Continuity Equation

Conservation principles are used to represent all physical transformations occurring in the universe, accordingly are also adopted to design runner conduit for thermoplastic melt injection. Conservation principles for thermoplastic melt injection through runner conduit are implemented by considering cylindrical co-ordinates system relevant to its geometrical configuration for deriving governing equations. While the continuity equation ensures volumetric conservation of thermoplastic melt, the momentum equation represents equilibrium of forces on thermoplastic melt injection through runner conduit. During an injection moulding cycle heat and work done energy transformations are balanced by implementing first and second law of thermodynamics. Thermoplastic melt state change through the runner conduit for a particular cycle is appreciated by heat conduction equation. Traditionally inertia and entropy contribution is neglected to skip rigorousness, nevertheless they continue to prevail....

Downpull on tunnel gates installed in the intake structure of a hydroelectric power plant was studied experimentally using a hydraulic model. The pressure distribution on the gate lip surface was measured, and the lip downpull was evaluated by surface-area integration of the measured pressure distribution. An easy to use lip downpull coefficient was defined as a function of the lip angle and gate opening. The lip downpull coefficient function is linked to a one-dimensional mathematical model of unsteady flow in the intake-penstock system. The model is based on the integral energy and continuity equations. Overflow through the gate spacings is also included in the model to compute the water level in the gate shaft and to evaluate the downpull component on the top face of the gate. Time-dependent calculation of the total downpull force acting on a closing gate is exemplified. The total downpull is also measured by the direct weighing method for fixed and closing gates. Predictions of the mathematical model compare favorably with the downpull obtained from the direct weighing method.

Vegetation and soil properties and their associated changes through time and space affect the various stages of soil erosion. The island of Ishigaki in Okinawa Prefecture, Japan is of particular concern because of the propensity of the red-soil-dominated watersheds in the area to contribute substantial sediment discharge to adjacent coastal areas. This paper discusses the application of remote sensing techniques in the retrieval of vegetation and soil parameters necessary for the distributed soil-loss modelling in small agricultural catchments and analyses the variation in erosional patterns and sediment distribution during rainfall events using numerical solutions of overland flow simulations and sediment continuity equations. To account for the spatial as well as temporal variability of selected parameters of the soil-loss equations, a method is proposed to account for the variability of associated vegetation cover based on their spectral characteristics as captured by remotely se...

Starting with the quantum Liouville equation, we write the density operator as the product of elements respectively in the left and right ideals of an operator algebra and find that the Schrodinger picture may be expressed through two representation independent algebraic forms in terms of the density and phase operators. These forms are respectively the continuity equation, which involves the

In this paper (Part I) we derive the basic equations governing the model, discuss the treatment of meteorological variables (inversion height, wind field, and turbulent eddy diffusivity), present a kinetic mechanism for photochemical smog, and describe the technique employed for ...

We demonstrate that the Navier-Stokes equation can be covariantized under the full infinite dimensional Galilean Conformal Algebra (GCA), such that it reduces to the usual Navier-Stokes equation in an inertial frame. The covariantization is possible only for incompressible flows, i.e when the divergence of the velocity field vanishes. Using the continuity equation, we can fix the transformation of pressure and density under GCA uniquely. We also find that when all chemical potentials vanish, $c_{s}$, which denotes the speed of sound in an inertial frame comoving with the flow, must either be a fundamental constant or given in terms of microscopic parameters. We will discuss how both could be possible. In absence of chemical potentials, we also find that the covariance under GCA implies that either the viscosity should vanish or the microscopic theory should have a length scale or a time scale or both. We further find that the higher derivative corrections to the Navier-Stokes equation, can be covariantized, only if they are restricted to certain possible combinations in the inertial frame. We explicitly evaluate all possible three derivative corrections. Finally, we argue that our analysis hints that the parent relativistic theory with relativistic conformal symmetry needs to be deformed before the contraction is taken to produce a sensible GCA invariant dynamical limit.

