1 MFGT 242: Flow Analysis Chapter 4: Governing Equations for Fluid Flow Professor Joe Greene CSU, CHICO

2 Governing Equaitons Introduction to Vector Analysis Mathematical Preliminaries Conservation of Mass Conservation of Momentum Conservation of Energy Boundary Conditions

3 Introduction to Vector Analysis Various quantities used in fluid mechanics –Scalars: needs only single number to represent it Temperature, volume,density –Vectors: needs magnitude and direction Velocity, force, gravitational force, and momentum –Tensors: needs two vectors to describe it (later) Vectors –Determined by its magnitude (length) and direction –Usually represented by boldface and shown with unit vectors, e x, e y, and e z with length of unity and point in each of the respective coordinate directions Coordinate system: RCCS (Rectangular Cartesian Coordiante System Vector, v, can be expressed as linear combination of unit vectors v = v x e x + v y e y,+ v z e z

4 Vector Operations Vector addition and subtraction If a vector u is added to another vector v, then the result is a third vector w whose components equals the sums of the corresponding components of u and v. w x = u x + v x Similar results holds for subtraction of one vector from another Vector products (Fig 5.2) –Multiplication of scalar, s, with vector, v, is a vector sv. –Dot product Dot product of two vectors yields a scalar –Which is the area of the rectangle with sides  u  and  v  cos  –Dot product can be expressed with the Kronecker delta symbol,  ij = 0 for i  j and  ij = 1 for i = j

5 Vector Operations Cross Product Cross product of two vectors, u and v, is a vector, whose –magnitude is the area of the parallelogram with adjacent sides u and v, namely, the product of the individual magnitudes and the sine of the angle  between them. –Direction is along the unit vector, n, normal to the plane of u and v and follows the right-hand rule. –Representative nonzero cross products of the unit vectors are: –Cross product may be conveniently reformulated in terms of the individual components and also as a determinate

6 Vector Operations Vector differentiation –Introduction of the (del or nabla) operator  In RCCS, if a scalar s=s(x,y,z) is a function of position, then a constant value of s, such as s=s 1, defines a surface The gradient of a scalar s at a point P, designated grad s (equal to  s) is a directional derivative. –Defined as a vector in the direction in which s increases most rapidly with distance, whose magnitude equals the rate of increase

7 Vector Operations Gradient is shown with unit vector r –Whose magnitude can readily be shown to be unity, –Vector  s –Component of in the direction of r is the dot product –Note: a differential change is s, and hence its rate of change in the r direction are given by –Then,

8 Physical Properties Density –Liquids are dependent upon the temperature and pressure Density of a fluid is defined as –mass per unit volume, and –indicates the inertia or resistance to an accelerating force. Examples, –Water: density = 1 g/cc = 62.3 lb/ft 3 –Steel: density = 7.85 g/cc; Aluminum: density: 2.7 g/cc –PP: density= 0.91 g/cc; HDPE: density= 0.95 g/cc –Most plastics: density = 0.9 to 1.5 g/cc Specific Gravity –Density of material divided by density of water (Unit-less) Examples, –Water: specific gravity = 1.0 –Most plastics: density = 0.9 to 1.5

9 Velocity Velocity is the rate of change of the position of a fluid particle with time –Having magnitude and direction. In macroscopic treatment of fluids, you can ignore the change in velocity with position. In microscopic treatment of fluids, it is essential to consider the variations with position. Three fluxes that are based upon velocity and area, A –Volumetric flow rate, Q = u A –Mass flow rate, m =  Q =  u A –Momentum, (velocity times mass flow rate) M = m u =  u 2 A

10 Mathematical Preliminaries Assumptions –Fluid is a continuous flow in a surrounding environment –The values of velocity, pressure, and temperature change smoothly and are differentiable Material Derivative –Some fluid properties change with position and time velocity, pressure, temperature, density Use chain rule for differentiation Then, –Material Derivative –Accounts for »motion of fluid »changing position with time

11 Compressible and Incompressible Fluids Principle of mass conservation –where  is the fluid density and v is the velocity For injection molding, the density is constant (incompressible fluid density is constant) Flux The flux v of an extensive quantity, X, for example, is a vector that denotes the direction and rate at which X is being transported (by flow, diffusion, conduction, etc.) (per unit area) Examples –Mass, momentum, energy, volume –(Volume transported per unit time pre unit area) or m/s

12 Basic Laws of Fluid Mechanics Apply to conservation of Mass, Momentum, and Energy In - Out = accumulation in a boundary or space Xin - Xout =  X system Applies to only a very selective properties of X –Energy –Momentum –Mass Does not apply to some extensive properties –Volume –Temperature –Velocity

13 Basic Laws of Fluid Mechanics Conservation of Mass –If V(t) is a material volume of fluid flowing continuously, then, the mass contained in V(t) does not change. –Mass is given by –Conservation says rate of change is zero. –For incompressible fluid, the density is constant and –For Material Derivative

14 Basic Laws of Fluid Mechanics Conservation of Momentum –Momentum is mass times velocity –Time rate of change of fluid particle momentum in a material, V(t), is equal to the sum of the external forces Force = Pressure Viscous Gravity Force Force Force Or rewritten by expanding the material derivative

15 Basic Laws of Fluid Mechanics Energy –Total energy of the fluid in a material volume V(t) is given by the sum of its kinetic and internal energies. Or expanded out as Energy = Conduction Compression Viscous volume Energy Energy Dissipation

16 Boundary Conditions Apply conservation of mass, momentum and energy to injection molding causes the appliction of the equations to specific problem –Example of injection molding surface –Pressure BC Pressure gradient in normal direction (90° from flow) is zero –The mold walls are solid and impermeable Melt flow rate, Q, or pressure, P, is specified at the inlet. The pressure is zero at surface or flow front. (Fountain effect) –Temperature BC Temperature profile through cavity is described as uniform at the injection point, Temperature at mold walls is initially constant and varies as the melt hit the mold wall and heats up. Mold walls are cooled by heat transfer fluid Flow Mold Wall Mold Wall Mold Wall