# Fluid Mechanics for Mechanical Engineers/Differential Analysis of Fluid Flow

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## Differential vs Integral Approach[edit | edit source]

Integral approach for a Control Volume (CV) is interested in a finite region and it determines gross flow effects such as force or torque on a body or the total energy exchange. For this purpose, balances of incoming and outgoing 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 , i.e describe 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. Therefore, we analyze an infinitesimal region of a flow by applying the RTT to an infinitesimal control volume, or , to a infinitesimal fluid system.

## Lagrangian versus Eulerian Approach: Substantial Derivative[edit | edit source]

Let be any flow variable (pressure, velocity, etc.). Eulerian approach deals with the description of at each location and time (t). For example, measurement of pressure at all defines the pressure field: . Other field variables of the flow are:

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

Oceanographic measurements made with floating sensors delivering location, pressure and temperature data, is one example of this approach. X-ray opaque dyes, which are used to trace blood flow in arteries, is another example.

Let be the variable of the particle (substance) P, this is called "substantial variable".

For this variable:

and

In other words, one observes the change of variable 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 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 , the 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.

## Conservation of mass[edit | edit source]

The conservation of mass according to RTT

or in tensor form

The differential volume is selected to be so small that density can be accepted to be uniform within this volume and it can be taken out of the integral. Since the volume is constant, the first integral in the above equation is:

where .The mass flow rate term (second integral term) in the equation of conservation of mass can be analyzed in groups:

Let's look to the surfaces perpendicular to

Similarly, the flux integrals through surfaces perpendicular to and are

Hence the integral of the mass flux reads;

The conservation of mass equation becomes:

Dropping the , we reach to the final form of the conservation of mass:

This equation is also called *continuity equation*. It can be written in vector form as:

For a steady flow, continuity equation becomes:

.

For incompressible flow, i.e. :

### Examples[edit | edit source]

#### Example 1[edit | edit source]

For a two dimensional, steady and incompressible flow in plane given by:

Find how many possible can exist.

For incompressible steady flow:

in two dimensions

Thus,

This is an expression for the rate of change of velocity while keeping

constant. Therefore the integral of this equation reads

Thus, any function is allowable.

#### Example 2[edit | edit source]

Consider one-dimensional flow in the piston. The piston suddenly moves with the velocity .

Assume uniform in the piston and a linear change of velocity such that at the bottom () and on the piston (), i.e.

Obtain a function for the density as a function of time.

The conservation of mass equation is:

For one-dimensional flow and uniform , this equation simplifies to

The same problem can be solved by using the integral approach with a deforming control volume.

## The differential equation of linear momentum[edit | edit source]

The integral equation for the linear momentum (2nd law of Newton) is:

For the first integral we assume and are uniform within , and is so small that:

Analyze the flow rate of the momentum terms on the faces perpendicular to each axis:

First consider the flow rate of (momentum per unit volume in *i*-direction) through the surfaces perpendicular to axis:

Similarly, the momentum flow rate through the surfaces in other directions read

,

.

Rearranging the equation for we obtain:

We can simplify further:

Hence

Let's look to the forces on the differential control volume:

Here, only gravitational force is considered as a body force. Thus,

Surface forces are the stresses acting on the control surfaces. can be resolved into three components. is normal to dA. are tangent to dA:

is a normal stress whereas is a shear stress. The shear stresses are also designated by .

Thus, the surface forces are due to stresses on the surfaces of the control surface.

A stress component is positive on a face, when it is in the same direction as the outwards normal vector as shown on ABCD or A'B'C'D'. Note that on A'B'C'D', the stresses are considered to be positive, tough the surface normal is in the direction.

The stresses on the surface are the sum of pressure plus the viscous stresses which arise from motion with velocity gradients:

has a minus sign since the force due to pressure acts opposite to the surface normal.

Let us look to the differential surface force in the direction:

Noting that and (4),

Thus in tensor form the differential surface forces in 'th direction can be written as

Note that is a symmetric tensor, i.e.

Hence, the diffential surface forces reads:

Inserting and into (2),

and canceling we obtain

Expanding the substatial derivative at the left hand side,

We obtain the the most general form of momentum equation which is valid for any fluid (Newtonian, Non-newtonian, Compressible, etc.). It is non-linear due to the term (Convective term) at the LHS. Efect of Newtonian and Non-newtonian properties appears in the formulation of the viscous stresses . will introduce also non-linearity when the fluid is non-Newtonian.

