![]() ![]() The only way this equation can be satisfied, therefore, is when the two are simultaneously equal to some constant, say : The former equation then tells us that the left hand side, namely, is just a function of, while the right hand side, namely, is a function of only. The latter equations imply that the pressure is only a function of the variable. Using these results in the incompressible Navier-Stokes equation, we get the following simple equation: To get the incompressible Navier-Stokes equation for this case, note that The incompressibility constraint,, implies that Thus, a general form of the velocity field for the present problem can be written as By symmetry, we do not expect any flow along the and directions. For our purposes, it can be taken to be an experimental fact. The origin of the no-slip boundary condition lies in the details of the interaction between the fluid and wall particles, and cannot be obtained from purely thermodynamic considerations alone. This cnidition requires that the velocity vanishes at the fluid-wall interface at. In addition to the pressure being specified at the two ends of the pipe, we will also need the no-slip boundary condition on the walls for the velocity field. To solve these equations, we need to augment them with the appropriate initial and boundary conditions. Let us start by writing out the mass and linear momentum balance equations for the steady state flow of an incompressible Newtonian viscous fluid: What we understand by a solution is the computation of the velocity field given all the information above. What does this mean? Recall that the only unknown in this case is the velocity field. Let us now see how to solve this problem using the incompressible Navier-Stokes equation. The steady flow in the pipe is caused by a pressure difference. Equivalently, this can be looked at as a a semi-infinite pipe with infinite extension along the out-of-plane direction. The geometry of the flow is illustrated below:Īs an idealization, we consider a two dimensional pipe of length and cross sectional width. The problem we will study is that of the steady flow of an inompressible Newtonian viscous fluid through a pipe of constant cross sectional area. This will illustrate how everything that we have studied thus far comes together. Let us now look at simple example, called a Poiseuille flow, of an application of the theory developed so far. You can become a millionaire and achieve a permanent place in the history of science if you crack this! Poiseuille flow Proving the existence of solutions for the incompressible Navier-Stokes equation is famously one of the Millenium Problems listed by the Clay Mathematics Institute see this page for more details. ![]() When supplied with the appropriate initial and boundary conditions, we can, in principle at least, solve the resulting set of partial differential equations. ![]() We saw in our previous discussin of constitutive models that this form of the constitutive model is objective provided the function $\mathcal$. Let us now consider the special case of a constitutive relation of the form This will be followed by a simple example, called Poiseuille flow, to illustrate how it all comes together. This will result in the famous Navier-Stokes equation in fluid dynamics. The worst case scenario is associated with a fast valve opening when a tiny air pocket exists in the pipeline.As a simple application of the constitutive theory developed so far, let us look into a special class of fluids called Newtonian fluids. Results show the pipeline drainage mostly occurs due to backflow air intrusion. The effects of the air pocket size, the percentage and the time of valve opening on the pressure variation have been studied. Also, this research demonstrates the ability of a computational fluid dynamic (CFD) model in the simulation of this event. This case has been studied experimentally and numerically in the current research considering objectives for a better understanding of: (i) the emptying process, (ii) the main parameters influencing the drainage, and (iii) the air-water interface deformation. Accordingly, some system malfunction and pipe buckling events have been reported in the literature. Keywords: Computational fluid dynamics (CFD) Emptying process Entrapped air simulation Experimental set-up Realizable k-ϵ turbulence model Sub-atmospheric pressure Volume of fluid (VOF) multiphase modelĪbstract: The occurrence of sub-atmospheric pressure in the drainage of pipelines containing an air pocket has been known as a major cause of several serious problems. Author(s): Oscar Enrique Coronado-Hernandez Helena Margarida Ramos Vicente Samuel Fuertes-Miquel Maria Teresa Viseu Mohsen Besharat
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