Fluid Mechanics
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 ?
Hint
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 ?
Hint
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 ?
Hint
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 ?
Hint
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.
Hint
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.
Hint
Solution
Question
Giving local expression of mass conservation law.
Hint
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.
Hint
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.
Hint
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 ?
Hint
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 quasipermanent ( varies slowly with time).
Question
What is called a Poiseuille flow ?
Define the hydraulic resistance.
Hint
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.