Navier-Stokes equation (derivation)

The Navier-Stokes equation is used to describe the motion of a viscous fluid and was named after Claude-Louis Navier, a French scientist, and George Stokes, an Irish scientist. To derive the equation, let’s consider the flow of an incompressible fluid through an infinitesimal cubic control volume* depicted in the diagram below.

* the control volume here refers to the volume of an object in the fluid.

The x-component of the velocity of the fluid is a function of x, y, z and t, i.e. ux (x, y, z, t) where t is time. The total differential of uis:

du_x=\frac{\partial u_x}{\partial x}dx+\frac{\partial u_x}{\partial y}dy+\frac{\partial u_x}{\partial z}dz+\frac{\partial u_x}{\partial t}dt

Dividing the equation by dt and letting dx/dtuxdy/dtuydz/dtuz, we have:

\frac{du_x}{dt}=u_x\frac{\partial u_x}{\partial x}+u_y\frac{\partial u_x}{\partial y}+u_z\frac{\partial u_x}{\partial z}+\frac{\partial u_x}{\partial t}\; \; \; \; \; \; \; (24)

Let’s analyse the hydrostatic and hydrodynamic forces acting on each face of the infinitesimal control volume in the x-direction (see diagram above where point C is the origin). They are summarised as follows:

Face

Force per unit area

Force

BDFH

σxx(x + dx) σxx(x + dx)dydz

ACEG

σxx(x) σxx(x)dydz

ABHG

τyx(y + dy) τyx(y + dy)dxdz

CDFE

τyx(y) τyx(y)dxdz

ABDC

τzx(z) τzx(z)dxdy

GHFE

τzx(z + dz) τzx(z + dz)dxdy

σxx = F/A , i.e. the force F acting on a unit area of the x-plane in the x-direction, thereby the double-x subscript. The notation σxx(x) refers to σxx evaluated at a distance of (x) while σxx(x + dx) is evaluated at a distance of (x + dx). A different notation τ is given to the forces that act parallel to the faces of the control volume, e.g. τyx(y + dy) is the force acting on a unit area of the y-plane in the x-direction that is evaluated at a distance of (y + dy).

With reference to the defined directions of the forces, Fthe sum of hydrostatic and hydrodynamic forces acting on the infinitesimal control volume in the positive x-direction is:

In addition to hydrostatic and hydrodynamic forces, an inertia force Fg (gravitational force) also acts on the infinitesimal control volume in the x-direction:

F_g=\rho g_xdxdydz

where ρ is the density of the fluid and gx is the x-component of the acceleration due to gravity.

The total force Facting on the infinitesimal control volume in the x-direction is therefore:

F_T=F_H+F_g\; \; \; \; \; \; \; (25)

Next, by i) substituting FH and Fg and eq24 in eq25; ii) dividing the substituted equation by dxdydz and iii) letting dx, dy, dz → 0 for an infinitesimal control volume, we have:

\rho\left ( u_x\frac{\partial u_x}{\partial x} +u_y\frac{\partial u_x}{\partial y}+u_z\frac{\partial u_x}{\partial z} +\frac{\partial u_x}{\partial t} \right )=\frac{\partial \sigma_{xx}}{\partial x} +\frac{\partial \tau_{yx}}{\partial y} +\frac{\partial \tau_{zx}}{\partial z} +\rho g_x\; \; \; \; \; \; \; (26)

Using the same logic and repeating the above steps, we have for the y-direction and the z-direction:

\rho\left ( u_x\frac{\partial u_x}{\partial x}+ u_y\frac{\partial u_x}{\partial y}+ u_z\frac{\partial u_x}{\partial z}+\frac{\partial u_y}{\partial t}\right )= \frac{\partial \sigma_{yy}}{\partial y}+\frac{\partial \tau_{zy}}{\partial z}+\frac{\partial \tau_{xy}}{\partial x}+\rho g_y\; \; \; \; \; \; \; (27)

