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Answer all parts of the questions below. a) What are the names and meanings of the following dimensionless groups? Use only one sentence to describe each dimensionless group. The symbols have their usual definitions as given in the lecture notes. i) Cp/k ii) /DAB ii) vb2/kT b) A heated sphere is suspended in a large motionless body of fluid in order to study heat conduction in the fluid in the absence of convection. Answer all the questions below. State all assumptions and show all work. i) Determine the temperature profile T(r) outside the sphere in the fluid in terms of the distance from the centre of the sphere r, the sphere radius R, the thermal conductivity k, the temperature at the surface of the sphere TR and the temperature in the fluid far from the sphere T. ii) Using the temperature profile derived in part (i), show that the Nusselt number, a dimensionless heat transfer coefficient, is given by Nu=khR=1 where h is the heat transfer coefficient defined by Newton's law of cooling q=hT. iii) Calculate how long it takes for the sphere to cool from 100C to 50C if the fluid temperature is at 25C. The thermal conductivities of the sphere and the fluid are equal to 450Wm1K1 and 0.65Wm1K1, respectively. The sphere radius is 10cm, the specific heat capacity is equal to 420Jkg1K1, and the density is equal to 8,050kgm3. Equation of motion in rectangular coordinates for a Newtonian fluid with constant and : x-component (tvx+vxxvx+vyyvx+vzzvx)=xp+(x22vx+y22vx+z22vx)+gx y-component (tvy+vxxvy+vyyvy+vzzvy)=yp+(x22vy+y22vy+z22vy)+gy z-component (tvz+vxxvz+vyyvz+vzzvz)=zp+(x22vz+y22vz+z22vz)+gz Components of the stress tensor for a Newtonian fluid in spherical coordinates rr=[2rvr32(v)]=[2(r1v+rvr)32(v)]=[2(rsin1v+rvr+rvcot)32(v)]r==[rr(rv)+r1vr]==[rsin(sinv)+rsin1v]=r=[rsin1vr+rr(rv)] Components of the stress tensor for a Newtonian fluid in cylindrical coordinates rr=[2rvr32(v)]=[2(r1v+rvr)32(v)]zz=[2zvz32(v)]r=r=[rr(rv)+r1vr]z=z=[zv+r1vz]zr=rz=[rvz+zvr] Components of the stress tensor for a Newtonian fluid in rectangular coordinates xx=[2xvx32(v)]yy=[2yvy32(v)]zz=[2zvz32(v)]xy=yx=[yvx+xvy]yz=zy=[zvy+yvz]zx=xz=[xvz+zvx] Equation of motion in spherical coordinates for a Newtonian fluid with constant and . r-component (tvr+vrrvr+rvvr+rsinvvrrv2+v2)=rp+[r21r22(r2vr)+r2sin1(sinvr)+r2sin2122vr]+gr -component (tv+vrrv+rvv+rsinvv+rvrvrv2cot)=r1p+[r21r(r2rv)+r21(sin1(vsin))+r2sin2122v+r22vrr2sin22cosv]+g -component (tv+vrrv+rvv+rsinvv+rvvr+rvvcot)=rsin1p+[r21r(r2rv)+r21(sin1(vsin))+r2sin2122v+r2sin2vr+r2sin22cosv]+g Equation of motion in cylindrical coordinates for a Newtonian fluid with constant and . r-component (tvr+vrrvr+rvvrrv2+vzzvr)=rp+[r(r1r(rvr))+r2122vrr22v+z22vr]+gr -component (tv+vrrv+rvv+rvrv+vzzv)=r1p+[r(r1r(rv))+r2122v+r22vr+z22v]+gz-component(tvz+vrrvz+rvvz+vzzvz)=zp+[r1r(rrvz)+r2122vz+z22vz]+gz