![]() ![]() ![]() ![]() This work presents a numerical study concerning the impact of operating temperature and pressure upon the physical properties and mixture dynamics within the SAS process, because in supercritical conditions the radius of the droplets formed exhibits great sensitivity to these variables. The knowledge of fluid dynamics behavior plays a key role in the search for such parameter combinations. However, a suitable combination of operating parameters is needed for each type of solute. Note also that the maximum velocity is attained at the center of the pipe.The Supercritical Antisolvent (SAS) technique allows for the precipitation of drugs and biopolymers in nanometer size in a wide range of industrial applications, while guaranteeing the physical and chemical integrity of such materials. The velocity profile is parabolic and vanishes at the walls of the pipe as required by the no-slip boundary conditions. The velocity profile across the cross-section of the pipe is shown below: This simply states that there is no flow if there is no external pressure gradient to drive it. Subject to the no-slip boundary conditions,Ī straightforward computation yields the desired solution,Īs a consistency check, note that when. Solving the equation, along with the boundary conditions that the pressure at is and the pressure at is, yields the following: 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. ![]() As a simple application of the constitutive theory developed so far, let us look into a special class of fluids called Newtonian fluids. ![]()
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