\rho\left ( u_x\frac{\partial u_x}{\partial x}+ u_y\frac{\partial u_x}{\partial y}+ u_z\frac{\partial u_x}{\partial z}+\frac{\partial u_z}{\partial t}\right )= \frac{\partial \sigma_{zz}}{\partial z}+\frac{\partial \tau_{xz}}{\partial x}+\frac{\partial \tau_{yz}}{\partial y}+\rho g_z\; \; \; \; \; \; \; (28)

From the articles on constitutive relation, \sigma_{ij}=-p\delta_{ij}+\mu\left ( \frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right ). We therefore substitute \sigma_{xx}=-p+2\mu \frac{\partial u_x}{\partial x} , \tau_{yx}=\mu\left ( \frac{\partial u_x}{\partial y}+\frac{\partial u_y}{\partial x}\right ) and \tau_{zx}=\mu\left ( \frac{\partial u_z}{\partial x}+\frac{\partial u_x}{\partial z}\right ) in eq26, noting that \frac{\partial u_x}{\partial x}+\frac{\partial u_y}{\partial y}+\frac{\partial u_z}{\partial z}=\nabla\cdot\textbf{\textit{u}}=0 (see eq14) to give:

\rho\left ( \frac{\partial u_x}{\partial t}+u_x\frac{\partial u_x}{\partial x} +u_y\frac{\partial u_x}{\partial y}+u_z\frac{\partial u_x}{\partial z} \right )=\rho g_x-\frac{\partial p}{\partial x}+\mu\left ( \frac{\partial^2 u_x}{\partial x^2} +\frac{\partial^2 u_x}{\partial y^2}+\frac{\partial^2 u_x}{\partial z^2}\right )\; \; \; \; \; \; \; (29)

Using the same logic and repeating the above step for eq27 and eq28, we have:

\rho\left ( \frac{\partial u_y}{\partial t} +u_x\frac{\partial u_y}{\partial x} +u_y\frac{\partial u_y}{\partial y}+u_z\frac{\partial u_y}{\partial z} \right )=\rho g_y-\frac{\partial p}{\partial y}+\mu\left ( \frac{\partial ^2u_y}{\partial x^2}+\frac{\partial ^2u_y}{\partial y^2} +\frac{\partial ^2u_y}{\partial z^2}\right )\; \; \; \; \; \; \; (30)

\rho\left ( \frac{\partial u_z}{\partial t} +u_x\frac{\partial u_z}{\partial x} +u_y\frac{\partial u_z}{\partial y}+u_z\frac{\partial u_z}{\partial z} \right )=\rho g_z-\frac{\partial p}{\partial z}+\mu\left ( \frac{\partial ^2u_z}{\partial x^2}+\frac{\partial ^2u_z}{\partial y^2} +\frac{\partial ^2u_z}{\partial z^2}\right )\; \; \; \; \; \; \; (31)

Eq29, eq30 and eq31 are the Cartesian form of the Navier-Stokes equations, which when multiplied throughout by the unit vectors \hat{x}, \hat{y} and \hat{z} respectively and summed, becomes the vector form:

\rho\frac{D\textbf{\textit{u}}}{Dt}=-\nabla p+\rho g+\mu \nabla^2\textbf{\textit{u}}\; \; \; \; \; \; \; (32)

where

\nabla^2\textbf{\textit{u}}=\nabla^2 u_x\hat{x}+\nabla^2 u_y\hat{y}+\nabla^2 u_z\hat{z}

\frac{D}{Dt} is the convective derivative defined as \frac{D}{Dt}=\frac{\partial }{\partial t}+\textbf{\textit{u}}\cdot \nabla

\nabla is the gradient defined as \nabla =\hat{x}\frac{\partial }{\partial x}+\hat{y}\frac{\partial }{\partial y}+\hat{z}\frac{\partial }{\partial z}

\nabla^2 is the vector Laplacian defined as \nabla^2\textbf{\textit{u}}=\nabla^2 u_x\hat{x}+\nabla^2 u_y\hat{y}+\nabla^2 u_z\hat{z}

\mu is the viscosity of the fluid

 

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