# FMM-fluid mechanics/ch1

Fluid Mechanics/Ch1

Contents   1 Fluid Kinematics 2 I.The Velocity Field 2.1 Eularian & Lagrangian Flow 2.2 Steady vs Unsteady Flow 2.3 Streamlines, Streaklines, Pathlines 3 II.Acceleration Field 3.1 Material Derivative 3.2 Unsteady and Convective Effects 3.3 Streamline coordinates 3.4 III. Reynolds Transport Theorem 4 Fluid Mechanics as a Subset of the Continuum Mechanics 5 Frames of Reference in Fluid Mechanics

Fluid Kinematics

In this section fluid motion will be described without concern with the actual forces necessary to produce the motion, hence kinematics of fluid motion.

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 as they move about, and determining how fluid properties vary as a function of time.

(-See below)

Steady vs Unsteady Flow]

Streamlines, Streaklines, Pathlines

II.Acceleration Field

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, concider a fluid particle A, moving along its pathline with velocity V_A

V_A = V_A(r_A,t) = V_A[x_A(t), y_A(t), z_A(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 derivs, since velocity a function with several variables. Changing the partial velocity components u = dx/dt, v = dy/dt, w = dz/dt in EQ. 1 we obtain:

 = dV/dt + u dV/dx + v dv/dy + w dv/dz   

Material Derivative

The material derivative is sometimes refered 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 occures 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 acounts for the variation in fluid property, be it velocity or temperature for example, due to the motion of a fluid partiacle in space where its values are different.

Streamline coordinates

III. Reynolds Transport Theorem

Fluid Mechanics as a Subset of the Continuum Mechanics

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.