TAMU CHEN 304 - L5 (6 pages)

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 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 x t is commonly referred to as the Eulerian derivative and is often simply t x f f written as or t t x Lagrangian derivative is commonly referred to as material derivative and denoted by Df As discussed in page 11 of the notes on Mathematical background Dt Vectors and Tensors 43 Df f v f Dt t x f x t f x t vi x t Xi t x i 1 3 49 Here the second equality applies for a Cartesian coordinate system and x t denotes X1 X 2 X 3 t 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 F ma m dv d mv dt dt 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 44 v n vs V t S 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 d rv dV dt V t 51 Net rate of linear momentum outflow through the control surface 2nd and 3rd terms rv v v n dS s 52 S t 45 Here v vs n denotes the component of the velocity of the material relative to the surface pointing in the direction of the outward normal v vs ndS is then the volumetric rate at which material leaves the control volume through a differential surface …