin"+ z ˆ cos!