Flow equations, nonlinear time-dependent

A closed form similarity solution for the nonlinear time-dependent slow-flow equations has been used as the basis for a simple, time-dependent, analytic model of localized ignition which requires minimal chemical and physical input (8). As a fundamental part of the model, there are two constants which must calibrated the radii, or fraction of the time-dependent simi-... 

Hicks (H6) and Frazer and Hicks (F3) considered the ignition model in which exothermic, exponentially temperature-dependent reactions occur within the solid phase. Assuming a uniformly mixed solid phase, the one-dimensional unsteady heat-flow equation relates the propellant temperature, depth from the surface, and time by the nonlinear equation ... 

A brief explanation of differential-algebraic equations (DAE) facilitates a further mathematical discussion of the stagnation-flow equations. In general, DAEs are stated as a vector residual equation, where w is the dependent-variable vector and the prime denotes a time derivative. For the discussion here, it is convenient to consider a restricted class of DAEs called semi-explicit nonlinear DAEs, which are represented as... 

You saw how the equations governing energy transfer, mass transfer, and fluid flow were similar, and examples were given for one-drmensional problems. Examples included heat conduction, both steady and transient, reaction and diffusion in a catalyst pellet, flow in pipes and between flat plates of Newtonian or non-Newtonian fluids. The last two examples illustrated an adsorption column, in one case with a linear isotherm and slow mass transfer and in the other case with a nonlinear isotherm and fast mass transfer. Specific techniques you demonstrated included parametric solutions when the solution was desired for several values of one parameter, and the use of artificial diffusion to smooth time-dependent solutions which had steep fronts and large gradients. 

Distributed parameter, nonlinear, partial differential equations were soloed to describe oxygen transport from maternal to fetal bloody which flows in microscopic channels within the human placenta. Steady-state solutions were obtained to show the effects of variations in several physiologically important parameters. Results reported previously indicate that maternal contractions during labor are accompanied by a partially reduced or a possible total occlusion of maternal blood flow rate in some or all portions of the placenta. Using the mathematical modely an unsteady-state study analyzed the effect of a time-dependent maternal blood flow rate on placental oxygen transport during labor. Parameter studies included severity of contractions and periodicity of flow. The effects of axial diffusion on placental transport under the conditions of reduced maternal blood flow were investigated. 

Equations 34, 35, 36, 37, 38, and 39 are a set of nonlinear partial differential equations that deseribe the time-dependent behavior of the deformation field /, the solvent flow field v, and the solvent pressure fieldp. The equations ean be linearized around an equilibrium value q>Q to deseribe gel dynamies for small deformation. Equation 36 is transformed to... 

The unsteady governing equations that apply in laminar flow also apply in turbulent flow but essentially cannot be solved in their full form at present In turbulent flow, therefore, the variables are conventionally split into a time averaged value plus a fluctuating component and an attempt is then made to express the governing equations in terms of the time averaged values alone. However, as a result of the nonlinear terms in the governing equations, the resultant equations contain extra variables which depend on the nature of the turbulence, i.e., there are, effectively, more variables than equations. There is, thus, a closure problem and to bring about closure, extra equations which constitute a turbulence model" must be introduced. 

---