Chapter 7

# To test the understanding of the lesson

## Question

• A Newtonian dynamic viscosity fluid, flowing with a velocity field of the type .

Give the expression of the shear stress exerted by the fluid located below the axis on the fluid are located above this axis.

• What is the volume expression of viscous forces acting on an incompressible viscous and Newtonian fluid ?

### Solution

• For unidirectional flow such that , the tangential surface force, known as shear force or viscous force, which is exerted through a area of surface normal to (it is the force exerted by the upper layer on the bottom layer) :

The viscosity effect in an unidirectional flow, accelerates the slow elements and slows down the fast elements.

It is therefore an internal transfer of momentum, which has the characteristics of diffusion of momentum.

• The volume force of viscosity is :

## Question

• Give the definition of the volume flow rate and mass flow rate of a flowing fluid.

• What is the relationship between them ?

### Solution

• The volume flow rate :

Called volume flow rate through a oriented surface ( ), the volume of fluid which crosses ( ) per unit of time, counted positively in the direction of the normal vector to the surface and negatively otherwise.

This flow rate is :

• Mass flow rate :

The mass flow rate corresponds to the mass of fluid passing through ( ) per unit of time, counted positively in the direction of the normal vector to the surface and negatively otherwise :

Where is the current density vector or vector density of flux of flow mass.

• If the fluid is incompressible and homogeneous, so if the density is constant and independent of the point, then :

It may be noted that these flow rates are equivalent to the electrical intensity, the heat flux and the particle flow rate (seen in diffusion).

## Question

• Give the definition and the expression of the Reynolds number. What is its dimension ?

• What is the relationship between the Reynolds number and the thickness of the boundary layer  ?

### Solution

• The Reynolds number is defined as the magnitude of the ratio between the convective term and the diffusion term :

Where is a characteristic size of the flow (the length of the obstacle, for example) and the velocity of the fluid away from the obstacle.

• The thickness of the boundary layer is expressed in terms of the Reynolds number :

## Question

• Give the definition of the vorticity vector of a flow fluid.

• What is the definition of a potential flow ?

### Solution

• The vorticity vector of a flow fluid is :

• If the vorticity vector is zero at any point in space :

We can then define a velocity potential, denoted , by :

The flow is said potential.

## Question

• Define an incompressible flow.

• For a potential flow, demonstrate the Laplace equation.

### Solution

• For an incompressible flow :

We can derive that the volume flow can be kept in a field tube and therefore evidently in pipeline.

In particular, if the speed is constant over a section of the pipeline :

Thus, when a flow field lines grow narrow, the norm of the vector speed increases.

• The connections between and leading to the Laplace equation (also seen in electrostatic) :

## Question

Give the expression of the particle acceleration in function of Eulerian field speeds.

## Question

Giving local expression of mass conservation law.

### Solution

This is a classic conservation equation :

With :

(current density vector or vector density of flux of flow mass)

## Question

• Write the Euler equation for a perfect flow in a Galilean frame of reference.

• Give the Navier - Stokes equation for viscous flow of an incompressible fluid in a Galilean frame of reference.

### Solution

• Euler equation for a perfect flow in a Galilean frame of reference :

Or again :

• Navier - Stokes equation for viscous flow of an incompressible fluid in a Galilean frame of reference :

## Question

• Give a Bernoulli equation on a current line in a perfect, homogeneous, stationary and incompressible flow.

• Give a Bernoulli equation in a perfect, homogeneous, stationary, incompressible and irrotational flow.

### Solution

• On a current line in a perfect, homogeneous, stationary and incompressible flow.

Between two points and of the same current line :

• For an irrotational flow :

Where is a constant (independent of the point in the fluid).

## Question

Emptying a container : give the Torricelli's formula . When does it apply ?

### Solution

Torricelli's formula is :

is the fluid velocity at the output of the hole and the height of liquid above the hole.

This formula rigorously applies when the height is constant.

It also applies when the flow is quasi-permanent ( varies slowly with time).

## Question

• What is called a Poiseuille flow ?

• Define the hydraulic resistance.

### Solution

• Poiseuille flow :

A incompressible viscous fluid of density flows through a cylindrical tube of length and radius .

The pressure at the input of the tube ( ) is . The pressure at the output of the tube is .

• The hydraulic resistance is :

Where is the volume flow rate of the Poiseuille flow.

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