Update analytical solution

src/lbmpy/phasefield_allen_cahn/analytical.py

@@ -15,23 +15,49 @@ def analytic_rising_speed(gravitational_acceleration, bubble_diameter, viscosity
     return result
 
 
-def analytical_solution_microchannel(reference_length, length_x, length_y,
+def analytical_solution_microchannel(reference_length, liquid_top_height, liquid_bottom_height,
                                      kappa_top, kappa_bottom,
                                      t_h, t_c, t_0,
                                      reference_surface_tension, dynamic_viscosity_light_phase,
                                      transpose=True):
     """
-    https://www.sciencedirect.com/science/article/pii/S0021999113005986
+    Analytic solution for thermocapillary-driven convection of superimposed fluids at
+    zero Reynolds and Marangoni numbers https://doi.org/10.1016/j.ijthermalsci.2010.02.003
+    The analytic solution is evaluated at the cell centers.
+
+                        L
+    +----------------------------------------+
+    |                                        |
+    |              Liquid 1                  | liquid_top_height
+    |                                        |
+    |========================================|  reference_length
+    |                                        |
+    |              Liquid 2                  | liquid_bottom_height
+    |                                        |
+    +----------------------------------------+
+
+    reference_length: the height is taken as reference length
+    liquid_top_height: the height of top liquid
+    liquid_bottom_height: the height of the bottom liquid
+    kappa_top: thermal diffusivity of the top liquid
+    kappa_bottom: thermal diffusivity of the bottom liquid
+    t_h: maximal temperature
+    t_0: minimal temperature
+    t_c: base temperature
+    reference_surface_tension: sigma_ref as defined in the original derivation of the analytic solution
+    dynamic_viscosity_light_phase: dynamic viscosity light phase
+    transpose: return solution arrays in transposed form
     """
     l_ref = reference_length
+    length_x = int(2 * l_ref)
     sigma_t = reference_surface_tension
 
     kstar = kappa_top / kappa_bottom
     mp = l_ref // 2
 
     w = pi / l_ref
-    a = mp * w
-    b = mp * w
+    a = liquid_top_height * w
+    b = liquid_bottom_height * w
 
     f = 1.0 / (kstar * sinh(b) * cosh(a) + sinh(a) * cosh(b))
     g = sinh(a) * f
@@ -51,7 +77,7 @@ def analytical_solution_microchannel(reference_length, length_x, length_y,
     umax = -1.0 * (t_0 * sigma_t / dynamic_viscosity_light_phase) * g * h
     jj = 0
     xx = np.linspace(-l_ref + 0.5, l_ref - 0.5, length_x)
-    yy = np.linspace(-(l_ref//2) + 0.5, (l_ref//2) - 0.5, (length_x//2))
+    yy = np.linspace(-(l_ref // 2) + 0.5, (l_ref // 2) - 0.5, l_ref)
     u_x = np.zeros([len(xx), len(yy)])
     u_y = np.zeros([len(xx), len(yy)])
     t_a = np.zeros([len(xx), len(yy)])