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Triangular Channel Flow (30pts) The velocity profile for flow of a Newtonian, incompressible fluid in a duct with cross section shaped as an equilateral triangular (shown in the Figure) has the form: vz=A(Hy23Hx2y3+3x2y) Show that this velocity profile satisfies the momentum balance and the no-slip boundary condition on all surfaces. What is the expression for the constant, A in terms of P, and L. (Bonus +5 pts) What is the "Hagen Poiseuille" equivalent expression for the pressure drop-flow rate through the triangular pipe? Equations of Continuity t+x(vx)+zy(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+vzzvx]=[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)+1rr+zrz]rp+gr[tv+vrrv+rvv+rvrv+vzzv]=[r21r(r2r)+r1+zz]r1p+g[tvz+vrrvz+rvvz+vzzvz]=[r1r(rrz)+r1z+zzz]zp+gz[tvz+vrrvz+rvvz+vzzvz]=[r1r(rrz)+r1z+zzz]zp+gz[tv+vrrv+rvv+rsinvv+rvrvrv2cot]=[r21r(r2rr)+rsin1(sin)+rsin1+rrrcot]r1ddp+g[tv+vrrv+rvv+rsinvv+rvrvrvvcot]=[r21r(r2r)+r1+rsin1+rr+2cot]rsin1ddp+g Stress Tensor Components for Generalized Newtonian Fluid in Cartesian, Cylindrical and Spherical Coordinates xy=yx=[vvx+xvy]r=r=[r1r(rv)+r1vr]r=r=[r1r(rv)+r1vr]yz=zy=[zvy+vvz]z=x=[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 Triangular Channel Flow (30pts) The velocity profile for flow of a Newtonian, incompressible fluid in a duct with cross section shaped as an equilateral triangular (shown in the Figure) has the form: vz=A(Hy23Hx2y3+3x2y) Show that this velocity profile satisfies the momentum balance and the no-slip boundary condition on all surfaces. What is the expression for the constant, A in terms of P, and L. (Bonus +5 pts) What is the "Hagen Poiseuille" equivalent expression for the pressure drop-flow rate through the triangular pipe? Equations of Continuity t+x(vx)+zy(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+vzzvx]=[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)+1rr+zrz]rp+gr[tv+vrrv+rvv+rvrv+vzzv]=[r21r(r2r)+r1+zz]r1p+g[tvz+vrrvz+rvvz+vzzvz]=[r1r(rrz)+r1z+zzz]zp+gz[tvz+vrrvz+rvvz+vzzvz]=[r1r(rrz)+r1z+zzz]zp+gz[tv+vrrv+rvv+rsinvv+rvrvrv2cot]=[r21r(r2rr)+rsin1(sin)+rsin1+rrrcot]r1ddp+g[tv+vrrv+rvv+rsinvv+rvrvrvvcot]=[r21r(r2r)+r1+rsin1+rr+2cot]rsin1ddp+g Stress Tensor Components for Generalized Newtonian Fluid in Cartesian, Cylindrical and Spherical Coordinates xy=yx=[vvx+xvy]r=r=[r1r(rv)+r1vr]r=r=[r1r(rv)+r1vr]yz=zy=[zvy+vvz]z=x=[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