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Momentum Balance

Momentum Balance. Steven A. Jones BIEN 501/CMEN 513 Friday, March 17, 2006. Momentum Balance. Learning Objectives: State the motivation for curvilinear coordinates. State the meanings of terms in the Transport Theorem

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Momentum Balance

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  1. Momentum Balance Steven A. Jones BIEN 501/CMEN 513 Friday, March 17, 2006 Louisiana Tech University Ruston, LA 71272

  2. Momentum Balance Learning Objectives: • State the motivation for curvilinear coordinates. • State the meanings of terms in the Transport Theorem • Differentiate between momentum as a property to be transported and velocity as the transporting agent. • Show the relationship between the total time derivative in the Transport Theorem and Newton’s second law. • Apply the Transport Theorem to a simple case (Poiseuille flow). • Identify the types of forces in fluid mechanics. • Explain the need for a shear stress model in fluid mechanics. The Stress Tensor. Appendix A.5 Show components of the stress tensor in Cartesian and cylindrical coordinates. Vectors and Geometry Louisiana Tech University Ruston, LA 71272

  3. Motivation for Curvilinear Coordinates Fully developed pipe flow (Poiseuille) Application: What is the flow stress on an endothelial cell? • Flow around a small particle (Stokes Flow) • Applications: • How fast does a blood cell settle? • What is the motion of a catalyzing particle? Louisiana Tech University Ruston, LA 71272

  4. Cylindrical Coordinates: Examples Fully developed pipe flow (Poiseuille) Cylindrical coordinates are simpler because of the boundary conditions: In cartesian coordinates, there are three velocity components to worry about. In spherical coordinates, one of these components is zero (u). r   Louisiana Tech University Ruston, LA 71272

  5. Stokes (Creeping Flow) In cartesian coordinates, there are three velocity components to worry about. To confirm the three components, consider the point (x, y, z) = (1, 1, 1). Slice parallel to the equator (say the equator is in the xz plane): Top View z z x x y This velocity vector has an x and z component (visible above) and a y component (visible to the left). z x x Louisiana Tech University Ruston, LA 71272

  6. Momentum Balance Consider flow entering a control volume: The rate at which momentum is generated in a chunk of fluid that is entering the control volume is governed by the Reynolds Transport Theorem Louisiana Tech University Ruston, LA 71272

  7. Momentum Balance Consider flow entering a control volume: The property  in this case is momentum per unit volume, = v. Both  andv are bold (vectors). Louisiana Tech University Ruston, LA 71272

  8. Momentum Balance It is useful to recall the meanings of the terms. Rate at which the momentum increases inside the sample volume (partial derivative) Flux of momentum through the surface of the control volume. Rate at which the momentum of the fluid passing through the sample volume increases (production of momentum). Louisiana Tech University Ruston, LA 71272

  9. Say What? The momentum of the car passing through the location of measurement is increasing. The momentum at the location of measurement is not increasing. Rate at which the momentum of the fluid passing through the sample volume increases (production of momentum). 40 mph Location of Measurement 25 mph Louisiana Tech University Ruston, LA 71272

  10. Momentum Balance & Newton The momentum balance is a statement of Newton’s second law. Production of Momentum (Force per unit volume). Eulerian form of the time derivative of momentum (i.e. ma per unit volume). Louisiana Tech University Ruston, LA 71272

  11. Momentum Balance & Newton Eulerian time derivative Eulerian form of the time derivative of momentum (i.e. ma per unit volume). Lagrangian time derivative Louisiana Tech University Ruston, LA 71272

  12. Momentum Balance It is also useful to note that this is three equations, one for each velocity component. For example, the v1 component of this equation is: But note that the full vector v remains in the last integral. Louisiana Tech University Ruston, LA 71272

  13. Momentum Surface Flux The roles of the velocity components differ, depending on which surface is under consideration. The velocity vector that carries momentum through the surface. The momentum being carried through the surface. In the figure to the left: The velocity component perpendicular to the plane (v2) carries momentum (v1) through the plane. v2 v1 Louisiana Tech University Ruston, LA 71272

