Various forms of N-S equations

1) Lamb’s form of N-S equation:-

The N-S equation of motion for a viscous incompressible fluid is

                    $\frac{D\overrightarrow{q}}{Dt}=\overrightarrow{F}-\frac{\nabla p}{\rho }+\upsilon {{\nabla }^{2}}\overrightarrow{q}$

    or, $\frac{\partial \vec{q}}{\partial t}+\left( \vec{q}.\nabla  \right)\vec{q}=\vec{F}-\frac{\nabla p}{\rho }+\upsilon {{\nabla }^{2}}\overrightarrow{q}$……(1)               

Where  $\vec{q}=(u,v,w)$

                $\vec{F}=(X,Y,Z)$

                 $\nabla ={{(}^{\partial }}{{/}_{\partial x}}{{,}^{\partial }}{{/}_{\partial y}}{{,}^{\partial }}{{/}_{\partial z}})$

               r = Pressure

               u= Kinematic viscosity

$or,\frac{\partial \vec{q}}{\partial t}+\nabla \left( \frac{1}{2}{{\overrightarrow{q}}^{2}} \right)-\overrightarrow{q}\times \left( \nabla \times \overrightarrow{q} \right)=\vec{F}-\frac{\nabla p}{\rho }+\upsilon {{\nabla }^{2}}\overrightarrow{q}$  [∵ By Lagrange’s vector identity]

or, , $\frac{\partial \vec{q}}{\partial t}+\nabla \left( \frac{1}{2}{{\overrightarrow{q}}^{2}} \right)-\overrightarrow{q}\times \left( \widetilde{w} \right)=\vec{F}-\frac{\nabla p}{\rho }+\upsilon {{\nabla }^{2}}\overrightarrow{q}$ 

where  $\widetilde{w}=\nabla \times \overrightarrow{q}$ as vorticity vector

Or, \[\frac{\partial \vec{q}}{\partial t}+\nabla \left( \frac{1}{2}{{\overrightarrow{q}}^{2}} \right)+\widetilde{w}\times \overrightarrow{q}=\vec{F}-\frac{\nabla p}{\rho }-\upsilon {{\nabla }^{2}}\overrightarrow{q}\]  which is known as Lamb’s form of N-S equation                    

2) If external forces form a conservative field of force then F=-ÑW, where W is potential 

     function.

The lamb’s form of N-S equation now becomes

\[or,\frac{\partial \vec{q}}{\partial t}+\nabla \left( \frac{1}{2}{{\overrightarrow{q}}^{2}} \right)-\overrightarrow{q}\times \widetilde{w}=-\nabla \Omega -\frac{\nabla p}{\rho }-\upsilon {{\nabla }^{2}}\overrightarrow{q}\]

\[or,\frac{\partial \vec{q}}{\partial t}-\overrightarrow{q}\times \widetilde{w}=-\nabla \left( \frac{1}{2}{{\overrightarrow{q}}^{2}}+\Omega +\frac{p}{\rho } \right)-\upsilon {{\nabla }^{2}}\overrightarrow{q}\]$$

But,

\[{{\nabla }^{2}}\overrightarrow{q}=\nabla (\nabla .\overrightarrow{q})-\nabla \times \left( \nabla \times \overrightarrow{q} \right)=0-\nabla \times \left( \nabla \times \overrightarrow{q} \right)=-\nabla \times \widetilde{w}\]for incompressible fluid.

The above equation reduces to

\[\frac{\partial \vec{q}}{\partial t}-\overrightarrow{q}\times \widetilde{w}=-\nabla \left( \frac{p}{\rho }+\Omega +\frac{1}{2}{{\overrightarrow{q}}^{2}} \right)-\upsilon \nabla \times \widetilde{w}\]which is another form of Lamb’s form of N-S equation under the external force is conservative.

