Classical Mechanics of Fluids and Stokess Stress Constitutive Equation - Assignment Example

FV2001 Assignment Brief – Re-sit Student Course Date Question1: Classical mechanics of fluids The Navier-Stokes Newton’s second law of motion by applying the pressure term as well as the stress in the viscous fluid to govern fluid flow in fires. The Navier-Stokes equations include the following. a. Stokes’s stress constitutive equation For elastic and incompressible solids the expression used is given as; In this expression; Referring to the strain-tensor rate Stokes’s stress constitutive equation for fluids that are viscous and incompressible is given as: In its convective form, the equation becomes; Where; Constitutive equation that’s is applied to elastic solids that consider linear stress is given as; In which; For elastic solids, the constitutive equation is given as; The constitutive equation for linear stress in thermal hydraulics is given as; where; Where µ and remain constant in the equation. The momentum equation is given as; In which, the bulk density is to be constant throughout. P is given as; The value of is given as zero. b. The Cauchy momentum equation The Cauchy momentum equation expresses the conservation form of the momentum. The equation is given as; Where; c. Momentum equation The moment equation is obtained by getting the product of mass and the velocity. The expression below is obtained by substituting in the mass conservation equation. Where; Given, Then the expression below is obtained; In the given equation, the left expression is always equated to zero and it refers to the mass continuity. Resulting to the expression given below; The expression can also be expressed as given below by applying the derivative operator and it refers to the Newton’s second law of motion; In the equation for momentum conservation, the following terms need turbulence modeling. i. The velocity of the fluid flow () ii. The flow expansion rate () iii. The fluid density (ρ) iv. The tensor product ( Turbulence models A turbulence model is very important in calculating the mean of glow directly without following the normal procedure of first considering the flow field that has full time-dependent properties. When considering the fluid mechanics, a vortex refers to a region in which the fluid flows around a line axis in a fluid. The line of axis in this case can either be curved or straight depending on the region considered. In turbulent flow, the vorticity as well as the velocity distribution are the major components used to determine the vortices characteristics. A figure that can be used to give a summary of the analysis of scales of vortices is given below; Figure 1: scales of vortices analysis Energy conservation equation Energy equation The energy equation is given below; Where; Question 1.2 Applying the Bernoulli’s equation Pressure drop is given by the equation given below; ) Where; But And; ) Then Therefore assuming; Given; Then substituting in the equation; 50.42 Then Then; Question 2 Dimensional analysis 2.1 Kolmogorov scale of velocity Using dimensional analysis, the relation among the given parameters: kinematic viscosity, specific dissipation rate and fluid density can be obtained. The Kolmogorov length scale is considered in this case. It is given by; Assuming that the rate of energy dissipation is the same at all the considered length of scale, the equation becomes; In which; Introducing the Reynolds number to the equation; It therefore indicates that; Referring to this dimensional analysis the relation among the three main parameters is obtained 2.2 Archimedes number Archimedes number is a dimensionless value that determines the fluid’s motion as a result of the differences in the density of the fluid. It majorly relates the external forces and the internal forces in the viscous fluid. The major variables used in the determination of the then number include; The acceleration due to gravity (g) m/s2 The fluid density () in kg/m3 The body’s density () in kg/m3 The dynamic viscosity () in kg/m3 The length of the body in m In obtaining the Archimedes number the ratio of the above parameters is obtained as follows; Therefore, (a) represents the correct expression for obtaining the Archimedes number. Question 3 Heat Transfer, Thermo chemistry and Fluid Dynamics of Combustion The equation for heat transfer by radiation is given as; Where; Assume; Therefore; 3.2 equivalence ratio Where; Then; Mass fraction of gasoline can be obtained as follows; Therefore; Question 4 4.1 Characteristics of Jet and Buoyant Flames & Fire Plumes A fire plume is a burnt gas column that is formed as a result of fire. The main characteristics of the fire plume are temperature and flow speed. Temperature in the fire plume is classified into three main categories which include; the mean temperature, temperature experienced at a radial distance measured from the centerline of the plume and the temperature at the centerline of the plume. The centerline temperature describes the developing fire plume. As the distance from the centerline of the fire plume increases, the temperature reduces. Therefore, the temperature experienced at a certain radial distance is less than the temperature at the centerline of the plume. The flow speed of the fire plume will majorly depend on the openings available in the building. The fire plume speed is high when a large number of openings are available. The induced air flow that reaches the plume increases the plume flow speed. The distance of the fire plume from the opening also plays a major role in influencing the fire plume speed. When the fire plume is near the opening then the flow speed experienced is higher than when the fire plume is very far from the opening. This is because the effect of the induced air will be less on the fire plume. The other types of flows encountered in the fire plume include; The plume mass flow rate: refers to the total mass flowing within the boundaries of the plume at a particular height in the upward direction. This flow rate increases with increase in the plume height steadily because the ambient air experienced is entrained continually over the height of the plume. Plume flow velocity: refers to the highest velocity experienced at the plume centerline. It increases with the plume height in a flame region that is continuous. Above the fire plume flame, the plume velocity decreases as the height increases because the plume is cooled as a result of the entrained ambient air. 4.2 Jet and buoyant flames In Buoyant flames temperatures are high at the upper side of the flame and continues decreasing with increasing height. The flow velocities are also influenced by the proximity to the openings as well as the number of the openings available. In jet flames, the temperatures are high throughout the flame and the flow velocity is still influenced by the number of openings and proximity of the flame to the openings. Works cited Fluid Mechanics 4th by Frank M. White Fluid mechanics 5th edition by John G.Douglas ,2005 Heskestad . G. Fire Plumes, Flame Height, and Air Entrainment. SFPE Handbook of Fire Protection Engineering. 4th ed,2008. Kreith, F.; Berger, S.A.; et. al. “Fluid Mechanics”Mechanical Engineering Handbook Ed. Frank Kreith Boca Raton: CRC Press LLC, 1999. Read More