The velocity q at which water flows in a porous medium eg gr

The velocity q at which water flows in a porous medium (e.g. groundwater aquifer) is given by q = k Pui where k is called the hydraulic conductivity and is piezometric head. If Phi (x, y, z) = 2xy2cos2 find the velocity vector at the point (2,1,2).

Solution

first of all value of k is not given,so answer will depend on value of k
now =2xy² cosZ

grad()=2y² cosZ i+4xycosZ j-2xy² sinZ k

and velocity q=-k*grad()

so velocity=-k*(2y² cosZ i+4xycosZ j-2xy² sinZ k)