We are interested in the distribution of field properties at each point in space. Therefore, we analyze an infinitesimal region of a flow by applying the RTT to an infinitesimal control volume, or , to a infinitesimal fluid system.
The differential volume is selected to be so small that density $\displaystyle (\rho )$ can be accepted to be uniform within this volume. Thus the first integral in 1 is:
Consider one-dimensional flow in the piston. The piston suddenly moves with the velocity $\displaystyle V_{p}$ . Assume uniform $\displaystyle \rho (t)$ in the piston and a linear change of velocity $\displaystyle U_{1}$ such that $\displaystyle U_{1}=0$ at the bottom ($\displaystyle x_{1}=0$) and $\displaystyle U_{1}=V_{p}$ on the piston ($\displaystyle x_{1}=L$), i.e.
$\displaystyle U_{1}={\frac {x_{1}}{L}}V_{p}$
Obtain a function for the density as a function of time.
The differential control volume dV and the flux of $\displaystyle \rho \;U_{i}$ (momentum per unit volume in i-direction) through the surfaces perpendicular to $\displaystyle x_{j}$ axis
The integral equation for the momentum conservation is
First consider the flux of $\displaystyle \rho \;U_{i}$ (momentum per unit volume in i-direction) through the surfaces perpendicular to $\displaystyle x_{1}$ axis:
Here, only gravitational force is considered as a body force. Thus,
$\displaystyle dF_{body\;i}=\rho \;dV\;g_{i}$
Differential surface forces
Surface forces are the stresses acting on the control surfaces. $\displaystyle F_{s}$ can be resolved into three components. $\displaystyle dF_{n}$ is normal to dA. $\displaystyle dF_{t}$ are tangent to dA:
$\displaystyle \sigma _{n}$ is a normal stress whereas $\displaystyle \sigma _{t}$ is a shear stress. The shear stresses are also designated by $\displaystyle \tau$.
Stresses on the surface of differential control volume
Thus, the surface forces are due to stresses on the surfaces of the control surface.
We define the positive direction for the stress as the positive coordinate direction on the surfaces (e.g. on ABCD) for which the outwards normal is in the positive coordinate direction . If the outward normal represents the negative direction (A'B'C'D'), then the stresses are considered positive if directed in the negative coordinate directions.
The stresses on the surface $\displaystyle (\sigma _{ij})$ are the sum of pressure plus the viscous stresses which arise from motion with velocity gradients:
We obtain the the most general form of momentum equation which is valid for any fluid (Newtonian, Non-newtonian, Compressible, etc.). It is non-linear due to the $\displaystyle 2^{nd}$ term at the LHS. Efect of Newtonian and Non-newtonian properties appears in the formulation of the viscous stresses $\displaystyle \tau _{ij}$. $\displaystyle \tau _{ij}$ will introduce also non-linearity when the fluid is non-Newtonian.
It should be noted that these formulations are based on stress conception which was thought to exist in fluids in motion. However it is known that $\displaystyle \tau _{ij}$ can be expressed as momentum transfer per unit area and time. Thus it can be considered as molecular momentum transport term. Derivations based on this concept requires a molecular approach (which is lengthy). The students should be aware that $\displaystyle \tau _{ij}$ causes momentum transport when there is a gradient of velocity.
Linear momentum equation for Newtonian Fluid: "Navier-Stokes Equation"Edit
For a Newtonian fluid, the viscous stresses are defined as:
When the velocity gradients in the flow is negligible and/or the Reynolds number takes very high values, the viscous stresses can be neglected:
$\displaystyle \displaystyle \tau _{ij}=0$
Since, the viscous stresses are proportional to viscosity:
$\displaystyle \tau _{ij}\propto \mu$
for flows, where $\displaystyle \tau _{ij}$ is neglected, the flow is called frictionless or inviscid, although there is a finite viscosity of the flow. Accordingly, the linear momentum equation reduces to
Differential control volume along streamline coordinates and the forces on it for a inviscid flow
Differential control volume along streamline coordinates and the forces on it for a inviscid flow
Euler's equation take a special form along and normal to a streamline with which one can see the dependency between the pressure, velocity and curvature of the streamline.
To obtain Euler's equation in s-direction, apply Newton's second law in s-direction in the absence of viscous forces.
which indicates that pressure increases in a direction outwards from the center of the curvature of the streamlines. In other words, pressure drops towards the center of curvature, which, consequently creates a potential difference in terms of pressure and forces the fluid to change its direction. For a straight streamline $\displaystyle R\rightarrow \infty$, there is no pressure variation normal to the streamline.
Bernoulli equation: Integration of Euler's equation along a streamline for a steady flowEdit
For a steady flow, Euler's equation along a streamline reads,
For incompressible flow $\displaystyle \rho ={\textrm {constant}}$ and after changing the notation as: $\displaystyle U=U_{s}$ and $\displaystyle z=x_{2}$, the integration results in
Note that due to the assumptions made during the derivation, the following restrictions applies to this equation: The flow should be steady, incompressible, frictionless and the equation is valid only along a streamline.
How do we measure pressure? When the streamlines are parallel to the wall we can use pressure taps.
If the measured location is far from the wall, static pressure measurements can be made by a static pressure probe.
The stagnation pressure is the value obtained when a flowing fluid is decelerated to zero velocity by a frictionless flow process. The Stagnation pressure can be calculated as follows:
Figure: a)Jet is impinging to the wall and stagnating at the point of impingment b)Schematics of a of a pressure tap in a channel c)Static pressure probe
If we know the pressure difference $\displaystyle P_{0}-P_{1}$, we can calculate the $\displaystyle U_{1}$ velocity.
Thus, measuring $\displaystyle P_{0}-P_{A}$ one can determine $\displaystyle U_{A}$.
Pitot-static tube for velocity measurement
However, in the absence of a wall with well defined location, the velocity can be measured by a Pitot-static tube. The pressure is measured at B and C; assuming $\displaystyle P_{B}=P_{C}$.