  14. Momentum Shell Balance Fully developed pipe flow (Poiseuille) dr vr dz • Assumptions: • Steady, incompressible flow (no changes with time) • Fully developed flow • Velocity is a function of r only (v=v(r)) • No radial or circumferential velocity components. • Pressure changes linearly with z and is independent of r. • Note: 3, 4 and 5 follow from 1 and 2, but it takes a while to demonstrate the connection. Louisiana Tech University Ruston, LA 71272

  15. The Control Volume The control volume is an annular region dz long and dr thick. We will be concerned with 4 surfaces: r rr Outer Cylinder rz Louisiana Tech University Ruston, LA 71272

  16. r rr rz The Control Volume Inner Cylinder Louisiana Tech University Ruston, LA 71272

  17. z zr zz The Control Volume Left Annulus Louisiana Tech University Ruston, LA 71272

  18. z zr zz The Control Volume Right Annulus Louisiana Tech University Ruston, LA 71272

  19. Continuity The mass entering the annular region = the mass exiting. dr dz Thus: This equation is automatically satisfied by assumption 3 (velocity does not depend on z). Louisiana Tech University Ruston, LA 71272

  20. Momentum in Poiseuille Flow The momentum entering the annular region - the momentum leaving= momentum destruction. (Newton’s 2nd law – F=ma) dr dz In fluid mechanics, we talk about momentum per unit volume and force per unit volume. For example, the force per unit volume caused by gravity is g since F=mg. (Units are g cm/s2). Louisiana Tech University Ruston, LA 71272

  21. Momentum Rate of momentum flow into the annulus is: Rate of momentum flow out is: Again, because velocity does not change with z, these two terms cancel one another. dr dz Louisiana Tech University Ruston, LA 71272

  22. Shearing Force Denote the shearing force at the cylindrical surface at r as (r). The combined shearing force on the outer and inner cylinders is: Note the signs of the two terms above. dr dz Louisiana Tech University Ruston, LA 71272

  23. Pressure Force The only force remaining is that cause by pressure on the two surfaces at r and r+dr. This force must balance the shearing force: dr dz Louisiana Tech University Ruston, LA 71272

  24. Force Balance Divide by 2 drdz: Louisiana Tech University Ruston, LA 71272

  25. Force Balance From the previous slide: Take the limit as Now we need a modelthat describes the relationship between the shear rate and the stress. Louisiana Tech University Ruston, LA 71272

  26. Shear Stress Model The Newtonian model relating stress and strain rate is: In our case, Thus, Louisiana Tech University Ruston, LA 71272

  27. Differential Equation So, with: and The equation is: Louisiana Tech University Ruston, LA 71272

  28. Differential Equation If viscosity is constant, Since the pressure gradient is constant (assumption 4), we can integrate once: , so C1= 0. By symmetry, Louisiana Tech University Ruston, LA 71272

  29. Differential Equation If viscosity is constant, Since the pressure gradient is constant (assumption 4), we can integrate once: , so C1= 0. By symmetry, Louisiana Tech University Ruston, LA 71272

  30. Differential Equation With Integrate again. The no-slip condition at r=R is uz=0, so Louisiana Tech University Ruston, LA 71272

  31. Review, Poiseuille Flow • Use a shell balance to relate velocity to the forces. • Use a model for stress to write it in terms of velocity gradients. • Integrate • Use symmetry and no-slip conditions to evaluate the constants of integration. If you have had any course in fluid mechanics before, you have almost certainly used this procedure already. Louisiana Tech University Ruston, LA 71272

  32. Moment of Momentum Balance It is useful to recall the meanings of the terms. Rate at which the moment of momentum increases inside the sample volume (partial derivative) Flux of moment of momentum through the surface of the control volume. Rate at which the moment of momentum of the fluid passing through the sample volume increases (production of momentum). Louisiana Tech University Ruston, LA 71272

