ch_1 - 1. Transport and mixing 1.1 The material derivative...

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1. Transport and mixing 1.1 The material derivative Let be the velocity of a fluid at the point and time . Consider also some scalar field such as the temperature or density. We are interested not only in the partial derivative of with respect to time, , but also in the time derivative following the motion of the fluid, . The latter is the so-called material derivative : . (1.1) In Cartesian coordinates, , and . (1.2) We will also be using spherical coordinates. The notation is conventional for the (eastward, northward, radially outward) components of the velocity field. In addition to the radial distance from the origin, , we use the symbol for latitude ( at the south pole, at the north pole) and for longitude (ranging for zero to ). The gradient operator in these coordi- nates is , (1.3) so that . (1.4) The material derivative of a vector, such as the velocity itself, is defined just as in 1.1. But care is required when considering the material derivative of a component of a vector if the unit vectors of one’s coordinate system are position dependent, as they are in spherical coordi- nates. For example, the radial component of the material derivative of the velocity is not equal to the material derivative of the radial component of the velocity; rather, . (1.5) The second term on the rhs is sometimes referred to as the metric term . After obtaining expressions for , etc., one arrives at the spherical-coordinate expression for the acceleration of a fluid parcel (see Problem 1.1): V x t , ( 29 x x y z , , ( 29 = t χ x t , ( 29 χ χ t d χ dt d χ dt χ x V x t , ( 29 t δ + t t δ + , ( 29 χ x t , ( 29 [ ] t 0 δ lim t δ = ∂ ∂ t V + ( 29χ = V u v w , , ( 29 = x y z , , ( 29 = d χ dt χ t u χ x v ∂χ y w ∂χ z + + + = u v w , , ( 29 r θ π 2 π 2 λ 2 π λ ˆ r θ cos ( 29 1 ∂ ∂λ ( 29 θ ˆ r 1 ∂ ∂θ ( 29 r ˆ ∂ ∂ r ( 29 + + = d dt ∂ ∂ t r θ cos ( 29 1 u ∂ ∂λ ( 29 r 1 v ∂ ∂θ ( 29 w ∂ ∂ r ( 29 + + + = V r ˆ d V dt ------- d dt ---- r ˆ V ( 29 V d r ˆ dt ----- = d r ˆ dt
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-1.2- . (1.6) Another case of interest is the material derivative of an infinitesimal material line segment. Suppose that and are infinitesimally close to each other, with the vector pointing from one point to the other, and let be the evolution of this vector assuming that its end- points move with the flow. Then, from the figure below, . To make sure that we understand this notation, suppose that the line segment is oriented vertically at the time in question. Then, in Cartesian coordinates, . We refer to the third term as stretching, as it lengthens or shortens the line segment, and the other
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This note was uploaded on 10/01/2011 for the course ME 509 taught by Professor Wereley during the Spring '11 term at Purdue.

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ch_1 - 1. Transport and mixing 1.1 The material derivative...

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