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The stress tensor and the Navier-Stokes equation

The velocity of a fluid will vary in a complicated way in space; however, we can still apply the above definition of viscosity to a bit of fluid of thickness tex2html_wrap_inline348 with an infinitesimal area tex2html_wrap_inline350 . The shear stress across the thickness tex2html_wrap_inline348 is


We can further generalize and define the xy component of the viscous stress tensor as


The notation is made clearer in Fig. 3.2. This definition ensures that for a uniform rotation of the fluid there are no viscous stresses. gif1

Figure 3.2: Stresses acting on a small cube.

We can apply the same arguments as above to other flow configurations, with the result that the viscous stress tensor tex2html_wrap_inline370 is given by


where i,j=(x,y,z). gif2. Now in a fluid we can have internal stresses due to the fluid pressure p in addition to the stress due to viscous forces; the total stress tensor tex2html_wrap_inline378 in the fluid is then


where tex2html_wrap_inline380 is the Kronecker delta, defined such that tex2html_wrap_inline382 if i=j, and tex2html_wrap_inline386 otherwise. Once we have the stress tensor, the force per unit volume of the fluid is obtained by considering the stresses acting upon an infinitesimal cube of the fluid, and calculating the net force on the cube by taking the differences in the stress acting on different faces (just like our calculation of the force arising from pressure gradients in the fluid). This results in derivatives of the stress tensor, so that the ith component of the force per unit volume is


If we work out this last sum, we find (assuming that the viscosity doesn't depend on position)


For an incompressible fluid tex2html_wrap_inline390 ; reinstating vector notation, we see that the sum is then just tex2html_wrap_inline392 , so that the force per unit volume is


This is the force which goes into Newton's Second Law in order to obtain the equation of motion for tex2html_wrap_inline394 :


This is the Navier-Stokes equation for an incompressible fluid. Along with the equation of continuity, it is the basic equation which governs the flow of Newtonian fluids.

To complete the description of the flow we need to specify the boundary conditions on the velocity field at a solid surface. In our dealings with nonviscous flow we imposed the condition that the normal component of the fluid velocity at a solid surface must be zero -- this just ensures that no fluid flows through the surface. Once we deal with real (viscous) fluids, we must also require that the tangential component of the velocity of the fluid be zero on the surface. This condition is often called the no-slip boundary condition, and is the result of experimental observations.


gif1 To see this, notice that a fluid in uniform rotation with angular velocity W will have a velocity field given by v = W×r, with r the displacement vector from the center of rotation. If the z axis is the rotation axis, then vx = -Wy, vy = Wx and we see that txy = 0.
gif2 This is the general result for an incompressible fluid; for compressible fluids there is a second viscosity, often labeled z, which will enter into the viscosity stress tensor. Since our discussion of fluid mechanics is confined to incompressible fluids for the moment, we will drop this term.

next up previous
Next: The Reynolds number Up: Real Fluids Previous: Viscosity

Vittorio Celli
Sun Sep 28 22:13:11 EDT 1997