MFGT 242: Flow Analysis Chapter 4: Governing Equations for Fluid Flow

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, ex, ey, and ez 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 = vxex + vyey,+ vzez

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. wx = ux + vx • 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=s1, 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/ft3 • 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 =  u2 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 Mold Wall Mold Wall Mold Wall Flow Mold Wall • 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