Let's say the source resistances are of the same magnitude.From a theoretical point of view (which seems to be the intent of the question), you can't say much about the current (except direction) unless you model the battery source resistances. You will end up with an equivalent circuit with two voltage sources and two resistors. You can then analyze this arrangement and get your answer.
OK, assume that each battery has a source resistance of Rs. In theory, the current would then be I=(10-5)/(2Rs), in the direction you indicated. Note that the 5V battery still has current, but it is a charging current (storing energy). The 10V battery is supplying current and discharging energy.Let's say the source resistances are of the same magnitude.
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