The three components of velocity in a flow field are given by u=x2+y2+z2 v=xy+yz+z2 w=−3xz−z22+4 (a) Determine the volumetric dilatation rate and interpret the results. (b) Determine an expression for the rotation vector. Is this an irrotational flow field?

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The three components of velocity in a flow field are given by  u=x2+y2+z2  v=xy+yz+z2  w=−3xz−z22+4  (a) Determine the volumetric dilatation rate and interpret the results. (b) Determine an expression for the rotation vector. Is this an irrotational flow field?

barcelodurazo0q · Accepted Answer

The rate of volumetric deformation is known as volumetric dilation rate is defined as the rate of increase of volume of a fluid element per unit volume. In an incompressible fluid, the volumetric dilation rate is zero because the fluid element volume cannot change without change in fluid density. Now we can use general equation to determine volumetric dilatation rate, where we have;  u=x2+y2+z2  v=xy+yz+z2  w=−3xz+z22+4  △⋅V→=dudx+dvdy+dwdz  △⋅V→=ddx(x2+y2+z2)+ddy(xy+yz+z2)+ddz(−3xz+z22+4)  △⋅V→=2x+x+z−3x−z  △⋅V→=0  Volumetric dilatation rate is equal zero, than flow is incompressible.  And now we will determine and expression for the rotation vector. The rotation, of the element about the one axis is defined as the average of the angular velocities and of the two mutually perpendicular lines.  W→=12((dwdx−dvdz)i^+(dudz−dwdx)j^+(dvdx−dudy)i^)  W→≠12(−2zi^+5zj^−yi^)  Since vector W is not zero everywhere the flow field is not irrotational.

The three components of velocity in a flow field are

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