Stokes flow through a rigid porous medium is analyzed in terms of the method of volume averaging.  The traditional averaging procedure leads to an equation of motion and a continuity equation expressed in terms of the volume-averaged pressure and velocity.  The equation of motion contains integrals involving spatial deviations of the pressure and velocity, the Brinkman correction, and other lower-order terms.  The analysis clearly indicates why the Brinkman correction should not be used to accommodate a no slip condition at an interface between a porous medium and a bounding solid surface.
The presence of spatial deviations of the pressure and velocity in the volume-averaged equations of motion gives rise to a closure problem, and representations for the spatial deviations are derived that lead to Darcy's law.  The theoretical development is not restricted to either homogeneous or spatially periodic porous media; however, the problem of abrupt changes in the structure of a porous medium is not considered.

A high-order accurate, finite-difference method for the numerical solution of the incompressible Navier–Stokes equations is presented. Fourth-order accurate discretizations of the convective and viscous fluxes are obtained on staggered meshes using explicit or compact finite-difference formulas. High-order accuracy in time is obtained by marching the solution with Runge–Kutta methods. The incompressibility constraint is enforced for each Runge–Kutta stage iteratively either

Abstract Second-order accurate projection methods for simulating time-dependent incompressible flows on cell-centred grids substantially belong to the class either of exact or approximate projections. In the exact method, the continuity constraint can be satisfied to machine-accuracy but the divergence and Laplacian operators show a four-dimension nullspace therefore spurious oscillating solutions can be introduced. In the approximate method, the continuity constraint is relaxed, the continuity equation being satisfied up to ...

The performance of a hollow-fiber membrane contactor in removing ammonia from aqueous solution was simulated. An unsteady state 2D mathematical model was developed to study the ammonia stripping in the hollow-fiber membrane contactor. Two sets of equations were considered for the membrane contactor and the feed tank. CFD technique was applied to solve the model equations in which concentration distribution was determined using continuity equation. Velocity field is also determined using Navier–Stokes equations for the contactor. The model was implemented in linked MATLAB–COMSOL Multiphysics. COMSOL software was applied to solve the model equations for the contactor while MATLAB software was employed to consider changes in the concentration of the feed tank. Predictions of the model were then validated against experimental data which were found to be in good agreement. The assumption of Knudsen diffusion for the transport of ammonia molecules through the membrane pores increased the accuracy of the model. The effect of different parameters including feed velocity, feed concentration and pH on the removal of ammonia was investigated. The results of simulation revealed that the developed model can be used to evaluate the effective parameters which involve in the ammonia removal by means of membrane contactors.► Ammonia removal from aqueous solutions using a hollow-fiber membrane contactor ► Development of a 2D mathematical model for prediction of ammonia stripping ► Excellent agreement between experimental and simulation results

We investigate the flow of various non-Newtonian fluids through three-dimensional disordered porous media by direct numerical simulation of momentum transport and continuity equations. Remarkably, our results for power-law (PL) fluids indicate that the flow, when quantified in terms of a properly modified permeability-like index and Reynolds number, can be successfully described by a single (universal) curve over a broad range of Reynolds conditions and power-law exponents. We also study the flow behavior of Bingham fluids described in terms of the Herschel-Bulkley model. In this case, our simulations reveal that the interplay of ( i) the disordered geometry of the pore space, ( ii) the fluid rheological properties, and ( iii) the inertial effects on the flow is responsible for a substantial enhancement of the macroscopic hydraulic conductance of the system at intermediate Reynolds conditions. This anomalous condition of "enhanced transport" represents a novel feature for ...

This paper analyzes the dynamic effects of pipe wall viscoelasticity on hydraulic transients. These effects have been observed in transient data collected from two polyethylene (PE) pipe systems. The first is a 270 m pipeline, 50 mm diameter, at Imperial College London, and the second is the world's longest experimental PE pipeline, 1.3 km long, 110 mm diameter, buried underground at Thames Water Utilities (London, UK). A mathematical model has been developed to calculate hydraulic transients in polyethylene pipe systems ...