Fluid Mechanics

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

  \(\vec F = \eta \frac{{\partial v}}{{\partial y}}S\;\vec u\)

  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 :

  \({\vec f_{vis}} = \frac{{d{{\vec F}_{vis}}}}{{d\tau }} = \eta \;\Delta \vec v = \eta \left| \begin{array}{l}\Delta {v_x} \\\Delta {v_y} \\\Delta {v_z} \\\end{array} \right|\)

• The volume flow rate :

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

  This flow rate is :

  \(D_v=\iint_{(S)}\vec v.\vec n \;dS\)

• Mass flow rate :

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

  \(D_m=\iint_{(S)} \mu \vec v.\vec n \;dS = \iint_{(S)}\vec j .\vec n\; dS\)

  Where \(\vec j=\mu \vec v\) 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 :

  \(D_m=\mu D_v\)

  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).

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

  \({R_e} = \frac{{\left\| {\rho (\vec v.\overrightarrow {grad} )\vec v} \right\|}}{{\left\| {\eta \Delta \vec v} \right\|}} \approx \frac{{\rho \frac{{U_\infty ^2}}{D}}}{{\eta \frac{{{U_\infty }}}{{{D^2}}}}} = \frac{{\rho D{U_\infty }}}{\eta }\)

  Where \(D\) is a characteristic size of the flow (the length of the obstacle, for example) and \(U_{\infty}\) the velocity of the fluid away from the obstacle.

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

  \(\delta \approx \frac{D}{{\sqrt {{R_e}} }}\)

• The vorticity vector of a flow fluid is :

  \(\vec \Omega = \frac{1}{2}\;\overrightarrow {rot} (\vec v)\)

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

  \(\overrightarrow {rot} (\vec v) = \vec 0\)

  We can then define a velocity potential, denoted \(\Phi\), by :

  \(\vec v = \overrightarrow {grad} \Phi\)

  The flow is said potential.

• For an incompressible flow :

  \(div (\vec v) =0\)

  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 :

  \(v_1S_1=v_2S_2\)

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

• The connections between \(\vec v = \overrightarrow {grad} \Phi\) and \(\;div( \vec v) =0\) leading to the Laplace equation (also seen in electrostatic) :

  \(\Delta \Phi = 0\)

\(\vec a = \frac{{D\vec v}}{{Dt}} = \frac{{\partial \vec v}}{{\partial t}} + (\vec v.\overrightarrow {grad} )\;\vec v = \frac{{\partial \vec v}}{{\partial t}} + \overrightarrow {grad} \left( {\frac{{{v^2}}}{2}} \right)\; + \;\overrightarrow {rot} (\vec v)\; \wedge \;\vec v\)

This is a classic conservation equation :

\(\frac{{\partial \rho (M,t)}}{{\partial t}} + div\vec j = 0\)

With :

\(\vec j = \rho \vec v\) (current density vector or vector density of flux of flow mass)

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

  \(\mu \;\frac{{\partial \vec v}}{{\partial t}} + \mu \;(\vec v.\overrightarrow {grad} )\;\vec v = - \overrightarrow {grad} P\; + {\vec f_v}\)

  Or again :

  \(\mu \;\frac{{\partial \vec v}}{{\partial t}} + \mu \;\overrightarrow {grad} \left( {\frac{{{v^2}}}{2}} \right)\; + \;\mu \;\overrightarrow {rot} (\vec v)\; \wedge \;\vec v = - \overrightarrow {grad} P\; + {\vec f_v}\)

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

  \(\mu \;\frac{{\partial \vec v}}{{\partial t}} + \mu \;(\vec v.\overrightarrow {grad} )\;\vec v = - \overrightarrow {grad} P\; +\eta\Delta \vec v +{\vec f_v}\)

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

  Between two points \(A\) and \(B\) of the same current line :

  \(\frac{1}{2}\mu \;v_A^2 + \mu \;g{z_A} + {P_A} = \frac{1}{2}\mu \;v_B^2 + \mu \;g{z_B} + {P_B}\)

• For an irrotational flow :

  \(\frac{1}{2}\mu \;{v^2} + \mu \;gz + P = C\)

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

Torricelli's formula is :

\(v=\sqrt{2gh}\)

\(v\) is the fluid velocity at the output of the hole and \(h\) the height of liquid above the hole.

This formula rigorously applies when the height \(h\) is constant.

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

• Poiseuille flow : 

  A incompressible viscous fluid of density \(\rho\) flows through a cylindrical tube of length \(L\) and radius \(R\).

  The pressure at the input of the tube (\(z = 0\)) is \(P_e\). The pressure at the output of the tube is \(P_S\).

• The hydraulic resistance is :

  \(R_{hyd}=\frac{P_E-P_S}{D_v}\)

  Where \(D_v\) is the volume flow rate of the Poiseuille flow.