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Microscopic Momentum Balance Use the equations of change to develop the design equations for the Double Gap Viscometer shown in the figure. It consists of an inverted cup mounted to a rotating shaft. The fluid to be tested fills the gap on either side of the cup faces. The dimensions are shown in the figure. Reduce the appropriate equation(s) from the formula sheet to determine the stress profile in the gaps. Use this to find the velocity profile for a power law fluid. If the lower plate rotates at 10rpm, the gap is 1mm and the measured torque is 0.30Nm, what is the viscosity of the fluid. (Bonus +5pts ) If the Torque at 20rpm is 0.424Nm, determine the power law constants m and n. Equations of Continuity t+x(vx)+y(vy)+z(vz)=0t+r1r(rvr)+r1(v)+z(vz)=0t+r21r(r2vr)+rsin1(vsin)+rsin1(v)=0 Equations of Motion in Cartesian Coordinates [tvx+vxxvx+vyyvx+vxzvx]=[xxx+yxy+zxz]xp+gx[tvy+vxxvy+vyyvy+vzzvy]=[xyx+yyy+zyz]yp+gy[tvz+vxxvz+vyyvz+vzzvz]=[xzx+yzy+zzz]zp+gz Equations of Motion in Cylindrical Coordinates [tvr+vrrvr+rvvrrv2+vzzvr]=[r1r(rrr)+r1rr+zrz]rp+gr[tv+vrrv+rvv+rvrv+vzzv]=[r21r(r2r)+r1+zz]r1p+g[tvz+vrrvz+rvvz+vzzvz]=[r1r(rrz)+r1z+zrz]zp+gz Equations of Motion in Spherical Coordinates [tvz+vrrvz+rvvz+vzzvz]=[r1r(rrz)+r1z+zxz]zp+gz[tv+vrrv+rvv+rsinvv+rvrvrv2cot]=[r21r(r2rr)+rsin1(sin)+sin1r+rrrcot]r1ddp+g[tv+vrrv+rvv+rsinvvvcot]rrr1]=[r21r(r2r)+r1+rsin1+rr+2cotr]rsin1ddp+g Stress Tensor Components for Generalized Newtonian Fluid in Cartesian, Cylindrical and Spherical Coordinates xy=yx=[yvx+xvy]r=r=[r1r(rv)+r1vr]r=r=[r1r(rv)+r1vr]yz=tzy=[zvy+vvz]z=z=[zv+r1vz]==[rsin(sinv)xz=zx=[zvx+xvz]rz=zr=[rvz+zvr]r=r=[rsin1vr+rr(rv)] Non-Newtonian Viscosity Models =mn1=(0)(1+()2)2n1ij=ji=yield[xjvi+xivj] Macroscopic Balance Equations 0=m=((v)A)0=(vv2m+pA)mg+F0=21vv3+gh+p+W^+Ev Friction Loss Coefficients, K, for Ev=21(v2)K Microscopic Momentum Balance Use the equations of change to develop the design equations for the Double Gap Viscometer shown in the figure. It consists of an inverted cup mounted to a rotating shaft. The fluid to be tested fills the gap on either side of the cup faces. The dimensions are shown in the figure. Reduce the appropriate equation(s) from the formula sheet to determine the stress profile in the gaps. Use this to find the velocity profile for a power law fluid. If the lower plate rotates at 10rpm, the gap is 1mm and the measured torque is 0.30Nm, what is the viscosity of the fluid. (Bonus +5pts ) If the Torque at 20rpm is 0.424Nm, determine the power law constants m and n. Equations of Continuity t+x(vx)+y(vy)+z(vz)=0t+r1r(rvr)+r1(v)+z(vz)=0t+r21r(r2vr)+rsin1(vsin)+rsin1(v)=0 Equations of Motion in Cartesian Coordinates [tvx+vxxvx+vyyvx+vxzvx]=[xxx+yxy+zxz]xp+gx[tvy+vxxvy+vyyvy+vzzvy]=[xyx+yyy+zyz]yp+gy[tvz+vxxvz+vyyvz+vzzvz]=[xzx+yzy+zzz]zp+gz Equations of Motion in Cylindrical Coordinates [tvr+vrrvr+rvvrrv2+vzzvr]=[r1r(rrr)+r1rr+zrz]rp+gr[tv+vrrv+rvv+rvrv+vzzv]=[r21r(r2r)+r1+zz]r1p+g[tvz+vrrvz+rvvz+vzzvz]=[r1r(rrz)+r1z+zrz]zp+gz Equations of Motion in Spherical Coordinates [tvz+vrrvz+rvvz+vzzvz]=[r1r(rrz)+r1z+zxz]zp+gz[tv+vrrv+rvv+rsinvv+rvrvrv2cot]=[r21r(r2rr)+rsin1(sin)+sin1r+rrrcot]r1ddp+g[tv+vrrv+rvv+rsinvvvcot]rrr1]=[r21r(r2r)+r1+rsin1+rr+2cotr]rsin1ddp+g Stress Tensor Components for Generalized Newtonian Fluid in Cartesian, Cylindrical and Spherical Coordinates xy=yx=[yvx+xvy]r=r=[r1r(rv)+r1vr]r=r=[r1r(rv)+r1vr]yz=tzy=[zvy+vvz]z=z=[zv+r1vz]==[rsin(sinv)xz=zx=[zvx+xvz]rz=zr=[rvz+zvr]r=r=[rsin1vr+rr(rv)] Non-Newtonian Viscosity Models =mn1=(0)(1+()2)2n1ij=ji=yield[xjvi+xivj] Macroscopic Balance Equations 0=m=((v)A)0=(vv2m+pA)mg+F0=21vv3+gh+p+W^+Ev Friction Loss Coefficients, K, for Ev=21(v2)K