  33. Types of Forces • External Forces (gravity, electrostatic) 2. Mutual forces (arise from within the body) • Intermolecular • electrostatic 3. Interfacial Forces (act on surfaces) Louisiana Tech University Ruston, LA 71272

  34. Types of Forces 1. Body Forces (Three Dimensional) • Gravity • Magnetism 2. Surface Forces (Two Dimensional) • Pressure x Area – normal to a surface • Shear stresses x Area – Tangential to the surface 3. Interfacial Forces (One Dimensional) e.g. surface tension x length) Louisiana Tech University Ruston, LA 71272

  35. Types of Forces 4. Tension (Zero Dimensional) The tension in a guitar string. OK, really this is 2 dimensional, but it is treated as zero-dimensional in the equations for the vibrating string. Louisiana Tech University Ruston, LA 71272

  36. The Stress Tensor The first subscript is the face on which the stress is imposed. The second subscript is the direction in which the stress is imposed. Louisiana Tech University Ruston, LA 71272

  37. The Stress Tensor The diagonal terms (normal stresses) are often denoted by i. Louisiana Tech University Ruston, LA 71272

  38. Exercise For a general case, what is the momentum balance in the q-direction on the differential element shown (in cylindrical coordinates)? dz dr dq Louisiana Tech University Ruston, LA 71272

  39. Look at the qr term Divide by dr, dq, dz Louisiana Tech University Ruston, LA 71272

  40. Contact Forces Diagonal elements are often denoted as  Stress Principle: Regardless of how we define P, we can find t(z,n) P B-P n t(z,P) Louisiana Tech University Ruston, LA 71272

  41. Contact Forces Diagonal elements are often denoted as  Stress Principle: Regardless of how we define P, we can find t(z,n) n P B-P t(z,P) Louisiana Tech University Ruston, LA 71272

  42. Cauchy’s Lemma Stress exerted by B-P on P is equal and opposite to the force exerted by P on B-P. t(z, n) P n BP P n BP t(z,n) Louisiana Tech University Ruston, LA 71272

  43. Finding t(z,n) If we know t(z,n) for some surface normal n, how does it change as the orientation of the surface changes? P n BP t(z,n) Louisiana Tech University Ruston, LA 71272

  44. The Tetrahedron We can find the dependence of t on n from a momentum balance on the tetrahedron below. Assume that we know the surface forces on the sides parallel to the cartesian basis vectors. We can then solve for the stress on the fourth surface. z2 A3 A1 n z1 A2 z3 Louisiana Tech University Ruston, LA 71272

  45. The Stresses on Ai Must distinguish between the normals to the surfaces Aiand the directions of the stresses. In this derivation, stresses on each surface can point in arbitrary directions. ti is a vector, not a component. z2 A3 A1 n z1 t1 A2 z3 Louisiana Tech University Ruston, LA 71272

  46. Components of ti Recall the stress tensor: Surface (row) z2 A3 A1 Direction of Force (Column) n z1 t1 A2 z3 Louisiana Tech University Ruston, LA 71272

  47. Total Derivative Lett1,t2, andt3be the stresses on the threeAi z2 A3 A1 n z1 A2 z3 Value is constant if region is small. Volume of the tetrahedron. Louisiana Tech University Ruston, LA 71272

  48. Body Forces Similarly, z2 A3 A1 n z1 A2 z3 Value is constant if region is small. Volume of the tetrahedron. Louisiana Tech University Ruston, LA 71272

  49. Surface Forces z2 A3 A1 n z1 A2 z3 Area of the surface. t does not vary for differential volume. Louisiana Tech University Ruston, LA 71272

  50. Find Ai Each of theAi is the projection of Aon the coordinate plane. Note that A projected on the z1z2 plane is just (i.e. dot n with the  coordinate). z2 A3 A1 n z1 A2 z3 Louisiana Tech University Ruston, LA 71272

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