CONSTITUTIVE RELATION FOR NEWTONIAN FLUID

1. CONSTITUTIVE RELATION FOR NEWTONIAN FLUID The Cauchy equation for momentum balance of a continuous, deformable medium combined with the condition of symmetry of the stress tensor yields the relation Further applying the condition of incompressibility ( = const., ui/xi = 0), it is found that (Why?)

2. H CONSTITUTIVE RELATION FOR NEWTONIAN FLUID But how does the stress tensor ij relate to the flow? 21 = 12 moving with velocity U u1 fluid x2 x1 fixed - 21 = - 12 Plane Couette Flow: shear stress 21 = 12 is applied to top plate, causing it to move with velocity U; bottom plate is fixed. Because the fluid is viscous, it satisfies the “no-slip” condition (vanishing flow velocity tangential to the boundary) at the boundaries: Empirical result for steady, parallel (u2 = 0) flow that is uniform in the x1 direction:

3. 21 = 12 moving with velocity U u1 fluid H x2 x1 fixed 21 = 12 CONSTITUTIVE RELATION FOR NEWTONIAN FLUID • Newton’s hypotheses: • for steady, parallel flow that is uniform in the x1 direction, the relation u1/U = x2/H always holds; • an increase in U is associated with an increase in 12; • an increase in H is associated with a decrease in 12. • The simplest relation consistent with these observations is: where  is the viscosity (units N s m-2).

4. u1 CONSTITUTIVE RELATION FOR NEWTONIAN FLUID Alternative formulation let FD,mom,12 denote the diffusive flux in the x2 direction of momentum in the x1 direction. The momentum per unit volume in the x1 direction is u1, and in order for this momentum to be fluxed downthe gradient in the x2 direction, where  denotes the molecular kinematic diffusity of momentum, []= L2/T]. We now show that so that the kinematic diffusivity of momentum = the kinematic viscosity. momentum source fluid momentum sink

5. u1 CONSTITUTIVE RELATION FOR NEWTONIAN FLUID x2 x2 x1 x1 Again, the flow is parallel (u2 = 0) and uniform in the x1 direction, and also uniform out of the page (u3 = 0) . Consider momentum balance in the x1 direction. The control volume has length 1 in the x3 direction, ,which is upward vertical. Momentum balance in the x1 direction ~ /t(u1x1x21) = net convective inflow rate of momentum + net surface force + gravitational force or equivalently /t(u1x1x21) = net convective inflow rate of momentum + net diffusive rate of inflow of momentum + gravitational force Now the net convective inflow rate of momentum is

6. x2 x1 u1 CONSTITUTIVE RELATION FOR NEWTONIAN FLUID FD,mom, 12 12 x2+x2 x2 x1 FD,mom, 12 12 The gravitational force in the x1 direction is 0. x2 Since p at x1 is equal to p at x1 +x1 (flow is uniform in the x1 direction), the only contribution to the surface forces is 21 = 12, so that net surface force = Equivalently, net diffusive rate of inflow of momentum = Thus /t(u1x1x21) = The only way that this could be true in general is if

7. CONSTITUTIVE RELATION FOR NEWTONIAN FLUID Generalization to 3D: where denotes the viscousstress tensor, According to the hypothesis of plane Couette flow, we expect a relation of the form However, note that Here ij denotes the rate of strain tensor and rij denotes the rate of rotation tensor. (See Chapter 8.)

8. CONSTITUTIVE RELATION FOR NEWTONIAN FLUID We relate the viscous stress tensor only to the rate of strain tensor, not rate of rotation tensor, in accordance with the hypothesis for plane Couette flow. Most general possible linear form: Consequence of isotropy, i.e. the material properties of the fluid are the same in all directions: where A is a simple scalar, (See Chapter 8)

9. CONSTITUTIVE RELATION FOR NEWTONIAN FLUID Set to obtain and thus the constitutive relation for a Newtonian fluid:

10. CONSTITUTIVE RELATION FOR NEWTONIAN FLUID The Navier-Stokes equation for momentum balance of an incompressible Newtonian fluid is obtained by substituting the Newtonian constitutive relation into the Cauchy equation of momentum balance for an incompressible fluid and reducing with the incompressible continuity relation (fluid mass balance) to obtain