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Navier-Stokes Equations REE 307 Fluid Mechanics II Lecture 5 November 8, 2017 Dr./ Ahmed Mohamed Nagib Elmekawy Zewail City for Science and Technology DIFFERENTIAL ANALYSIS OF FLUID FLOW 2 The fundamental differential equations of fluid motion are derived in this chapter, and we show how to solve them analytically for some simple flows. More complicated flows, such as the air flow induced by 2 a tornado shown here, cannot be solved exactly. 3 Objectives • Understand how the differential equation of conservation of mass and the differential linear momentum equation are derived and applied • Obtain analytical solutions of the equations of motion for simple flow fields 3 4 LAGRANGIAN AND EULERIAN DESCRIPTIONS Kinematics: The study of motion. Fluid kinematics: The study of how fluids flow and how to describe fluidmotion. There are two distinct ways to describe motion: Lagrangian and Eulerian Lagrangian description: To follow the path of individual objects. This method requires us to track the position and velocity of each individual fluid parcel (fluid particle) and take to be a parcel of fixedidentity. With a small number of objects, such In the Lagrangian description, one as billiard balls on a pool table, must keep track of the position and individual objects can be tracked. velocity of individual particles. 4 5 • A more common method is Eulerian description of fluid motion. • In the Eulerian description of fluid flow, a finite volume called a flow domain or control volume is defined, through which fluid flows in and out. • Instead of tracking individual fluid particles, we define field variables, functions of space and time, within the control volume. • The field variable at a particular location at a particular time is the value of the variable for whichever fluid particle happens to occupy that location at that time. • For example, the pressure field is a scalar field variable. We define the velocity field as a vector field variable. Collectively, these (and other) field variables define the flow field. The velocity field can be expanded in Cartesian coordinates as 6 6 • In the Eulerian description we don’t really care what happens to individual fluid particles; rather we are concerned with the pressure, velocity, acceleration, etc., of whichever fluid particle happens to be at the location of interest at the time of interest. • While there are many occasions in which the Lagrangian description is useful, the Eulerian description In the Eulerian description, one is often more convenient for fluid defines field variables, such as mechanics applications. the pressure field and the velocity field, at any location • Experimental measurements are and instant in time. generally more suited to the Eulerian description. 7 7 CONSERVATION OF MASS—THE CONTINUITY EQUATION The net rate of change of mass withinthe control volume is equal to the rate at which mass flows into the control volume minus the rate at which mass flows out of the control volume. To derive a differential conservation equation, we imagine shrinking a control volume to infinitesimal size. 8 8 Derivation Using an Infinitesimal Control Volume At locations away from the center of the box, we use a Taylor series expansion about the center of the box. A small box-shaped control volume centered at point P is used for derivation of the differential equation for conservation of mass in Cartesian coordinates; the blue dots indicate the center of each face. 8 9 1 0 10 1 1 Conservation of Mass: Alternative forms • Use product rule on divergence term V u i v j w k i j k x y z Continuity Equation in Cylindrical Coordinates Velocity components and unit vectors in cylindrical coordinates: (a) two- dimensional flow in the xy- or r휃-plane, (b) three-dimensional flow. 12 13 Conservation of Mass: Cylindrical coordinates 1 5 The divergence operation in Cartesian and cylindrical coordinates. 1 6 Special Cases of the Continuity Equation Special Case 1: Steady Compressible Flow 1 7 Special Case 2: Incompressible Flow 1 8 1 9 2 0 2 1 THE DIFFERENTIAL LINEAR MOMENTUM EQUATION Derivation Using Newton’s Second Law If the differential fluid element is a material element, it moves with the flow and Newton’s second law applies directly. Acceleration Field The equations of motion for fluid flow (such as Newton’s second law) are written for a fluid particle, which we also call a material particle. If we were to follow a particular fluid particle as it moves around in the flow, we would be employing the Lagrangian description, and the equations of motion would be directly applicable. For example, we would define the Newton’s second law applied to a fluid particle’s location in space in terms particle; the acceleration vector (gray arrow) of a material position vector is in the same direction as the force vector (black arrow), but the velocity vector (red (xparticle(t), yparticle(t), zparticle(t)). arrow) may act in a different direction. 