3) When there exist a functional relation between pressure and density, the N-S equation of motion for compressible viscous fluid is

$\frac{D\overrightarrow{q}}{Dt}=\overrightarrow{F}-\int{\frac{\partial \rho }{\rho }}+\upsilon {{\nabla }^{2}}\overrightarrow{q}+\frac{\upsilon }{3}\nabla (\nabla .\overrightarrow{q})$….(1)

Where  $\vec{q}=(u,v,w)$

                $\vec{F}=(X,Y,Z)$

                 $\nabla ={{(}^{\partial }}{{/}_{\partial x}}{{,}^{\partial }}{{/}_{\partial y}}{{,}^{\partial }}{{/}_{\partial z}})$

  $\frac{D\vec{q}}{Dt}=\frac{\partial \vec{q}}{\partial t}+\left( \vec{q}.\nabla  \right)\vec{q}$

                   $=\frac{\partial \overrightarrow{q}}{\partial t}+\nabla \left( \frac{1}{2}{{\overrightarrow{q}}^{2}} \right)-\overrightarrow{q}\times \left( \nabla \times \overrightarrow{q} \right)$   

   [∵$\left( \overrightarrow{q}.\nabla  \right)\overrightarrow{q}=\nabla \left( \frac{1}{2}{{\overrightarrow{q}}^{2}} \right)-\overrightarrow{q}\times \left( \nabla \times \overrightarrow{q} \right)$]     \[and\nabla \times \left( \nabla \times \overrightarrow{q} \right)=\nabla \left( \nabla .\overrightarrow{q} \right)-{{\nabla }^{2}}\overrightarrow{q}\Leftrightarrow {{\nabla }^{2}}\overrightarrow{q}=\nabla \left( \nabla .\overrightarrow{q} \right)-\nabla \times \left( \nabla \times \overrightarrow{q} \right)\]  

Then the equarion (1) reduces to

$\frac{\partial \overrightarrow{q}}{\partial t}+\nabla \left( \frac{1}{2}{{\overrightarrow{q}}^{2}} \right)-\overrightarrow{q}\times \left( \nabla \times \overrightarrow{q} \right)=\overrightarrow{F}-\int{\frac{\partial \rho }{\rho }}+\upsilon \left\{ \nabla \left( \nabla .\overrightarrow{q} \right)-\nabla \times \left( \nabla \times \overrightarrow{q} \right) \right\}+\frac{\upsilon }{3}\nabla (\nabla .\overrightarrow{q})$

$or,\frac{\partial \overrightarrow{q}}{\partial t}+\nabla \left( \frac{1}{2}{{\overrightarrow{q}}^{2}} \right)-\overrightarrow{q}\times \left( \nabla \times \overrightarrow{q} \right)=\overrightarrow{F}-\int{\frac{\partial \rho }{\rho }}-\upsilon \nabla \times \left( \nabla \times \overrightarrow{q} \right)+\frac{4\upsilon }{3}\nabla (\nabla .\overrightarrow{q})$

Relation between Cartesian (or rectangular) components of stress

Consider the motion of a small rectangular parallelepiped in viscous fluid with centre at P(x,y,z) and it’s edges of length dx, dy, dz parallel to the fixed rectangular axes.

The mass (density´volume) i.e rdxdydz of the element remains constant and the element is assumed to move along with fluid. In relation to the surface at right angle to the axis of x, the force components per unit area (or stress component) at P(x,y,z) are P_xx, P_xy, P_xz.

The corresponding force components at the centre

                                                         of the face of area dydz far from the origin are 

The corresponding force components at the centre

                                                           of the face of area dydz near from the origin are 

acting in the fluid in the opposite sense of former.

1)The force component on this pair of parallel faces of area dydz through P₁ & P₂ at right angle to the axis of x may be compounded into a single force having components

                                 

acting at P(x,y,z) in the direction parallel to ox, oy & oz respectively together with couple of moments -p_xydxdydz  and +p_xydxdydz about oy and oz respectively.

2) The force component on this pair of parallel faces of area dxdz through P₁ & P₂ at right angle to the axis of y may be compounded into a single force having components

                           

   

acting at P(x,y,z) in the direction parallel to ox, oy & oz respectively together with couple of moments -p_yzdxdydz  and +p_yzdxdydz about oz and ox respectively.

3) The force component on this pair of parallel faces of area dxdy through P₁ & P₂ at right angle to the axis of z may be compounded into a single force having components

                             

acting at P(x,y,z) in the direction parallel to ox, oy & oz respectively together with couple of moments -p_zxdxdydz  and +p_zxdxdydz about oy and ox respectively.

