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.,  u i /  x i = 0), it is found that (Why?)

2 CONSTITUTIVE RELATION FOR NEWTONIAN FLUID But how does the stress tensor  ij relate to the flow?  21 =  12 -  21 = -  12 x2x2 x1x1 u1u1 moving with velocity U fixed fluid 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 (u 2 = 0) flow that is uniform in the x 1 direction: H

3 CONSTITUTIVE RELATION FOR NEWTONIAN FLUID Newton’s hypotheses: for steady, parallel flow that is uniform in the x 1 direction, the relation u 1 /U = x 2 /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:  21 =  12 x2x2 x1x1 u1u1 moving with velocity U fixed fluid H where  is the viscosity (units N s m -2 ).

4 CONSTITUTIVE RELATION FOR NEWTONIAN FLUID Alternative formulation let F D,mom,12 denote the diffusive flux in the x 2 direction of momentum in the x 1 direction. The momentum per unit volume in the x 1 direction is  u 1, and in order for this momentum to be fluxed down the gradient in the x 2 direction, where denotes the molecular kinematic diffusity of momentum, [ ]= L 2 /T]. We now show that so that the kinematic diffusivity of momentum = the kinematic viscosity. u1u1 fluid momentum source momentum sink

5 CONSTITUTIVE RELATION FOR NEWTONIAN FLUID u1u1 x2x2 x1x1 x1x1 x2x2 Again, the flow is parallel (u 2 = 0) and uniform in the x 1 direction, and also uniform out of the page (u 3 = 0). Consider momentum balance in the x 1 direction. The control volume has length 1 in the x 3 direction,,which is upward vertical. Momentum balance in the x 1 direction ~  /  t(  u 1  x 1  x 2 1) = net convective inflow rate of momentum + net diffusive rate of inflow of momentum + gravitational force  /  t(  u 1  x 1  x 2 1) = net convective inflow rate of momentum + net surface force + gravitational force or equivalently Now the net convective inflow rate of momentum is

6 CONSTITUTIVE RELATION FOR NEWTONIAN FLUID u1u1 x2x2 x1x1 x1x1 x2x2 The gravitational force in the x 1 direction is 0. net diffusive rate of inflow of momentum = net surface force = Equivalently, The only way that this could be true in general is if Since p at x 1 is equal to p at x 1 +  x 1 (flow is uniform in the x 1 direction), the only contribution to the surface forces is  21 =  12, so that Thus  /  t(  u 1  x 1  x 2 1) = x2x2 x2+x2x2+x2  12 F D,mom, 12

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

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