22 Local Advective (convective) acceleration acceleration The components of the acceleration vector in cartesian coordinates: Flow of water through the nozzle of a garden hose illustrates that fluid particles may accelerate, even in a steady flow. In this example, the exit speed of the water is much higher than the water speed in the hose, implying that fluid particles have accelerated even though the flow is steady. 24 Acceleration Components The components of the acceleration are: VVVV Vector equation: a u v w particle t x y z x- component u u u u a u v w x t x y z v v v v Y-component a u v w y t x y z w w w w a u v w z-component z t x y z Question: Give examples of steady flows with acceleration Incompressible Steady ideal flow in a variable-area duct Conservation of Momentum Types of forces: 1. Surface forces: include all forces acting on the boundaries of a medium though direct contact such as pressure, friction,…etc. 2. Body forces are developed without physical contact and distributed over the volume of the fluid such as gravitational and electromagnetic. • The force F acting on A may be resolved into two components, one normal and the other tangential to the area. Body Forces Positive components of the stress tensor in Cartesian coordinates on the positive (right, top, and front) faces of an infinitesimal rectangular control volume. The blue dots indicate the center of each face. Positive components on the negative (left, bottom, and back) faces are in the opposite direction of those shown here. 27 28 Surface Forces 휏푖푗 called the viscous stress tensor For fluids at rest, the only stress on a fluid element is the hydrostatic pressure, which always acts inward and normal to any surface. 3 0 Sketch illustrating the surface forces acting in the x- direction due to the appropriate stress tensor component on each face of the differential control volume; the blue dots indicate the center of each face. 3 1 If the differential fluid element is a material element, it moves with the flow and Newton’s second law applies directly. 3 2 Newtonian versus Non-Newtonian Fluids Rheology: The study of the deformation of flowing fluids. Newtonian fluids: Fluids for which the shear stress is linearly proportional to the shear strain rate. Newtonian fluids: Fluids for which the shear stress is not linearly related to the shear strain rate. Viscoelastic: A fluid that returns (either fully or partially) to its original shape after the applied stress is released. Rheological behavior of fluids—shear stress as a function of shear strain rate. Some non-Newtonian fluids are called shear thinning fluids or pseudoplastic fluids, because the In some fluids a finite stress called the more the fluid is sheared, the less yield stress is required before the viscous it becomes. fluid begins to flow at all; such fluids Plastic fluids are those in whichthe are called Bingham plastic fluids. shear thinning effect is extreme. 3 3 Derivation of the Navier–Stokes Equation for Incompressible, Isothermal Flow The incompressible flow approximation implies constant density, and the isothermal approximation implies constant viscosity. The Laplacian operator, shown here in both Cartesian and cylindrical coordinates, appears in the viscous term of the incompressible Navier–Stokes equation. 34 The Navier–Stokes equation is an unsteady, nonlinear, second order, partial differential equation. Equation 9–60 has four unknowns (three velocity components and pressure), yet it represents only three equations (three components since it is a vector equation). Obviously we need another equation to make the problem solvable. The fourth The Navier–Stokes equation is the equation is the incompressible continuity cornerstone of fluid mechanics. equation (Eq. 9–16). 28 35 Navier-Stokes Equations u u u u p u 2 ( u v w ) g [(2 .V )] t x y z x x x x 3 x- v u w u [( )] [( )] (a) momentum y x y z x z v v v v p v u ( u v w ) g y [( )] t x y z y x x y y- v 2 v w momentum [(2 .V )] [( )] (b) y y 3 z z y w w w w p w u ( u v w ) g z [( )] t x y z z x x z z- v w w 2 momentum [( )] [(2 .V )] (c) y z y z z 3 Navier-Stokes Equations • For incompressible fluids, constant µ: • Continuity equation: .V = 0 u 2 v u w u [(2 .V )] [( )] [( )] x x 3 y x y z x z u v u w u { [(2 )] [( )] [( )]} x x y x y z x z 2u 2u 2u 2u 2v 2w ( ) ( ) x2 y 2 z 2 x2 xy xz 2u 2u 2u u v w ( ) ( ) x2 y 2 z 2 x x y z 2u 2u 2u ( ) 2u x2 y 2 z 2 Navier-Stokes Equations • For incompressible flow with constant dynamic viscosity: Du p 2u 2u 2u • x- momentum
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