Method to Use Conservations Laws in Fluid Flows…… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Mathematics of Reynolds Transport Theorem

Deformation of a differential volume at different instant of time

The Field quantity The field quantity f(x,t) may be a zeroth, first or second order tensor valued function. Namely, as mass, concentration, velocity vector, and stress tensor. A fluid element of given initial volume (dV 0 ) mya change its volume and/or change it surface (ds 0 ) with a given time, while moving through the flow field. This is due to various experiences by the element namely, dilatation, compression and deformation. Let us consider the same fluid particles at any time and therefore, it is called the material volume.

The volume integral of the quantity f(x,t): This is a function of time only. The integration must be carried out over the varying volume V(t). The material change of the quantity F(t) is expressed as: Since the shape of the volume V(t) changes with time, the differentiation and integration cannot be interchanged.

This analogy permits the transformation of the time dependent volume V(t) into the fixed volume V 0 at time t = 0 by using the Jacobian transformation function:

With this operation it is possible to interchange the sequence of differentiation and integration: The chain differentiation of the expression within the parenthesis results in

Introducing the material derivative of the Jacobian function and simplifying based on derivatives of transformation functions This equation permits the back transformation of the fixed volume integral into the time dependent volume integral.

The chain rule applied to the second and third term yields: The second volume integral in above equation can be converted into a surface integral by applying the Gauss' divergence theorem:

This Equation is valid for any system boundary with time the dependent volume V(t) and surface s(t) at any time. Also valid at the time t = t 0, where the volume V = V C and the surface s = s C assume fixed values. We call V C and s C the control volume and control surface. These control surfaces can be inlets or exits.

Thermodynamic form of RTT

Steady State Steady Flow Thermodynamic Model

Uniform State Uniform Flow Thermodynamic Model

Mass Flow Balance in Stationary Frame of Reference The conservation law of mass requires that the mass contained in a material volume V=V(t), must be constant: Consequently, above equation requires that the substantial changes of the above mass must disappear: Mass contained in a material volume

Using the Reynolds transport theorem, the conservation of mass, results in: This integral is zero for any size and shape of material volume. Implies that the integrand in the bracket must vanish identically. The continuity equation for unsteady and compressible flow is written as: This Equation is a coordinate invariant equation. Its index notation in the Cartesian coordinate system given is:

Continuity Equation in Cylindrical Polar Coordinates Many flows which involve rotation or radial motion are best described in Cylindrical Polar Coordinates. In this system coordinates for a point P are r,  and z. The velocity components in these directions respectively are v r,v  and v z. Transformation between the Cartesian and the polar systems is provided by the relations,

The gradient operator is given by, As a consequence the conservation of mass equation becomes,

Continuity Equation in Cylindrical Polar Coordinates Spherical polar coordinates are a system of curvilinear coordinates that are natural for describing atmospheric flows. Define  to be the azimuthal angle in the x-y - plane from the x-axis with 0   < 2 .  to be the zenith angle and colatitude, with 0   <  r to be distance (radius) from a point to the origin. The spherical coordinates (r, ,  ) are related to the Cartesian coordinates (x,y,z) by

or The gradient is As a consequence the conservation of mass equation becomes,