Question

Derive Navier0Stokes equations for the 'agitated vessel'.

Answer 1

if we start with the general way of the derivation of the Navier-Stokes equations we start with the basic laws of physics.

• lets consider a general flow field which is represented in the given Figure

• lets consider a closed control volume, which is within the flow field. hence the control volume will be kept fixed in space and the fluid is flowing through it. now the control volume will occupy considerably large finite region of the given flow field.

• A control surface , A₀ is basically defined as the surface which is having bounds with the volume .

• now from the Reynolds transport theorem, which states that "the rate of change of momentum for a system will be equal the total sum of the rate of change of momentum which is inside the control volume and the rate of efflux of momentum which is across the control surface".

• therefore the rate of change of momentum for a system (which is in our case, the value of control volume boundary and also is the system boundary which are eventually the same) and is also equal to the net external force which is acting on it.

        so from now , we will transform these statements into basically a equation by taking into account for each term,

as We know the relationship which is nothing but the general form of mass conservation equation (which is popularly known as the continuity equation), and is valid for both compressible and incompressible flows.

• substituting this relationship in Eq. previously obtained

or      

• This equation what is referred to as Cauchy's equation of motion . Now, In this given equation, is the stress tensor,
• After having substituted we get

also we know that from Stokes's hypothesis we get,

by using the above mentioned two relationship we have

This is what is referred to as the most general form of Navier-Stokes equation.