TAMU CHEN 304 - L5 (6 pages)

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CBE341 Lecture 5 9 22 Momentum balance Lagrangian and Eulerian perspectives A useful definition of a fluid is that it is a material that deforms continuously when subjected to a shear stress Whereas a solid placed under shear stress will assume a new static position indicative of elastic response a fluid will continue to change its shape indefinitely Thus for a fluid it is not the amount of deformation but the rate of deformation which is crucial in the mechanics see Lecture 1 notes For a material to deform there must be motion of one point in the material relative to another The concept of material points is useful in the description of fluid kinematics A material point is an infinitesimal mass which moves always at the local fluid velocity and which retains its identity as it changes position In other words a material point is a small particle of fluid which may be tracked over time Let us suppose that each material particle in a fluid is labeled by a position x it occupied at some reference time say t 0 As time evolves this material particle will occupy different locations x We write x x x t This equation says The material point which was located at the position x at t 0 is located at position x at time t We write any property of this material point f as f x t Some examples the density r at time t of the material point which was located at the position x at t 0 is written as r x t its velocity at t is written as v x t Clearly v x t dx dt x fixed What does f x fixed t mean x fixed means that we are following the fluid element i e material point and watching how the property f of this fluid element 42 changes with time Such a view of transport problems is known as the Lagrangian viewpoint While a Lagrangian observer moves with the material particle an Eulerian observer remains in a location x and observes variation of a quantity of interest say f as a function of time In the presence of flow clearly this observer will see different material particles at different time From observations made by many such observers located at different x we can construct f x t Such a view of properties of interest is known as the Eulerian viewpoint If the Eulerian observer can read the label of the material points as they pass through x i e x then one can construct x x t which can be used to convert f x t to f x x t t so that Eulerian observations can be mapped to yield Lagrangian observations and vice versa In other words one can always recast f x t where the independent variables are x and t into some new function f x t where the independent variables are now the moving material points and time