# Setting the pressure derivative and the stress

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Setting the pressure derivative and the stress derivatives in the x- and z-directions to zero, one findsdτyzdy=ρgorτyz=ρg(yH),where it has been used thatτyz=0aty = H.The fluid begins to flow when the magnitude of the stress (regardless of the sign) exceeds the yieldstress, which will happen first at the wall where the shear stress is greatest.The paint begins to flowwhenτyx>τ0or whenH>τ0ρg.The same conclusion can be reached by performing a force balance on a section of fluid as shown in thediagram below.The shear stress acting on an area LW must balance the weightρgLHW,τyzLW+ρgLWH=0.The paint begins to flow when the shear stress becomes equal to the yield stress, or when
H=τ0ρg.4.Whitaker 5-15.Use coordinate axes such that x points down, y points from the solid/water surface toward the solid/airsurface, so that y=0 is the water/solid surface and y=hIis the air/liquid interface.Neglect the density ofthe air, and require that the velocity vxand shear stressτyxbe continuous at the air/water interface.a)Derive the velocity profiles in both the air and the water.b)Identify a criterion that must be satisfied for the velocity profile in the water to yield a shear stressof zero at the air/water interface.SolutionyzρgτyzLH
Also, for the y- and z-directions,py=0andpz=0.The equations from the y- and z-directions imply the pressure only depends on x.But pressure doesnot depend on x at the air/water interface, and pressure is independent of y and z, so the pressure isconstant everywhere in this problem.The flow is driven purely by gravity.We haved2vxIdy2=ρwgμwso integrating twice givesvxIy( )=ρwgμwy22+C1y+C2.The same analysis applies in the air, but the density in the air is negligible (this assumption isconsistent with the problem statement, in which pressure variations in the air are neglected).Flow inthe air is driven by friction (i.e., the no-slip surface) between the water and air.In the air we havevxIIy( )=C3y+C4.The no-slip condition at y=0 yields C2=0, so in the watervxIy( )=ρwgμwy22+C1y0yhI.At the air/wall interface the no-slip condition isvxII=0aty=hI+hIIso0=C3hI+hII()+C4orvxIIy( )=C41yhI+hIIforhIyhI+hII.Two constants remain and for those we use the conditions at the air/water interface. We require that theshear stress be continuous, so
μwdvxIdy= μAdvxIIdyaty=hIand the velocity is continuous sovxI=vxIIaty=hI.These two conditions yield the equationsρwgμwhI+C1=μAμwC4hI+hIIandρwgμwhI2+C1=C4hI1hIhI+hII.

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