42 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 (,t)=xxx 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 (,t)r x, its velocity at t is written as (,t)v x. Clearly, fixedd(,t)dtö=÷øxvx = x 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 element43 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. The converse is also true. Although we will not actually do this transformation in this course the point I wish to get across is that one can, at least conceptually, go back and forth between Lagrangian and Eulerian viewpoints. It is simply the same set of data presented in two different ways. ( )f,tt¶æöç÷¶èøxx is commonly referred to as the Eulerian derivative, and is often simply written as ft¶æöç÷¶èø or ft¶æöç÷¶èøx. Lagrangian derivative, is commonly referred to as 'material derivative', and denoted by DfDt. As discussed in page 11 of the notes on “Mathematical background - Vectors and Tensors”,44 ( )( ) ( )( )ii1,3iDf f fDt tf,t = v , t f , ttX=¶æö=+×Ñç÷¶èø¶æö¶+ç÷¶¶èøåxxvxxx (49) Here, the second equality applies for a Cartesian coordinate system and ( ),tx denotes ( )123,,,XX Xt. Conservation of Linear Momentum Newton’s first law states that a body at rest remains at rest, or a body in motion remains in motion at the same velocity along a straight path when the net force acting on it is zero. Therefore a body tends to preserve its momentum (mv). Newton’s second law states that the acceleration of an object is proportional to the net force acting on it and is inversely proportional to its mass (a = F/m). For a rigid body of mass Newton’s second law can be expressed as: dd(m)mmdt dt== =vvFa Where F is the net force acting on the body, a is the acceleration, and v is the velocity. Therefore, Newton’s second law can be restated as the rate of change in momentum of a body is equal to the net force acting on it. Along this same line, the momentum of a system remains constant when the net forces acting on the system sum to zero. This is known as the conservation of momentum principle. Let us consider an arbitrary control volume V(t), enclosed by a surface S(t). Newton’s second law applied to the material inside this control volume takes the following form:45 v n vs S(t) V(t) {rate of momentum accumulation}={rate of momentum in}-{rate of momentum out}+{sum of the total forces} (50) Often in undergraduate courses, this equation is used as the starting point for the application of Newton’s second law to fluid flow problems. It is not immediately obvious as to how equation (50) follows from the statement of Newton’s second law “F = ma” that one sees in elementary physics courses! When we write “F = ma”, we really imply the following: If we follow the motion of an object of mass m, and if at any instant its acceleration is a, then the net force acting on it must be equal to ma. A key part of this statement is “if we follow the motion of an object” – to do this, we must track the object. When we apply it to a solid such as a projectile, it is easy to follow the object. However, when we consider a fluid, it is hard to track a packet of fluid! Yet, to write F = ma, we must track a packet of fluid. When we write equation (50), we are not tracking a packet of fluid – if we did, the first and second term on the right hand side of equation (50) would be zero! Newton’s second law when applied to an arbitrary volume bounded by a surface through which material is going in and out becomes equation (50). Let us now construct mathematical expressions for each of these terms: Rate of accumulation of momentum inside the control volume = ( )V(t)ddVdtròv (51) Net rate of linear momentum outflow