concerned with individual fluid parcels or their trajectories Moreover the flow

Concerned with individual fluid parcels or their

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concerned with individual fluid parcels or their trajectories. Moreover, the flow velocity will now be measured directly at these locations rather than being deduced from the time rate-of-change of fluid parcel location in a neighborhood of the desired measurement points. It is fairly clear that this approach is more suitable for practical purposes, and essentially all engineering analyses of fluid flow are conducted in this manner. On the other hand, such a viewpoint does not produce “total” acceleration along the direction of motion of fluid parcels as needed for use of Newton’s second law. It is worth noting that, because of this, physicists still typically employ a Lagrangian approach. 3.1.3 The substantial derivative The disadvantage of using an Eulerian “reference frame,” especially in the context of deriving the equations of fluid motion, is the difficulty of obtaining acceleration at a point. In the case of the Lagrangian formulation, heuristically, one need only attach an accelerometer to a fluid parcel and record the results. But when taking measurements at a single (or a few, possibly, widely-spaced) points as in the Eulerian approach, it is more difficult to produce a formula for acceleration (and for velocity as well) for fluid elements in their direction of motion—which is what is needed to apply
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50 CHAPTER 3. THE EQUATIONS OF FLUID MOTION Newton’s second law. With respect to the mathematical treatment of the equations of motion, this difficulty is overcome by expressing accelerations in an Eulerian reference frame in terms of those in a Lagrangian system. This can be done via a particular differential operator known as the substantial (or material ) derivative which can be derived by enforcing an equivalence of motion in the two types of reference frames. We first state the formal definition of this operator, after which we will consider some of the physical and mathematical details. Definition 3.1 The substantial derivative of any fluid property f ( x, y, z, t ) in a flow field with velocity vector U = ( u, v, w ) T is given by Df Dt ∂f ∂t + u ∂f ∂x + v ∂f ∂y + w ∂f ∂z (3.1) = ∂f ∂t + U · ∇ f . We again recall (see Eq. (2.13) that the operator is a vector differential operator defined as ∇ ≡ ( ∂/∂x, ∂/∂y, ∂/∂z ) T , so that in our subscript notation used earlier f = ( f x , f y , f z ) T . It is worthwhile to consider some details regarding the substantial derivative. First, it is easily derived via a straightforward application of the chain rule and use of the definitions of the velocity components. For example, for a general function f which might represent any arbitrary fluid property, we can write (for a fluid parcel) f ( x, y, z, t ) = f ( x ( t ) , y ( t ) , z ( t ) , t ) if we recall that in a Lagrangian system the spatial coordinates of fluid particles are functions of time—so, any property associated with that fluid particle would also, in general, change with time. Now differentiate f with respect to t using the chain rule: df dt = ∂f ∂x dx dt + ∂f ∂y dy dt + ∂f ∂z dz dt + ∂f ∂t dt dt = ∂f ∂t + u ∂f ∂x + v ∂f ∂y + w ∂f ∂z ,
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