* red indicate just changing the term

Thus taking into account, the surface forces on all six faces at parallelepiped, we see that they reduced into single force at P(x,y,z) having components

                                   

    together with a vector couple having components

                                          (p_yz - p_zy) dxdydz about ox

                                          (p_zx - p_xz) dxdydz about oy

                                          (p_xy - p_yx) dxdydz about oz

Note that the fact has been used here is that the moment of forces about an arbitrary axis must be zero when contracting the element to a point (i.e making edges shrink to zero), for instance

                                          (p_yz - p_zy) dxdydz =0

                                          (p_zx - p_xz) dxdydz =0

                                          (p_xy - p_yx) dxdydz =0

       Þ                               (p_yz - p_zy) =0

                                          (p_zx - p_xz) =0

                                          (p_xy - p_yx) =0

       Þ                               p_yz = p_zy

                                          p_zx = p_xz

                                          p_xy = p_yx

Hence nine rectangular components of stress at a point reduced to six.

Equation of continuity in euler' s form.

     

 let ρ be the density of the fluid at p(x,y,z)  and let q=(u,v,w)be the velocity. 

Construct a small parallelopiped taking the point P as centre with edges of length h,k and l parallel to  the coordinate axes which is occupied by fluid.

the mass of fluid that flows across the unit area parallel to yz plane through  P in time δt                                                                        =ρuδt

                                                  =f(x,y,z)δt….say

                                                                         

      

we know that ,mass=volume×density                        and here volume=Area×distance                     velocity= →distance=velocity×δt=uδt and area=1 unit               ∴volume=1×uδt=uδt , so mass=uδt×ρ=ρuδt      since ρ and u depend upon the variable of x,y and z so they both can be considered as f(x,y,z)

                      

     let P₁ and P₂  be the projection of P at the faces kl near to and away from the origin respectively where the coordinate of

it' s because P is the centre of the parallelopiped and the length along x axis is h and following figure

             

   due to only projection of P ,P₁&P₂ do not change the coordinate of y and z it only change x coordinate here P₁ lies on faces kl near to origin and P₂  lies on faces kl but away from origin

                      the x coordinate of P₁=ON-MN

                                                        

                 the x coordinate of P₂=ON+NA

                                                

To find the flow across the face kl of the parallelopiped nearest to origin ,take a point          

 on the surface having area dη.dξ at this point 

then the mass of fluid that flows across the area dη.dξ in time δt 

   [∵using Taylor^' s expansion for several variables neglecting higher order]      

    The mass of the fluid that flows across the faces kl near to the origin in x↑direction in time δt, 

The mass of the fluid that flows across the faces kl away from the origin in x↑direction  in time δt, 

 

therefore the increase in mass inside the parallelopiped in time δt due to the flow across these two faces in x↑direction is

similarly the excess of mass of fluid that flow in across the hl faces near to the origin over the mass that flows out across the hl faces away from the origin in y↑direction in time δt 

 and the excess of mass of fluid that flow in across the kh faces near to origin over the mass flows out across the kh faces away from the origin in z↑direction in time δt,

  

hence the total mass of fliud that flows in the parallelopiped over the mass that flows out of it in time δt,

on the other hand ,the initial mass of the fluid inside the parallelopiped=ρhkl 

and the increase in mass of fluid inside the parallelopiped in time δt

 now by principal of continuity,(1)and (2)are equal. i.e

    which is the required equation of continuity in euler' s form.                                                              

Information about fluid

Fluid :-

          Fluid is a substance which is capable to flow.

Classification of fluid :

                                                                  

Isotropic substance

                    Fluid is treated as isotropic substance which implies that physical property(i.e pressure, density etc) is same in all direction.

Anisotropic substance

                    Fluid is treated as anisotropic substance if that  property is not same in all direction.

Newtonian fluid

The fluid which obeys Newtonian relationship between shearing stress and gradient of velocity given by

                        

where  
 is the velocity gradient and τ is shearing stress.

Ideal fluid

Ideal fluid is a Newtonian fluid in which there exist no shearing or tangential stress but exist only normal or direct stress between two contacting layers.

Note that in nature ideal fluid doesn’t occur. They are only as mathematical concept.

Real  fluid

Real fluid is a Newtonian fluid in which there exist both tangential and normal stress between two contacting layers.