2. A Mathematical Frame Work to Create Fluid Flow Devices…… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Conservation Laws for Viscous Fluid Flows

3. Viscous Fluid Flow is A Control Volume Flow

4. Steady Viscous Flow If the density does not undergo a time change (steady flow), the above equation is reduced to:

5. Continuity Equation in Cartesian Coordinates 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:

6. 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,

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

8. 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

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

10. Balance of Linear Momentum The momentum equation in integral form applied to a control volume determines the integral flow quantities such as blade lift, drag forces, average pressure. The motion of a material volume is described by Newton’s second law of motion which states that mass times acceleration is the sum of all external forces acting on the system. These forces are identified as electrodynamic, electrostatic, and magnetic forces, viscous forces and the gravitational forces: For a control mass This equation is valid for a closed system with a system boundary that may undergo deformation, rotation, expansion or compression.

11. Balance of Momentum for Flow In a flow, there is no closed system with a defined system boundary. The mass is continuously flowing from one point to another point. Thus, in general, we deal with mass flow rather than mass. Consequently, the previous equation must be modified in such a way that it is applicable to a predefined control volume with mass flow passing through it. This requires applying the Reynolds transport theorem to a control volume.

12. The Preparation The momentum balance for a CM needs to be modified, before proceeding with the Reynolds transport theorem. As a first step, add a zero-term to CM Equation.

13. Applying the Reynolds transport theorem to the left-hand side of Equation Replace the second volume integral by a surface integral using the Gauss conversion theorem

14. Viscous Fluid Flows using a selected combination of Forces Systems only due to Body Forces. Systems due to only normal surface Forces. Systems due to both normal and tangential surface Forces. –Thermo-dynamic Effects (Buoyancy forces/surface)….. –Physico-Chemical/concentration based forces (Environmental /Bio Fluid Mechanics)