
FLUID MECHANICS  THEORY



The differential form of the linear momentum equation (also known as
the NavierStokes equations) will be introduced in this section. Euler's
equations for inviscid flow is also discussed. 





NavierStokes Equations

Newton's Second Law
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The integral form of the linear momentum equation was discussed
in the Linear Momentum Integral Equation section. Recall, Newton's second law on a differential fluid element is
δF = δm a
where δF is the resultant force acting on the fluid element (mass =
δm). a is the acceleration
of the fluid element, and it is given by
Expanding into its Cartesian components yields
There are two types of forces acting on the fluid element: body force
(δF_{B})
and surface force (δF_{S}).
δF = δF_{B} +
δF_{S}
The only body force considered here is the weight of the
fluid element. That is,
δF_{B} = δm g = δm
(g_{x} i + g_{y} j + g_{z} k)
Generally, gravity only acts in one direction, but since the coordinate system is not set, all three terms are included for the general case. Other body forces, such as those due to the magnetic and electric fields, can be fit into this framework, but they are not covered in this eBook. 



Notations for the Stresses
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Surface Forces in the xdirection


The surface forces are due to the stresses exerted on the sides of the
fluid element. There are two types of stresses
applied on the surface: normal stress (σ_{ij})
and shear stress (τ_{ij}).
Normal stress acts perpendicular to the surface while shear stress
is
tangential
to the surface. The subscript i refers to the axis normal to the
surface, and the subscript j represents the direction of the stress.
The notation
of the stresses is illustrated further in the animation shown on the
left.
All surface forces acting in the xdirection on the fluid element
are shown in the figure. A summation of the surface forces in the xdirection
yields
Note that the stresses are multiplied by the area to obtain the surface
forces. Similarly, the total surface forces in the y and zdirections
(not shown in figure) are obtained as
The resultant surface force is then given as
δF_{S} = δF_{Sx} i + δF_{Sy} j + δF_{Sz} k
The mass of the fluid element can be expressed in terms
of its volume and fluid density (δm = ρ δxδyδz),
so that the linear momentum equation in Cartesian coordinates reduces to 









For Newtonian fluids, such as water, oil and air, the
shear stress field is symmetric, and it is related to the rate of shear
strain in a linear fashion. The results are (derivation details not given here),
where μ is the viscosity of the fluid. Notice, the pressure term, p, only acts normal to the surface for each element face. Also, the pressure is assumed to be the same on all three faces, i.e. hydrostatic pressure.
For
incompressible flow, the term is
zero based on the continuity equation. The linear momentum equations thus
become 









The above equations are generally referred to as the NavierStokes equations,
and commonly written as a single vector form,
Although the vector form looks simple, this equation is the core fluid mechanics equations and is an unsteady, nonlinear, 2nd order, partial differential equation. It is extremely hard to solve, and only simple 2D problems have been solved. Computational Fluid Dynamics (CFD) is most often used to solve the NavierStokes equations. 





Euler's Equations



For inviscid flow (μ = 0), the NavierStokes
equations reduce to
The above equations are known as Euler's equations. Note that
the equations governing inviscid flow have been simplified tremendously
compared to the NavierStokes equations; however, they still cannot be
solved analytically due to the complexity of
the nonlinear terms (i.e., u ∂u/∂x, v ∂u/∂y, w ∂u/∂z,
etc.). Hence, in the study of fluid mechanics, numerical methods such as
the finite element and finite difference methods (along with the use of computers)
are often used to approximate the fluid flow problems. Euler's equations
can be written in vector form as
Euler's equations can be simplified further to obtain
Bernoulli's equation, which is applicable to
steady, incompressible, inviscid flow along a streamline. A more detailed
discussion on this subject is given in the solutions for the case study
presented
in this section. 



