In this section fluid motion will be described without concern with the actual forces necessary to produce the motion. The principles of conservation of mass and conservation of momentum permit some patterns of fluid motion, and exclude others. Many important real-world situations can be analyzed using this approach, without considering friction.
I.The Velocity Field
Fluids tend to flow easily, which results in a net motion of molecules from one point in space to another point as a function of time. Using the continuum hypothesis, fluids are broken down into fluid particles, which are composed of numerous fluid molecules. These particles interact with one another and with their surroundings. Thus the motion of a fluid, using a Eulerian model (continuum hypothesis), can be described in terms of the acceleration or velocity of fluid particles and not in terms of molecular motion.
The continuum assumption allows for any fluid property (density,pressure, velocity, acceleration, ...) to be described as a function of fluid location. The representation of fluid parameters in terms of spacial coordinates is called a field representation.
Ex: T = T(x,y,z,t)
Eularian & Lagrangian Flow
Eulerian analysis uses the field concept, derived from the continuum assumption.
Lagrangian analysis involves following individual fluid particles as they move about, and determining how fluid properties vary as a function of time.
Steady vs Unsteady Flow
When the flowing particle velocity, temperature and pressure do not change with respect to time is called steady flow. It should be noted that the conditions are different at different points, but at a particular point it is constant.
Streamlines, Streaklines, Pathlines
If in a flow field a curve is drawn in such a way that the tangent to the curve at any point represents the velocity to that point. Such imaginary curve in the flow field is known is stream line. the curve along which a particle moves in a flow field for a given period of time is known as path line. the curve drawn by the particles in a flow field that passes through the same point for a given period of time is known as streak line
for a steady flow stream line, path line and streak line will be the same.
The acceleration of a particle is the time rate of change of its velocity. Using the an Eulerian description for velocity, the velocity field V = V(x,y,z,t) and taking deriving it with respect to time, we obtain the acceleration field.
Now, consider a fluid particle A, moving along its pathline with velocity VA
VA = VA(rA,t) = VA[xA(t), yA(t), zA(t), t]
Differentiate to obtain the acceleration: (chain rule)
(1) = dV/dt + dV/dx dx/dt + dv/dy dy/dt + dv/dz dz/dt
Note: these are partial derivatives, since velocity is a function with several variables. Changing the partial velocity components u = dx/dt, v = dy/dt, w = dz/dt in EQ. 1 we obtain:
The material derivative is sometimes referred to as the substantial derivative, ...
Unsteady and Convective Effects
The material derivative, as seen above, contains two types of terms. Those involving the time derivative d()/dt and those involving spatial derivatives d()/dx, ... , ... . The time derivative portion is denoted as the local derivative, and represents the effects of unsteady flow. The local derivative occurs during unsteady flow, and becomes zero for steady flow.
The portion of the material derivative represented by the spatial derivatives is called the convective derivative. It accounts for the variation in fluid property, be it velocity or temperature for example, due to the motion of a fluid particle in space where its values are different.
III. Reynolds Transport Theorem
Fluid Mechanics as a Subset of the Continuum Mechanics
Rate of continuity= AV
Frames of Reference in Fluid Mechanics
There are two different frames of reference that are commonly used in the analysis of fluid mechanics problems: Fixed (Eulerian) reference frames and Material (Lagrangian) reference frames.
- In fixed reference frames fluid motion is defined with respect to a coordinate system that does not vary with time or fluid motion.
- In material reference frames fluid motion is defined with respect to a coordinate system that follows the motion of the fluid so that the same volume of fluid remains enclosed throughout the analysis. The Reynold's Transport Theorem (to be defined later) is used to solve fluid mechanics problems when the frame of reference is Lagrangian.