It should be noted that these formulations are based on stress conception which was thought to exist in fluids in motion. However it is known that can be expressed as momentum transfer per unit area and time (diffusive momentum flux). Thus it can be considered as molecular momentum transport term. Derivations based on this concept requires a molecular approach. The students should be aware that causes momentum transport when there is a gradient of velocity.

### Closure Problem[edit | edit source]

The conservation of mass and momentum equation form a system of equations. At isothermal conditions, there are 11 unknowns involved in this 4 equation system namely where and . The equation system is not closed, i.e. it is impossible to solve this equation. Extra equations are necessary for the closure. This was achieved by Navier and Stokes by relating the stress term to the deformation rate of the fluid, i.e. to the velocity gradients and, consequently, introducing the viscous effect into the momentum equation.

## [edit | edit source]

For a Newtonian fluid, the viscous stresses are defined as:

Note that derivation of this relation is beyond the scaope of this course.

Thus, the momentum equation becomes

or when the LHS(substantial derivative) is expanded:

For a flow with constant viscosity ():

since,

then,

For an incompressible flow

hence assuming that the viscosity is constant, it can be easily shown that the momentum equation reduces to

or when the LHS(substantial derivative) is expanded:

## Euler's equation: Inviscid flow[edit | edit source]

When the velocity gradients in the flow is negligible and/or the Reynolds number takes very high values, the viscous stresses can be neglected:

Since, the viscous stresses are proportional to viscosity:

for flows, where is neglected, the flow is called frictionless or inviscid, although there is a finite viscosity of the flow. Accordingly, the linear momentum equation reduces to

### Euler's equation in streamline coordinates[edit | edit source]

Euler's equation take a special form along and normal to a streamline with which one can see the dependency between the pressure, velocity and curvature of the streamline.

To obtain Euler's equation in s-direction, apply Newton's second law in s-direction in the absence of viscous forces.

Omitting would deliver

Since

then the Equler's equation along a streamline reads

For a steady flow and by neglecting body forces,

it can be seen that decrease in velocity means increase in pressure as is indicated by the Bernoulli equation.

To obtain Euler's equation in n direction, apply Newton's second law in the absence of viscous forces and for a steady flow.

Since,

Then,

For a steady flow, the normal acceleration of the fluid is towards *the center of curvature* of the streamline:

Hence,

For an unsteady flow,

For steady flow neglecting body forces, the Euler's equation normal to the streamline is

which indicates that pressure increases in a direction outwards from the center of the curvature of the streamlines. In other words, pressure drops towards the center of curvature, which, consequently creates a potential difference in terms of pressure and forces the fluid to change its direction. For a straight streamline , there is no pressure variation normal to the streamline.

## Bernoulli equation: Integration of Euler's equation along a streamline for a steady flow[edit | edit source]

For a steady flow, Euler's equation along a streamline reads,

If a fluid particle moves a distance ds, along a streamline, since every variable becomes a function of :

Integration of the Euler equation between two locations, 1 and 2, along reads

For incompressible flow and after changing the notation as: and , the integration results in

or in its most beloved form:

In other words along a streamline:

Note that due to the assumptions made during the derivation, the following restrictions applies to this equation: The flow should be steady, incompressible, frictionless and the equation is valid only along a streamline.

### Different forms of Bernoulli equation[edit | edit source]

The common forms of Bernoulli equation are as follows:

**Energy form (per unit mass)**

**Pressure form**

**Head form**

### Static, stagnation and dynamic pressures[edit | edit source]

How do we measure pressure? When the streamlines are parallel to the wall, we can use pressure taps.

If the measured location is far from the wall, static pressure measurements can be made by a static pressure probe.

The *stagnation pressure* is the value obtained when a flowing fluid is decelerated to zero velocity by a frictionless flow process. The Stagnation pressure can be calculated as follows:

when

where the last term is the dynamic pressure.

If we know the pressure difference , we can calculate the velocity.

The stagnation pressure is measured in the laboratory using a probe that faces directly upstream flow.

Such a probe is called a *stagnation pressure probe* or *Pitot tube* . Thus, using a pressure tap and a Pitot tube one can measure local velocity:

Thus, measuring one can determine .

However, in the absence of a wall with well defined location, the velocity can be measured by a *Pitot-static tube*. The pressure is measured at B and C; assuming .

Hence,

### Unsteady Bernoulli equation[edit | edit source]

The Euler's equation along a streamline is:

along ds,

hence,

Integration between two points along a streamline is:

For incompressible flow, , thus the integral reads

The unsteady Bernoulli equation involves the integration of the time gradient of the velocity between two points.: