# Analysis of Fluid Flow

## Methods and Differential Analysis

Abstract

Who is Fluid Mechanics Analyst?

The Fluid Mechanics Analyst supports engineers and customers who validate the design requirements of complex products by means of the extremely accurate simulation technologies for flow calculation. Fluid mechanics analyst allows seamless transition among applications so that user that users can entirely concentrate on the product: They will always use the right tool at the right time in the respective workflow processes. This results in a considerably higher degree of flexibility and cooperation in the design and simulation processes. Changes in the geometry are automatically updated and are immediately available on the same platform so that the results retrieved by analysts are always up-to-date fluids, have two state; static or dynamic. The state of static for fluids in much simple than dynamic state. There are many approaches and methods to analysis of fluid mechanics problems.

* Hydrostatics

There is a relation dp/dz = -ρg where dz is the change is the direction of the gravitational field (usually in the vertical direction). This relation gives the pressure distribution in a fluid under gravity. The relation is quite straightforward to get the relations for arbitrary fields too, for instance, the pseudo field due to rotation.

We know that the pressure in a fluid acts equally in all directions. Thus, when the pressure comes in contact with a surface, the force due to the pressure acts normal to the surface. There is a relation for magnitude of the force acting on an area. It means F = PA. Therefor, the force on a small area dA is given by pdA where the force is in the direction normal to dA. The total force on the area A is given by the vector sum of all these infinitesimal forces.

Differential Analysis of Fluid Flow

* Differential vs Integral Approach

Gross flow effects such as force or torque on a body or total energy exchange usually determine by integral approach for a control volume (CV). Integral approach for a CV is interested in a finite region. Balance of incoming and outcoming flux of mass, momentum and energy are made through this finite region. It gives very fast engineering answers, sometimes crude but useful.

Differential approach seek solution at every point (x1, x2, x3). In better words, it means describing the detailed flow pattern at all points. In other words, when we use differential relations, we are interested in the distribution of field properties at each point in space. Therefor, we analyze an infinitesimal region of a flow by applying the RTT (Reynolds Transport Theorem) to an infinitesimal control volume, or, to an infinitesimal fluid system.

* Lagrangian versus Eulerian Approach: Substantial Derivatives

Assume that α be any variable (pressure, velocity, etc.). Eulerian approach deals with the description of α at each location(xi) and time (t). For example, measurement of pressure at all xi defines the pressure field: P (x1, x2, x3, t). Other field variables of the flow are:

Lagrangian approach tracks a fluid particle and determines its properties as it moves.

One example of this approach is oceanographic measurements made which made with floating sensors delivering location, pressure and temperature data. X-ray opaque, dyes, which are used to track blood flow in arteries, is another example. Let’s αp be the variable of the particle (substance), P, this αp is called “substantial variable”. For this variable:

In other words, one observes the change of variable α for a selected amount of mass of fixed identity, such that for the fluid particle, every change is a function of time only. In a fluid flow, due to excessive number of fluid particles, Lagrangian approach is not widely used. Thus, for a particle P finding itself at point xi for a given time, we can write the equality with the field variable:

Along the path of the particle:

Hence:

Using Taylor series approximation for α ((xi+∆xi)p, t+∆t), the dαp can be written as:

The local change in time is the local time derivative (unsteadiness of the flow) and the change in space is the change along the path of the particle by means of the convective derivative.

The substantial derivative connects the Lagrangian and Eulerian variables.