Conservation of Mass

The first equation we can write is the conservation of mass over time. Consider a system where mass flow is given by dm/dt, where m is the mass of the system. The term on the left denotes the rate of change of the mass of the system. The first term on the right describes the amount of flow across the control surface. The second term on the right refer to the condition within the control volume. We have,

${\displaystyle {\dot {m}}=\int _{CS}\rho \left(\mathbf {V} \cdot \mathbf {n} \right)dA+{\frac {d}{dt}}\int _{CV}\rho dV}$

However, by definition of a system the mass of which is a constant; thus the left-hand side of the above equation equals to zero and it could be rewritten as:

${\displaystyle \int _{CS}\rho \left(\mathbf {V} \cdot \mathbf {n} \right)dA=-{\frac {d}{dt}}\int _{CV}\rho dV}$

For steady flow, the time derivative of the quantity is zero, we have

${\displaystyle \int _{CS}\rho \left(\mathbf {V} \cdot \mathbf {n} \right)dA=0}$

And for incompressible flow, we have

${\displaystyle \int _{CS}\left(\mathbf {V} \cdot \mathbf {n} \right)dA=0}$

If we consider flow through a tube, we have, for steady flow,

${\displaystyle \rho _{1}A_{1}V_{1}=\rho _{2}A_{2}V_{2}}$

and for incompressible steady flow, A₁V₁ = A₂V₂.

Conservation of Momentum

Law of conservation of momentum as applied to a control volume states that

${\displaystyle \sum F={\frac {d}{dt}}\left(\int _{CV}\!\mathbf {V} \mathbf {\rho } \,dV\right)+\int _{CS}\mathbf {V} \mathbf {\rho } \left(\mathbf {V} \cdot \mathbf {ng} \right)dA}$

where V is the velocity vector and n is the unit vector normal to the control surface at that point.

The sum of the forces represents the sum of forces that act on the entirety of the fluid volume (body forces) and the forces that act only upon the bounding surface of a fluid (surface forces). Body forces include the gravitational force.

Conservation of Energy

The law of Conservation of Energy in fluid mechanics is a specific application of the First Law of Thermodynamics.

${\displaystyle {\frac {d\mathbf {Q} }{dt}}+{\frac {d\mathbf {W} }{dt}}={\frac {d}{dt}}\left(\int _{CV}e\mathbf {\rho } \,dV\right)+\int _{CS}e\mathbf {\rho } \left(\mathbf {V} \cdot \mathbf {n} \right)dA}$

where e is the energy per unit mass.

Equation of Continuity

• A differential mass balance relating density change to velocity.

${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \left(\rho \mathbf {V} \right)=0}$

• For incompressible fluids the equation of continuity reduces to:

${\displaystyle \rho \left(\nabla \cdot \mathbf {V} \right)=0}$

since

${\displaystyle {\frac {\partial \rho }{\partial t}}=0}$

for all incompressible fluids

Euler's Equation

• applies conservation of momentum to inviscid, incompressible flow.

${\displaystyle \rho \mathbf {g} -\nabla p=\rho {\frac {d\mathbf {V} }{dt}}}$

Stokes' Equation

• applies conservation of momentum in creeping flow limit (low Reynold's Number)

${\displaystyle \nabla p=\mu \nabla ^{2}\mathbf {V} }$